3 1822 01187 9392 iiillii'iiiiBRA WILGZYNiSKl AND SLAUGHT LIBRARY UNIVERSITY Of CALIFORNIA SAN DIESO r^ "-7 UNIVERSITY OF CAUFORNlf Y|| iri'lllli I 11 iilllli , 3 1822 01187 9392 SSE i Call This I one \, COLLEGE ALGEBRA WITH APPLICATIONS BY E. J. WILCZYNSKI, Ph.D. THE UNIVERSITY OF CHICAGO EDITED BY H. E. SLAUGHT, Ph.D. THE UNIVERSITY OF CHICAGO ALLYN AND BACON 23oston NebJ gork Ctirago COPYRIGHT, 1916, BY E. J. WILCZYNSKl Noriuooti Prtss J. S. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE Nothing causes more trouble to young inatliematicians than the traditional method of introducing complex numbers into algebra. The student knows that the square root of minus one is neither a positive nor a negative number, and that it is not equal to zero. Nobody has ever told him that there are any numbers which are neither positive, nor negative, nor zero. In spite of this, square roots of negative numbers are introduced for him to reckon with. Of course he does not know what they mean and he becomes sus- picious. Even later, if a concrete representation of these so- called " imaginary " numbers is ever presented to him, he never quite gets over his first suspicions ; the uncomfortable feeling re- mains within him that, somehow, he has been cheated, that imag- inary numbers are impossible and without meaning. It is historically correct to introduce imaginaries in this way. But the student's doubts are also justified historically. His sus- picious attitude toward imaginary numbers is merely a repeti- tion of the position taken by the whole mathematical world from the fifteenth to the nineteenth century. The historical order of presentation is not ahvays the best peda- gogic order, and the subject of complex numbers is one of the most striking instances ichere the historical order should be avoided. In Chapter I of the present book, the imniber system of algebra, up to and including the system of complex numbers, is developed in a concrete and convincing way by means of the geometry of directed line-segments or vectors. No student will feel any doubt concerning the legitimacy of complex numbers, after mastering this chapter, unless his suspicions have been previously aroused in his first course in algebra. In a sense, Chapter I may be regarded as devoted to the foundations of algebra. It is not, however, a chapter on foundations in the formal technical sense in which that word is being used at the present time. Chapter I also serves the purpose of revieicing the most ele- iv PREFACE mentary parts of arithmetic and algebra. A revieio, tvJiich is a mere repetition of material covered once before, is a great ivaste of time. For this reason no formal review work is offered at the be- ginning of the book. The review work is scattered through the various chapters. It is placed wherever it is needed for the pur- poses in hand, and it is always illuminated by the discussions which precede and follow it. That everything really essential is covered by this kind of a review is guaranteed by the nature of the book which, while intended for use in college, is built up from iirst principles, so that no reference to more elementary books on algebra is ever necessary. A fairly mature mind might begin his study of algebra with this book. The principal criticism of " college algebra " as a course has been its lack of unity. This criticism seems to be exceedingly well founded, if we examine the various texts on the subject which have appeared to date. The present book ivas vritten pri- marily for the purpose ofshoiving, that the undoubted scrajjpiness of the traditional course in college algebra is due to poor arrangement and piresentation, and not to any intrinsic defect of the subject itself. The following sketch of the contents of the book will show how the desired unification has been accomplished. The function concept is the central notion of the book. Chapter I, as has been mentioned, is devoted to the number system, a neces- sary preliminary. Chapter II begins with a general discussion of the function concept and takes up in detail the simplest cases, namely the subject of variation, and linear functions of a single variable, the graphs of such functions being introduced at an early stage. Chapter III deals with quadratic functions and equations. Chapter IV with integral rational functions of any order, the corresponding equations, and the numerical calculation of their roots (theory of equations) ; this treatment is completed in (Jhapter V by a discussion of the algebraic calculation of their roots, the fundamental theorem, and so on. Chapter VI deals with fractional rational functions, in particular with their expres- sion as a sum of partial fractions. Chapter VII discusses the simplest irrational functions and formulates the distinction be- tween algebraic and transcendental functions. Chapter VIII follows with the general power function, the exponential func- tion, and logarithms. In Chapter IX functions of more than one variable are intro- PREFACE V (lueed, linear functions to begin with, leading to determinants of the second and third order. In preparation for the general theory of determinants of the ?ith order, we interpolate a chapter (Chapter X) on permutations and combinations, followed by Chapter XI on probability. We are now ready for Chapter XII, which discusses linear functions of oi variables and determi- nants of the nth order. Chapter XIII goes on with the discussion of quadratic functions of two independent variables, and simul- taneous quadratics. So far the functions considered have been functions of contin- uous variables. If, instead, we restrict the variable to integral values, every function gives rise to a sequence. Thus a linear function gives rise to an arithmetic progression and, for this reason, arithmetic progressions are included in Chapter II. Har- monic and geometric progressions, suggested by analogy, are also treated in Chapter II. In Chapter XIV, however, the discontinuous variable is em- phasized, and results in a more extensive discussion of sequences and series with a finite number of terms. Chapter XV and XVI, on limits and series, now follow naturally from the suggestions of Chapter XIV and earlier chapters. Tlie applications are scattered through the entire hook and form an integral part of it. They are discussed with as much care as though the book had been written for their sake. Only ajjpli- cations of real and general importance have been included, making the course in algebra a valuable adjunct to the courses in physics and chemistry. These applications include such subjects as the meas- urement of length, time, and mass ; the theory of the vernier, slide rule, logarithmic paper, and of scales in general ; the notions of velocity, acceleration, density, specific gravity, force, uniform motion, uniformly accelerated motion, pressure of gases ; the principle of Archimedes, the motion of a projectile, Doppler's principle, the theory of dimensions in physics, and indirect analy- sis in chemistry. They also include a discussion of compound interest, annuities, and life insurance ; the comj)Oun(l interest law and its applications to dampened vibrations, transmission of light, pressure in the atmosphere, and cooling bodies. Professor A. C. Lunn was kind enough to give the author the benefit of his criticisms on some of these topics, and we take this opportunity of expressing our indebtedness to him. vi PREFACE Each of these applications is discussed as carefully as though the book ivere a treatise on chemistry or physics. The student is never asked to do examples involving applications which he cannot understand because the fundamental principles are not explained. Of course, these applications are here classified from the mathe- matical point of view, and therefore they appear in an order different from that which would result if the point of view were primarily physical or chemical. But the student can only gain by having the same subject appear in such a different way in several of his courses. AVe have explained the general policy of the book. Much more might be said about specific details in which it differs from other books ; we hope that the reader will discover these for himself, and thus make it unnecessar}' for us to unduly expand this preface. We close with the request that all prospective users of the book read the suggestions to the instructor and to the student given on pages vii and ix. An answer book will be supplied to those classes whose instructors wish this to be done. E. J. WILCZYNSKI. H. E. SLAUGHT, Editor. SUGGESTIONS TO THE INSTRUCTOR The material incliuled in this book probably contains everything ever given under the title " College Algebra" in any American college. But it includes more than can be given profitably in any one course of about fifty recitations. It will usually be necessary to make a selection. The following sample outlines of courses are intended to be helpful in this connection. But it should be understood that, although they have been considered carefully, many different selections might be made by the instructor, who probably knows the special needs of his class better than the author. Another possible way of using the book would be to consolidate the courses in College Algebra and Analytic Geometry, supplementing the book with explanations by the teacher on such topics in analytic geometry as are not included. It may also be used as a basis for a course in higher algebra. Course A Short course. Emphasis upon the applications. Articles 16-75, 78-113, 126-132, 135-146, 148-151, 156-182, 185-212. Much of this work has been covered in the student's high school course and may be reviewed in a brief time. Arts. 167-17.5 (on calculation with logarithms) may be transferred to the course in trigonometry. Arts. 80, 195, 190 may be omitted. The subjects included in Course A may be treated in the order given in the book, or else in the order : Chapters I, II, IX, III, IV, etc. ; or Chap- ters I, II, III, IX, IV, etc. Course B To Course A add articles 213-217, 219-227, 238-240, 243-252, 256, on permutations, combinations, probability, and simultaneous quadratics. Course C To Course A add articles 257-297, on summation of series, limits, and infinite series. viii SUGGESTIONS TO THE INSTRUCTOR Course D Short course. Emphasis upon the purely mathematical aspects. Articles 1-38, 50-78, 82-111, 114-120, 126-145, 147-184, 197-210. See remarks under Course A. Course E To Course D add articles 213-217, 219-227, 238-240, 243-252, 256, on permutations, combinations, probability, and simultaneous quadratics. Course F To Course D add articles 257-297, on summation of series, limits, and infinite series. SUGGESTIONS TO THE STUDENT Material The student should provide himself with a pair of compasses, a ruler divided decimally into centimeters and millimeters, and a quantity of cross-section paper (preferably millimeter paper), for facilitating his graphic work. General Directions for Study 1. Carefully read the assigned lesson. 2. Test your understanding of the j)rinciples involved as follows : (a) Convince yourself that you are really able to follow the logic of the argument by stating, explicitly, a reason in justification of every one of its steps. (b) Try to discover a fundamental idea which runs through the argument and illuminates it. This, together with (a), will enable you to master the whole argument, so as to reproduce it and to use similar reasoning, else- where, under similar circumstances. (c) See whether you can appreciate the practical importance of the subject under discussion by finding some appli- cations of it. 3. If this test of yourself, recommended in No. 2, turns out to your own satisfaction, proceed to the examples. 4. If your self-examination -has had an unsatisfactory result, find out whether your difficulty should be classified under 2a, 2b, or 2c. If it falls under 2a, try to find out just where the argument begins to become unintelligible for the first time, and fix your at- tention on the statements made at that ])oint. Look up, in review, the definitions of the terms which are used in those state- ments. Nothing causes more trouble in mathematics than lack of appreciation of just what the definitions say. If you find X SUGGESTIONS TO THE STUDENT that your trouble is not with the definitions, search your memory and your text for passages dealing with the subjects that cause the difficulty. Use the index and the table of contents. Finally, if you fail to conquer the difficulty, formulate it in writing as a question to your instructor. Often you will be able to answer your question yourself after having put it into written form. If your difficulty is not with 2a, but with 26, or 2c, tr}^ the ex- amples. They will probably helj) you. 5. Make your oral and written statements clear and unambig- uous. As a test of clearness, imagine yourself addressing a person of intelligence who is properly prepared to follow your argument, but do not expect him to do any mind-reading. Ask yourself this question : Can such a person be expected to under- stand what I have said ? In written work give everything that is essential, and try to make no statement which is not literally true. Good paper, good ink, and neatness in form are of great assistance in orderly thinking, 6. Formulate in writing questions concerning the lesson. Pre- sent them to your instructor. But try first to answer them for yourself. 7. Some students rush to the examples without reading the text. You will fail if you adopt this plan. Even if you should succeed in solving the examples, you will probably fail to under- stand the principles which they are intended to illustrate. It is not the purpose of a course in algebra to train you to merely fol- low mechanically certain rules of procedure. Make the rules your own by practice and memory ; but more important, make them truly and permanently your property by means of a thorough understanding of the principles upon which they are based. CONTENTS CHAPTER I The Number System of Algebra. 7. 8. 9. 10. 11, 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. The positive integer Addition, subtraction, and multiplication of positive integers Factors or divisors of integers . Division of integers The greatest common divisor of two integers The prime factors of an integer . Geometric representation of positive integers Rational numbers Some properties of rational numbers Addition and subtraction of rational numbers Multiplication of rational numbers . Geometric construction for the product of two positive rational numbers ...... Division of positive rational numbers A further property of rational numbers . The existence of irrational numbers . Negative numbers and zero Directed lines and directed line-segments . Addition and subtraction of positive and negative numbers Multiplication of positive and negative numbers Division by a positive or negative divisor .... Division by zero The monotonic laws for positive and negative numbers . Directed line-segments in a plane ..... The complex numbers Equality of two complex numbers ..... Vectors, vector addition, and addition of complex numbers Subtraction of vectors and complex numbers . Multiplication of a complex number by a positive or negative number Multiplication of a complex number by i . PAGE 1 1 3 4 5 6 7 8 9 11 13 13 14 16 17 20 21 24 26 28 29 29 31 34 36 36 40 41 42 Xll CONTENTS 30. Polar form of a complex number 31. Multiplication of two complex numbers 32. Division of complex numbers 33. Real and imaginary numbers 34. Conjugate complex numbers 35. Validity of the fundamental laws for complex numbers 36. History of the number system of Algebra CHAPTER II Linear Functions and Progressions. 37. Constants and variables 38. Variation 39. Application to concrete problems 40. Measurement of length 41. Length, area, and volume 42. Time 43. Mass 44. Density and specific gravity 45. Velocity 46. Acceleration . . 47. Uniformly accelerated motion . 48. Falling bodies .... 49. The importance of dimensional symbols . 50. Graphical representation of a pair of numbers 51. Graphical representation of variation 52. Graphical representation of the function y = nix + b 53. Slope of a straight line .... 54. Equation of a straight line 55. The zero of a linear function 56. Arithmetic progressions .... 57. Insertion of arithmetic means . 68. Harmonic progressions .... 59. Geometric progressions .... 60. Sum of a geometric progression of n terms 61. Geometric means 62. Geometric progressions with infinitely many ter 63. Periodic decimals CHAPTER III Quadratic Functions and Equations. 04. Standard form of a quadratic function 65. Graph of a quadratic function . 94 94 CONTENTS Xlll 66. Tlie maximum or minimum of a quadratic function 67. Graphic determination of tlie zeros of a quadratic function 68. Calculation of the real zeros of a quadratic function 69. Another method of deriving the formulpe for the roots of a quadratic equation 70. Complex roots of a (juadratic equation 71. Various methods of solving a quadratic equation or of factor ing a quadratic function .... 72. Special forms of quadratic equations 73. Equations of higher degree solvable by means of (juadratics 74. Rational and irrational roots of a quadratic equation 75. Quadratic surds 76. The square root of an expression of the form a + h\/d 11. Application of the monotonic laws of Algebra in numerical calculations involving quadratic surds 78. Interpretation of negative, fractional, and complex roots in concrete problems 79. Uniform motion along a straight line 80. Force 81. Motion of a projectile under the influence of gravity PAGB 96 99 100 102 104 106 109 110 111 112 115 117 118 120 121 126 CHAPTER IV Integral Rational Functions of the nth Order, of their Real Zeros. Numerical Calculation 82. Calculation of the numerical values of an integi-al rational function ..... 83. The functional notation 84. The factor theorem .... 85. The remainder theorem 86. Synthetic division .... 87. The slope of the tangent . 88. The binomial theorem 89. The derivative of an integral rational function 90. Derivatives of higher order .... 91. Taylor's expansion 92. Diminishing the roots of an equation 93. Arrangement of the calculation 94. Multiplication of the roots of an equation by m 95. Changing the sign of the roots .... 96. Continuity of integral rational functions . 97. Newton's method of approximation . 98. Geometric significance of Newton's method 99. The method of false position (Regula falsi) 129 131 132 134 135 136 141 146 147 148 149 151 153 154 155 157 158 158 xiv CONTENTS PAGE 100. An example of Newton's method 160 101. Horner's method 164 102. Abbreviated calculation 165 103. Negative roots 165 104. Computation of more than one root 166 105. Upper limit for the positive roots of an equation . . . 166 106. Descartes's rule of signs 168 107. Maxima and minima of an integral rational function . . 172 108. Rolle's theorem . . .175 109. Multiple roots 175 110. Rational roots of an equation with rational coefficients . . 177 111. Summary of the operations required in solving an equation with given numerical coefficients 181 112. Application of cubic equations to floating spheres . . . 182 113. Application of cubic equations in trigonometry . , . 185 CHAPTER V Integral Rational Functions of the nth Order. The Problem of the Algebraic Determination of their Zeros, and their General Properties. 114. Distinction between the algebraic and numerical solution of an equation ......... 187 115. The equation x" — 1 = 187 116. The nth power of a complex number 188 117. The complex roots of unity 189 118. Numerical expressions for the complex roots of unity for n = 2, 3, 4 191 119. Construction of regular polygons 193 120. The equation x" — a = 194 121. The cubic equation 197 122. Discussion of the roots 202 123. The ratios of the coefficients of the general cubic equation expressed in terms of its roots 202 124. The equation of the fourth order 204 125. The equations of higher order 207 126. The fundamental theorem of Algebra 208 127. Application of the fundamental theorem to functions with real coefficients ......... 212 128. Use of the factored form of f(x) in plotting .... 215 129. Form of the graph in the case of real and distinct factors . 216 130. Form of the graph in the case of real factors some of which are repeated 217 CONTENTS XV PAGE 131. Form of the graph when some of the linear factors are imagi- nary 219 132. Relations between the roots and the coefficients of an alge- braic equation 219 133. Symmetric functions 221 134. Vanishing and infinite roots 222 CHAPTER VI Fractional Rational Functions. 135. Definition of a rational function 226 136. Proper and improper rational fractions 226 137. Reduction of a rational function to its lowest terms . . 228 138. Zeros of a rational function 230 139. Poles of a fractional rational function 230 140. Graph of a fractional rational function 231 141. General form of a rational function in terms of its zeros and poles 233 142. Partial fractions 234 143. Resolution into partial fractions, when the poles are not simple 237 144. Modified form of the partial fractions in the case of imaginary poles 240 145. Fractional rational equations 241 146. Pressure exerted by gases 244 CHAPTER VII Irrational Functions. 147. Existence of irrational functions 248 148. The function y = 'Vx and its principal value .... 250 149. The graph oiy = y/x 251 150. The function y = ^x^ .253 151. Properties of radicals 254 152. The square root of a rational function 256 1.53. Functions which involve the square root of a rational function and no other irrationality 258 154. Irrational equations of the simplest type . . . .261 155. The general algebraic function 262 XVI CONTENTS CHAPTER VIII Fractional and Negative Exponents. The General Power Function. The Exponential Function and Logarithms. 156. Fractional, negative, and vanishing exponents 157. The index laws 158. The principle of permanence . 159. The case of an irrational exponent 160. The power function . 161. The exponential function 162. Graphs of exponential functions 163. Properties of a-^ 164. Definition of logarithm 165. Graph of a logarithmic function 166. Properties of logarithms . 167. Common logarithms 168. Characteristic and mantissa 169. Properties of the mantissa 170. Determination of the characteristic 171. Arrangement and use of the table of logarithms 172. Extraction of roots by means of logarithms 173. Logarithmic calculations which involve negative number 174. Principles used in logarithmic calculations 175. Arrangement of the calculation 176. The logarithmic or Gunter scale 177. The slide rule 178. The general notion of a scale . 179. Relation between the logarithms of two different systems 180. Selection of a standard logarithmic curve 181. The derivative of the logarithmic function 182. The numerical value of e 183. Exponential equations 184. The calculation of a table of logarithms . 185. Applications of logarithms 186. Simple interest . 187. Compound interest 188. Annuity 189. Interest compounded more than once annually 190. The compound interest law ..... 191. Dampened vibrations ...... 192. Variation of density and pressure in the atmosphere 193. Transmission of light by imperfectly transparent media 194. Cooling bodies 195. Semi-logarithmic paper ... 196. Logarithmic paper CONTENTS xvii CHAPTER IX Linear Functions of More than One Variable. Linear Equations and Determinants of the Second and Third Order. PAGE 319 320 321 322 197. Functions of two variables 198. Linear functions of two variables ..... 199. Linear equations ........ 200. Simultaneous linear equations ...... 201. General foruiul;? for the solution of two simultaneous linear equations with two unknowns ...... 324 202. Determinants of the second order 325 203. Homogeneous linear equations with two unknowns . . 326 204. Discussion of the solutions of two linear equations with two unknowns 329 205. Properties of determinants of the second order . . . 331 20(5. Determinants of the third order 332 207. Cof actors 337 208. The principal properties of determinants of tlie tiiird order . 338 209. Solution of a system of three simultaneous linear equations with three unknowns ....... 342 210. Homogeneous equations 344 211. An application of linear equations in Chemistry . . . 346 212. Generalization to systems of n linear equations with n unknowns 350 CHAPTER X Permutations and Combinations. 213. The notion of order 352 214. Permutations 352 215. The number of permutations of n elements taken k at a time 354 216. Circular arrangements 356 217. Permutations when all of the elements are not distinct . . 357 218. Two classes of permutations 359 219. Combinations 361 220. Independent combinations 363 221. The binomial theorem 364 222. Total number of combinations 365 CHAPTER XI Probability. 223. Definition of probability . . . • 366 224. Compound events 368 xviii CONTENTS PAGE 225. Repeated trials 372 226. Application to life insurance 373 227. Other applications of the theory of probability . . • 376 CHAPTER XII Determinants of the nth Order and Systems of Linear Equations with n Unknowns. 228. Definition of a determinant of the «th order .... 376 229. Another method for determining the sign of a term of the determinant 378 230. Properties of determinants 380 231. Minors 382 232. Cofactors 384 233. Solution of a system of n linear eciuatioiis for n unknowns . 386 234. Homogeneous equations 387 235. Systems of linear equations with more equations than un- knowns .......... 388 236. Systems of linear equations with fewer equations than un- knowns .......... 388 237. Application of determinants to the theory of elimination . 388 CHAPTER XIII Quadratic Functions of Two Independent Variables and Simultaneous Quadratic Equations. 238. Integral rational functions of two independent variables . 391 239. Quadratic function of x and y 392 240. Composite and non-composite quadratic functions . . . 392 241. The values of a quadratic function ...... 398 242. The existence of solutions of a quadratic equation . . . 399 243. Graph of a function defined by a quadratic equation in x and y 401 244. Solution of a system of simultaneous equations one of which is linear and one of which is quadratic .... 407 245. Simultaneous quadratics 409 246. Equivalent systems of simultaneous equations . . . 409 247. Normalization 411 248. Existence of four solutions 414 249. Special cases of two simultaneous quadratics . • • 416 250. Case I. H = If = F = F' - G =G' = 0. Neither equation contains a first degree term or a term inxy . ■ ■ 416 251. Case II. F=F'^G=G' = 417 CONTENTS XIX PAGE 252. Case III. Both equations contains x and y in symmetric fashion, so that the equation is left uualtered if x and y are interchanged 420 263. Case IV. When at least one of the given equations is com- posite ........■•• 423 254. A new method for the general case of simultaneous quad- ratics .......•••• 423 255. The method of small corrections 425 256. Applications which involve simultaneous quadratics . , 427 CHAPTER XIV Sequences and Series with a Finite Number of Terms. 257. Continnous and discontinuous variation .... 428 258. Definition of a sequence 429 259. Higher progressions 430 260. Geometric progressions 432 261. Series 433 262. Summation of series by mathematical induction . . . 433 263. General characteristics of the method of mathematical induction .......... 434 264. The summation sign 437 265. Summation of a series whose ^•th term is an integral rational function of fc 438 266. Summation of some other simple series 441 CHAPTER XV Limits. 267. Limits suggested by series 443 268. Definition of a limit 444 269. Infinity 447 270. Infinitesimals 448 271. Variables which remain finite 448 272. A theorem about infinitesimals ...... 449 273. Theorems about limits 450 274. Limit of a quotient of two variables 451 275. Limit of the nth power of a positive number as n grows beyond bound 454 276. Continuity of a function • 455 277. Continuity of a fractional rational function .... 458 278. Indeterminate forms 460 XX CONTENTS CHAPTER XVI Infinite Series. 279. Non-terminating geometric progressions . 280. Some other non-terminating series . 281. Convergence and divergence of infinite series 282. Fundamental criteria for convergence 283. Series all of whose terms are positive 284- Comparison tests ..... 2^5. Some convenient comparison series . 28(5. Ratio test 287. Ratio of corresponding terms of two series 288. Series with positive and negative terms . 289. Conditionally convergent series 290. Alternating series 291. Series whose terms are functions of X 292. Power series ...... 293. Equality of two power series . 294. Expansion of functions as power series . 295. Expansion of rational functions 296. Expansion of some ii'rational functions . 297. The expansion of (1 -f a;)" 298. Exponential series 299. Logarithmic series PAGB 465 465 466 468 471 471 472 474 479 482 484 484 485 486 488 488 489 492 493 495 496 APPENDIX Table 1 . Four Place Logarithms of Numbers Table 2. American Experience Table of Mortality 498 500 COLLEGE ALGEBRA WITH APPLICATIONS CHAPTER I THE NUMBER SYSTEM OF ALGEBRA 1. The positive integer. The most fundamental notion of arithmetic, that of tlie positive integer^ is obtained by the process of counting and becomes familiar to us in our early childhood. The names and symbols used for the positive integers are well known to all. The following fact is also familiar, but its importance justifies an explicit formulation at this time. There exists a first positive integer, namely unitg, but there is no last. 2. Addition, subtraction, and multiplication of positive in- tegers. If the letters a, 5, c, etc., are used to denote given positive integers, we know from our early training in arith- metic how to form their sums, differences, and products. In fact, we know that the sum, a + 5, and the product, a x b or ab, is always again a positive integer. This will also be true of the difference, a — b, provided that a is greater than 6, that is, (a > b). The following laws sum up the most essential properties of positive integers, and may be regarded as justified by our experience with a large number of special cases : I. If we add a positive integer b to another positive integer a, we always obtain a uniquely determined positive integer c= a -\- b, which is called the sum of a and b. II. Addition is commutative; that is, a + b = b + a. 1 2 THE NUMBER SYSTEM OF ALGEBRA [Art. 2 III. Addition is associative ; that is, a+(b + c) = (a + J) + c. IV. Addition is monotonic ; that is, if a is greater than b, then a + is greater than b + c. In syynbols, if a > b, then a + c > b -\- c. V. If zve midtiply a positive integer a by another positive integer b, tve alivays obtain a uniquely determined positive ^'^^^9er c = axb = a. b = ab, tvhich is called the product of a by b. VI. Multiplication is commutative ; that is, a x b = b x a. VII. Multiplication is associative; that is, a x(b X c') = (a X b') X c. VIII. Multiplication is monotonic; that is, if a > b, then a x c> b x c. IX. Multiplicatioyi is distributive with respect to addition; ^^"<'^''^' c(a+6) = ca + 6'5. Subtraction may be clelined as follows. If a and b are positive integers, such that a > b, then there exists a positive integer x, tvhich added to b will give a as a sum; that is, if a "> b, there exists a positive integer xfor which b + x= a. This integer x is denoted by x= a — b, and is called the difference between a and b. Moreover, a and b are called minuend and subtrahend respectively. The laws of subtraction, which are familiar, may be derived from this definition and the laws I . . . IX. In Algebra we wish to make general statements, applicable to all integers. While we can easily verify that the above laws are true of a great mnn)j integers, all of those less than ten, for instance, we cannot, by the method of actual test, assure ourselves that they are true of all Akt. 3] FACTORS OR DIVISORS OF INTEGERS 3 integers. For we should never get through with the task of testing them all, since there is no last integer. (See Art. 1.) Thus it is impossible to justifij the general valididj of the above laws on the basis of experience alone. On the other hand, a lofjicul proof of these laws could, at best, be only partially successful. For sucli a logical proof would consist in showing that they are necessary consequences of certain other law's, and then the latter would have to remain without a strictly logical proof, however plausible it might seem that they should be true. Since the laics I . . . IX can never be tested completely by experience, and since they can never receive a complete proof by the method of loyic, any statement that these laws are universally true is of thr nuturr- of an assumption. We therefore speak of these laws as fundamental assumptions * of Algebra. They are called fundamental for two reasons. First, because they are used in all of the elementary operations of Arithmetic, and second, because it can be shown that they are of the greatest importance throughout all of Algebra, as will become apparent very soon. To il- lustrate the first point, let us multiply 8 by 13. We reason as follows : 8 X 13 = 8(10 + 3) = 8 • 10 + 8 • 3 = 80 + 21 = 80 + (20 + 4) = (80 + 20) + 4 = 100 + 4 = 104, where we have used laws V, IX, III, I. This example illustrates the following general statement: To perform a calculation with positive integers ice need to know, in the first place, the results obtained by the addition, subtraction, and multiplication of any two integers less than ten (the addition and multiplication tables^; and in the second place, we must know how to apply the nine fundamental laws. 3. Factors or divisors of integers. Since some positive integers may be obtained by multiplying together two or more others, it becomes important to understand the fol- lowing definitions : If a positive integer n can he expressed as a product of two or more positive integers., each of the latter integers is said to he a factor or divisor of n. If an integer has no integral divisors excepting itself and unity., it is said to he a prime number. All other integers are said to he composite. The lowest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. * We do this in spite of the fact tliat it is possible to build up Algebra on fewer assumptions than we have here listed. 4 THE NUMBER SYSTEM OF ALGEBRA [Art. 4 4. Division of integers. Let B and d be two positive in- tegers, D being greater than d. \i D has c? as a factor, we may write (1) D = dq, where q, the second factor, is also a positive integer. It is customary to write (1) in the equivalent form (2) . = f, and to speak of q as the quotient obtained by dividing D (the dividend) by d (the divisor). Thus, we may write 14 = 7 • 2 or 2 = ^. In this case Z) = 14, rf = 7, q = 2. But suppose that d is not a factor of 2>, and let us ex- amine the successive multiples d,2d, 3 c?, etc., of d. Let qd be the largest integral multiple of d which is less than Z>, so that the next multiple (jq -^l^d will be greater than D. We shall then have qdD, and the difference D — qd will be a positive integer r which is less than d ; that is, (3) D — qd = r, or D = qd -\- r, r q = D divided hy d = the fraction— • Consequently the symbol for a fraction D/d may be read : D divided hy d. Since we have attached a concrete meaning to the operation of dividing one integer by another, the fol- lowing definition is now intelligible. Any number which may he regarded as the quotient of one positive integer divided by another is called a j^ositive rational number. Of course all integers are included among the rational nurnl)ers, since any integer may be regarded as being the quotient obtained by dividing that integer itself by unity. If D and d are any two positive integers, we may therefore always write (2) I)-^d = ^=q where q is either an integer or a fraction, and this relation may be regarded as equivalent to (3) D = dq. 9. Some properties of rational numbers. It is clear that the process of introducing intermediate points on the scale by making use of fractions will enable us to label many of these points in more than one way. Thus, the point which corresponds to the symbol ^ is the same as that which corre- sponds to |, and the lengths of the corresponding line-seg- ments are equal. We therefore agree that the fractions | and I shall be regarded as equal. With this introduction, the following statements will be easily understood. 10 THE NUMBER SYSTEM OF ALGEBRA [Art. 9 (a) A rational number p/q^ where p and q are integers, is said to he in its lowest terms, if the integers p and q have no common divisor except unity, or in other words, if p and q are relatively prime. (5) Any rational number may be reduced to its loivest terms by dividing both numerator and denotninator by their greatest common divisor. (c) Two rational nu^nbers are equal to each other if they re- duce to the same number when both are expressed in their lowest terms. Thus we have -t- = -J-' q mq More gCDerally, if p' /q' is equal to p" /q", both of these frac- tions will reduce to the same fraction p/q when written in their lowest terms. We shall therefore have p'=m'p, q' = m'q, p" = m"p, q" =m"q, where m' and m" are integers, and consequently whence p'q" = m'm"pq, p"q' = m'm"pq, (1) p'q"=p"q'. This relation must hold if the fractions p' /q' and p" /q" are equal. Conversely, if (1) is satisfied, we may obtain from (1), by dividing both of its members by q'q", the equation pi _p" q' q" We may therefore replace statement (c) which involves the definition of equality of two rational numbers, by the following : (c7) Ttvo rational numbers p' /q' and p" /q" are equal, if and only if , ,, ,, , ^ •' p q =p"q' . If two rational numbers, p/q and p' /q', are not equal, the points, P and P', which represent them on the scale of AuT. 10] ADDITION AND SUBTRACTION 11 rational numbers, will not coincide. We shall say that ^ is less than ^ , or ^ precedes ^ , if P is closer to the origin than P' . (See ,12 p/g p/g' Fisr. 3.) In symbols we express this rela- o a b p p' ,. '' , ^ .,/ ^ Fig. 3 tion by writing , q q' The equivalent relation -, > - may be read jo'/^'. is greater i\vA\\p/q^ ov p' /q' follows jo/^'. We easily recognize the validity of the following numerical test : (e) The rational number p/q is greater than p' /q' ^ if and only if the integers p, q^ p', q' are such that pq'>p'q. Similarly, p/q is less than p' /q', if and only if pq' I since 3 • 3 > 2 • 4. To prove statement (e) it suflBces to re- duce the two fractions which are to be compared to a common denomina- tor qq'. Thus we have f = t\ and f = r\, so that J > f because 9 > 8. (/) The tests given under ((i) and (*') alivays enable us to decide whether two given positive rational numbers are equal or not, and, in the latter case, to decide ivhich is the greater. {g} If no tivo of the three rational numbers p/q, p' /q'-, p" /q^' are equal, and if either p / q <:: p' / q' p' /q' > p" /q" -, then 2^' /q' is said to be between p/q and p" /q". 10. Addition and subtraction of rational numbers. Let a— p/q be any rational number and let A be the corre- sponding point of the number scale. (See Fig. 4.) Then OA is a units long and we shall call the origin and A the 12 THE NUMBER SYSTEM OF ALGEBRA [Art. 10 terminus of this line-segment OA. Let a'=p'/q' be any other rational number represented by the line-segment OA', whose origin is and whose k — -—a^a-i-a ^^^ >, terminus is A' . We now define 5 li A' 'Y' addition as follows. To add a' to a ive first place the origin of the line-segment OA, which represents a, upon the origin of the number scale. We then place the origin of the line-segment OA', which represents a', 7ipon A, the terminus of OA. Let A" he the position which the terminus of OA' will then occupy. The liiie-segment OA" will then represent the sum a + a' . The arithmetic rule for forming the sum of the two rational numbers, ^/^ and p'/q', follows from this definition. On account of statement (c) of Art. 9, we may write (1) a=E = ^, a' = 4 = ^' q qq' q' qq' thus reducing the two fractions to a common denominator qq' . If we divide our original unit of length into qq' equal parts, equations (1) tell us that a con.tains pq' and that a' contains p'q of these smaller units. Thus their sum con- tains jo^-' +jt?' 5- of these smaller units, so that (2) p_^p[^ pq' + p'q^ q q' qq' since each of the smaller units is equal to the original unit divided by qq'. On the basis of the above definition for addition, or of the equivalent formula (2), we can now easily verify that the fundamental laws /, //, ///, IV of Art. 2 are true, not merely when the symbols there used stand for positive integers, but also if these symbols represent any positive rational numbers. The geometric construction for subtraction and the corre- sponding formula are so immediate as not to call for a separate discussion. Arts. 11,12] MULTIPLICATION OF RATIONAL NUMBERS 13 11, Multiplication of rational numbers. To multiply a fraction 'pjq by a positive integer jo' means merely to take the fraction ip' times, so that ' ^P^P'Z. P To multiply p/(i by l/q means the same thing as to divide p/q by q\ that is, to divide the line-segment of length p/q into q' equal parts and to take one of these parts. Thus q' q q'q We combine these two special cases into the following definition : To multiply a rational number p/q {the multiplicand') hy the rational number p' /q' (the multiplier) means to find a third rational number p"/q" (the product) such that (1) p1 = pL^p^i^. q q q q q In other words, multiplication of two fractions is carried out by multiplying the two numerators and the two denom- inators. It is easy to show, as a consequence of this definition, that the laws V, VI, VII, VIII, IX of Art. 2 will be true for positive rational numbers. If we combine this result with the statement at the end of Art. 10, we see that the nine fundamental laws of Art. 2 apply not only to positive integers but also to all positive rational numbers. 12. Geometric construction for the product of two posi- tive rational numbers. The following familiar construction enables us to find the product of two rational numbers as a line-segment when the numbers themselves are given as line-segments. On OX (Fig. 5) lay oE OA = a units. Through draw any line OY, not coinciding with OX, and on OY lay o£E 14 THE NUMBER SYSTEM OF ALGEBRA [Art. 13 0U= 1 unit and OA' = a' units. Join U to A and draw a line parallel to AU through A'. Let P be the point in which the latter line intersects OX. Then OP will contain aa' units, so that OP will represent the product aa'. In fact, the triangles OAU and OPA' are similar, so that Fig. 5 OP: 0A= OA':OU, OP:a = a':l, OP = aa'. or whence, This construction becomes especially convenient when OY is taken at right angles to OX. 13. Division of positive rational numbers. According to Art. 11, the product of two rational numbers is again a rational number. The problem of finding one of these factors, when the other factor and the product are given, constitutes division. Let d = p/q be the given factor (divisor) and let I) = p'/q' be the given product (dividend). We wish to find a rational number x = p"/q" such that (1) E q q" q' We may reduce ^and^ qq" q to a common denominator, writing Jr.Ji £PL = PJJEL and £- = ^^l^il, qq' ([qq q qqq so that (1) becomes pq'p" _ p'qq" qq' q" qq'q" ART. 13] DIVISION OF RATIONAL NUMBERS 15 But these two fractions, with equal denominators, can be equal only if their numerators are equal, that is, if (2) (pq')p"=(p'q)q"- But, according to Statement (c?), Art. 9, equation (2) implies that the fractions j»"/r^" and p'q/pq' are equal. Therefore (3) a: = P^ = P^ = ^xi. q pq q p We have obtained the familiar rule for division of fractions, (4) pL^p^pL^i, q' q q' p which is usually expressed as follows : To divide hy the fraction p/q is equivalent to multiplyijig hy the reciprocal frac- tion q/p. For two fractions, such as p/q and q/p, whose product is equal to unity are said to be reciprocal to each other. It is easy to give a geometri- cal construction for division. To divide a by a' we lay off (see Fig. 6) OA = a units, OA' —■ a' units, 0U=1 unit. We then join A to A' and through C/" draw a line parallel to AA', intersecting OX in P. Then OP = x = — units. a In fact, the similar triangles OPU and OAA' yield the proportion OP^OA , OP ^ a. OU OA'' °' 1 a'' whence follows OP = — • a This construction, as well as our arithmetical argument, shows that division is unique; that is, two positive rational numbers, D and d, determine a unic^ue rational number as their quotient. 16 THE NUMBER SYSTEM OF ALGEBRA [Art. U EXERCISE II Perform the following indicated operations arithmetically and geo- metrically. 1. 1 + ^. 3. fxf. 5. (i-i)^(i-2). 6 2« — 1 4-I- — i. 'oii' • 9 3- ^- 6 • 3- ^ + i Reduce the following expressions to simpler forms and state which of the nine laws of Art. 2 you are using. 7. (6o + 3 i + 5/)5 r/. 11. (3 ac + ade + uf+ a) ^ «. 8. (a + b)(c + (1). 12. (ab + ac)^{b + c). 9. (a + b + c)(d + e+f). 13. (ac + hc + ad + bd) -^ (a+ b). 10. (x + a)(x + b)(x+ c). 14. (xx + 2 xy + yy) ^ (x + y). Reduce the following expressions to the form of simple fractions, 11 1,1 « ^ 21. ^-Jx- + ^^+l. p^r qs ps + rq 16.^4-^- '? ' 7 *■ 1+11 + 1 P2 ^ ?! _£ / 17. ^ + ^ + ^. r-T • 1 + 1" 11 23. i • 18. - X - b 1 + 1 + 19. 1±J!^ '^ 1 +c ^+7 24. ^ 1 + 20. ^L±i^l±/ " ■ l+x + ^^ c + d g + h 1 + X 25. Arrange the following fractions in order of magnitude : 14. A further property of the rational numbers. Let a =p/q and a' = j)' /q' be two rational numbers and let a < a'. ^Moreover, let OA and OA' (Ficr. 7) be the I III, X . V c^ / o ABA' corresponding line-segments on OX, so that (1) OA = a=^, OA' = a'=^, AA' = a' - a. q q' Akt. 15] THE EXISTENCE OF IRRATIONAL NUMBERS 17 Let B be the point which bisects AA'. Then 0B= OA+IAA' = a + i(a' - a) or, if we denote by h the length of OB, (2) OB=h=l{a+a'^. But by (1) we find that 2 -iKq q'J 2qq' is again a rational number, and it is obviously between a and a' . In the same way we may find another rational number between a and h, still another between a and 5', etc. Thus, hetiveen any two rational numbers, no matter hoiv close together they may be, there are always infinitely many other rational numbers. Therefore, the rational numbers give rise to infinitely many points infinitely close together on the number scale of OX, and in any interval on OX there lie infinitely many points (^called the rational points of the scale^ ivhose distances from are represented by rational numbers. This is often expressed by saying that the rational points of the scale form a dense set. 15. The existence of irrational numbers. It is not true, however, tliat the distance from to every point of OX can be represented by a rational num- ber. To show this, let us con- struct a square (Fig. 8) on OA as a side, where OA = 1 unit, and 6 a b draw the diagonal ON. Then ^^^- ^ 6N^ =6A^ + AN"^ =1+1 = 2. With as a center and ON as radius, strike an arc inter- secting OX at B. Then OW = 2. 18 THE NUMBER SYSTEM OF ALGEBRA [Art. 15 Thus we have constructed a line-segment on OX whose length is equal to V2 units, if we use the customary notation for a square root. But we can easily show that V2 is not a rational number. For, if it Avere, we could write (1) v^ = ^ where p and q are positive integers without a common divisor, since we may think of the rational number to which V2 is hypothetically equal as being expressed in its lowest terms. jSIoreover q cannot be equal to unity. For if it were, (1) would require V2 to be an integer, and clearly there exists no integer whose square is equal to 2.* The prime factors (see Art. 6) of p are all different from those of q since, by hypothesis, p and q have no common divisor. The prime factors of p^ are the same as those of p, each of them occurring twice as often inp"^ as in p. Similarly for (f. Consequently every one of the prime factors of p^ must be different from every prime factor of 7), German. ^ Pythagor.as and his puiuls, tlio Pythagoreans, flourished five or .six centuries before Christ in various countries then under Greek influence. 20 THE NUMBER SYSTEM OF ALGEBRA [Art. 16 4. Review from your Geometry the method of constructing a line- segment of lengtli ^ ah if the line-segments of length a and h are given. Apply this method to the construction of V^, Vo, yJk. 5. Choose a unit of length. Construct V3 and V5, V3 + \/5, V5- V3, V3 X V5, V.5 ^ V.5, V5 - V3. 6. Observe that the constructions for a + />, a — i, «ft, a/6, and Va can all be performed by using ruler and compass. Show that if a, h, c, d, etc. are given line-segments, it is possible to find a ruler and compass con- struction for any line-segment which can be obtained from a, h, c, d, etc. by a finite number of additions, subtractions, multiplications, divisions, and extractions of square roots. 16. Negative numbers and zero. If we prolong the line OX backward from in the direction OX' (see Fig. 9), we may lay off line-segments, such as OC x' < , I, I ' "^1 \ "^' X and OB' , in either of two opposite p,j^ q directions. These segments differ, not only in length, but also in direction. We agree to indicate this difference hi direction hy the use of the signs + ayid — . Thus we say that the number which corresponds to OC is -|- 3 and that which corresponds to OB' is — 2. The new numbers (such as — 2) which are introduced in this way are called negative numbers. If we agree further that the line-segment 00, of no length, which joins to itself, shall be represented by the symbol or number (zero), every line-segment of X' X which has as origin ivill determine a number, positive^ zero, or negative, according as the terminus of this line-segment is to the right of 0, coincides with 0, or is to the left of 0. Every point P of XX' may be thought of as the terminus of a line-segment OP which has as its origin. If we label the point P with the positive or negative number whicli cor- i-esponds to tlie segment OP, we obtain a scale of positive and negative numbers, as in Fig. 9. A familiar instance of such a scale is furnished by an ordinary thermometer. It makes no essential difference on which side of the scale we mark the Art. 17] DIRECTED LINES 21 positive numbers, provided we use the opposite side for the negative numbers. We have iutroduced positive and negative numbers to indicate opposi- tion in direction, such opposition as is expressed in ordinary language by the terms right and left, above and below, north and south, east and icest. But there are many other instances where a similar kind of opposition is to be indicated, although not of geometric character, and where the use of negative numbers is of great importance. A few examples of this are indicated by the terms, before and after, temperatures above and below zero, profit and loss, credit and debit. If one of the members of any of these pairs be represented by a positive number, the other member may be represented by a negative number. 17. Directed lines and directed line-segments. When a line XX' has been provided with a scale of positive and nega- tive numbers, as in Art. 16, we shall henceforth speak of it as a directed line, its positive sense being that one which is directed from the origin toward the side which represents the positive numbers. In our figures we shall, from now on, indicate the positive sense of a directed line by placing a + sign near the end of that portion of the line which actually appears in the figure. (See Fig. 10.) A line-segment is a finite portion of a line and may be described by naming its end-points, such as AB in Fig. 10. But if we think of it as a directed line-segment we must dis- tinguish between AB and BA. We may think of a directed line-segment as being generated by the motion of a poinU which moves from one of its end-points toward the other without ever changing the direction of its motion. That end of the directed line-segment from which the generating point starts its motion is called the origin of the directed line-segment. The other end-point is called its terminus. In speaking of a directed line-segment we shall name its origin first. Thus, in Fig. 10, AB is the directed line-segment which has A as origin and B as terminus. The directed line-segment BA has the same length as AB, l)ut the opposite direction. If a directed line-segmeiil lies on a directed line, its direc- tion or sense may be the same as that of the directed line or 22 THE NUMBER SYSTEM OF ALGEBRA [Art. 17 else opposite. A directed line-segment, which lies on a di- rected line, shall be regarded as having a positive or negative number as its measure according as it has the same direction as the directed line or the opposite direction. Thus ill Fig. 10, if the length of AB is 5 units, we may write AB = 5 and BA — — 5. The measure of vIjB is 5, that of BA is — 5. We shall use the same symbol ^.6 to represent both the line-segment itself and its measure. The absolute value of a line-seg7nent, or its numerical value, is always a positive number expressing how many units there are in its leyigth. We use the notation \ AB \, placing AB between two vertical lines, to represent the absolute value of AB. Thus in Fig. 10, \AB\ = 5, and \BA\^ 5. The first of these state- ments is read : the absolute value oi AB is equal to 5. A directed line-segment on a directed line is said to be in its standard position if the origin of the line-segment coincides with the origin of the scale. In Fig. 11, the directed line-segment AB, whose measure is + 3, is not in its standard position. If it be placed in standard position it will coincide with OC. Its measure will still be + 3. This remark also illustrates the following statement. Two directed line-segments on the same directed line have the same measure if their lengths and senses are the same. Two such segments will henceforth be regarded as equal whether their origins coincide or not. From what has been said it is clear that, if A and B are two points on a directed line, the measures of the directed line-segments AB and BA will be numerically equal, but opposite in sign, that is, (1) AB=- BA. The following theorem is also important. If A, B, C are any three points on a directed line, then (2) AB + BC=AC, Art. 17] DIRECTED LINES 23 where AB^ B (7, and A are the measures of the corresponding directed line-segments. Proof. The symbol + in equation (2) is used to indicate addition as defined geometrically on page 12, and again on page 24. According to this definition, to obtain the sum of two directed line-segments we place the origin of the second segment on the terminus of the first, and then join the origin of the first segment to the terminus of the second. If we apply this definition to the sum AB + BO we obtain AC, no matter in wliat order the three points A, B, and O are ar- ranged on the line. Thus the theorem is proved. Figure 12 shows three of the six possible arrange- ments of the three points, namely those three in which the measure of AC is positive. Fig. 12 (a), in which AB and BC also have positive measures, teaches us nothing new. In Fig. 12 (i), AB and CB are positive, but BC is negative. According to our theorem, which is true in all cases, we have AB +BC ^AC. But Fig. 12 (i) shows that AB — CB = AC. Therefore we obtain the same result whether we add to AB the negative segment BC or subtract from AB the positive segment CB. suit from Fig. 12 (c). The following is a simple corollary of the above theorem. J^ A, B, (7, JD are any four points of a directed line., tve have the relation (3) AB+BC+ OB = AI) between the measures of the directed line-segments AB., BC, CD, and AD. In fact, by the theorem just proved, we have AB + BC=AC AC+ CD = AD, so that we find by addition AB + BC-{-AC+CD = AC+AD which reduces to (3) if we subtract AC from both members. Similar remarks re- 24 THE NUMBER SYSTEM OF ALGEBRA [Art. 18 It may now be proved by mathematical induction (see Art. 263) that, if A, B, C, . . . M, N are any finite 7iumber of points on a directed line, then (4) AB + BC-{-- -\-MN=AK 18. Addition and subtraction of positive and negative num- bers. The fundamental operations of arithmetic have been defined, so far, only for positive numbers. These definitions may, in part, be applied without essential change to the case where some or all of the numbers involved are negative. But since nothing was said, at the time, concerning such cases, it becomes necessary to re-define these operations so as to take into account all of the possibilities. We define addition geometrically, as follows : To add a number a' to a number a, we first construct two directed line-sec/ments OA and O'A' on a directed line I, such that the measures of OA and O'A' are equal to the numbers a and a' respectively. We then place the origin 0' of O'A' upon the terminus A of OA. The directed line-segment OA', which joins the origin of the first segment to the terminus of the second, will have a -{- a' as its measure. In each of the cases represented in Fig. 13, OA' represents the sum of OA and O'A'. Jn all three cases OA is positive. In the first case O'A' is also positive. In the second case O'A' is nega- +1 tive but of absolute value less than OA, so that the +1 sum OA' is positive. In the third case O'A' is +i negative and numerically greater than OA, so that the sum OA' = OA +- O'A' is negative. This definition enables us to prove easily that addition of numbers (positive or negative) still satisfies the first three fundamental laws of Art. 2, that is, addition gives a unique result, it is commutative and associative. The question whether addition is also monotonie will be discussed a little later. If the sum s of two numbers and one of the numbers a are given, the problem to find the other, a', may be expressed o A a' o' a' o A Fig. A 13 Art. 18] ADDITION AND SUBTRACTION 25 as follows: wliat number added to a will give the sum «? We usually write (1) a' = s—a and speak of the process of finding a' as subtraction. « is called the minuend, a the subtrahend, and a' the difference. As long as we dealt with positive numbers only, subtraction was impossible when the subtrahend was greater than the minuend. The introduction of negative numbers frees us from this restriction. Thus in Fig. 13, let the directed line-segment OA' repre- sent the minuend, and OA the subtrahend. Then, according to (3) of Art. 17, we have in all cases OA + AA' = OA' or OA + O'A' = OA' so that we must add O'A' to OA (the subtrahend) in order to obtain OA' (the minuend) as a sura. Therefore O'A' is the required difference, that is, (2) O'A' = OA' - OA. After a slight change of notation we may formulate this re- sult in the form of a geometric rule for subtraction. To subtract a number a' (subtrahend^ from a number a (minuend) we first construct two directed line-segments OA and O'A' on a directed line I, such that the measures of OA and O'A' are equal to a and a' respectively . We then place the termini of these two segments so that they shall coincide. The directed line-segment which theri joins the origin of the minuend to the origi^i of the subtrahend will represent the difference iii magnitude and sign. These constructions easily lead to the following familiar remarks. 27ig addition of a negative number is equivalent to the sub- traction of a positive number of the same absolute value., that is., (1) a + (-5)=a-6. 26 THE NUMBER SYSTEM OF ALGEBRA [Art. 19 The subtraction of a negative number is equivalent to the addition of a positive number of the same absolute value, that is, (2) a-(-5)=a+5. Since subtraction of any number may tlierefore always be regarded as addition of a number of the same absolute value but of opposite sign, we may from now on suppress any ex- plicit mention of the laws of subtraction. They are included in the laws of addition. The following statements are also immediate consequences of our geometric definitions : (3) a -\- = a, a — = a, a — a = 0, « + (— a)=0. 19. Multiplication of positive and negative numbers. We have, so far, defined multiplication only for the case where both factors are positive. In our attempt to formulate an appropriate definition for multiplication for the case where one or both factors are negative, we return for a moment to the simplest case of all when both factors are positive in- tegers. In that case multiplication reduces to a repeated addition. Thus, for instance, (1) . 5 X 3 = 5 -h 5 -f- 5 = 15. If the multiplicand is negative, equal to — 5 for instance, while the multiplier is a positive integer, we naturally ex- tend this by saying that — 5 multiplied by 3 shall mean — 5 — 5 — 5, that is, (2) (_ 5) . 3 = - 5 - 5 - 5 = - 15. If the multiplier is negative, it becomes impossible to think of multiplication as repeated addition. But we know that if both factors are positive, multiplication is commutative so that 5 • 3 = 3 • 5. If multiplication is to be defined in suph a way as to remain commutative, even if one factor is negative, we must liave (3) 3.(-5) = (-5).3 Aht. 19] MULTIPLICATION OF NUMBERS 27 and therefore, on account of (2) (4) 3.(-5) = -15. Let us now think of these numbers as the measures of directed line-segments. In cases (1) and (2) the multiplier is positive (equal to + 3), and the ec^uations show that the multiplication by the positive multiplier 8 merely stretches the line-segment which represents the multiplicand in the ratio of 3 : 1 without altering its direction. For a positive multiplier, the product has the same sign as the multiplicand. But (4) shows that multiplication by — 5 not merely stretches the line-segment in the ratio 5 : 1 but also reverses its direction. These remarks suggest the following defini- tion of multiplication for positive or negative numbers : To multiply a directed line-segment a, of a directed line, by a positive line-segment b, we merely stretch a in the ratio of 5:1. To mtdtiply a by a negative line-segment 6, we stretch a in the ratio of \b\'. 1 and also turn the resulting line- segment around through tivo right angles, thus reversing its direction. We may re-formulate the substance of this definition with- out any reference to Geometry as follows : T/ie numerical value of a product of two factors is equal to the product of the numerical values of the factors. The sign of the product is plus if both factors have the same sign, and minus if the signs of the ttvo factors are opposite. This statement includes the familiar rules expressed by the symbolic equations (5) ( + )(+)= -f, (-)(-)=+, ( + )(_) = _, (-)( + ) = -. It should be noted that these rules have not been proved. They are a part of the definition of multiplication. Our definition of multiplication is completed by consider- ing the case where one factor is equal to zero. 28 THE NUMBER SYSTEM OF ALGEBRA [Art. 20 If one of the factors of a product is equal to zero, the product is also equal to zero, that is, (6) a X = X a = 0. If a product is equal to zero, at least one of its factors- must vanish. For our definition of multiplication gives us a non-vanishing product whenever both factors are different from zero. Consequently both factors cannot be different from zero if the product is zero. The geometric construction for multiplication given in Art. 12 becomes applicable also to the case where one or both factors may be negative, if we think of the lines OX and Oy as directed lines and lay off the line-segments which represent the factors in one direction or another upon OX and OY according to their signs. It will usually be most convenient for this purpose to draw OX and OY at right angles to each other. 20. Division by a positive or negative divisor. As in the case of positive numbers, we think of division as the opera- tion inverse to multiplication. That is, if D is the dividend, d the divisor, and q the quotient, the problem involved in the statement ._, , V ^ d = q is that of finding a number q such that d q = D, where d and D are given. We have seen how to solve this problem, both arithmetically and geometrically, if D and d are both positive. From what was said in Art. 19 about multiplication it appears immediately that the following rule will hold : To divide D hy d where either or both numbers may be nega- tive, first proceed as though both were positive. Then give the quotient the plus or minus sign according as D and d have the same sign or opposite signs. Arts. 21,22] THE MONOTONIC LAWS 29 21. Division by zero. To divide I) hy d means to find a number q such that (1) D = dq. If D is not zero and t? = 0, this equation involves a contra- diction. For, if rf = 0, the right member of (1) will have the value zero, no matter what may be the value of q^ and this contradicts the assumption that D is not zero. Con- sequently, the notion of a quotient formed from a non-vanish- ing dividend and a zero divisor is self -contradictor i/ , This contradiction disappears if dividend and divisor are both equal to zero. In this case, however, equation (1) will be satisfied by any number q whatever, so that the quo- tient is entirely undetermined if both D and d are equal to zero. Thus, the notion of a quotient formed from a zero divi- dend and a zero divisor, while not self -contradictory^ is useless since it gives no determinate result. In neither case can we make use of division by zero. For these reasons, division by zero must be rigorously excluded from algebra. Whenever, in any algebraic argument, we perform a division we must always either prove that the divisor is different from zero, or else we must state explicitly that our conclusion has been proved only for those cases in which the divisor is not equal to zero. Neglect of these precautions easily leads to the most absurd results. 22. The monotonia laws for positive and negative numbers. The monotouic laws for positive numbers are concerned with inequalities and contain the symbols < and >. Before we can speak about the monotonic laws for negative numbers, we must extend the significance of these symbols so as to make them applicable to negative as well as positive num- bers. In order to do this, let us think of the scale of positive and negative numbers, and let tis think of a point ivhich moves along this scale in the direction from the negative toward the positive numbers. We shall then sag that a <. b {a precedes b, or a is less than 6) if this moving point reaches a before it reaches b. 30 THE NUMBER SYSTEM OF ALGEBRA [Art. 22 Moreover, if a a, that is, h follows a, or h is greater than a. In accordance with tliis definition we have, for instance, 3 < 5, — 2 < 0, — 5 < — 3, although 5 is greater than 3. It is now easy to see that, if we use the signs of inequality in accordance with this definition, addition will still be monotonia even though one or all of the numbers concerned may be negative. That is, if a> b, then a -\- c > b + c, whether a, 5, c are positive or negative. The monotonic law for multiplication.^ however., undergoes a slight but essential modification. If a > b, theti ac > be only if c is positive. If c is negative the conclusion to be drawn from a> b is, ac < be. Thus, from — 4 < — 3 follows — 8 < — 6 if yve multiply both mem- bers of this inequality by + 2. But if we multiply by — 2 we must change the sense of the inequality sign, since + 8 > + 6. The monotonic laws are essential in all questions involv- ing inequalities. Except for this slight change in the monotonic law for imdti- plication, the nine fundamental laws of Art. 2 hold for all posi- tive and negative ^lumbers. We have already proved this for the first three laws in Art. 18. The general validity of law IV has just been proved. The truth of laws V, VI, VII, and IX for the case of positive and negative numbers follows at once from our definition of multiplication, and we have just seen how the eighth law must be modified so as to remain valid in the case of a negative multiplier. EXERCISE IV Perform the following indicated operations arithmetically and also graphically. 1. 5 + (-3). 4. 6- (-8). 7. (-8) -(-2). 2. 5 -(-3). 5. (_7)(-4). 8. [5 + (- 3)][- 7 + 9]. 3. (-6) -7. 6.-8-4. 9. [5 + (- 3)] -^ [- 7+ 9]. Art. 23] DIRECTED LINE-SEGMENTS IN A PLANE 31 Simplify the following expressions. Justify each step of the trans- formation by appealing to one of the fundamental laws. 10. (a + b)(c-d). "• iriy^-'^' 11. (a-h)(c-(0- 14. (£_lj(^ + iy 16. Prove that | « + ft | = | a | + | 6| if « and h have the same sign, but that |a-f-6|<|n|+|ft|if the signs of a and h are not alike. Consequently we shall always have , , , ^ , •^ \'-i + fj\<[a\ + \b\, where the symbol < is read is not greater than. 17. Show that a- + h- >2ah\i a and h are not equal. Solution. If a and b are not equal, a — b may be positive or negative, but {a -hy= {a-b)(a - b) will be positive. (See Art. 19.) There- ^°''® «2 -Oah + b^ > 0. According to the monotonic law of addition, we may add '2 ab to both members without changing the sense of the "inequality. Therefore a- + b- > 2 ab as was to be proved. 18. Show that a- + b- + c'->ab + (ic + be. 19. Show that > -^^-^ if a and b are positive. 2 a + ft 20. Show that (/ + - > 2 if '/ is a positive number not equal to either a 1 or 0. Why this last restriction ? 21. Point out the error in the following argument. Let x = a. Then we find successively x^ = ax, x- — a^ = ax — a-, (x — (i)(x + a) — a(x — fl), X + rt = a. But since x = a, this gives 2 a — a and therefore 2 = L 23. Directed line-segments in a plane. We have repre- sented positive and negative numbers by directed line-seg- ments on a directed line, which we shall henceforth call the r-axis. From the point of view of Plane Geometry, however, the directed line-segments of this particular line, the ^--axis, are of no greater importance than the directed line-segments 32 THE NUMBER SYSTEM OF ALGEBRA [Art. 23 of any other line. We propose therefore to generalize the notion of a directed line-segment, by admitting that such a segment may be situated anywhere in the plane, not neces- sarily upon the a;-axis. We shall regard tivo directed line-segments (such as AB and A' B' in Fig. 14) as equal, if they have the same length, if they are on the same or parallel lines, and if they have the same sense. In applying this definition of equality, *-* we must take into account all of the three characters mentioned. Thus, in Fig. 14, AB and B'A ' are not equal although they have the same length, and are on parallel lines. Their senses are opposite. As in Art. 17 we may think of the directed line-segment as generated by a point which starts from one end-point and moves without reversing its direction, until it reaches the other end-point. Thus, we reach the notion of the origin and terminus of such a directed line-segment. In Fig. 14 the origin of AB is A, its terminus is B. . We may always construct a directed line-segment OP with its origin at the origin of the number-scale of the a^-axis, and equal to any given directed line-segment AB. (See Fig. 14.) We then say that the line-segment AB has been placed in its standard position OP. The length r of the line-segment is called its modulus or absolute value and is always a positive number ; the angle 6 which it makes with the 2;-axis is called its amplitude or argument. We see that two numbers (r and ^) must be given before we can regard the directed line-segment OP, or its equal AB, as known. There is a second way of describing such a directed line-segment. We introduce a second scale of numbers, just like the a:-scale, on a line perpendicular to the rc-axis through the point 0. We place the origin of this second scale (the y-scale) also at and place its positive end, as in Fig. 15, so as to be above the a;-axis. If now we project the directed +y .v y .r F ^ y •c J I i+y R F / E -V A\ > N -yP M C D Art. 23] DIRECTED LINE-SEGMENTS IN A PLANE 33 line-segment OP on the aj-axis and also on the y-axis, we obtain two directed line-segments, OM and OiV, to each of which will correspond as its measure a positive or negative number, on its own scale. We shall call these numbers x and t/ respectively. Their numerical values give the lengths of the segments OM and OiV respectively, while their signs indicate whether M is to the right or left of 0, and whether iVis above or below 0. (In Fig. 15, x and y are both posi- tive.) The numbers x and y are called the components of the directed line-segment OP. It is clear that any directed line-segment AB, which is equal to OP but which is not in its standard position, will have the same components as OP. See Fig. 16, where OP and AB are equal directed line-segments, and where the components CD and UF of AB are equal to the com- ponents OM and OiV of OP respectively. If both of the components of OP are given, in magnitude and sign, we can clearly find OP itself by a simple construc- tion. Therefore the components of a directed line-segment determine it just as completely as its modulus and amplitude. If we are acquainted with Trigonometry we can easily compute the components when the modulus and amplitude are given.* From Fig. 15 we see at once that (1) X = r cos 6, y = r sin 6 We have agreed already that the modulus r shall always be regarded as a positive number. It is desirable to define the amplitude a little more precisely than we have done so far. The amplitude 0, of a directed line-segment OP in its standard position, is the angle generated hy a line which origi- nally coincides with the positive x-axis and tvhich rotates around as a center., in a counter-clocku'ise direction (toivard the posi- tive y- axis')., until it coincides with OP. * Students who have not studied Trigonometry may omit those parts of this chapter iu whicrh Triijonometry is used. Alternative treatments are given not involving a knowledge of Trigonometry. 34 THE NUMBER SYSTEM OF ALGEBRA [Art. 24 Therefore the amplitude need not be an acute angle. Thus, in Fig. 17 the amplitude of OP is obtuse, that of OP' is in the third, and that of OP" in the fourth quadrant. For some purposes it is important to remember that the detinition of ampli- tude just given, does not determine the amplitude without ambiguity. The last „ words of the definition do not read : until Fig. 17 it coincides with OP for the first time. Consequently we may add any integral multiple of 360° to the amplitude of a directed line-segment, and the resulting angle may still be regarded as the amplitude of that line-segment. Obviously we may now look upon the negative numbers of the a;-axis as corresponding to directed line-segments of amplitude 180° or tt radians. Thus the modulus of — 4 is 4; its amplitude is 180°. The truth of equations (1) for the case where 6 is an acute angle is evident from Fig. 15. But these equations are also true if is in any quadrant. That this is so will become apparent if we compare these equations with those which are used for the purpose of defining the sine and cosine of a general angle. Either from (1) or directly from Fig. 15, we find (2) r = + V.r2 + ^2, tan 9 = •^, thus enabling us to compute the modulus and amplitude of a directed line-segment when its components are given. However, the second equation alone does not suffice to deter- mine 6 unambiguously as to its quadrant, since there are two angles in the first four quadrants (differing by 180°) which have the same tangent. However, since x and y are given, and r is positive, equations (1) tell us by inspection the sign of sin 6 and cos ^, and therefore the quadrant of 6. 24. The complex numbers. We have seen that a directed line-segment, such as OF in Fig. 15, is determined in mag- nitude and direction by jneans of its components, x and y. Art. 24] THE COMPLEX NUMBERS 35 •2 -1 o +y +3t In particular, if ?/ = 0, OP becomes a directed line-segment on the rr-axis and may be regarded as representing a positive or negative number, as in Art. 16. If a: = 0, OP becomes a directed line-segment on the y-axis, and we might again associate with such a line-segment a positive or negative number. Unless we adopt some notation, therefore, which will enable us to distinguish at a glance between directed line-segments on the a:-axis and directed line-segments on the y-axis, the number + 3 might be thought of as a line-segment on either axis. In order to avoid this ambiguity, we label the scale of numbers on the ?/-axis with tlie symbols + i, +2 i, + 3 z, etc., — i, — 2 z, — 3 i, etc. (See Fig. 18.) The four directed line-seg- ments OA, OB, 00, OD in Fig. 18, all of length 2 but having different directions, may now very briefly be denoted by -h 2, -f 2 2, — 2, — 2 i respectively. More generally ive may noiv represent any directed line-seg- ment OP, ivhose componeyits are x and y, hy the symbol z = X + yi. Thus in Fig. 18, OP will be represented by 3 + 2 /, OQ by — 1 + 2 1. + 1 +2 +3 +4 -2i Fig. 18 A symbol of the for7n x -f yi, lohich may he regarded as representing a directed line-segment of the plane, is called a complex number. This complex number is said to have the positive or negative numbers x and y as its components. Thus a complex number is really a symbol involving a pair of ordinary numbers x and y. The numbers which we have considered so far might, by way of contrast, be called simple numbers. We may regard the symbol x + yi as a description of a path -which leads from to P. Thus, in Fig. 18, the symbol 3 + 2 « tells us to start at 0, to go 3 units in the direction of the positive ar-axis, and then 36 THE NUMBER SYSTEM OF ALGEBRA [Arts. 25, 26 2 units in the direction of the positive 7/-axis. If the x-axis points east and the y-axis north, we may regard the symbol i as an abbreviation for north and — i for south. 25. Equality of two complex numbers. We have intro- duced the complex numbers x + ^i as symbols for directed line-segments of the plane, and we have agreed that two such line-segments are to be regarded as equal if they are of the same length, if they are on parallel lines, and if they have tlie same sense. The symbols for two equal directed line-segments will, therefore, be identical, since equal directed line-segments have their corresponding components equal. (See Fig, 16.) It is also evident that two line-segments which are not equal, in the sense of the above definition, will not have their corresponding components equal. These remarks lead us to the following definition of equality of two complex numbers: Two complex numbers, x + yi and x' + y'i, are said to he equal, if and only if their corresponding components are equal, that is, if ayid only if , _ , _ X — X, y — y. In particular, if x + yi = 0, then x= 0, y = 0. We see, therefore, that a single equation between two com- plex numbers implies two equations between their components. The discussion in Art. 23 shows us that a complex num- ber may also be said to have a inodulus and an amplitude. It follows immediately that two equal complex numbers have equal moduli while the amplitudes of two equal complex num- bers 7ieed not be equal; it suffices that the difference between them be an integral multiple of 360°. 26. Vectors, vector addition, and addition of complex numbers. It remains to justify the use of the sign + in the symbol x -f yi. If we write the number x alone, we may think of a; as a directed line-segment OM on the a;-axis. (See Fig. 10.) We may, a little more concretely, think of OMnH a displacement which has the effect of moving a point from to M, and which moves every other point of the Art. 26] VECTORS, VECTOR ADDITION 37 +y X X f A y V X U Fig. 1!) plane through a distance equal to OM along- a line parallel to the a;-axis. Similarly we may think of yi as the symbol for a displacement which moves every point of the plane through a distance equal to ON along a line parallel to the y-axis. Now let us think of these two displacements as being made in succession. The first displacement, symbolized by x^ would move a point from to M', the second displacement would move tliis point, which is now at iHf, from M to P. Since the same result would have been obtained if the point had been moved directly from to P along the straight line path OP, we may say that the resultant of these two dis- placements is that one which is represented in magnitude and direction by OP. Conversely, the two displacements OiUfand OiVare called rectangular components of OP. It is customary to speak of the resultant of two displacements as their geometric sum. We may therefore actually regard X + yi as the sum (in this geometric sense) of x and yi. The actual relation between the magnitudes of the resultant and of the components is given by oP=6m^^-on^, as is evident from the figure. These considerations suggest the following generalization. Let OP and OQ (Fig. 20) represent two displacements. By virtue of the first displacement every point of the plane would move along a line parallel to (9P, through a distance equal to the length of OP, in the sense from toward P. The directed line-segment OQ represents in similar fashion the character- istic properties of the second displacement. It is clear that the resultant of the two displacements made in succession is again a single displacement, represented by Oi2, where OB, is that diagonal of the parallelogram deter- mined by OP and OQ which passes through 0. 38 THE NUMBER SYSTEM OF ALGEBRA [Art. 26 If again we use the word sum, instead of resultant, we may say that the sum of the displacements OP and OQ is, the displacement OR. Let us now represent these displacements by complex numbers, z and z' . If the components of OP are x and y^ and those of 0'^ are x' and y', we have (1) z = X + yi, z' = x' + y'i. We see from Fig. 20 that the components of OR are x -\- x' and y -\- y' . Consequently we obtain the following result : If the ivord sum, ivlien applied to two displacements., he regarded as synonymous with the word resultant, and if the complex numbers x + yi and x' + y'i are used as symbols for two displacements ivhose components are (a:, ?/) and (x' , y'^ respectively, then the sum of these displacements is a displace- ment ivhose symbol ivill be the complex number X + x' +(y i- y')i. We are thus led to define x -\- x' -\- (^y + y'~)i as the sum of X + yi and x' +y'i, and we ivrite (2) X -{- yi + (x' + y'i')= x + x' + Cy + y')i. There are many instances of quantities which combine into a resultant in accordance with the parallelogram law illustrated in Fig. 20. If a steamer is moving north with a velocity of 800 feet per minute, and a passenger is crossing the deck walking eastward at the rate of 300 feet per minute, his actual velocity in space may be obtained as, in Fig. 21, by laying off a line-segment OB, 800 units long toward the north, a line-segment OA, 300 units long toward the east, and completing the paral- lelogram which, in this case, is a rectangle. The number of units in OC (obtained by measurement or b}'" calculation) will give us the number of feet per minute which represents the passenger's speed in space, and the direction of 00, measured by the angle AOC, will give us the direction of the Art. 26] VECTORS, VECTOR ADDITION 39 resulting velocity. Thus, velocities are compounded according to the parallelogram law. It is a familiar fact that the resultant of two forces attacking a body at the same point is also found by applying the parallelogram law, and there are many other instances of such quantities. Directed quantities, such as displacements, velocities, and forces, which combine in accordance with the parallelogram law, are called vectors.* By a proper choice of the units every vector may be represented by a directed line-segment, or, what amounts to the same thing, by a displacement. We shall, from now on, use the word vector instead of directed line-segment. We may then say, referring again to Fig. 20, that (3) vector OF + vector OQ = vector OR. Of course this does not mean that the length of OP -j- the length oi OQ is equal to the length of OM. In fact the figure shows that the length of OR is always either less than this sum or at most equal to it, the case of equality presenting itself only if the Vectors OP, OQ, and OR have the same direction. This simple remark may be formulated alge- braically as follows : Let z = X -\- yi and z' = x' + y'i be the complex numbers which correspond as symbols to the vectors OP and OQ, and let r and r' be their respective moduli or absolute values. Then r and r' are the lengths of these vectors, and r = Va-^ + y^, r' = Va:'^ + y'^. It is customary to uae the symbol \ x -|- yi \ for the absolute value or modulus of a complex mimber. The observation just made, that the length of OR is at most equal to the sum of the * From the Latin vector, meaning one who carries or conveys. The student should observe that the parallelogram law may be regarded as a generalization of the geometric definition for addition wliich was given in Art. 18, for the case where the line-segments OA and O'A' mentioned in Art. 18 are not situated on the same or on parallel lines. 40 THE NUMBER SYSTEM OF ALGEBRA [Art. 27 RJx-t-yi) lengths of OP and OQ, may therefore be written as follows; (4) \x + i/i + x' + i/'i\ <\x + ^i\ -\- \x' + y'i \ where the symbol < is to be read either " is not greater than,'''' or "is less than or at most equal to.^^ It is apparent from Fig. 20 that tlie length of OM, that is, the absolute value of a; + i/i + a;' + y'i is exactly equal to (5) -y/{x + xy + iy + i/'y. 27. Subtraction of vectors and complex numbers. If we have vector OP + vector 0^= vector OR, we say that Vector 0<^ = vector 0^ — vector OP, thus defining sub- traction of vectors. We regard the diagonal OM, and one of the sides OP of the parallelogram as given ; the problem of subtraction consists in finding the other side OQ of the parallelogram. The geometric so- lution is so simple as to make ex- plicit directions- unnecessary. See Fig. 22. The corresponding for- mula for the subtraction of complex numbers is just as simple. We find (1) (x + i/i) - (x' + y'i) = x — x' + (y — y')i, where x and y are the components of OR the minuend, and x', y' are the components of the subtrahend OP. The absolute value of the difference, that is, the length of OQ or PR, is Fig. 22 (2) \x + yi - (2.-' + y'i-) ' = -^^(^x-x')''-\-(^y-y') ' \2 EXERCISE V* Plot the vectors represented by the following complex numbers, and find their moduli and arguments approximately by measurement: 1. 1 + 1. 3. -3 - 2«. 5. -7. 2. -2 + 3i. 4. 5 - 2 I. 6. 8 i. * An arithmetical solution may be obtained, where a graphic solution is called for in these examples, by those students who liave studied Trigonometry. Art. 28] MULTIPLICATION OF COMPLEX NUMBERS 41 Plot the following vectors whose moduli and arguments are given. Find their components approximately from the figure by measurement, and write the corresponding complex numbers. 7. ,=1,^ = 45°. 10. r = 4, ^ = 315°. 8. 7- = 3, ^ = 135°. 11. r = 5,d= 180°. 9. r = 2.6= 225°. 12. r = 7,d = 90°. What must be the values of x and y in order that the following equa- tions may be true? 13. X + ij + i(x - y) = 2 + 4 i. 14. 2 x + 7 1/ + i(S X - 2y) = - 3 - i. Perform the following operations algebraically and graphically. 15. 1 + £ + (2 + 3 0- 19- 7 + 8 I - (5 + 6 i). 16. 1 + t- (2 + 3 0- 20. - 1 +t + (2 + 3 - (1+5 /). 17. 3 - 5 I + (3 + 5 t). 21. 5.6 + 7.8 i + ( - 3.2 + 4.7 i). 18. 3-5 i - (3 - 5 0- 22. 5.6 + 7.8 i - (3.2 + 4.7 i)- 23. A horizontal force of 10 pounds and a vertical force of 24 pounds are acting simultaneously on a point. Represent the resulting force as a complex number. 24. A schooner is sailing due west at the rate of 6 miles per hour. A sailor is crossing the deck, from south to north, at the rate of 3 miles per hour. Represent the resulting velocity of the sailor as a complex number. 25. Two forces are represented by the complex numbers 3 + 5 i and — 4 + 6 I respectively. Find the complex number which represents their resultant. If the x-axis is horizontal, the y-axis is vertical, and if the unit of force used is a pound, find approximately, by a graphic solu- tion, the magnitude and direction of the resultant. 28. Multiplication of a complex number by a positive or negative number. //' m is an// positive or negative number or zero, ive define the product m{x + yi) to he equal to mx +- myi. If m is positive, the corresponding product vector will be m times as long as the vector x + yi, but its direction will be the same. If m is negative, the direction of the product vector will be opposite to that oi x + yi. This definition is a necessary consequence of Art. 2fl in the case where m is a positive integer, if multiplication by a positive integer be interpreted as repeated addition. This same definition must also be 42 THE NUMBER SYSTEM OF ALGEBRA [Art. 29 adopted in all other cases if we wish to define multiplication of a com- plex number by m iu such a way as to have it resemble multiplication of two positive or negative numbers in so ftir as to preserve the validity of the commutative and distributive laws iu this case also. 29. Multiplication of a complex number by /. Tlie num- bers on the positive rr-axis were denoted by +1, +2, +3, etc. ; those on the positive y-axis by + i, +2 i, + 3 i, etc. This notation suggests the possibility of regarding 3 i as a product obtained by multiplying i by 3, a point of view which is entirely consistent with Art. 28. But we may also think of 3 i as the product obtained by multiplying + 3 by i ; in fact we must do so, if we wish multiplication by i to be commutative, that is, if we wish 3 x i to be equal to i x 3. If we adopt this point of view, we see that multiplication by i of any vector on the positive a;-axis, such as OM in Fig. 23, will produce a vector ON of equal length on the positive ?/-axis. We may express this by saying tliat mul- tiplication by i has the effect of turning the vector OM through a right angle in the counter-clockwise direction. If we wish to make a consistent use of this idea, we should agree that, to multiply ON by ^, means to turn ON +y also through a right angle in the counter- clockwise direction. But this operation would convert ON into OM', a vector of the M' o ~M^ same length as OM but having the opposite Fig. 23 direction. If OM is of length x units, we shall have these three vectors represented by the following complex numbers (see Fig. 23): OM by X, ON by xi, OM' l^y - x. But, since OM' was also obtained from ON by multiplica- tion with 2, we may also represent OM' by the symbol (xi)i or xi^, and the two symbols for OM' , x^ and —a;, will be identical if, and only if we agree that, (1) ^^ = ^•2 = -l. Let us adopt this agreement, and let us further agree that, in all cases, multiplication of a; + yi by i shall obey the cora- N' Art. 29] MULTIPLICATION OF COMPLEX NUMBERS 43 mutative, associative, and distributive laws of multiplication, so that (2) i{x + yi) = ix + ii/i = ix -{-yi^ = ix — y = — y + xi. Let OP (Fig. 24) be tlie vector whose components are a: = 6>if, y = MP. Let OP' be a vector of the same length as OP, obtained by turning OP through an angle of 90° in the counter-clockwise direction. Then OP' will make the same angle witli the y-axis which OP makes with the 2:-axis, and the triangles' OP'M' and 0PM will be equal. Therefore OM' = x and P'M' = y. The components of the vector OP' are M' P' and OM' respec- tively. If we denote them by x' and ?/', we shall have M'P' = x' = -y, OM'=y' = + x. so that the complex number which represents OP' will be ^ -\- y'i = — y + xi, which, according to (2), is the same as i(x-{-yi). We have therefore the following result : If we assume that the operation of multiplying a complex number x + yi by i obeys the commutative, associative, and dis- tributive laivs, then i^ ivill be equal to —\ and such a multi- plication is geometrically equivalent to rotating the vector represented by x -f- yi through a right angle in the counter-clock- wise direction. EXERCISE VI Perform arithmetically and graphically the following multiplications. Multiply : 1. 2 i by 3. 6. (4 + 3 0'- 2. 2i by -3. 7. (-4 - 3 0'- 3. 3-0 i by 2. 8. [2 + 5 / - (3 + 2 /)] t'. 4. 3 - 5 i by - 2. 9. [(- 2 + 3 Oi]3. 5. 3 - 5i by t. 10. {[4 -I- i -(-2-30]t}(-2). 44 THE NUMBER SYSTEM OF ALGEBRA [Arts. 30, 31 30. Polar form of a complex number.* Let r and 6 denote the modulus and amplitude of the complex quantity x + yi. We shall then have x=. r cos 6, y = r sin 6 (see Art. 23, Fig. 15), so that we may write (1) X -\- yi = r(cos 6 -{- i sin ^). Every complex number can be written in the form (1), wliich is called the polar form, or the trigonometric form of the complex number. The polar form is especially convenient when two or more complex numbers are to be combined by multiplication or division. 31. Multiplication of two complex numbers. Let x + yi and x' + iy' be any two complex numbers, and let us form their product, assuming that the commutative, associative, and distributive laws are valid, and that i"^ is equal to — 1. These are the same assumptions which we made in Art. 29. We find , , (x + yi^Qx + y'i^ = xx' + xy t + yix + yiy i = xx' + xy'i + x'yi — yy', or finally (1) (x + yi) {x' + y'i) = xx' - yy' + (xy' + x'y)i. But this result may be expressed in a very simple form if we make use of the polar form of the complex numbers involved. t Let r and r' be the moduli, and 6 and 6' the amplitudes of the two complex numbers x + yi and x' + y'i respectively, so that in accordance with (1) of Art. 30, X + yi = r(cos 6 + i sin ^), x' -f y'i = /(cos 6' + i sin 6'). We find (2) (x + yi) (x' + y'i) = rr' [cos cos 0' - sin sin 0' + ^(sin cos 0' + cos sin 0' )] . * To be omitted by students who have not studied Trigonometry t Students who have not studied Trigonometry will tind an alternative treat- ment of this subject in fine print toward tlie end of tliis article. Art. 31] MULTIPLICATION OF COMPLEX NUMBERS 45 But, according to the addition formulae, of the trigonometric functions, we have cos cos 6' - sin 6 sin 0' = cos(^ + 0')* sin cos 0' + cos sin 0' = sin((9 + 0'). Consequently we may write (2) as follows : (3) (x+7jiy(x' + i/'i)=rr'[co^{0 + 0')+ /sin (0+0'-)], so that tlie product is a complex quantity whose modulus is rr' and whose amplitude is ^ + 0'. We have proved the following theorem : Theorem I. If the associative, commutative, and distrib- utive laws of multiplication are assumed to hold for all com- plex numbers, the modulus of the product of two complex numbers is equal to the product of the moduli of the factors, and the amplitude of the product is equal to the sum of the ampli- tudes of the factors. We may express this in a different way. Let us speak of X + yi as the multiplicand, and of x' + y'i as the multiplier. According to (3), the product will be represented by a vector of length rr' making an angle + 0' with the x-axis. Now the multiplicand was represented by a vector of length r making an angle with the a:-axis. We shall therefore obtain the product vector from the multiplicand vector by rotating the latter through an angle 0' and stretching it in the ratio of r' : 1, where r' and 0' are the modulus and ampli- tude of the multiplier. Thus we find Theorem II. The product vector is obtained by rotating the multiplicand vector through an angle equal to the amplitude of the multiplier, and at the same time stretching the multipli- cand vector in the ratio r' : 1, ivhere r' is the modulus of the multiplier. * Wilczynski-Slaught. Plane Trif/onometry and Applications (Allyn and Bacon), p. 184. Hereafter this book will be referred to as Plane Trigonometry and Applications, 46 THE NUMBER SYSTEM OF ALGEBRA [Art. 31 In Fig. 25, OP represents the multiplier, OP' the multi- plicand, and OQ the product. Let U be at unit distance from on the positive a:-axis. Then the angles UOP and P' OQ are both equal to 6. and OQ: OP' = OP: OU, since 0U=1, OP = r, OP' = /, OQ = rr'. Consequently the triangles OUP and OP'Q are similar. This remark may be formu- lated as follows : Theokem III. If each of Uvo complex numbers he repre- sented hy a vector^ the vector ivhich represents their product may he found hy the following construction. Construct the vector Z7, of unit lengthy on the positive x-axis, the vector OP repre- senting the multiplier x -\- yi, and the vector OP' representing the multiplicand x' + y'i. Construct a triangle OP' Q similar to the triangle OUP. Then OQ will he the vector which represents the product (x + yi^(x' + y'i^- This last theorem may also be proved without making use of Trigo- nometry. Equation (1) shows that the product of x + iji and x' + y'i is a vector OQ whose components are x" = xx' — yy', y" — xy' + x'y. Let us compute the length | OQ \ of this product vector OQ. We have 1 OQ I '^ = x"'^ + 7/"2 = {xx' — yy'y + {xy' + x'y)'^ = x'^x''^ — 2 xyx'y' + y'^y''^ + x'^y'^ + 2 xyx'y' + x'^y"^ = x'^x''^ + y^y'^ + x'^y''^+ x''^y'^ = x'^{x'^ + y''^) + y\x'^ + y/'2) = (x2 + 2/2) (x'2 + 2/'2), whence (4) \0Q\=^ Vx2 + y^ Vx'-^ + y'-\ But we have (5) \0P\= Vx2 + if, I OP' I = Vx'2 + y'\ SO that (4) becomes (6) 1 OQ I = I OP I • I OP' I, telling us that the modulus, \ OQ |, of the product is equal to the product of the moduli, \ OP | and \ OP' |, of the factors. Akt. 32] DIVISION OF COMPLEX NUMBERS 47 To prove Theorem III we now refer to Fig. 25. The symbol of the vector OU h 1 + • i and we have | 0U\ = 1 ; we may therefore write (Q) as follows : ^ ^ |(>(^1:10P| = \0P'\ : I Of/I. To prove that the triangles OUP and OP'Q are similar, it will suffice to show further that (7) \P'Q\:\UP\ = \OP'\:\OU\. Making use of the notion of adding of vectors (see Art. 2G, equation (3)), we have vector OP = vector OU + vector UP, vector OQ — vector OP' + vector P'Q, and therefore, vector UP = vector OP — vector OU = x + i/i — 1, vector P'Q = vector OQ — vector OP' — x" + y"i — {x' + y'i) = x" - x' +{u" - u')i, so that (8) \UP\= \^{x - 1)2 + ,/, \P'Q\= V(a:" - x')2+(?/" - >j'y. The components x" and rj" of the product vector were given by x" = xx' — yij', y" = xy' + x'//, so that x" — x' = (x — l)x' — yy', y" — y' =(x — l)y' + x'y. Consequently we find from (8) I P'Q I = V(x - 1)V2 - 2 x'y'{x - \)y + ijhf^ + (x - l)Y2+2 x'y'{x-\)y + x'-y''- = \/(x - l)2(x'2 + ?/'2) + y2(x'2 + y'-'^) = Vx"-2 + ^'2V(x - 1)^ + y% which reduces to (9) I P'Q I = I OP' \ ■ \ UP\ on account of (5) and (8). But (!>) is equivalent to (7) since | 0U\ = 1. Consequently we have now proved Theorem III. Theorems I and II are merely diffei-ent ways of expressing the same relations and are immedi- ate consequences of Theorem III. 32. Division of complex numbers. If we define division as the operation inverse to multiplication, Theorem I of Art. 31 gives us the following result without any calculation. Theorem. The modulus of a quotient h equal to the modu- lus of the dividend divided hy the modulus of the divisor ; the 48 THE NUMBER SYSTEM OF ALGEBRA [Art. 32 amplitude of the quotient is equal to the amplitude of the divi- dend minus the amplitude of the divisor. We may liowever prove the same theorem by the follow- ing calculation. We have r(cos 6 + i sin 6) _ r(cos 6 -\- i sin ^)(cos 6' — i sin 6' ) r'(cos^' + zsin(9') ~ ?7(cos 6' + i sin <9')(cos 6' - i sin 9') _ r cos 6 cos 6' + sin 6 sin 6' + i(sin 6 cos 6' — cos 6 sin ^') r' cos^ 6 + sin^ ^ But cos d cos 6'' + «in 6 sin ^' = cos((9 - (9'), sin ^ cos 6' - cos ^ sin 6' = sin(^ - 6'^, cos2 (9 + sin2 e = \* Consequently we find ,-1^ rCcos ^ + « sin ^) r r- .n /j^x , • • ,a m^-i <^1) -77 ^, . . - ^, =-[cos(^-6'^) + ^Sln(6'-^0], r (cos a' + z sm c'') r and this formula is equivalent to the above theorem. If the complex quantities are given in the form x + yi and x' + y'^, instead of in the polar form, their quotient may be found as follows. We have ,.-,. X + yi __ X -{- yi x' — y'i _ xx' + yy' + (x' y — xy')i x' + y'i ~ x' + y'i x' - y'i ~ x'^ + y'^ This formula shows that the quotient is again a complex number x" + y"i, whose components are <-qN ,, ^ xx' +yy' f, ^ x'y-xy' ^^ x'^ + y'^' ^ x'^+y'^ Of course x" and y" may be positive or negative. The}' are always definite numbers obtained from x, y, :c', and ?/', by (3), unless x' and y' are both equal to zero, that is, unless the divisor x' + y'i is equal to zero. Therefore, division by zero is excluded from the algebra of complex numbers just as ynuch as from the algebra of real numbers. Two complex numbers * Plane Trigonometry and Applications, pp. 148 and 186. AuT. ;5_>] OPERATIONS WITH COMPLEX NUMBERS 49 always have a uniquely determined complex number for their quotient unless the divisor is equal to zero. The laws of multiplication and division of complex num- bers are used in many parts of applied mathematics. It is not feasible, however, to present any of these applications in this book because tlie preliminary notions needed from Physics are so complicated as to- necessitate a lengthy dis- cussion. EXERCISE VII Perform the following multiplicatiou.s and divisions both algebraically and graphically : 1. (:5 + 5 0C2+3/). 6. (3 + 5 /) - (2 + 3 0- 2. (3 - 5 /) (2 - ;} 0- 7. (3 - 5 - (2 - 3 i). 3. (-4 + 20(1 + 0. 8. (-4 + 20-(l + 0- 4. (2 + 3 i)-. 9. 1 ^ (2 + 3 0^. 5. (1 + 0-" 10. 1 -^(1 + ly. Reduce the following expressions, in whicli n, h, c. d, etc. represent positive or negative numbers, to the form of a complex number A + Bi: 11. (a + Ih)(c - di). 14. (rt + hi) ^ (c - di). 12. (a - hi){c + di). 15. {a - bi) ^ (c + di). 13. (a + hi) (a - bi). 16. (a + bi) ^ (a - bi). Write the following complex numbers in the form x + yi:* 17. ;)(cos 30° + i sin 30°). 21. 3(cos 90° + i sin 90°). 18. 3(cos 150° + / sin 150°). 22. 2(cos 180° + i sin 180°). 19. rj(cos 225° + i sin 225°). 23. 3(cos 270° + i sin 270°). 20. 4(cos 0° + i sin 0°). 24. 4(cos 360° + i sin 360°. Write the following complex numbers in the polar form : 25. 1 + /. 29. I + I y/■^ i. 33. 2 - 3 i. 26. - 1 + /. 30. - f + f V.^ ;. 34. - 5 i. 27. - 1 - /. 31. - f - f \/3 I. 35. 5-4 i. 28. \-i. 32. + 1 - ^ >/3 i. 36. -3 + 4 i. * Examples 17-40 presuppose knowledge of Trigonometry. 50 THE NUMBER SYSTEM OF ALGEBRA [Art. 33 Perform the following multiplications and divisions by first reducing the complex numbers concerned to their polar form. 37. (3 + V3 (2 + 2 /) . 39. ( - J + i 2 v'3)8. 38. (1 + 0(2 - 2 V3 /)• 40. (1 + ^ (2 + |V3 O- 33. Real and imaginary numbers. The simple numbers, that is, the positive and negative numbers and zero, are often spoken of as real numbers. Of course, a complex number, x-\-yi^ reduces to such a real number when «/= 0, so that the real numbers are included among the complex numbers. Tliose complex numbers x + yi in which y is not equal to zero are frequently called imaginary numbers. The reason for this peculiar nomenclature is not hard to understand. No real number, positive, zero, or negative, has a negative square. Therefore, in the domain of real num- bers, an equation such as (1) a:2 = - 1 has no solution. Mathematicians, several centuries ago, found it convenient, nevertheless, to treat an equation like a;2 = — 1 according to the same rules as were used for equa- tions, such as x^ = 2, which do have real solutions. This led to such symbols as V— 1. It was clear to them, of course, that the symbol V— 1 could not represent a real number (that is, a positive or negative number), and it seemed to them that this symbol could not be regarded as a number at all. To express all of these doubts these symbols, whose significance was not understood, were called imaginary num- bers, and the word still persists. From our present point of view, however, any complex number represents a real thing, namely a vector having a definite length and direction. And from this point of view the equation x^ = — 1 has two solu- tions. In fact we found in Art. 29 that the vector one unit long in the direction of the positive ^/-axis, which we called i, is such that i^ = — 1 and it is apparent that (— z')^ is also equal to — 1. Thus we see that the equation (1) is satisfied by '' X = I QV X = — I. Arts. 34, 35] VALIDITY OF FUNDAMENTAL LAWS 51 We now see further that we may identify our symbol i with the symbol V— 1. The symbol V— 1 was, from the old point of view, not a real number, but something imaginary, and the letter i which we still use for it is accounted for by this fact. In the technical sense we shall still say that i is an imaginary number, meaning that it is not a positive or negative number nor zero. But it is, in our interpretation, just as real a thing as the numbers 4- 1» + 24, or — 3. Every complex number is of the form x + yi- We shall, henceforth, speak of a: as its real part and of yi as its imagi- nary part. If the real part of a complex number is zero, the number is said to be purely imaginary. 34. Conjugate complex numbers. Ttvo complex numbers such as , • J • X + yi and x — yi ivhose real parts are the same, and for which the coefficients of i are numerically equal but opposite in sign, are said to be conjugate. The truth of the following statements is evident : The sum and product of two conjugate complex quantities are both real. In fact we have (1) (x+ yi^ + (x — yi) = 'lx, {x + yi)(x - yi) = x^-^ y^. The difference between two conjugate complex numbers is purely imaginary. (2) {x + yi) -i^- yi) = 2 yi. 35. Validity of the fundamental laws for complex numbers. It is easy to verify that, with the exception of IV and VIII, the fundamental laws of Art. 2 are all satisfied, if the symbols there used to denote positive integei's are now regarded as standing for complex numbers. Laws IV and VIII, the monotonic laws of addition and multiplication, are not ap- plicable to complex numbers for the simple reason that complex numbers cannot be arranged in a simply ordered 62 THE NUMBER SYSTEM OF ALGEBRA [Art. 36 sequence as is the case for real numbers. In other words, the fundamental notions " greater than " and " less than " have no simple significance for complex numbers. As a consequence of the facts just noted, we may manipu- late equations involving complex numbers according to the same rules as though the numbers concerned were real. For the monotonic laws have nothing to do with equations ; they are concerned with inequalities only. 36. History of the number system of Algebra. The successive generalizations which have enriched the number concept, beginning with the primitive idea of a positive integer and leading up to the general notion of a complex number, were not obtained by sudden inspiration, but as a result of the painstaking work of mathematicians throughout thousands of years. Even our present method of writing numbers, the conven- ience of which can be appreciated only by contrasting it with the clumsy methods used by the Greeks and Romans, was the result of a long proc- ess of evolution. The most essential feature of this system, its positional character, was rendered possible only by the invention of a symbol for zero. It is usually conceded that the Hindu mathematicians of the sixth century a.d. took this step, although there seems to be some evidence that the Babylonians also had a symbol for zero. The characters 0, 1,2, etc. which we use nowadays are also supposed to be of Hindu origin. We usually speak of them as Arabic figures, because they were trans- mitted to the nations of western Europe through the Arabs, who had previously received them from the Hindus. The oldest mathematical manuscript with which we are acquainted, the so-called Rhind papyrus, written by the Egyptian Ahmes about 1700 B.C., contains many calculations which involve fractions. P^xcept for the fraction 2/3 all of the fractions used by Ahmes were unit fractions, that is, such fractions as 1/2, 1/3, 1/4, etc., whose numerators are equal to unity. The Babylonian astronomers introduced the system of sexagesimal frac- tions, which was far superior, for purposes of practical reckoning, to the system of unit fractions used by Ahmes. In this system every unit is divided into sixty equal parts, each of these is divided into sixty smaller parts, etc. The sexagesimal system was adopted by Ptolemy of Alexandria (about 1.50 a.d.) in his famous treatise on Astronomy, the Abnar/esi, and remained in general use for scientific purposes until the sixteenth century, when it was replaced by tlie modern decimal system. We still have important traces of the sexagesimal system in our reckoning of time and angles. Thus we divide an hour into sixty minutes, and a minute into sixty seconds; we also divide a degree of arc into sixty minutes, and a minute of arc into sixty seconds. Art. 36] HISTORY OF THE NUMBER SYSTEM 63 As was mentioned in Art. 15, the discovery of the distinction between rational and irrational quantities is usually ascribed to Pythagoras. But Pythagoras and the Greek mathematicians who followed him made this distinction in an exaggerated fashion. They refused to regard irra- tional quantities as numbers at all, and did not introduce them into their Arithmetic and Algebra. Further progress in this direction was made possible only by the Hindus, especially by Bhaskara (1114 a.d.). The Hindus did not perhaps appreciate the fundamental character of this dis- tinction as clearly as the (ireeks, but they showed, by actual trial, that it was possible to use irrational numbers, and to calculate with them ac- cording to the same rules that hold for rational numbers. It was re- served for the nineteenth century to justify the procedure of the Hindus by a strictly logical theory of irrational numbers such as would have been acceptable to the Greeks. This theory is due to Uedekxnd, Cantor, and Weierstrass. Negative numbers also seem to have appeared first among the Hindus. Much later they gradually forced themselves also upon the attention of occidental mathematicians. Thus, we find that Leonardo of Pisa or Fibonacci (1180-12.50) accepts negative solutions of an equation in all cases where these negative numbers are capable of being interpreted as debts. The great Italian mathematicians of the Renaissance, Cardano, BoMHELLi (sixteenth century a.d.), and others followed Leonardo in this practice. They also began to use imaginaries in their calculations, although they regarded them with much suspicion. The geometric repre- sentation of positive and negative numbers as directed line-segments on a directed line was contained siibstantially in the famous Ge'ometrie of Descartes (1596-1650), one of the founders of what we now call Ana- lytic Geometry. As a result of this interpretation, it was generally recognized that negative numbers had established their citizenship in the republic of numbers. In the same way, complex numbers were regarded as mere phantoms and of no real significance, until an adequate geometric representation was found for them. This was accomplished independently in 1797 by Caspar Wessel, a Dane ; in 1806 by J. R. Argand, a French mathema- tician ; and finally in 1831 by C. F. Gauss.* The influence of the latter, who was probably the greatest mathematician of the nineteenth century, and the important applications which he made of complex numbers, finally established the complex numbers in the place which they have occupied ever since, as the only completely adequate number system of Algebra. * This is the representation upon which we have based our theory of complex numbers in Arts. 23-.34, and is frequently referred to as the Argand diagram. This representation defines a one-to-one correspondence between the points of the plane and the values of the complex variable x + yi. Consequently, a plane, to each of whose points there corresponds in this way a complex number, is often called the plane of the complex variable, or the Gauss plane. CHAPTER II LINEAR FUNCTIONS AND PROGRESSIONS 37. Constants and variables. When we have solved a numerical problem by the methods of elementary Arithmetic, we usually recognize that there are other numerical problems of the same kind which may be solved by applying the same methods. This leads to the formulation of certain rules, which state that in certain problems certain numbers must be added, subtracted, multiplied, or divided. Algebra, by introducing the notion that any number whatever may be represented by a letter, such as a, 6, p constructed to enable us to read any kind of subdivisions not indicated on tlie principal scale. Moreover, the two scales may be circular; the ]n-inciple remains uncihanged. Consequently verniers may also be employed in tlie measurement of angles, and, in fact, this is one of their most important applications. 41. Length, area, and volume. The result of measuring a length is expressed by saying that the required distance d contains x units of length, or briefly (1) d = x- L, if L denotes the unit of length. In this equation x is an abstract number and the product xL represents a length only on account of the presence of the factor L. Areas and volumes are usually determined, not by direct measurement, but by calculation from certain lengths. As unit of area we usually select a square, represented by I?, whose side is the unit of length ; and the numerical measure of any area is an abstract number which tells us how many times the unit area is contained in the given one. Thus, any area may be regarded as a product of the form (2) A=x-U where x is an abstract number and where X^, the unit of area, is a square whose side is X, the unit of length. Similarly, any volume Fmay be regarded as a product, (3) V=x-L\ of an abstract number x and a unit of volume i^, a cube whose side is equal to the unit of length. Formulae (1), (2), (3) may be summed up by stating that the dimensions of a length, area, and volume are i, L\ L^ respectively. 62 LINEAR FUNCTIONS AND PROGRESSIONS [Arts. 42-44 These dimensional symbols make it very easy to change from one unit of measurement to another. Thus we may convert an area of 3 square feet into square inches as follows : g ^^^ ,^2 ^ g^^2 in.)2 = 3 x 144 (in.)^ = 432 (in.)^. 42. Time. The fundamental and natural unit of time is the day. It is determined by the uniform rotation of the earth around its axis. The shorter units, hours, minutes, and seconds, are measured by the use of clocks, whose ap- proximately uniform motion is guaranteed by their mechani- cal construction. We shall use the symbol T to represent a unit of time. 43. Mass. If two bodies exactly balance each other when placed on the two scales of a balance with equal arms, they are said to have equal masses. If any body be chosen as a standard, and if it takes m of these standard bodies to bal- ance another body, the latter is said to contain m mass-units, or else its mass is said to be equal to m. The unit of mass most frequently used in scientific measurements is called a gram. A gram is the mass of a cubic centimeter of water at a temperature of 4° Centigrade. 1000 grams or 1 kilo- gram is equivalent to 2.2046 English pounds. We shall use the symbol Mior tlie unit of mass. 44. Density and Specific Gravity. The mass of a body de- pends upon the nature of its material and its volume. A body is said to be homoc/eneous if all of its parts are exactly alike. If a unit volume of such a homogeneous body con- tains p mass units, v unit volumes will contain pv mass units, so that we shall have (1) m = pv, if m denotes the total mass and v the complete volume of the body. The number p, which expresses the number of mass units in a unit volume, and ivhich is different for different bodies, is called the density of the body. Art. 45] VELOCITY 63 Equation (1) expresses the fact that the mass of a homo- geneous body varies as its volume ; the factor of proportion- ality in this case is the density. Since we may write (1) as follows (2) P = -, and since the symbols for unit of mass and unit of volume are iHf and L^ respectively, we find the symbol M/L^ for the unit of density. Let us use the centimeter as unit of length and the gram as unit of mass. A cubic centimeter of water has (by definition of the gram) the mass of one gram. Therefore the application of equation (2) to a cubic centimeter of water teaches us that the density of water is (8) p-l_g£am. (cent.)^ To find the density of water in terms of the units inch and pound, we use the relations 1 gram = 0.0022 lb., 1 centimeter = 0.3937 in., so that (1) becomes o-ram 0.0022 lb. ^ao^i lb. P = 1 a'-^- ^ ^•^"" "'• = 0.0361 (cent.)s (0.3937)3 (in.)^ (in.)^ The ratio of the density of any substance to the density of water is called the specific gravity of that substance. Since this is, by definition, a ratio of two quantities of the same kind, the specific gravity of a substance is independent of the units of length and mass. It is a pure number and therefore has no dimension. 45. Velocity. If a train makes a run of 120 miles in 4 hours, we say its velocity is 120/4 or 30 miles per hour. More generally ; if a uniformly moving body describes a distance of s length units in t time units, it would, at that rate, describe s/t length units in a single time unit, and we call the number (1) ?=z>, the velocity or speed of the body. This formula gives v = \ when s = 1 and t = \. Therefore this definition of velocity 64 LINP:AR functions and progressions [Art. 46 includes also a definition of a unit of velocity. The unit of velocity is the velocity of a body which moves at the rate of one length unit per time unit. Therefore the unit of velocity will change if the unit of length or the unit of time or both are changed. Since, in (1), s is a length and t a time, the symbol for a unit of velocity is L/T. Thus we may write .,^ mi. .,^5280 feet ..,, 5-J8() feet ..p. n feet , , feet hour 60 min. 60 min. min. sec. From (1) we find (2) 8 = vt, that is, the distance described by a uniforndy moving body varies as the time. The factor of proportionality is v, the velocity of the body. 46. Acceleration. Consider the motion of a train which has not yet reached its full velocit}'. At a certain moment let its velocity be 6 feet per second ; and five seconds later let its speed be 28 feet per second. Then the velocity of the train has increased 22 feet per second in 5 seconds, or at the rate of 4.4 feet per second every second. This is ex- pressed by saying that the average acceleration of the train in this interval of time is 4-4 f^-^^ P^^ second ])er second. In general, if the velocity of a body is measured by v velocity units at a certain time, and by v' velocity units after t time units have elapsed, the velocity has changed v' — V velocity units in t time units, that is, at the rate of 0) ^" velocity units per time unit. This quotient is called the average acceleration of the body during this interval of time. If the change in velocity is the same for every second of this time interval, the average acceleration is the same as the actual acceleration, and is said to be constant. If a body moves in such a way that its velocity increases by a single velocity unit in a single time unit, it is said to Akt. 47] UNIFORMLY ACCELERATED MOTION 65 have unit acceleration. In fact, the expression (1) reduces to unity if v' —v is equal to a unit of velocity and if t is equal to a unit of time. If the units of length and time are a foot and a second, the unit of acceleration is that of a body whose velocity increases by one foot per second every second. Thus the train in the abuve example has an accel- eration of 4.4 acceleration units. Since a unit of velocity has the dimension L/T (Art. 45), and, since according to what we have just seen, a unit of acceleration is obtained by dividing a unit of velocity by a unit of time, the symbol for a unit of acceleration is L/T^. If velocities in a certain direction are regarded as posi- tive, those in the opposite direction will be regarded as nega- tive. If, in (1), V and v' are both positive and if v' is greater than v, the acceleration computed by (1) will be positive. If v' is less than v, (1) will give a negative result. Thus, for a positive velocity a negative acceleration has the significance of a retardation. 47. Uniformly accelerated motion. Let us consider the case of a uniformly accelerated body, that is, one whose velocity changes by the same amount in equal intervals of time. Let us count time in seconds from a certain instant on, say 6 a.m., and let the velocity of the body be Vq feet per second, tQ seconds after 6 A.M. At any other time, t seconds after 6 A.M., let the velocity be v feet per second. Then .^. V — Vn change in velocitv (1) a = -0 = f : r-^ t — t^ time elapsed will be the average acceleration, expressed in feet per second per second, during the interval t — ^q. If the acceleration is constant, we sliall obtain the same quotient a from (1) no matter what value we use for t, provided v represents the veloc- ity at that instant. Consequently we find that the equation (2) v-v, = a(t-t,\ obtained from (1) by clearing of fractions, will be true for all values of t as long as the acceleration a remains constant. 66 LINEAR FUNCTIONS AND PROGRESSIONS [Arts. 48,49 That is, the change in the velocity of a hody^ which moves with a uniformly accelerated motion^ varies as the time which has elapsed. The constant of variation in this case is a, the con- stant acceleration. 48. Falling bodies. The simplest instance, in nature, of uniformly accelerated motion is that of a falling body. The constant acceleration of such a body is usually denoted by ^, and simple experiments, with the Atwood Machine for instance, show that the value of g is approximately (1) ^=32.2 ^^^\ „ (second)'' if the foot and the second are used as units of measurement. Let us begin to count time from the moment the body begins its motion, and let v^ be its initial velocity. Then we may put ^^ = 0, a = g, in equation (2), Art. 47, giving the equation (2) v = v^+gt for the velocity of a falling body at the erid of t seconds if its initial velocity (velocity at the time t = 0} is v^ feet per second. In this equation v and Vq are positive for downward and negative for upward velocities. If the body falls from rest, Vq is equal to zero. If it is thrown downward, v^ is positive ; if it is thrown upward, Vq is negative. 49. The importance of dimensional symbols. — The illus- trations just given will suffice to justify the following re- marks. Algebra strictly speaking deals only with abstract numbers. Algebra may be applied to the discussion of con- crete problems only by introducing for every concrete quan- tity a certain quantity of the same kind as unit. In dealing with concrete problems it is very important to specify what units have been employed. Since it is often convenient to change from one system of units to another, we must learn how to make such transformations, and this is done most conveniently by means of the dimensional symbols. Art. 49] IMPORTANCE OF DIMENSIONAL SYMBOLS 67 An equation between two concrete quantities is really an equation between the numerical measures of these quantities. • Such an equation implies equality between the corresponding concrete quantities only if they are quantities of the same kind, and if they are measured in terms of the same unit. Thus it is obviously meaningless to say that 40 feet = 40 square feet. We have here two concrete quantities whose numerical measures are equal. But this does not imply equality of the two concrete quantities themselves, because the latter are different in kind, that is, have different dimensions. We may also express this as follows : An equation between concrete quantities has a concrete (^not merely a numerical^ significance, only if all of the terms of the equation have the same dimension. This fact often enables us to find the dimension of a quantity which may otherwise not be apparent, as in the following example. A familiar law of Mechanics (see Art. 48) states that a body, falling from rest, acquires after t seconds a velocity of v feet per second, where (1) (■ = :32.-2/. The left member (a velocity) has the dimension L/T and the right member seems to have the dimension T. But we have just seen that both members ought to have the same dimension. The cause for this apparent contradiction lies in thinking of 32.2 as an abstract number. The equation itself tells us that 32.2 is equal to v/t, and therefore is not an abstract number. It has the dimension of a velocity divided by a time or L/T-. A more adequate \\a,y of w-riting (1) would be (2) V - yt, where g is the numerical measure of a quantity of the dimension L/T^, and where the numerical value of g becomes equal to 32.2 only if the units of length and time are the foot and the second respectively. In fact g is the acceleration of a falling body as explained in Art. 48. The dimension of g now being known, we may find the value of g in terms of the units centimeter and minute, or in terms of any other units whatever. 68 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 49 EXERCISE IX 1. Devise a vernier for making readings to ^^ of an inch on a foot- rule which is divided so as to read directly to one-fourth of an inch. 2. Show how to construct a vernier which shall enable the observer to read angles to a minute of arc, if the circle is divided into half- degrees. 3. The area A of a, circle varies as the square of its radius r. Ex- press this as an equation. What is the value of the factor of propor- tionality? Is it an abstract number? If not, what is its dimension? 4. A glass beaker has a graduated scale upon it enabling an ob- server to read oft' the volume of fluid which it contains. By immersing spheres of various sizes in the fluid, it is found experimentally that the volume F of a sphere varies as the cube of its radius, and that V = 4.19 cu. in. when r = 1 in. Express F as a function of r. 5. The distance d (in miles) traveled by a train varies as the time t (in hours) counted from the moment of its departure, and d = 150 when t = 5. Express this by an equation. What is the dimension of the con- stant of variation, and what is its physical significance? How will the equation be modified if the distance is expressed in feet and the time in minutes? 6. The number of feet s which a body falls (from rest) in / seconds varies as fi, and s = 64.4 when t = 2. State this as an equation. What is the dimension of the constant of variation, and how is the equation modified if s is expressed in yards and t in minutes? 7. The volume of a right circular cylinder varies jointly as the square of the radius of its base, and its altitude. Write this as an equation. 8. The specific gravity of cast iron is 7.2. Find the mass (in grams) of a rectangular block of iron of dimensions 2 centimeters x 3 centimeters x 4 centimeters. 9. A certain amount of air is inclosed in a cylinder which has a movable piston. The volume of the inclosed air may be changed by moving the piston. Show that the density of the inclosed air varies inversely as the volume. 10. The volume of a sphere varies as the cube of its radius. If three spheres of radius 2, 3, and 4 inches respectively be melted and formed into a single sphere, what is the radius of the latter? (Solve without making use of the actual value of the constant of variation.) 11. A solid glass sphere 2 inches in diameter is melted and blown into a hollow spherical shell whose outer diameter is 4 inches. Find the thickness of the shell. Ai:t. :)0] GRAPHrCAL REPRESENTATION 69 12. The safe load of a horizontal beam supported at both ends varies jointly as the breadth and the square of the depth, and inversely as the length between the supports. Plan some experiments to prove this Taw and state it in the form of an equation. Determine the factor of pro- portionality for a certain kind of wood, if the maximum safe load for a beam 15 feet long, 3 inches wide, and 6 inches deep is 1800 pounds. State explicitly what units must be employed in applying the residting equation. Find the safe load of a beam made of the same material, 18 feet long, 4 inches wide, and 4 inches deep. 13. The planets describe approximately circular orbits around the Sun. The time required for a planet to make one revolution in its orbit is called its period. For the Earth this period is one year. By compar- ing the distances and periods of the various planets, Kepler * discovered the following law, usually called Kepler's third law : in the solar system the square of the period of a planet is proportional to the cube of its distance from the Sun. Express this law in the form of an equation, and deter- mine the factor of proportionality if the Earth's distance from the Sun be taken as unit of distance, and one year as unit of time. What is the distance of Jupiter from tlie Sun if Jupiter's period is 11.86 years? 14. At the earth's surface a body falls 193 inches in the first second. The number of inches which a body not at the earth's surface, would fall in the first second is inversely proportional to the square of the dis- tance of that body from the earth's center.f If the distance from the earth's center to the moon is sixty times the earth's radius, how far would a body fall in the first second if it were as far away as the moon? 50. Graphical representation of a pair of numbers. When- ever z/ varies as a;, we have a rehition such that to a given value of X corresponds a definite value of i/. The numbers, * JoHANN Kepler (1571-1(530), a famous German astronomer. Kepler's greatest achievement was the discovery of the three fundamental laws of planet- ary motion, the la^^t of wliich is quoted above. Tlie first of tliese laws asserts tliat a i>lanet moves in an ellipse of wliirh the Sun occupies one locus, and the second law states that the planet's radius vector sweeps over equal areas in equal times. Kepler obtained these laws by discussing the observations of the great Danish astronomer, Tyiho BKAnt:. Newton showed later that all these laws are con- sequences of tlie law of gravitation. t This is one way of stating the law of gravitation. The calculation indicated in this example was first performed by Newton. The result of this calculation did not agree as well as it shoulil with the observed motion of the moon on ac- count of the inadequate notions curreiU at that time in regurd to the dimensions of the earth. Newton therefore gave up the theory that gravitation varies ac- cording to the law of inverse squares uiUil a few years later, wlicii a new survey showed him a complete agreement with this theory. (See Grant's Hhtory oj I'liii.siciif As/riinn)ni/. p. 'Jl.) M Fig. 27 To LINEAR FUNCTIONS AND PROGRESSIONS [Art. 50 X and y, which are so related may be thought of as belonging together and constituting a pair. We shall now show how such a pair of numbers may be represented geometrically. Let us draw two lines, unbounded in length and perpen- dicular to each other. We shall usually think of one of them as horizontal and call it the x-axis, and call the other, which is vertical, the y-axis. The point 0, in which the two axes intersect, is called the origin of coordinates. We adopt a unit of length, and denote the distances from any point P to these two axes by X and ?/ respectively. In Fig. 27, we have NP=OM=x, MP=ON=y, where the notation is chosen in such a way that X is measured on or parallel to the a;-axis, and y on or parallel to the ^-axis. We call X the abscissa and y the ^^ ordinate of the point P. Both numbers together are called the coordinates of P. -j^ If we take into account only the magnitudes and not the direc- tions of lines OM^ ON, etc., that is, if X and y are regarded as numbers without sign, there will be four points which have the same co5rdinates. For instance, the points P, P', P", P'", in Fig. 28, would all corre- spond to X = 3, y = 2. In order to avoid this inconvenience, we introduce the convention that the abscissas of all points to the right of the ^-axis shall be positive, and of those to the left negative ; that the ordinates of all points above the a;-axis shall be positive, and of those below negative. The coordinates of the four points in Fig. 28 are now different from each other. The coordinates of P are x =+ 3, y = + 2, The coordinates of P' are x = — B, y = + 2, The coordinates of P" are x — — S, y — — 2, The coordinates of P'" are x — + 3, y — — 2. 1 Fig. 28 Art. 5(1] GRAPHICAL REPRESENTATION 71 The positive directions of the x- and yaxis, which have now been deiined, tvill hereafter he indicated by a plus sign (as in Fig. 28). The introduction of a coordinate system enables us to estab- lish a one-to-one correspondence between the points of the plane (^objects of G-eometry^ and pairs of real numbers (objects of Arithmetic^. To every loair of real numbers there corresponds one and only one point of the plane, namely that one which has the given real numbers as coordinates ; and to every point of the plane there corresponds a single pair of real numbers, namely the coordinates of that point. This method of representing a pair of real numhers by a point is, of course, closely allied to our former method of representing a single com- plex Dumber by a point. See Art. 24.* In fact a single complex number, z = X -{■ yi, determines a pair of real numbers {x, y). It is merely a qnestion of convenience whether we wish to think of the points of the plane as geometrical representatives of a sbigle complex number, or of a pair of real numbers. We can not, however, represent a joair of complex numbers, or even a pair of numbers one of which is complex, by means of the points of a single lihine. If, therefore, in the solution of a problem which involves the coordinates of a point in a plane, we find that either of these coordinates receives a comi>lex value, we conclude that the required point does not exist. This in no wise contradicts the fact that a point of the plane may be represented by a complex number x + yi, for, in this symbol also, a; and y are supposed to be real. EXERCISE X Plot the points whose coordinates are given in examples 1-4: 1. (+3, +4). 3. (-:3, -4). 2. (-3, +4). 4. (+3, -4). 5. If we know nothing about a point except that x = 0, what can we say about its location ? 6. Write down an equation, involving one or both of the coordinates of a point P, which will be satisfied provided that P is anywhere on the y-axis. Will this equation be satisfied by any point which is not on the y-axis? * We represented a complex number x + yi hy a, vector, whose components were z and y. If this vector is put into its standard position, its terminus will be a point whose coordinates are x and y. 72 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 51 7, If ^/ = 0, what can we conclude about the position of Pi 8. Write down an equation which will be satisfied it' and only if P is on the X-axis. 51. Graphical representation of variation. If y varies as x. that is, a y ■= mx^ there are intinitely many pairs of values, x and y, which satisfy this relation. If we plot a large number of these pairs, we find points which seem to be on a straight line through the origin of coordinates. ^ I Example. Let y = 2 x. For x = + 1, this gives y = + 2 ; for X = + 2, y = + 4 ; etc. For x = — 1, ^ = — 2 ; for x = — 2, ?/ = — 4 ; etc. These results are collected in the adjoining table. If we plot these pairs of numbers (—3, — 6), (—2. — 4), etc , we obtain the points P_3, P_.^, P_i, P^, P^, P^, P^ of Fig. 29, which seem to be on a straight line through the origin. The idea naturally suggests itself that not merely the few pairs of numbers which we have calculated, but that all number pairs which satisfy the equation ij = '2x will give rise to points on We shall actually />?-ot'e that this is so in Art. 54. It often happens, especially in concrete problems, that m is so large or so small as to render the resulting graph practically useless. This may be avoided by choos- +x ing different scale units on the two axes. These scale units should always be specified in concrete problems, so as to make evident the concrete significance of such a diagram. -8 -6 -2 -4 -1 _ 2 + 1 + 2 + 2 + 4 + 3 + 6 this line. Fig. 29 EXERCISE XI By the method of Art. 51 plot the graphs of the following equations: 1. y = 3 X. 4. y = — x. 2. y — — 'ix. 5. 2 y — 3 x = 0. 3. // = X. 6. 2 y + 3 x = (». 7. Represent graphically the equations obtained in examples 1, 2, 3 of Exercise VIII. 8. Represent graphically the equation obtained in example 5 of Exercise IX. Arts. 52, 53] SLOPE OF A STRAIGHT LINE 73 52. Graphical representation of the function y = mx + b. The metliod explained in the preceding article applies just as well to any equation of the form y = mx + h. We observe, in this case, that the locus of the points whose coordinates satisfy such an equation, again appears to be a straight line ; but this line does not pass through the origin unless h is equal to zero. EXERCISE XII Draw the graphs of the following equations: "i.. y — X -\- \. 5. y = — .r + 1. 2. y = 2x + 3. 6. ^ = - 2.r + 3. Z. y = 3x + 2. 7. // = - 3x - 2. 4. // = 4x - 2. 8. j^ := - 4,r + 2. 53. Slope of a straight line. Let Pj and F^ be two points on a straight line, and let the coordinates of these points be called (a:j, y-^ and (x^^ y^ respectively. F'igure 30 shows that a point P, in mov- ing from Pg to Pj, will move x^ — x^ units toward the right and yi — y^, units upward. The ratio (1) m =^1 ~ ^2 F^«-^ is called the slope of the line. Thus, if a railroad track rises 3 feet in a horizontal distance of 200 feet, its slope is = 3/200 = 1.5/100. It rises 1.5 feet in 100 feet; it lias a slope or grade of 1.5%- If the line P^P^ makes an angle 6 with the x-axis, Fig. 30 shows that (2) y±ZJb = tan 6. Consequently, any one who has studied Trigonometry, may say that the slope ofn line in the tangent of the angle which the line makes tvith the x-axis. It is important to note that the definition of the slope, as given by (1), is independent of the choice of the particular 74 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 54 points, Pj and P^, provided both of them are on the line. If we take any two other points of the line, Pg and P^, the value of m given by will be exactly the same as that given by (1). That this is so follows from familiar properties of "similar triangles. Moreover, we might write, instead of (1), (3) m = UiLZLli^ interchanging the order of the two points Pj and Pg. Since Vi- y\ = - iy\ - ^2)' 2^2 - ^1 = - (^1 - ^2) the values of m given by (1) and (3) are the same. Formula (1) may give a positive, zero, or negative value for the slope w of a line. The slope will be positive if the line rises from left to right as in Fig. 30. The slope is negative if the line falls from left to right. Tlie slope is zero if the line is parallel to the x-axis. In that case we have y,^ = y^, and we shall have (^) y = yi for every point of the line. A line parallel to the y-axis cannot he said to have any slope. For such a line we should have x^ = x^^ so that formula (1) which defines the slope becomes useless, since division by zero is excluded from Algebra. (See Art. 21.) It is clear however that /or every poiiit of such a line., the equation (5) x = x^ will he satisfied. 54, Equation of a straight line. Let Pj (Fig. 31) be a given point and let us con- struct the line AP which passes through Pj, and which has the given number m as -+a; its slope. If (a^j, y^ are the codrdinates ■P^G' ^^ of Pj, we may then regard 2;^, ?/j, and m, as Art. 54] EQUATION OF A STRAIGHT LINE 75 given numbers. If we denote by (.c, ^) the coiJrdiuates of any other point on AP^ we must have (1) . y^lJh = m, X — T-^ since the slope of AP miiy be computed, by formula (1) of Art. 53, in terms of the coordinates of any two of its points. Thus, if we regard x^^ y^, m as given numbers, equation (1) will be satisfied by the coordinates (x^ «/) of every point, different from P^, of the line AP. There are no other points, except those on AP^ whose coordinates satisfy (1). For, let P' (Fig. 31) be any point not on AP. If we substitute iU coordinates for x and y in the left member of (1) we shall obtain, not w, the slope of AP., but some other number equal to the slope of P^P' . Consequently, the equation obtained from (1) by clearing of fractions, namely (2) y-y\ = ^(^ - a^i) t's satisfied hy the coordinates of every point on the line AP * ; and conversely^ every point whose coordinates satisfy this equa- tion is on AP. We express this more briefly by saying that (2) is the equation of the line AP. Thus, the equation of a line which passes through the point (a:^, y^ and ivho> is a straight line, ever g function of the form mx + h is called a linear function. 55. The zero of a linear function. The graph of the linear function , 7 y = mx + is a straight line of slope m. This line will meet the a^-axis in a point A unless m happens to be equal to zero. (See Fig. 33.) The ordinate of A is equal to zero; we wish to find its abscissa. Since A is on the line AB, the coordinates of A must satisfy the equation (1) y = mx + b, which is satisfied by the cocirdinates of every point on AB ; and, since the ordinate of A is zero, the abscissa x oi A must satisfy the equation (2) = mx + b, obtained from (1') by substituting in it y = 0. But (2) gives (3) x=-^ m 78 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 55 provided that m^O.* If m = 0, equation (1) reduces to 1/ = b, the corresponding line is parallel to the 2;-axis, and there is no point of intersection. The particular value of x, namely/ x = — h/m, which causes the function mx + b to assume the value zero, is called the zero of the function. The zero of the function mx + ^ is the same number as the solution or root of the equation mx + 6 = 0. Moreover, the zero of the function mx + b is equal to the abscissa of the point in which the graph of the function crosses the X-axis. EXERCISE XIII 1. Draw the lines for which ?h = 3, 6 = 2 ; m = — 2, 6 = 1 ; m — — \, J = — 1 ; and write their equations. Find the zeros of the corresponding functions both by measurement from the figure and by calculation. 2. The equation .3a: + 5^ + 4 = defines i/ as a linear function of x. Find this function and its zero, draw the corresponding line, calculate its ^/-intercept and slope. 3. What is the slope of a line which joins the points (3, 4) and (5, 7,) ? Of the line which joins ( - 1, - 2) and (- 4, 3) ? 4. A line of slope 2 passes through the point (2, 3). Draw the line and find its equation : What are the .r- and ?/-intercepts of this line ? 5. Find the zeros of the following functions : 3 X - 4, 7 X + 9, 8 2- - 4, 4 X + 8. Solve the following equations : (Ex. 9 to three decimal places.) 6. 8 x - 5 = 13 - 7 X. 7. 12 + 3 .r - 6 - — = — - 51. 3 4 * 8. ^ + J + ?=.7x-734 + f. 2 3 4 5 9. 3.2.50 .r - 5.007 - x = 0.200 - 0.340 a:. 10. -^ ^ = 1. 11. -^ ^ dc = hx- ac. a — b a + b h — c * The symbol ^ is read, is different from. Art. 55] THE ZERO OF A LINEAR FUNCTION 79 12. A and B go into partnership. A contributes twice as nuich as B to their joint capital of 138,700. What is the contribution of each ? 13. A man has f 1.500 in the bank drawing simple interest at 3 % a year. Express the amount at the end of n years as a function of n, and represent this function graphically. What is the financial significance of the slope of the resulting line? Do the ordinates which correspond to negative values of n have any significance ? What does the ordinate mean which corresiionds to n = 0? 14. (Continuation of 13.) A second man has '11200 in the bank, drawing simple interest at 4 % per annum. Express the amount of this sum at the end of n years by a formula, and represent this function graphically, making use of the same axes of reference as in Ex. 13, What is the financial significance of the point of intersection of the twa lines ? 15. A sum of money •*$ P is put out at simple interest at the rate of r% annually. Find a formula for the amount A at the end of n years. 16. Making use of Ex. 15, find a formula for the sum of money P, which will yield an amount of ^^4 at tlie end of ?i years, at simjile inter- est of r % a year. 17. A bicyclist starts from a certain place at 8 a.m. and rides at the rate of 10 miles per hour. An automobile, going with a velocity of 35 miles per hour, follows the same road, starting at 2 p.m. At what time will the automobile catch up with the bicycle? Solve this problem both numerically and graphically. 18. The following abstract problem contains all examples like 17 as special cases. A starts from a certain place and travels at the rate of v miles per hour. B starts h hours later than A and travels along the same road at a rate of v' miles per hour. In how many hours will A and B be together? Solve this general problem, and discuss the solution. In particular, distinguish between the three cases v' > v, v' = i% v' < v. 19. The distance between two cities, A and B, is 180 miles. An auto- mobile starts from A toward B at 8 a.m. with a speed of 30 miles per hour. A second automobile starts from B toward A at 8 : 30 a.m. travel- ing 35 miles per hour. When and where will they meet? Solve this problem both numerically and graphically. 20. Formulate and solve a general problem of which 19 shall be a particular case. Discuss your solution. 21. An observer is stationed on a rowboat which is at rest. He notices that n wave-crests pass the boat in one second, and that each ■wave is propagated with a velocity of Ffeet per second. He then starts to row in the direction of the waves with a velocity of v feet per second. 80 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 56 How many wave-crests will pass the boat per second while it is in motion ? Answer the same question for the case when the boat is mov- ing with a velocity of v feet per second against the direction of the waves. 22. Sound is produced by waves or vibrations of the air, and the pitch of a tone depends upon the number of vibrations which strike the ear in one second. Making use of Ex. 21, explain the following phenomenon. When a train passes a source of sound (a bell or a whistling engine), the pitch of the sound changes. (Doppler's principle for sound.) 23. A train with a velocity of 47 feet per second is passing a whistling engine at rest. The velocity of sound is about 1131 feet per second. Find the ratio of the number of sound vibrations which strike the ear of an observer on the moving train before it reaches the engine to the cor- responding number of vibrations after it has passed. (Read Examples 21 and 22 to assist you in solving this example.) 24. Light is produced by the vibrations of the so-called "ether." The number of vibrations which strike the eye per second will be modified by motion in the line of sight either of the observer or of the source of light. The number of vibrations which strike the eye per second may be meas- ured (l)y means of a spectroscope). Explain how these facts make it possible to measure the velocity with which a star is approaching or receding from the earth. (Doppler's principle for light.) 25. The centigrade thermometer scale is obtained by marking the freezing point of water 0°, the boiling point of water 100°, and dividing the interval into 100 equal parts. On the Fahrenheit scale the freezing- point and the boiling point are marked 32° and 212° respectively. Ob- tain a formula for reducing temperatures expressed in Fahrenheit degrees to centigrade degrees, and make a graph of this formula. 56. Arithmetic progressions. If we compute the values of a linear function 7nx + b, not for all values of x, but only for a; = and for positive integral values of x, we find a set of numbers (1) b, m ->f b, 2 w + b, 3 m + b, 4 m + b, etc. each of which differs from the preceding one by the same amount. These numbers form an ordered set or a sequence, since we are thinking of them as being arranged in a detinite order. A finite set of numbers is called an ordered set or a sequence^ if the numbers of the set are thought of as being arranged in a definite order. Art. jG] ARITHMETIC PROGRESSIONS 81 If we wish to discuss an ordered set of niiinbers, it does not suffice to know the value of every number of the set. ^Ve must also know which is to be regarded as first, which is the second, etc. The numbers of an ordered set are said to form an arithmetic progression, if the difference hetiveen any number of the set, after the first, and the one which precedes it is the same for all numbers of the set. The numbers of the set are called the terms of the pro- gression, and the difference between any term and the pre- ceding one is called the common difference. The numbers (1) form an arithmetic progression whose first term b is any number ; and whose constant difference m is any other number. Consequently, any arithmetic pro- gression may be represented by the formula (1). We find the following result: Tlie values which a linear function mx + b assumes, when x assumes in order the values 0, 1, 2, 3, etc., form an arithmetic progression; and converseli/, with any arithmetic p7'Of/ressioji there may be associated a linear function mx + b whose values coincide ivith the terms of the progression when x is equated in order to 0, 1, 2, 3, etc. The notation used in (1) is not the one Avhich is usually employed in the theory of arithmetic progressions. Let us call the first term a instead of b, the common difference d instead of m, and let i denote the number of any term, so that i replaces x. We then have the customary expressions (2) a, a + d, a -f- 2 (?, a -f 3 rZ, ••• for the terms of an arithmetic progression. The ith term will be (3) a+(i- l)d. If there are n terms altogether, and if we denote the last (wth) term by I, we shall have (4) l=a + {n- l)f?. 82 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 56 The principal problem in the theory of arithmetic pro- gressions is that of finding a formula for S, the sum of all of its n terms. Of course we have (5) >S=a+ (a + d) + (a + -2d) + ... +1. But, in order to compute the value of jS by this formula, we should have to compute each of the terms separately and that would be very laborious when the number of terms is large. In order to find a more convenient formula for S, let us first rewrite S by beginning with the last term, that is, by inverting the order of the terms of the progression. We shall then have also (6) S=l+(l-d-) + (l-2d)+ ... +a. If we add the corresponding members of (5) and (6) we 2S={a + l) + {a + l) + {a + l)+ ... + (a + 0, and the right member will contain the binomial a+^ as many times as there are terms in the progression, namely n times. Consequently and (7) S=n'^- This formula is very easy to remember, if we agree to call |-(a. -f Z), which is half of the sum of the first and last terms, the average term of the progression. For, we may then ex- press the content of (7) as follows : The sum of an arithmetic progression of n terms is equal to n times its average term. If we substitute (4) in (7), Ave find (8) S = '^^[2a+in-l)dl for the value of S in terms of the first term, the common difference, and the number of terms. Art. 56] ARITHMETIC PROGRESSIONS 83 If three of the five quantities a, d^ n, I, S are given, the other two may be found from (4) and (7), or from (4) and (8). It is necessary, however, to remember that n must be a positive integer, while a, d^ I, and S may be positive or negative, integral or fractional, rational or irrational, real or complex. EXERCISE XIV Find the last term and the sum of the following arithmetical pro- gressions. 1. 1 +2 + 3 + ••• + 14. 2. 2 + 11 + 20 + ••• to 10 terms. 3. - 3 - 8 - 13 - ••• to 19 terms. Compute I and S in the following cases. 4. a — 2.(1 = 3, n = 17. 5. o = 2.5, d = 1/3, n = 100. 6. o = 1/2, f/ = - 1/8, n = 20. 7. a = -10,d = -'2,n = 6. In the following examples, compute those of the five quantities, a, d, I, n, S, which are not given. 8. a = 3, n = 333, S = 166,833. 9. a = 3/4, n = iO, S= 517.5. 10. d = S,n = 16, S= 440. 11. d = 2/7, 71 = 32, S = 160. 12. a = 1700, d = o,l = 1870. 13. In Examples 1-7 find the sum when the number of terms n is arbitrary. 14. Find a formula for d when a, n, and / are given. 15. Find a formula for I when a, n, and S are given. 16. Find a formula for n when a, I, and S are given. 17. Find a formula for n when d, n, and I are given. 18. Find a formula for a when li, it, and S are given. 19. Find the sum of the first n odd numbers. 20. A ball rolling down a plane inclined at an angle of 30° to the horizon, rolls 8 feet in the first second, and in every second thereafter it rolls 16 feet more than in the preceding second. How far will it roll in 9 seconds? 8-t LINEAR FUNCTIONS AND PROGRESSIONS [Arts. 57, 58 57. Insertion of arithmetic means. To insert n arith- metic means between two given numbers, a and J, means to find n numbers, a^, a,^, a^^ •-•, a„, such that the w + 2 numbers (1) a, «i, ^2, •••, «„, h shall form an arithmetic progression. There are n + 2 terms in this progression, a is the first term and b the last. Hence we have from (4) of Art. 56, h = a + (7i + 2 — l}d = a + (n + l)c?, where d denotes the common difference. Consequently we find 7 d=^-'' n+ 1 and therefore /-ON , b — a , ,-.b — a , b — a ^ ^ ^ n + 1 71+1 n + 1 are the n arithmetic means required. In particular, if n = 1, we find a.^^ = a-\ — =a-\-l-b— la=la+^b = |-(a + i). Therefore, to insert a sinr/le arithmetic mean between a and 6, ive need merely compute half the sum of a and b. This fact accounts for the name arithmetic mean which is usually given to half the sum of two numbers. EXERCISE XV 1. Insert six arithmetic means between 3 and 8. 2. Insert five arithmetic means between 3 and — 2. 3. The two end posts of a fence have been placed at two points 300 feet apart. The fence is to have 35 other posts. How far apart must they be placed if all equally spaced? 58. Harmonic progressions. A sequence of numbers a^, a^, rtg, etc. is said to form a harmonic progression, if the recip- rocals of these numbers form an arithmetic progression. Art. 59] HARMONIC PROGRESSIONS 85 Consequently the general ex[)ression for the terms of a harmonic progression of n terms is given by (^ 1 1 1 a a + cC a + '2.d' ' a+(n—V)d The most familiar illustration of such a progression arises if in (1) we put a = d=\ ; this gives the harmonic progression r9>. 1111 ... 1. ^•^^ 1' 2' 3' 4' ' n If we wish to insert n harmonic means between two given numbers, a and 6, we may first insert ii arithmetic means be- tween 1/a and 1/h. The reciprocals of these arithmetic means will be the harmonic means between a and h. EXERCISE XVI 1. Insert three harmonic means between 4 and 8. 2. Find an expression for the harmonic mean between a and b. 3. Show that the values of the function which correspond to mx + b 3: = 0, 1, 2, 3, etc. form a harmonic progression. 4. Prove that if all of the terms of an arithmetic or harmonic progression are multiplied by the same number, the result is again a progresi^ion of the same kind. 59. Geometric progressions. The ancients were familiar with the essential properties of arithmetic and harmonic progressions. They also considered progressions of the form (1) a, ar, ar^ ar^^ ar^^ •••, in which, the exponents of r, namely 1, 2, 3, 4, etc., are in arithmetic progression, and spoke of such sequences as geometric progressions. If we divide any term of the sequence (1), excepting the first, by the term which im- mediately precedes it, we always obtain the same quotient, namely r. We may therefore define a geometric progres- sion as follows: 86 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 60 Let us divide each term of a sequence (^excepting the first) by the term which immediately precedes it. If all of the quo- tients obtained in this ivay have the same value r, the sequence is called a geometric progression, and r is called the common ratio of the proyression. Let a be the first term of the geometric progression and let r be the common ratio. Then the various terms of the progression will be represented by the expressions (1). The first term is a, the second is ar, the third is ar^, and so on. The ^th term is ar^~^. If we use the notation a. (not to be confused with the symbol a^ for the ith power of a), to stand for the ^th term of the progression, we have ftj = a, ^2 = ar, a^ = ar\ •••, and in general, for any value of the integer i, (2) ai = ar'-\ If the progression contains w terms altogether, the last or nth term will be (3) «„ = ar"-\ Sometimes the last term is denoted by I, so that we have also (4) I = ar'^-\ 60. Sum of a geometric progression of n terms. Let us denote by S^ the sum of a geometric progression of n terms, so that (1) Sn= a + ar+ ar^ + ar^ + • • • + ar"*'^. If we multiply both members of (1) by r, we find (2) rSn = ar -{■ ar'^ ■\- ar- -\- • • • + ar"" ^ + ar"^. If we subtract the members of (2) from the corresponding members of (1), we find (3) ^; - r/S; = a - ar''. Art. 61] GEOMETRIC PROGRESSIONS 87 since all of the other terms in the right members of (1) and (2) are eliniinated by this subtraction. We may write (3) as follows : (l-r)>S'„ = a(l-r"), so that we obtain finally (4) ^, = <1^^ 1 — r provided that r is not equal to unity. Formula (4) enables us to compute the sum S^ of n terms of a geometric progression ivhose first term is a and whose com- mon ratio r is different from unity. If r = 1, the final step in the process used for deducing (4) is not per- mitted, since we must never divide by zero, and since 1 — r would be equal to zero in that case. It is very easy, however, to find the formula for S„ in the case when r = 1. We then liave (5) S^ = a + a + ■•• + a — na. The following alternative expressions for 8^ follow im- mediately from (4) (6) 8^ = ^ zr-^ = 1- = ^, r— 1 r— 1 r— 1 where I = ar""^ is the last or nth term of the progression. We may also write ^rrx a a — ar^ a ar^ (J) *^n = -^ - r 1 — r 1 — r 61. Geometric means. If three numbers a, 5, c form a geo- metric progression, b is said to be a geometric mean of a and c. Since we shall then have b^c a b we find b^ = ac, and therefore (1) b = ± -JTc. 88 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 61 Thus, two positive numbers a mid c have two geometric means ivhich are equal to + Vac and — 'Vac respectively. Usually the positive square root of ac is called the geometric mean of a and c. To insert n geometric means between a and c we must find n numbers a^, a^, ■■■ «„, such that the n + 2 numbers shall form a geometric progression. Let r denote the common ratio of such a progression. Then a will be the first term, and c will be the (w + 2)th term. Therefore we must have (Art. 59, equation (3) where n is to be replaced by n -\- 2), c = ar""^^, whence, (2) . .= V„-- and ftj = ar, ag = ar^ • • •, a„ = ar", enabling us to compute all of the desired geometric means. EXERCISE XVII In examples 1-4 compute S^ and I from the given quantities. 1. a = 1, r = 2, n -7. 3. a = 8, r = i, n = 15. 2. a = 4, r = 3, n = 10. 4. a = 3^, r = f, n = 8. 5. The third term of a geometric progression is 3, and the sixth term is 81. Find the tenth term. 6. What is the sum of the first five terms of a geometric progres- sion whose second term is 2 and whose fourth term is 8? 7. Insert one geometric mean between 7 and 252. 8. Insert two geometric means between 2 and 250. 9. Find a formula for I in terms of a, r, and S^. 10. Find a formula for I in terms of r, n, and S^. 11. Find a formula for «S„ in terms of r, n, and /. 12. Find a formula for a in terms of r, n. and /. 13. Find a formula for a in terms of r, n, and S,^, 14. Find a formula for a in terms of r, /, and 5„. 15. Find a fornmla for r in terms of a, I, and S„. Art. (52] INFINITE GEOMETRIC PROGRESSIONS 89 16. If a, n, and 5„ are given, show that r must be a root of the equation ., „ '■ S,, S, — a ^ r" -r + -^ = 0. a a 17. If n, I, and 5„ are given, show that r must be a root of the equa- tion S„ -I S,- I 18. Prove the theorem : if all of the terms of a geometric progres- sion are multiplied by the same number, the products also form a geo- metric progression . 19. Prove the theorem : if the corresponding terms of two geometric progressions are multiplied together, the products also form a geometric progression. 20. The story is related that the inventor of the game of chess de- manded the following reward ; one grain of wheat on the first field of the chess board, two on the second, four on the third, eight on the fourth, and so on. How many grains of wheat would he be entitled to altogether, there being sixty-four fields on the chess board V 21. The directors of a certain charity devised the following plan for raising money. They sent a letter to one hundred people ask- ing each of them to contribute one dollar and to write to three friends making the same request of them. The original one hundred letters are marked 1, those sent out by the original one hundred contributors are marked 2, and so on. The chain is to be broken when the mark 5 has been reached. If all of the persons respond and if there are no dupli- cates, how much will the charity receive? 22. Show that the geometric mean of two numbers is also the geo- metric mean of their arithmetic and harmonic means. 62. Geometric progressions with infinitely many terms.* Let us consider the geometric progression (1) l + ^+l+i+-' for which a = 1, and r = |. According to formula (7) of Art. 60, the sum of n terms of such a progression is C'7\ ,e _ _J}: (2^" _ O _ 0/ 1 -xn _ 9 _ ^ 2 2 "" * The discussion of this subject may be postponed until the chapters on limits and series have been reached. Or else, the theory of limits (See Chapter XV) may be inserted at this place in the course. 90 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 62 If we allow w, the number of terms, to increase indefinitely, the number 2"~i will grow large very rapidly, and the recip- rocal of this number, l/2"~^ will become very small for large values of n. It can be shown that l/2'*~i can be made as small as may be desired by merely choosing the integer n large enough. We express this by saying that the fractioji l/2"~^ approaches zero for its limit, when n grows beyond all hound, or in symbols lim _L = 0. ^^00 2"-i Consequentl}^ >S'„, as given by (2), will approach 2 as its limit when n grows beyond all bound, or in symbols, lim ^„ = 2. We may explain this situation by the following geometric representation. In Fig. 34, let the line segment OA be two units long, and let OPi = l, 0^2=1 + 1, OP^ = l + l+\, 0^4 = 1 + 1-^1 + ^, etc. It is clear that each of these line-segments is obtained from the preceding one by adding to it just half of what would be required to make it exactly two units lonsf. ' ""^ Consequently the line-segment obtained by a large number of such operations will never be exactly equal to two units ; but the difference between 2 and the length of such a segment may be made as small as we please by repeating the operation often enough. In other w^ords, the length of the wth line-segment of this sequence, 0P„, Avill approach the limit 2. We may argue in this way whenever the common ratio r is a positive or negative proper fraction. For we shall prove later (See Art. 275) that the nth. power of any proper fraction approaches the limit zero when n grows beyond all bound, that is, lim r" = 0, whenever i r I < 1. Art. 0:5] PERIODIC DECIMALS 91 Consequently, formula (7) of Art. 60, that is, r shows us that S^ will approach the limit 1 — r when n grows beyond all bound, since the second term of (3) Avill have zero for its limit if r is a proper fraction. We obtain the following theorem : If the common ratio r of a geometric progression is a positive or negative proper frac- tion, the sum of the first n terms of the progression will approach a finite limit, namely (4) S=-^, 1 — r . if the numher of terms n is allowed to increase beyond all bound. The limit S, obtained in this way, is usually called the sum of the progression with infinitely many terms. There is no harm in using this terminology. But it is necessary to remember that this involves an extension of the notion of a sum, since the original definition of a sum was applicable only to the case of a finite number of terms. Strictly speak- ing, iS is not a sum at all, but it is the limit which a certain sum approaches when the number of terms is increased beyond all bound. 63. Periodic decimals. A periodic decimal is a geometric progression whose common ratio is the reciprocal of a power of ten. Thus q q ^ In this case a = 3/10 and r = 1/10. Therefore formula (4) of Art. 62 gives 3 ^ ^ 10 ^ 92 LINEAR FUNCTIONS AND PROGRESSIONS [Art. 63 furnishing a proof of the fact that the decimal fraction .333 • •• approaches the value 1/3 as its limit as the number of places grows beyond all bound. The value of any repeating decimal may be found in this way, even if the repeating part of the decimal is preceded by several figures which are not repeated. Thus 1.7414141 ••• = 1.7 + .041 + — + -^^^ + ... 100 (lOO)-^ ^ 17 .041 ^17 .041 ^17 41 1 - ik 10 .99 10 990' These examples illustrate the following general theorem: Every periodic decimal is equal to a rational number^ whose value may he obtained by finding the sum of a geometric pro- gression with infinitely many terms. Of course every terminating decimal is also equal to a rational number, that is, to a quotient of two integers. In the case of a terminating decimal, the denominator is a power of ten. Thus 1.376 is a rational number, since it is equal to 1376 -=- 1000. Thus, every terminating and every periodic decimal repre- sents a rational number. The converse of this theorem is also true. That is, if any rational number is expressed as a decimal., this decimal must either terminate., or else it must be periodic. In this statement, given without proof, it is understood, of course, that there may be any number of digits in a period, and that the periodic portion of the decimal may be preceded by a finite number of digits which do not repeat. Thus we shall speak of 327.56431431431 .-.as a periodic decimal although the first period (consisting of three digits) is preceded by five digits wliich are not repeated. Therefore, if an irrational number is expressed as a decimal., this decimal cannot be periodic, nor can it terminate. This theorem offers an excellent illustration of the great power of a mathematical argument, and its immense superiority over mere numerical Art. 63] PERIODIC DECIMALS 93 calculation. Thus we have seen in Art. 14 that a/'J is an irrational number, and we have learned in elementary algebra how to express v'2 as a decimal. Xo amount of calculation, even if carried to thousands of places, could ever show us whether this decimal expression for V2 is periodic or not. But our theorems enable us to assert positively, with- out any calculation whatever, that the decimal expression for y/2 is not periodic. EXERCISE XVIII Find the sum of the progressions with infinitely many terms given in Examples 1 to 6. 1- 1 + ^ + i + i + •••• 4- 3 - 1 + 1 - ^ + .... 2. 1 - 1 + ^ - I + .... 5. 9 + V + ¥ + If + •••• 3. ;3 + 1 + I + 1 + .... 6. 40 + ifa + 3^6^o + .... Find the values of the following repeating decimals : 7. 0.1111.... 10. 0.8333... 8. 0.868686.. .. 11. 0.00207900207900.... 9. 0.142857142857.... 12. 0.00854700854700.-.. 13. Find the sum of 1 1 h ••• to infinity if n n + 1 (n + 1)2 (1 + n)3 ^ is a positive number. What can we say about this sum if n is zero? If n = - 1 ? If n = - 2 ? 14. A pendulum is brought to rest by the resistance of the air, each swing being one tenth less than the pi-eceding one. If a certain point on the pendulum describes a path 15 inches long in the first swing, what will be the total length of the path which it describes before it finally comes to rest? CHAPTER III QUADRATIC FUNCTIONS AND EQUATIONS 64. Standard form of a quadratic function. Any expression of the form (1) ax^ 4- Sa; + c, where «, ft, e are constants, and where re is a variable, is called a quadratic function of x. Since such a function would reduce to a linear function if a were equal to zero, we shall henceforth assume a^O. Aside from this restriction upon a, the coefficients a, 6, c may be any real or complex numbers, but we shall usually confine our discussion to the case where a, ft, c are real. In particular, ft or 0. These two zeros coincide (^become identical) if 6^ — 4 a 0. Similarly y will be negative for all real values of a; if 5^ — 4 ac < and a < 0. In neither case will there exist a real value of x which makes y equal to zero. The values of x which satisfy the equation (4) ar^ + bx + c=0 are called its roots. From this definition it follows that the zeros of the quadratic function ax^ + bx -\- c are also the roots of the quadratic equation (4), this equation being obtained from the function by equating the function to zero. Thus the roots of (4) are given by (3). Let us denote these roots by x^ and x^, that is, let us put .r. _ - ^ + V&2 _ 4 ag _ - ^ _ V^2 _ 4 a g ^'^^ ""1" 2a ' "^2- l~a 102 QUADRATIC FUNCTIONS [Art. 69 Then we may write (2) as follows: (6) y = ax^ -\- hx -\- c = a(x — x-^(x — x^. Consequently we find the followmg relation : If the quadratic function ax^ ■\- hx + c has the factors x — x^ and X — x^, then the quadratic equation ax^ + hx + c = Q has the roots x^ and x^ ; and conversely. Clearly, then, the problem of factoring the quadratic function ax^ + hx + c and the problem of solving the quad- ratic equation a:x? -\-hx ■\- c =■ ^ are so closely allied as to be regarded as equivalent. The solution of either problem implies that of the other. If we add the two roots x^ and x^ as given by (5), we find (7) a^i + 0^2 = - -, and if we multiply these roots we find (8) ^^^^ = _Ljj2_^52_4^,)j=^. That is : the sum of the two roots of a quadratic equation ax^ + hx -{- c = ^ is equal to — b/a, and the product of the roots is equal to c/a. Thus, the product and sum of the roots of a quadratic equation may be obtained by mere inspection although the determination of the roots themselves requires the extraction of a square root. 69. Another method of deriving the formulas for the roots of a quadratic equation. The formulas (3) or (5), of Art. 68 may also be obtained as follows. If the equation (1) ax^ + bx+ c= is given, we divide both members by a and transpose the constant term. This gives 2 , ^ 2 _ 4 ac = 0, that is, ax^ -\- hx -\- c = Va( x-\- ~— 2a H^^i:)] Hence, the quadratic function has two linear factors in this case also, even if they are both the same. Since the fac- tors of the function are so closely related to the roots of the equation, we shall say that the corresponding equation has two roots, but that these roots are equal to each other or coincide. With this terminology now perfected, we have actually proved the following important theorem : The quadratic equation with real coefficients (1) ax? + hx + c=0 always has two roots, namely ,-,. -J + V62-4a6' _5_V^>2_4a6' (2) x^=- , x^= la la These roots are real and distinct if b^ — 4: ac > 0, they are real and coincident if h^ — 4: ac = 0, and they are conjugate complex numbers if b^ — 4 ac + V b^ — ac _ — ^ — ^^^ — <^c IZJ X^ ■, Xn • a a EXERCISE XXIV Solve the following ten quadratic equations. Always check your re- sults by substitution or by computing the sum and product of the roots and comparing with the values which the sum and product should have according to Art. 68. 6. X- + X + 1 = 0. 1. x2 + 12 X + 35 = 0. 5. 6 X - 30 = 3 j;2. 2. 20,748 - 1616 x + 21 x^ = 0. 3. x2 - 8 X = U. 4. 3 a:2 + X = 7. 7. 8 x^ - 7 x + 34 = 0. 3 ;-2 01 V. _ 077 SO 8. 80 X -I- ^^ + ^^ — ^^^^^ = 18.-)9i - 3 x2. 4 12 9. x2 - ^-±^x + 1=0. 10. (1 - e^)x-^ - 2 mx + m^ = 0. ab Find quadratic equations whose roots are the given complex numbers. 11. 1 + /, 1 - i. 12. -h + \ J V3, -l-i tV3. 13. ^5(7 + iVl039), yV(7 - J V1039). 14. 1 + 3 i, 1 - 3 i. 108 QUADRATIC FUNCTIONS [Art. 71 Without solving the following equations, discuss the nature of their roots, that is, state whether they are real and distinct, coincident, or iniaginai'y. 15. 2-2+ llx+ 30 = 0. 19. 4x2- 9x = 5x2-255| -8x 16. 622 X = 15 x"- + 6384. 20. 3 a:2 - 6 x + 30 = 0. 17. x'^ - X + 1 = 0. 18. 3 x2 + 24 X + 48 = 0. 21. 18x2 + 24x + 8 = 0. 22. 3x-- X - 4 = 0. In examples 23 to 29, find what value or values k must have in order that the quadratic equation may have its two roots equal. 23. x2 + 3 ^•x + ^- + 7 = 0. Solution. In order that the roots may be equal, the discriminant must vanish, that is, we must have 62 - 4 ac = 9 F - 4(A: + 7) = 0. Solving this quadratic equation for k gives k = 2 or — ^^-. Verification. For k = 2, the equation becomes x2 + 6x+9=0, which actually has two equal roots, each being — 3. For k — — 4/9, the equation becomes x2 - -V ^ + ¥ = 0, which has two equal roots, each being — 7/3. 24. + A:x + 4 = 0. 29. 30. 26. 3x2 + 4x + ifc = 0. 25. 4x2 + (l + A:)x+ 1 = 0. 27. 4x2+ ^x + ^2=0. 28. x2(l + ni-) + 2 kmx + k^ - r^ = 0. (fi(inx + ^")2+ h-x- — a%'^. Verify that 1 + i is a root of the equation x2 — 2x+ 2 = 0, by making use of the graphic interpretation of the complex quantity given in Art. 24. Solution. The complex quantity x = 1 + i is represented in Fig. 37 by the vector OP whose length is equal to V2 and which makes an angle of 45° witli the x-axis. The square of this vector may be constructed by the method of Art. 31. It is rep- resented by the vector OQ of length 2 and am- plitude 90°. Clearly OP' represents 2x, and OP" represents — 2x. The parallelogram construction (see Art. 26) gives OP'" as representative of x2 — 2 X, and OR represents the number + 2. But the sum of the vectors OP"' and OR is clearly equal to zero. 31. Verify that 1 — i is a root of the equation x2 — 2x + 2 = 0, by means of the graphic representation. Fig. 37 Art. 72] SPECIAL FORMS OF EQUATIONS 109 32. Verify graphically that — ^ + ^ i VS and — I — \ t V3 are roots of the equation x"'^ + x + 1 = 0. 33. If the equation x- + ^x — 14 = has one root equal to 7, what is the other root and the value of k ? Hint. Use (7) and (8) of Art 68. 34. Find the value of k and the second root of x- + x + 8A,- = 0, if one root is 4. 35. Find the value of k and the roots of x^ — x — /.• = 0, if the differ- ence between the roots is equal to 9. 36. What relation must there be between a, b, and c if one root of rtx^ + 6x + c = is twice as great as the other? Hint. Use (7) and (8) of Art. 68, write Xg = 2 Xj, and eliminate x\ and Xg between the three equations obtained in this way. 72. Special forms of quadratic equations. The special cases which arise if one or more of the coefficients a, b, c of the quadratic equation ax^ + bx + c = are equal to zero, should be mentioned explicitly. 1 . /f a =it 0, 6 ^fc 0, c = 0, the resulting quadratic, namely ax^ + 6a; = 0, has one of its roots equal to zero. 2. If a=^0, b = 0, c^O, ice obtain a so-called pure quadratic ax^ + c= 0, whose two roots are numerically equal but opposite in sign. The case a = cannot properly present itself here, since the equation would then cease to be a quadratic. We shall, however, later, consider a case closely connected with this, namely the case where a is regarded as a variable whose value approaches zero as a limit. 3. // a ^ 0, 6 = c = 0, the quadratic reduces to ax^ = 0, both of whose roots are equal to zero. It is easy to show that the converse of each of the three statements 1, 2, 3, is also true. The proof is left for the student. 110 QUADRATIC FUNCTIONS [Art. 73 EXERCISE XXV Determine the value or values which k must have in Examples 1-3 so that the equations may have one root equal to zero. What will be the value of the second root in each case ? 1. 8x2-7A•a; + 2^-- 16 = 0. 2. 2x2 - 5x + ^- - 4 = 0. 3. x2 - kx + F _ 4 ^. + 3 = 0. Determine the value or values which k must have in Examples 4-G so that the corresponding equations may have their roots numerically equal, but opposite iu sign. Solve the resulting equations. 4. 3x2 -2x+ Tix - 6 = 0. 5. 2 kx"^ - (5 k + 26)x + k^ = 0. 6. 7 x2 - (A;2 _ 6 X + 5)x - 3 = 0. What values must k and I have in order that the equations in Examples 7-9 may have both roots equal to zero? 7. 5 x^-16lx + kx- il + k + Q = 0. 8. 3 x2 + (A- + /)x + ^• - Z - 1 = 0. 9. 4 x2 + (3 k + l)x + k- 31-2 = 0. 73. Equations of higher degree solvable by means of quad- ratics. It often happens that an equation of higher degree than the second may be solved by a succession of quadratic equations. Every problem of geometry, for instance, which can be solved by means of ruler and compasses leads to such equations.* While it is not always easy to recognize equa- tions of this kind, the following examples will furnish some illustrations. EXERCISE XXVI 1. Solve the equation x* - 13 x2 + 36 = 0. Solution. We may regard this as a quadratic equation for x2. We find x^ = 4 or 9 and consequently x = ± 2 or x = ± 3. Each of these four values of x satisfies the given equation. 2. x" - 74 x2 = - 1225. I. (a: + iy" + 4x+* = 12. \ x/ X * In this connection see Ex. 6, Exercise III. Art. 74] RATIONAL AND IRRATIONAL ROOTS 111 4. X* + 2 x8 - z2 - 2 x - 3 = 0. Hint. Regard x^ + x = z a.s the unknown quantity. 5. ax^n + ix" = c. 6. x8 - 8 = 0. Hint. Observe that x = 2 is one root of this equation so that x — 2 is a factor of x^ — 8. 74. Rational and irrational roots of a quadratic equation. We have discussed carefully the character of the roots of a quadratic ec^uation from the point of view as to whether they are real or complex. But there is another, more subtle, distinction which is also important, namely the dis- tinction between rational and irrational roots. Let (1) ax'^-\-bx + c = be a quadratic equation whose coefficients a, b, c are ra- tional numbers. (Cf. Art. 8.) Each of these coefficients may then be expressed as a quotient of two integers, that is, as a fraction. If these three fractions be reduced to a common denominator p^ we may write I 7 m n a = -, A = — , c = -, F P P so that the equation (1) becomes -x^ ■] — x+ -= 0. p p p But this equation has the same roots as the equation (2) Ix^ + mx 4- w = 0, whose coefficients are integers. (See Art. 67.) The roots of (2) are ^o\ — m + Vw^ — 4ln — 711 — Vwz^ — 4 In (6) 2*1 = ■ , 3*0 = , 11 21 and they are clearly rational numbers if m^ — 4 /w is a perfect square, so that Vw^ — 4 ^Ai is an integer. If ni^ — 4 ?n is pos- 112 QUADRATIC FUNCTIONS [Art. 75 itive and not a perfect square, its square root will be irra- tional. We may prove that this statement is true by the method which was used in Art. 15 to show that V2 is irrational. From the first equation of (8) we find, by clear- ing of fractions and transposing, 2 Ix-^ -\- m = Vw'^ — 4 In. If x^ were rational, the left member would be a rational number, while the right member is irrational. This is a contradiction, so that a-j must be irrational. Similarly it follows that X2 must be irrational. We have proved the following theorem : The solution of a quadratic equation with rational coefficients may he reduced to the solution of an equivalent equation., Ix^ + mx -\- n = with integral coefficients. Tlie roots of this equation ivill he rational., if and only if the discriminant mn? — \ In is a perfect square. EXERCISE XXVII Apply this criterion to Examples 1 to 8 of Exercise XXIV. 75. Quadratic surds. The roots of the quadratic equation (2) of Art. 74 are irrational if m^ — 4 Zw = c^ is positive and not a perfect square. This depends essen- tially upon the fact that the expressions (3) of Art. 74 for these roots contain the square root of d. An irrational num- ber, such as V5, the square root of an integer which is posi- tive and not a perfect square, is called a quadratic surd. The same name is sometimes also applied to irrational numbers of the form Vc^ where d., instead of being an integer, is a ra- tional fraction which is not a perfect square. We shall use the term in this more general sense. The following theorem is fundamental in dealing with such surds : Art. 75] QUADRATIC SURDS 113 //' a, b, a\ h\ and d are rational numbers^ and if d is posi- tive and not a perfect square, so that y/d is irrational, then an equatio7i of the form (1) a + h^d = a' + h'-^d can subsist only if (2) a = a' and b = b'. In fact, from (1) follows (3) a-a' = ib' - b)^d. If b' were not equal to b, b' — b would be different from zero and it would be permissible to divide both members of (3) by b' - b, giving a — a / 7 _ = V t«. b' -b But this equation involves a contradiction, since one of its members is a rational number, while the other is irrational. Therefore b' —b cannot be different from zero ; that is, we must have b' = b. But this condition, together with (1), shows that we must also have a' = a. Consequently the theorem is established. We know that irrational numbers obey the same funda- mental laws of addition and multiplication (Laws I to IX of Art. 2) which were originally observed to be true for positive integers. Let us then examine the sura, difference, product, and quotient of two numbers of the form a + bVd and a' + b'Vd, where a, b, a', 6', d are rational numbers and where Vo? is irrational. We have immediately (a + b^d) + (a' + bWd) = a + a' + (b + //) V^, ^ ^ (a + bVd) - (a' + bWd) = a-a' + (b- b')Vd, if we make use of the commutative and associative laws of addition and multiplication, and also of the distributive law of multiplication. Again we find in similar fashion (5) (a + bVd)(a' + bWd) = aa' + bb'd + i^ab' + a'b)Vd, 114 QUADRATIC FUNCTIONS [Art. 75 and also, if a! and h' are not both equal to zero, ^a\ a + bVd a + h^d a' — b'^d (6) ■ = = = X r a'+bWd a' + bWd a'-bWd _ aa' — bb'd + (a'b — ab'^-Vd _ aa' — bb'd a'h — aV /-^ ~ a'2 _ 5/2;^ - - a'2 _ 5^2^ + a'^-b'H ' where the denominator a'^ — b''^d cannot be equal to zero. For, if it were, we should have d='' contrary to our assumption that d is not a perfect square. Each of the right members in (4), (5), and (6) may be rewritten in the form A + B^d, where A and B are rational numbers. We see, therefore, that the sum, differ- ence, product, and quotient of two numbers of the form a + hVd, a' + b' Vd is again a tiwnber of the same kind, the usual excep- tion, which excludes division by zero, being made in the case of the quotient. This property of the numbers a + b^d is often expressed by saying that they form a field. The process indicated in (6) is usually called rationalizing the denominator, and is of great importance in dealing with surds. The auxiliary quantity- a' — b'^d, used in this pro- cess, is often called the conjugate of a' +b'Vd. If an expression involves more than one quadratic surd, it may be simplified by treating separately the several surds which occur in it by the method here indicated. EXERCISE XXVIII Simplify the following expressions involving surds : 1. y/2i + VM -VQ. 2. 2V8 -7v'l8 + 5\/72. 3. Vl8a563 + V50a3?A ^Ili^. 8. V.5 - 1 V3-V2 7. ini}/!. 8. ^ + ^- 4. (.3. 4-V5)(2- V5). 5. (7 ■f2V6)(9 -5^/6). 6. (9- -7VI3)(^ . - 6V13). q 2 1 10 '^ + V3 V8- 2 Art. 76] SQUARE ROOT OF a + bVd 115 76.* The square root of an expression of the form a + by/d. Let us again consider a niuuber of the form a + hVd where a, b, and d are rational, but where d is a positive rational number which is not a perfect square. The square root of a + bVd will not, in general, be expressible as a sum of two quadratic surds. There are some cases, how- ever, in which this may be done, and we propose to answer the question as to what cases these are. Let us suppose that there exist two positive rational numbers, z and y, such that (1) Va + hVd = \/x± Vy. To avoid ambiguity we shall assume here, as elsewhere in this book, that the symbol y/k stands for the positive square root of k whenever k itself is a positive number. We shall assume, moreover, that a + bVd is a positive number, so that it has a positive square root. This assump- tion does not prevent one of the rational numbers, a or b, ivom being negative ; but they may not both be negative. One of the two terms in the right member of (1) maybe negative, but not both. Since the right member as a whole must be positive, equation (1) implies that, if there is a minus sign at all in the right member, the notation has been so chosen that, of the two positive numbers z and y, the greater has been called X. If (1) hold.s, we must have (2) a + bVd =(y/x ±\/]/y^ = X + y ±2Vxy. Since a, b, x, and y are rational, while y/d is not, the quantity Vxy cannot be rational. For if it were, a + bVd would, according to (2), be rational. According to Art. 75 we therefore conclude, from (2), (3) X + y = a, ±2 Vxy = bVd, whence (4) x + y = a, xy = I b^d. We can easily form a quadratic equation of which x and y shall be the roots, namely (see Art. 68 and Exercise XXIII, Ex. 19), (z-x)(z-y)=0, ^'^ z' - (x 4- y)z + xy = 0, which becomes, on account of (4), (.5) z^ - az + \ b-d = 0. If this quadratic has irrational roots, equation (1) will be impossible, since we assumed x and y, which are the roots of (5), to be rational. Now the discriminant of (5) is a- - b'-d and the coefficients of (5) are ♦Article 76 and Exercise XXIX may be omitted without destroying the conti- uuity. 116 QUADRATIC FUNCTIONS [Art. 76 rational. Therefore (see Art. 74) the roots of (5) are irrational unless fl2 — Ifid is the square of a rational number. Thus, if ifi — }P-d is not a perfect square, (1) is impossible. If cfi — IM is a perfect square both of the roots of (5), namely 0+ Va2-62rf a-^/a^ - hM x = ^ , y = 2 , will be rational. Moreover, if a is positive, both x and y will be positive. This is evident as far as x is concerned; y will be positive in this case because Va- — h'^d is less than Va^ = a. If a is negative, y will be nega- tive, so that this case (o < 0) is excluded, since we have assumed that both X and y shall be positive. If we have a > and a- — V^d a perfect square, x and y will be positive rational numbers which satisfy equations (4). Both Vx and ^y will then be real positive numbers, and we may choose the sign + or — in (8) according as h is positive or negative. After this choice of sign has been made, equations (3) will be satisfied, and we shall actually have the positive square root of a + h^/d expressed in the form (1), with the + or — sign according as h is positive or negative. Our complete result may be summarized as follows: Let a, b, d, x, and y be positive rational numbers, such that y/d is irrational. It is possible to write y^ (6) Vn + bVd = Vx + \^y, Va - by/d = y/x -\/, if and only if a? — bM is a perfect square. The values of x and y will then be given by the expressions a + y/cfi - bM a - Va^ - ¥d (0 ^ = 7i » y^ o EXERCISE XXIX 1. Examine the possibility of expressing v 3 + 2V2 in the form y/x -\- yfif. Solution. In this case a = 3, b = 2, d — 2, so that o- — b-d = 1, a perfect square. Moreover, a, b, and d are positive. Therefore we find from (7) 3 + 1 o '^ - 1 1 X = — -— = 2, // = -— — ^ 1, so that V3 + 2 V2 = V2 + Vl := 1 + V2. Of course the solution may also be obtained without using the formulic (7), by applying to this particular case the process by means of which these formulas were derived. The student should do this to help him understand the general process. Art. 77] APPLICATION OF THE MONOTONIC LAWS 117 2. Examine the possibility of expressing V 4 + 2\/2 in the form Vj; + Vy. Solution. In this case a^ — b^d = 16 — 8 = 8, which is not a perfect square. Therefore such an expression is impossible for V 4 + 2V2. Examine in the same way the following numbers, and find their square roots in the form y/x ± Vy whenever possible. 3. 7 + 4V3. 6. 87 - r2V42. 4. 5 - \/24. 7. I + V2. 5. 28 + 5v'12. 8. 2 - V4 - 4 a^. 77.* Application of the monotonic laws of Algebra in nu- merical calculations involving quadratic surds. The student has learned in his first course in Algebra how to calculate the value of a quadratic surd, that is, of a square root, to as many decimal places as may be desired. Let x be the exact value of the surd ( V2 for instance), and let x„ be the approximate value found for it by carrying out the process of extracting the square root to n decimal places. Then x„ will be a decimal fraction with n digits to the right of the decimal point, and we shall have x„, z =g {x)= ^ x^ - x + 5.. Find f\jg (x)]. 5. Given /(x) = x^ + 5, ^(x)= 3 x'^ - x + 5. Find /[^^(x)] and 6. Show that a linear function of a linear function of x is again a linear function of x. 7. Show that a linear function of a quadratic function of x is a quad- ratic function ; and that a quadratic function of a linear function of x is a quadratic function of x. 8. Show that a quadratic function of a quadratic function of x is a function of the fourth degree. 84. The factor theorem. We have already noticed that a quadratic function which has a: — a as a factor vanishes for 2J = a, and conversely, that if a quadratic function vanishes for x= a^ then it has x — a as a factor. (See Art. 68.) We shall now show that a corresponding theorem, known as the factor theorem, holds for integral rational functions of any order. If X — a is a factor of an integral rational function f (x^, the function will assume the valUe zero when x is equated to a. Con- versely^ if such a function f (^x) becomes equal to zero for x =■ a^ thenf(x) has x — a as a factor. The proof of the first part of this theorem is immediate. If /(a;) has a; — a as a factor, we may write f(x) = {x-a>^g(x), where ^(2;), the other factor of /(a;), is an integral rational function of x whose degree will be less by a unit than that oif(x). If in this equation, we put a: = a, the factor x — a becomes equal to zero, and the other factor becomes equal to^(«), which will be some finite number. Consequently the product will vanish, so that /(a) = 0, as we wished to prove. To prove the converse, let n ) fix) = Ax^ + 5a:"-i + 6V-2 + ... ^Lx^^Mx^N. Art. 84] THE FACTOR THEOREM 133 According to the hypothesis, the value of this function for x= a^ that is,/(rt), is equal to zero. Therefore we have (2) /(a)=^a" + ^a"-i+ Ca"-2 + ... +La^ + Ma + N=Q. We may therefore write (3) /(2-)=/(2^)-/(a)=^(2;»-a») + 5(x»-i-a'»-i)4- ••. + Lix^ - 0^)+ M(x- a). Each of the last two binomial terms of (3) obviously has a: — a as a factor: We shall show immediately that the same thing is true of each of the other binomial terms of (3). Consequently, f(x^ has a: — a as a factor, as was to be proved. In order to complete the proof of this theorem it only re- mains to show that, for every value of the positive integer n, z" — a" has x — a as a factor. We know that this is true for n = 1 and for n = 2, since a: — a = (a; — a) • 1 , x^ — a^ = (x — a)(x + a^. It may be verified easily that it is also true for 7i = 3, since x^ — a^ = (^x — a) {x^ + ax + a^), as may be seen by performing the multiplication on the right member and simplifying. To prove that this theorem is true for all values of w, we first prove the following lemma, or auxiliary theorem. If x^ — a^ ha^ x — a as a factor^ so does x^""^ — a'^'^^. Proof. \Vc may write ^k¥\ _ ^A+l _ j.k + 1 _ ^kj. _|_ ,^kJ. _ ^A + l __ ^^J^k _ (ikyy _j_ (ik^j. _ ^-^^ Since the last term, rt*(a- — a), has x — a as a factor, the whole right member, which is equal to x^ ^ — a''"'\ will have x—a as a factor if x'^ — a''' has such a factor, lint this remark proves tlie lemma. Now 2;2 — a^ has a- — a as a factor. A first application of the lemma (for ^ = 2) sliows that ;r^+^ — a-'*"^ or .x^ — a^ also 134 INTEGRAL RATIONAL FUNCTIONS [Art. 85 has X— a as a factor. A second application of the lemma (to the case A; = 3) shows that a;* — a* has a; — a as a factor. We may proceed in this way until we reach re" — a", thus proving that x^ — a" has a; — a as a factor. The method of proof just employed is called the method of mathematical induction, and is very important in all parts of mathematics. We shall have occasion to apply this method frequently during this course. 85. The remainder theorem. The following theorem, known as the remainder theorem, includes tXvd factor theorem as a special case. If an integral rational function f(^x) be divided hy x — a until a remainder independent of x is obtained^ this remainder is equal tof(^a^, the value of the function f(^x^ for x = a. Proof. Carry out the process of dividing /(a^) hjx — a until we reach a remainder M independent of x, and let the quotient obtained by this division be called Qi^x). Accord- ing to the definition of the terms division, quotient, and remainder (see Art. 4), this means that we shall have (1) flx)=Q(:xX^-a)+B, where i^ is a constant whose value does not depend upon x, and where Q^x} will be of degree w — 1 if /(a^) is of degree n.* Let us now put ;r = a in (1). Since Q{x^ is an integral rational function of ;r, (?(«) is a finite number, and we find from (1) (2) /(a)= Q(a) .0 + B=E. But (2) is nothing more or less than the remainder theorem which we wished to prove. In the particular case when M=0, we obtain from (1) and (2) the factor theorem ; saying that if /(a) = 0, then /(a;) has X — a as a factor, and conversely. Thus we have found incidentally a new proof of the factor theorem. * This process shows that /(x) may be written in the form (1) and does not involve any actual division until we write Art. SG] synthetic DIVISION 135 86. Synthetic division. Tlie remainder theorem shows us lliut if we divide f(^x) by a; — a, the remainder is equal to /(a). But in Art. 82 we found a convenient method for calculating the value of /(a). We may therefore use this same method for calculating the remainder in the division oif(x) by X— a. But this calculation will at the same time give us the value of the quotient. To see this, let us apply the ordinary process of long division to the problem of divid- ing Ax^ + Bx^ + Cx + D by j- — a, and then let us compare with Art. 82. We find Ai^ + Bx'^ + Ox + D x-a Ax^ - A ax^ I Ai^ + Ex + F Ux^'+ Ox Bx^- - Eax Fx + D Fx - Fa Fa + B=a = R The term Fx^ is first « obtained in- the form Aax^ -{- Bx^ = {Aa + B}x'^, but according to the notation used in (8) Art. 82, this is the same as Fx^. Similarly the term Fx arises from (Fa + 0)x which, according to (4) Art. 82, is equal to Fx. If now we write down once more the scheme for the calculation explained in Art. 82, namely A B a B [a Aa Fa Fa A F F B~=f(a), we notice that the first three numbers of the last line, A, E, and F, are the coefficients of x\ x, and 1, in the quotient, whereas M is the remainder. As applied to the example f(x) = 3 x^ — i x'^ + 7 x — 2, which was used at the end of Art. 82 as an illustration, we see from the numbers obtained there that the quotient obtained, when 3 x^ — 4 x^ + 7 x — 2 is divided by x — (- 2) = x + 2, will be S x^ - 10 x + 27, and the remainder will be — 56. This calculation has been written out in just this form in Art. 82. 136 INTEGRAL RATIONAL FUNCTIONS [Art. 87 This method of dividing an integral rational function by a; — a is far more convenient than the ordinary method. It is known as the method of synthetic division. Synthetic division may be performed according to the following rule. 1. To divide f(x) by x — a, arrmige f(x') in descending powers of X. 2. Write the coefficients of /(x) on a horizontal line, in the order ivhich corresponds to the arrangement specified in No. 1 of this rule. If any power of x is missing inf(x)., supply that power with a zero coefficient. 3. Multiply the first coefficie7it A by a, write the product beloiv the second coefficient B, and add. Multiply this sum E by a, ivrite the product below the third coefficient (7, and add. 4. Proceed in this umy until all of the places in the third row except the first are filled up, and ivrite down the first coef- ficie7it A of f(x') in the first place of the third roiv. The last number of the third row will be the retnainder., and the other numbers in the third row will be the coefficients of the quotient obtained when f(x) is divided by x — a. EXERCISE XXXIV Find the quotient and remainder in the following divisions. Use the method of synthetic division. 1. Divide 2 x^ + 5 a;^ - 7 x + 1 by a; - 3. 2. Divide .3 a.-^ - 7 a;^ + 4 a: - 5 by x + 3. 3. Divide 2 a;^ + x + 1 by a' - 2. 4. Divide x^ — 1 by x + 2. 87. The slope of the tangent. When we have drawn the graph of an integral rational function y = f{x) by the method of computing the coordinates of a large number of its points, we can draw the tangent to the curve at any one of its points with some degree of approximation. We wish, however, to be in a position to draw the tangent with greater accuracy, and this desire leads us to adopt a precise detinition for a tangent and, in this way, to seek a precise method for its determination. Art. 87] THE SLOPE OF THE TANGENT 137 We define a tangent to a curve as follows : Let Pj be any point on a given curve {see Fig. 40), and let P^ be a second point (^distinct from Pj) of the same curve. As Pg approaches P^ as a limit, the line PiP^ (sometimes called a secant of the curve} will turn around Pj as a center. If the secant approaches a limiting position P^T as P^ approaches Pj, this limiting position of the secant is called the tangent of the curve at P^, and J\ is called its point of contact. This detiuitiou ie(iiiires a few words of explanation. In the fir.st place it presiii)poses the notion of limits which the student has discussed to some extent in his earlier courses in elementary algebra and geometry. A more detailed discussion of this important subject will be given later in this book. (See Chapter XV.)* In the second place it is essen- tial to remember that the line PiP., is regarded as unbounded. We are not talking about the line-segnient Px^r ^^^li^'l' [- 1. Akt. 88] THE BINOMIAL THEOREM 145 Thus we have actually proved that if the rth term of the expansion of (a: + hy is c^iven by fornri'ula (2) for n = k, then the rth term in (x -{■ A)'-+^ will be given by this same formula for n = k-lt-^- Since the rth term represents ani/ term of the expansion, we liave actuall}'' proved our lemin:i : if the binomial formula is correct for n = k, it will also he correct for n = k-\-'^. But we know that the binomial formula is correct for n= 1, 2, 3, 4, 5. Oar lemma allows us to conclude, without actual test, that it will also be true for n = G. A second application of the lemma shows the formula to be true for w = 7, and so on, for all positive integral values of 7i. Formula (2) gives the rth term of the expansion of (x + A)"- The formula for the (r + l)th term, which may be called the rth term after the firsts is a little easier to re- member ; it is equal to ^\ 1.2.3...r EXERCISE XXXVI Write out the following espausious by use of the formula : 1. {x + hy. 7. (2a-3/>)e. 12. (1 + J^) 2. {X - hy. 3. {a + hy. 4. {a -by. ' \ ■ xJ ,^_ /^^l 7. (2 a - '^ly. 8. (4x-5//)5. 9. hi)' 10. ii + \y- 13. 1 + ,!)■ 5. (-a + l>y. 6. (a' + l/^y. 11. (1+i)'- 15- (^-^0" 16. Use the binomial theorem to compute lOP. Hint. Put 101 = 100 + 1. Use the binomial theorem to compute the following powers. 17. 1026. 18. 996. 19. (1.1)^*' to four .significant figures. 20. (1.01)1"° to four significant figures. 21. Find the eleventh term in the expansion of (x +h)^'. Hint. Use the formula (2), Art. 88, with n - 17, r = 11. 146 INTEGRAL RATIONAL FUNCTIONS [Art. 89 22. Find the fourth term of (a - 4 by^. 23. Find the fifth terra of (3x - 2 >jy^ 24. Find the sixth term of ( x + - j . 25. Find the rth term of (■^^r■ 26. Find the middle term of {x + /0^°. 27. Find the two middle terms of (x + /i)"- 28. Equations (1), Art. 88, seem to indicate that, in the expansion of (x + hy, the numerical coefficients equidistant from the ends are equal. Prove that this is so by making use of formula (2) of Art. 88. 29. Equations (1), Art. 88, seem to indicate that the middle term when n is even, and the two middle terms, when n is odd, have the greatest numerical coefficients. Prove that this is so. 30. Write out the expansions for (a + ft + c)^ and {a A- h + c)*. Hint. Use the binomial formula with x = a A- h, and h = c. 89. The derivative of an integral rational function. It is now an easy matter to find the derivative of the function (1) fQx) = Ax^ + Bx^~^ + 6V-2 + ...^Lx + M. According to the binomial theorem we have /(^ + ^0 = A{x+hy->rB(x->rhy-'^-\-CQx-\-hy-''-+-- +L{x + h)+M + (7[.?:"-2^(?i-2):C"-3/i+ ..■]-!- ... + L{x + K)+M, where the terms which contain /i^ or a higher power of h as a factor have not been written down, but are merely indi- cated by dots. If we collect those terms which contain no h at all, and those which contain h as a factor, we find f(x + h) = Ax"" + 5a;"-i + Ox^-"^ + ■.- +Lx + M (2) + h[nAx--^ + (w - l)5.c"-2 + (n - 2) Ox^-^ + •.. + i] + terms each of which contains h^ as a factor. Akt. 90] DERIVATIVE OF AX INTEGRAL FUNCTION 147 From (1) and (2) we find, by subtraction and division bv h, f(x + h)-f(x) h ^^^ = wAx»-i + (w - l)5a:"-2 + (m - 2) (7a;"-3 + • • • + i^ + terms each of which has A as a factor. The derivative of f(x) is the limit which (3) approaches when h approaches zero (Art. 87). That part of (3) which has been written out will remain unchanged as h approaches zero, because it contains no h. All of the other terms of (3) will have the limit zero since each has A as a factor. Therefore, the derivative of the function /(a:), de- fined by (1), is (4) fix) = nAx^-^ + (n - l}Bx"-^ + (»i - 2) Cx'^-^ + ■■• + L. Observe that the law according to which f (x) is obtained from f(jjc) is very simple. Each term of /(^) produces a corresponding term of f'(x) by applying the following rule: Multiply any term of f(x) by the exponent of the power of x which occurs in it, and afterward reduce this exponent by unity. This rule may even be regarded as applying to the last terra M of /(a:), which at first sight seems to be an excep- tion, inasmuch as it produces no corresponding terra in f {x). For we may think of M as the coefficient of aP = 1. (See Art. 88 No. 8 of the fine print.) If we do, the above rule will give as the term oi f (x^ which corresponds to the term M\nf(x). EXERCISE XXXVII 1. Solve a second time the examples of Exercise XXXV, making use of formula (4), Art. 89, for the purpose of computing the derivatives. 2. Prove directly that the derivative of a constant is equal to zero. 90. Derivatives of higher order. We now know how to find the derivative f'(x) of any integral rational function f(x). \i f(x') is of degree n,f'{x^ is an integral rational function of degree n~l. The derivative of /'(a;), denoted 148 INTEGRAL RATIONAL FUNCTIONS [Art. 91 by/" (a:), will therefore be an integral rational function of degree w— 2, and is called the second derivative of /(a;). We may continue in this way. Clearly the kth derivative of /(a;) will be an integral rational function of degree 71 — k. In particular the nth derivative of /^a;) will be a constant, that is, it will contain no x. The {n + l)th deriva- tive and all derivatives of order higher than this will be equal to zero. (See Ex. 2, Exercise XXXVII.) EXERCISE XXXVIll Compute the derivatives of higher order for all of the functions men- tioned in Exercise XXXV. 91. Taylor's expansion. By making use of the higher derivatives we can now write the expansion of f{x+ K) in a very simple form. We shall give the details of the proof only for the case where /(a;) is of the third order, so that (1) / (x) = Ax^ + Bx^ + Gx + i), but the same method would apply to an integral rational function of ;iny degree. Let/' {x),f" (x),f"' (a;), etc., represent the first, second, third derivatives, etc., of/ (x). Then we find from (1), fix) = 3Ar2 + -lBx+ 0, fix) ^6 Ax + 2 B, ^"^^ . /'"(a:) = 6A /4^(a:)=/'5>(a:)= ••. =0. Again, we find from (1) fix + h) = A:r"-3 + . . . = and (2) Ax'^ - Bx'--^ + CV"-2 - Dx'""'^ + . . . = are so related that, to every root of (1) there corresponds a root of (2) of the same magnitude but of opposite sign. This transformation is very useful when tve are attempting to calculate a negative root of (1), since it enables us to calculate instead the corresponding positive root of (^2). EXERCISE XLI In Examples 1-4 find the equation whose roots are obtained from those of the given equation by multiplication with the number indicated in parenthesis. 1. X3+ 2x2 + 1 X+3 = 0. (^(;) 3. .7-''- 2x3+ 7x2-1 = 0. (2) 2. 2 x3 + 7 X - 1 = 0. (;3) 4. x3 _ i x- - | = 0. (.5) 5. From each of the equations in Examples 1-4 find another one whose roots shall be numerically equal to those of the given equation, but opposite in sign. Use the transformation of Art. 94 to obtain from each of the follow- ing equations another one all of whose coefficients shall be integers, and for which the coefficient of the highest power of the unknown shall be equal to unity. Use the smallest value of ?/* which will accomplish the purpose. Aht. !)()] CONTINUITY OF INTEGRAL FUNCTIONS 155 6: ,r^- I .c2 + 1 X + ', = 0. 8. L> ./•* + 3 x^ + 2 x + 3 = 0. 7. 3 ./:3 + X- + 7=0. 9. .r3 - J x2+ ^ x^ + *| = 0. 10. It may happen that the transformation of Art. 95 does not change the equation at all ; that is, the coeHicients of (1) and (2), Art. 95, may be identical. Under what circumstances will this happen, and what can you conclude concerning the roots of such an equation? 96. Continuity of integral rational functions. In calculat- ing a real root of an e(jiuitioii of the form (1) f(x) = Ax" 4- i^x"-! + -'. + Lx + 31= 0, we may proceed as follows. We first make a graph of the function (2) ^=/(^)- The real roots of (1) will be the abscissas of the points in which the graph crosses the a;-axis. Suppose that, in mak- ing the calculations required for drawing this graph, we happen to strike a value of x which makes f{x) exactly equal to zero. Then, of course, this value of a; is a root of the equation. Ordinarily, however, this will not occur ; usually none of the values of X, say a, b, c, d, etc., which are used in these calculations will be roots of the equation ; so that the values of /(«), f(J>'), /(c), etc., will, all of them, be different from zero. Suppose /(a) < 0, and f{h) > 0, a case illustrated in Fig. 43, where MF =f(a-) < 0, NQ =f{b) > 0. The figure leads us to conclude that there will be at least one point, between M and iV, where the curve crosses the a;-axis. If a^j is the abscissa of such a point, a-^, which lies be- tween a and b, will be a root of the equation (1). Although this argument sounds very plausible we have, so far, no guarantee for its correctness. If the graph of the function /(x) is indeed an unbroken (continuous) curve as indicated in Fig. 43, the conclusion will be correct. But if 156 INTEGRAL RATIONAL FUNCTIONS [Akt. 96 the graph were broken ((liscontiiuious) as in Fig. 44; the conclusion might be erroneous. We shall see later that such graphs as that of Fig. 44 are not at all uncommon, but they never present them- \ selves wlien f {x) is an integral function. M 1^ ^ It can be shown that the graph of an I * integral rational function is always con- ^ tinuous. Fig. 44 In order that we may express these con- ditions somewhat more precisely, we formulate the following definition : A function f (x) is said to be continuous in the neighborhood of a particular value of x^ say x = p^ if the difference f(p+h)-f(p) approaches zero for its limit, whenever h approaches zero in any manner, through positive or negative values, continuously, or by jumps. If the function is continuous in the neighborhood of all values of x which lie between a and b, the function is said to be continuous in the interval from a to b. The graph of a continuous function will be an unbroken (continuous) curve, such as is illustrated in Fig. 43. This figure suggests the following theorem. I. If the function f (x) is continuous in the interval from a to b, and if the values off(^a^ andf(^l)} are opposite in sign, there will exist at least one value of x, between a and b, for which the function f (^x^ will become equal to zero. The argument which we attempted to make, at the be- ginning of this article, will therefore be justified if the following statement is correct. II. An integral rational function of x is continuous in any finite interval. Theorems I and II are both correct. We shall not however attempt to prove them now, since the proofs are somewhat Art. 97] NEWTON'S METHOD OF APPROXIMATION 157 abstract. The discussion just given, and the experience which we are gradually gaining in the drawing of graphs, will serve to make both theorems seem plausible. 97. Newton's method of approximation. Let a; be a root of the equation f{x)=0 ; let a l)e an approximate value of this root, obtained perhaps by inspection of the graph, and let us put re = a + A, so that h is the correction whicli must be added to a to obtain the true value of the root. According to Art. 92, tliis unknown correction h will be a root of the equation (1) /(a) +/'(«)/.+ l/"(a)A2+ ... =0. Now, if a is a fairly good approximation to x, h will be a comparatively small fraction of a so that the ratios h^/a\ h^/a^, etc., will be small as compared with h/a. In most cases, therefore, we may expect to get a good approximation to the correction h by neglecting the terms which involve A^, Jfi, etc., in (1). If we do this, (1) reduces to an equation of the first degree for h, namely /(a) + /'Ca)A=0, which gives (2) A = _ZO!l. If A is computed by this formula, a + A = a^ will usually be closer to the true value of x than was a ; but, in general, a^ will still not be the exact value of a:, but only an approxima- tion. If now we repeat this process, using a^ instead of a, we may compute the quantities ''' = -/(S' ''2 = ''. + ^.- and usually a^ will be a still better approximation to the root. In most cases this process of approximation, if repeated often enough, will enable us to obtain the value of x with any desired degree of accuracy. The process is known as Newton's method. 158 INTEGRAL RATIONAL FUNCTIONS [Arts. 98, 99 98. Geometric significance of Newton's method. Let A (Fig. 45) be the point of the graph of y=if(x) which cor- responds to the value of x, x = a = OM. Ihen MA^fCa), and the slope of the tangent ^7^ is equal to /'(a). (See Art. 87.) Let H be the point in which the tangent crosses the a:-axis, and let a + 7i be the abscissa of IT. Then the coordi- nates of if will be (a -I- A, 0), while those of A are (a, /(«))• Therefore (see Art. 53), the slope of J. 2^ will be equal to /(«)-Q ^ /(«), a — (a + A) /* Since the same slope is also equal to /'(a), as we have seen above, we must have wlience h = — ■ /'(«) But this is precisely the value of h given by equation (2) of Art. 97. Therefore Newton's method of approximation consists in replacing the curve ABO by the straight line AT^ which is tangent to the curve at the point A whose abscissa is equal to a. The approximate value of x obtained by a single applica- tion of Newton's method is the abscissa a^ of the point H (Fig. 45). A second application of the method would con- sist in replacing that part of the curve between A^ and B by the tangent at Ay The figure shows how very nearly the intersection of this tangent with the aj-axis would coincide with B. 99. The method of false position (Regula falsi). A second method of approximation, also suggested by geometry, is as follows. If /(a) and/ (6) are opposite in sign, we know that Art. 99] THE METHOD OF FALSE POSITION 159 Fig. 46 there is at least one root of the equation /(.c) = between a and b. In Fig. 46 we have a = 031 /(a) = 3IA < 0, b=ON, f(h) = NB>0, and the root of /(a')=0, which lies be- tween a and b, is the abscissa of the point R. li A and B are close together, we may regard the straight line AB as a reasonably close approxi- mation to the curve ABB, and the point S, in which AB intersects the 2:-axis, as a fairly good approximation to the point R. The abscissa of jS will therefore be an approxima- tion to that root of the equation /(a;) = which corresponds to tlie point B. We shall now show how to calculate this approximate value of the root. The slope of the line AB is /W-/(«) (1) m = (Art. 53) since the coordinates of A and B are (a, /(a)) and (^,/(i)) respectively. The equation of a straight line which passes through the point A (a, /(a)), and whose slope is equal to 771 IS Therefore (2) .. . f(b^-f(a). b— a a) is the equation of the line AB. To find the abscissa of the point S in wliich AB intersects the aj-axis, we must put ^ = in (2) and compute the corre- sponding value of X, which we shall call x-^. Thus we find -/(«)= /W-/('') (.,-a). b — a whence or (3) X-, — a — x, = a -f(a)(b-a) f(a)(b-a^ 'fib)-f(a^ 160 INTEGRAL RATIONAL FUNCTIONS [Art. 100 The value of x^ given hy (3) is the abscissa of the point S^ and will in general furnish a close approximation to the desired root if a and b are close enough together. By repeating this process, using x^ and 6, or a and a:^, instead of a and 6, we may obtain a second approximation, and so on. This method may be regarded as the geometrical equiva- lent of the following arithmetical argument. If a and b are close together, and if x is between a and 5, the change in the function /(a;) will be approximately proportional to the change in the variable x. Now as x changes from a to a:^, f(x) will change from /(a) to f{x^); as x changes from a to 6, f(x) will change from /(a) to f(h^. Therefore we shall have approximately : (4) f(h)-fia) b-a If x^ is approximately a root of the equation f{x) = 0, we shall have very nearly /(a-j) = 0. If we substitute /(a;j)= in (4), we find -fjo-^ ^ x^ — a which again gives the value (3) when solved for x^ One may usually obtain a very close approximation by combining Newton's method with the method of false position. For the true value of the root usually lies be- r tween the two approximate values obtained by these different meth- ods. Thus, in Fig. 47, the point M lies between the points H and aS'. 100. An example of Newton's method. We wish to find a real root of the equation (1) f(x') = x^ + J^-Sx^- x-4 = 0. We find /(I) =- 6, /(2) =+ 6. Art. 100] AN EXAMPLE OF NEWTON'S METHOD 161 Consequently (Art. 96, Theorems I and II), there is a real root between x = 1 and x = 2. Therefore we put a: = 1 4- 2/, or 7/ = 3- — 1, where ?/ is a positive proper fraction ; ?/ is the correction which must be added to 1 so as to make 1 + y a root of (1); it is the quantity which was denoted by h in Art. 97. According to Art. 92, t/ will be a root of the equation ob- tained from (1) by diminishing its roots by 1. We perform the calculation by synthetic division : 1 -3 1 +2 -4|1 _ 2 2 1 -1 -2 + 3 +2 -6* 3 1 + 2 + 4 0* 4 1 + 6* (2) 1* 5* Thus, the correction y will be a root (between and 1) of the equation (3) g(l/) = y' + f>f + ^f + 0-i/- 6 = 0. It frequently happens that Newton's method fails to furnish a good approximation at the first stage of the calculation even if it works satisfactorily at the later stages. It fails entirely at the first stage of tliis example because the co- efficient of J/ in (3) is equal to zero. But the method of false position (Art. 99) may be used to advantage. If we use formula (3) of Art. 99, putting a = 1, 6 = 2, we find ' 6 -(-6) -2 thus suggesting 1.5 as an approximate value for x, or 0.5 as an approximate value for ^. 162 INTEGRAL RATIONAL FUNCTIONS [Ai:t. 100 To locate the root a little more precisely we calculate ^(0.5) by synthetic division. This gives 15 6 -6 |Q.5 (4) 0.5 2.75 4.375 2.1875 5.5 8.75 4.375 -3.8125 We have thus found ^(0.5) = — 3.8125. Since we have i/CO) =/(l) = - <3, ^(1) =/(2) = + (3, the true value of the correction i/ must lie between 0.5 and 1.0. Let us try y = 0.8 next. Sjaithetic division gives 15 6 -6 |0.8 (5) 0.8 4.64 8.512 +6.8096 5.8 10.64 8.512 + 0.8096 so that ^(0.8) =+ 0.8096. Since ^(0.5) = -3.8125, the root of (3) for which we are looking lies between 0.5 and 0.8. Since the value of ^(0.8) is small, it is apparent that the approximation 0.8 for i/ is fairly close and we shall there- fore continue our calculation based upon the approximations 1.8 for a: or 0.8 for y, by putting (6) ^ = 0.8 + 2. To determine the equation which the correction z must satisfy, we must therefore diminish the roots of (3) by 0.8. We have already performed a part of this calculation in (5). We now complete this transformation. 5 0.8 4.64 8.512 -6 |0. 6.8096 5.8 0.8 10.64 5.28 8.512 12.736 (7) + 0.8096* 6.6 0.8 7.4" 0.8 1* 8.2* 15.92 5.92 21.84* 21.248* Art. 100] AN EXAMPLE OF NEWTON'S METHOD 163 Thus z is a root of the equation (8) h(z) = s4-f 8.2 z3 + 21.84 s2 + 21.248 z+ 0.8096 = 0. Newton's method, which consists essentially in neglecting z^, z3, s* in the above equation, leads us to regard CQ\ 0.8006 ^ ^ 21.248 as an approximate value of z. It is not safe to assume that this value will be correct to more than its first significant figure. We therefore use as our approximate value of z, suggested by (9), z = — 0.04, giving j/ = 0.8 — 0.04 = 0.76 and a; = 1.76 as the approximate value of a root of (1) probably correct to the second decimal place. We now continue this process so as to get a still closer approximation. Since (8) has a root approximately equal to — 0.04, we put (10) z = -0.04 + ^ that is, we diminish the roots of (8) by — 0.04 (or increase them by + 0.04). This gives rise to the following calcu- lation: 0.8096 0.04 -0.04 - 0.3264 -0.860544 -0.81549824 8.16 -0.04 21.5136 - 0.3248 20.387456 -0.847552 - 0.00589824* 8.12 -0.04 21.1888 -0.3232 19.539904* (10) 8.08 -0.04 20.8656* 1* 8.04* This leads to the approximate value - 0.00589824 t 19.539904 = 0.000302, where only the last figure is uncertain. For it can be shown that, if the quotient fia)/f'{a} begins ivitli k zeros when ex- 164 INTEGRAL RATIONAL FUNCTIONS [Art. 101 pressed as a decimal fraction, then the best approximation is obtained by carrying out the divisioyi to '2 k decimal places. Thus we have found a; =1.76 + ^ = 1.760302, the last figure only being in doubt. When the student has become familiar with the process, the discussion given in the text of this article may be omitted, and the actual calculation to be exhibited will con- sist only of the numbers in the schemes (2), (7), (10). Since the problem of finding the nth. root of a given num- ber a is equivalent to the solution of the equation a:" — a = 0, Newton's method is also applicable to this problem. In fact, even in the case of a square root and cube root, the calcu- lation by Newton's method is far more convenient than the method taught in elementary algebra. 101. Horner's method.* Horner's method is very similar to Newton's method. The principal difference consists in the fact that, in Horner's method, negative corrections are avoided. In our example in Art. 100 we found 1.8 as an approximation to the root and then proceeded to improve our knowledge of this root by starting out in our calculations with this approximate value. If we had been using Horner's method, we should not have used 1.8 as an approximation ; for 1.8 is greater than the required root, as is brought out by our calculation, since the correction turns out to be negative. Consequently by Horner's method, at this stage, we should proceed as follows. As soon as we have dis- covered that the root is less than 1.8, we try the value 1.7 for X. We should then find that 1.7 is the correct approxi- mation to use for x in tlie application of Horner's method ; since the root actually lies between 1.7 and 1.8, the first two significant figures of the root are 1.7 and the correction to this will be positive. We then proceed as in Newton's * W. G. Horner, London Philosophical Transactions, 1819. Arts. 102, 103] NEGATIVE ROOTS 165 method, using 1.7 as our approximate value of x instead of the value 1.8 whicli we actually used. In this example 1.8 was actually a closer approximation to the root than 1.7. Consequently in this case Horner's method would not be quite as advantageous as Newton's. In general, we may expect to obtain a result, correct to a given number of significant figures, more rapidly by New- ton's than by Horner's method. F'or Newton's method allows us the privilege of using that one of two numbers between which the required root is known to lie, whicli is the closer approximation. The only advantage of Horner's method is that all of the corrections are positive. 102. Abbreviated calculation. It often happens, especiall}"- in problems of applied mathematics, that only a certain num- ber of significant places are required. In fact, in most such problems, the coefficients of the equation are themselves not known with absolute accuracy. Their values depend upon certain measurements, and owing to the uncertainties of such measurements, their values are only known with an accuracy of a certain number of decimal places. The exact application of Newton's or Horner's method, however, usu- ally introduces decimal places far beyond these. But these additional decimal places add nothing to the accuracy of the desired result from the practical point of view. Much labor may therefore be avoided by not calculating these higher decimal places at all. The following is a good prac- tical rule. If in a practical problem a root of an equation is to be found correct to k decimal places, abbreviate all the num- bers ivhich occur in the calculation to k -\- 1 decimal places, and abbreviate the final result to k decimal places. 103. Negative roots. Negative roots may be obtained by the same process as positive roots. It is usually preferable, however, to first make the transformation of Art. 95, thus reducing the problem of computing a negative root of the given equation to that of computing a positive root of the transformed equation. 166 INTEGRAL RATIONAL FUNCTIONS [Arts. 104, 105 104. Computation of more than one real root. After one root a of an equation has been determined, the problem of computing a second real root (if th^e is a second) may, of course, be treated in the same way. It is simpler, however, to first divide the left member of the equation hy x — a^ and to continue the calculation with the equation of lower degree obtained in this way. This equation is called the depressed equation. Moreover, if Newton's or Horner's method was used for calculating the first root, the coefficients of the depressed equation have already been determined in the last synthetic division, so that everything is prepared for the determination of the next real root. EXERCISE XLII 1. Find to two decimal places the root of x^ + ?> x — 20 = which lies between 2 and 3. 2. Find to three decimal places the root ofa;^ — 2x — 5 = which lies between 2 and 3. 3. Find to four decimal j^lfices the root of x^ + 5 .r — 7 = which lies between 1 and 2. 4. Find to two decimal places the root of a;^ + 3x- — 2 a; — 5 = which lies between 1 and 2. 5. Find a positive root of x^ = 63 correct to two decimal places. 6. Find the real fifth root of 37 correct to two decimal places. 7. Find to two decimal places tJie root of x^ — '2x'^ — 'd x + Q — which lies between — 1 and — 2. 8. If the coefficients of the highest and lowest powers of x which occur in/(a;) have opposite signs, the equation f(x)= has at least one positive root. Prove this statement. Hint. Discuss separately the cases : I, when the equation has no zero roots; II, when it has zero roots. In case I, observe that/(.r) will assume values opposite in sign for x = and for x sufficiently large and positive. Then apply Art. 96. 105. Upper limit for the positive roots of an equation. Let us turn our attention once more to the example of Art. 100, and more specifically to the last line of the synthetic division (5) of that article. The numbers which occur in this line Art. 10.-.] IPPEU LTMTT FOR THE POSITIVE ROOTS 1G7 are all positive, and the last one, + 0.8096, is the value of ^(7/) for 2/ = 0.8, or of f(x) for x = 1.8. From the method of calculation it is evident that any value of x greater than 1.8 would only serve to* increase the value of f(x^. Conse- quently no root of the equation can be as great as 1.8, and we may say that 1.8 is an upper limit for positive roots of this equation. But what was the significance of the numbers in the last line of the synthetic division (5) of Art. 100 ? According to Art. 86 this synthetic division teaches us that the quotient obtained in dividing .r* + a:^ — 3 a;^ — 2. — 4 by a; — 1.8 is a;3+ 5. 8a'^ + 10.64a; + 8.512, and that the remainder is + 0.8096. Putting these two things together, our example suggests the following theorem. Let f(x^ be an integral rational function, in which the coefficient of the highest poiver of x is positive. Let a he a positive number, let Q(jc) be the quotient, and let R be the remainder obtained when we divide /(a:) by X — a. If R is positive and if none of the coefficients of Q(^x) are negative, then a is an upper limit for the positive roots of the equation f (x) = 0. Tliat is, no positive root of this equation can be as great as a. The proof is very simple. We have f(x~) = Q(x} (x - a) + R. (Art. 85) Since R is positive and Q{x} has no negative coefificients the right member will be positive for all values of x as great as, or greater than, a. That is, no number as great as, or greater than, a can be a root of the equation /(.r)= 0. Of course the value of R and the coefficients of Q{x^ are best obtained by synthetic division. We may therefore re- state our theorem in the following form which is especially well adapted for practical application. If' in the synthetic division of f(x^ by x — a, a being positive, none of the results in the third line are negative, tJien a is an upper limit for the positive roots of the equation f {x^=- 0. 168 INTEGRAL RATIONAL FUNCTIONS [Art. 106 In applying this critei'ion it is again presupposed that the highest power of X in/(x-) has a positive coefficient. If this coefficient should happen to be negative, change the sign of all of the coefficients before proceeding farther. The knowledge which results in a specific case from this statement will save us from wasting time in searching for roots of the equation. For we need not test an}'^ values of x as large as the upper limit. Thus in our example, if we wish to find a second real root of the equation we know that it cannot be as large as 1.8. For other methods of obtaining upper limits see Dickson's Elementary Theory of Equations, page 57. Since all questions concerning negative roots may be con- verted into questions concerning positive roots of another equation, by means of the transformation of Art. 95, we need not expressly formulate the corresponding criterion for the lower limit of the negative roots of an equation. 106. Descartes's rule of signs. It is evident that an equation (1) / (a:) = Ax^ + 5:r«-i + .-. +Lx + M=^ can have no positive roots if all of the coefficients have the same sign, that is, if there are no variations of sign among the coefBcients. For the value of f(x) will obviously be positive for all positive values of x if all of its coefficients are positive ; it will be negative for all positive values of x if all of the coefficients are negative. In neither case can it become equal to zero for a positive value of x. The question arises whether there may not be a more general connection between the number of variations of sign among the coefficients, and the number of positive roots. That such a relation exists was discovered by Descaktes. Let us, as usual, arrange the terms of the equation accord- ing to descending powers of the unknown quantity, and let (2) A, B, C, i), ... i, M be the coefficients of a;", x"~^, ••• x, 1 in this order. If A were negative, we might consider the equivalent equation Art. 106] DESCARTES'S RULE OF SIGNS 169 — j(^x)= 0, whose A would then be positive. We may therefore assume that A is positive. We now inspect the coefficients (*2), beginning with A^ in the order written. If they are all positive, we say there is no variation. If, on the other hand, we observe k changes of sign among the coefficients (2), as we read them in the order written, we say that there are k variations. Zero coefficients may be omitted in counting variations. Descartes's rule says that an equation with k variations has at most k positive roots. To prove this rule we proceed as follows. Let a:^, x^, ••• x^ be all of the positive roots of the equation /(a:)= U. Then, according to the factor theorem (Art. 84), x — x^, x — x^^ •••, x — x^ are factors Q)if{x^. Consequently the division of/(x) by {x — x^{x — Xc,^ ••• {x—x^) will be exact, and the (quo- tient g{x) will be an integral rational function of degree n —r such that the equation g{x) = has no positive roots. The polynomial g{pc) may have some variations or else it may have none ; that is a matter about which we profess ignorance. We put this ignorance into evidence by letting Fo represent the number of variations in ^(:c), it being under- stood that Vf;^ may be equal to zero or some positive integer. Let us now write down the signs of the coefficients of ^(a:), as follows : (8) g{x) = +••• + + ••• + + ••• +, where the dots indicate any number of terms, whose coeffi- cients have the same sign as those between which the}^ are placed. As has been stated already, V^ denotes the number of variations in g{x). We now attempt to determine the number of variations in the product (^x — x^ )g(^^ where x-^ is a positive number. If we perform the multiplication in the usual manner, however, writ- ing down only the signs of the various partial products, we find (a: - x^)g(x) = + + ••• 4-- - + +- -+ ••• + - -+ +- -+ +- - + ± ••• ±-± ••• ± + ± ••• ±-± ••• ± + ± ••• ±-, 170 INTEGRAL RATIONAL FUNCTIONS [Art. 106 where the ambiguous sign ± indicates those terms of the product concerning the sign of whose coefficients we can say- nothing definite. Since in any particular case some of these ambiguous signs may be replaced by + and others by — , it is evident that in some cases the product {x — x^)g(x} may have man?/ more variations than ^(2;). But in all cases this product will contain at least one more variation than ^(x). For we shall certainly not be overestimating the number of variations in the product if we replace all of the ambiguous signs of a group by the sign of the term which just precedes the group. If we do this the signs of the product are as follows : + +•••+-- + ••• + +•••+-. But this arrangement of signs is the same as in g(^x} except- ing the last, which gives rise to an extra variation. Thus, if Fj denotes the number of variations in the prod- uct (x — x{)g(^x^, Fj is at least greater by one unit than V^. That is Ti^Fo + l- Let us now multiply (x — x{)g{x') by a: — x^, where x^ is a second positive root of the original equation /(a;) = 0. Let V^ denote the number of variations in the product {x — x^{x— Xc^^g(x). By the same argument we find and therefore V^ ^ Vq + 2. Let us proceed in this way until we have multiplied gix) by the product of x — Xy, x — x^^ •" x — x^. The complete prod- uct is equal to/(a;). If Vr denotes the number of variations in /(a;), we have therefore or r ^ Vr - To- Since Vq may be equal to zero or a positive integer, we shall certainly have (4) r < Vr. Art. 106] DESCARTES'S RULE OF SIGNS 171 Now the equation f {x) — has precisely r positive roots and Vr variations. Therefore the inequality (4) states in symbols precisely what Descartes's rule states in words, namely : The number of positive roots of an equation f (a:) = cannot exceed the number of variations among the coefficients of the equation. It slioiild be noted that Descartes's rule does not state that an equation has as many positive roots as it has variations. It merely says that the equation can have no more positive roots than variations. It may have that nuiny positive roots or it may have fewer. Thus, in the simplest case when there are no variations, we may say at once tliat there are no positive roots. But an equation may have variations and still have no positive roots. Thus the quadratic equation a;2 _ X + 1 = has two variations but no positive root, both of its roots being imaginary as may be verified by solving the equation. We may state Descartes's rule in a more definite form on the basis of the following remarks. The equation g(x) = was assumed to have no positive roots. We made Use of this assumption tacitly when we wrote down the signs of g(x) in (3), by using a group of -|- signs for the last group in (3). For, if the last group of signs in (3) had consisted of — signs, the equation g(x^= would have to possess at least one pos- itive root. (See Ex. 8, Exercise XLII.) Since the first and last group of signs in gCx) both consist of + signs, g(x) can contain only an even number of variations, if it has any vari- ations at all. Thus Vq is either equal to zero or an even integer. Again, when we were estimating the smallest number of variations which the product (x — x{)g(x) could possibly have, we replaced the first group of signs in the product, -\- ±± ••• ±, which was followed by a — sign, by the group -I- + -I- ... -(- followed by a — sign. Any one of the terms of this group except the first might, however, have a negative coefficient. But if we change any one or several of the -f- signs in the group -h + + ••• +» except the first, which is 172 INTEGRAL RATIONAL FUNCTIONS [Art. 107 not doubtful, to — , the number of variations in the group will change by an even number or not at all, never by an odd number. Similarly for each of tlie other groups. Consequentl}" the true number of variations of tlie product (x — x^g(x) can differ from Vq-{- 1 only by an even number. If we combine these remarks with our former argument, we obtain the following more precise statement of Descartes's rule : The equation f (2) = has either as maiiy positive roots as there are variations among the coefficients off(x)^ or else fewer hy an even number. Descartes's rule may also be used in discussing negative roots. The transformation of Art. 95 enables us at once to make the following statement : An equation f(x') = has either as many negative roots as there are variations among the coefficients of f(^— x), or else fewer by an even number. EXERCISE XLIII In the following examples find the niaxiniuni number of positive and negative roots by applying Descartes's rule. 1. a;3 + 5 X - 7 = 0. 2. x3 + 2 X- + 8 = 0. 3. x3 + 1 = 0. 4. x" + 1 = 0. 5. x" - 1 = 0. 11. Instead of counting the variations among the coefficients as in Art. 106 (2) we may count the number of permanences, that is, the number of times that the coefficients of /(x) written in the order (2) of Art. 106, fail to change sign. A complete equation is an equation of the form f(x) = in which no term of /(x) has a zero coefficient. Prove that a complete equation has no more negative roots than the number of permanences among its coefficients. 107. Maxima and minima of an integral rational function. We have learned how to compute tlie slope of the straight line which is tangent to the graph of y =/(2;) at the point 6. X* + 3-3-3 X- + X - 3 = 0. 7. .H + j;-2 -1=0. 8. 2 x3 + 7 x2 - 5 = 0. 9. .0. x^ + x« -1 = 0. xs + x3 - 7 X + 1 = 0. Art. 107] MAXIMA AND MINIMA 173 (x, y). Tilt' slope of this tangent is equal to the derivative, f'(x), of f{x). (See Art. 87.) We have also seen (in Art. 53) that a line slopes upward from left to right or downward from left to right according as its slope is positive or negative. Consequently the course of a curve y =/(a;) will he upward from left to right in the neighhorhoo'1 of a point whose abscissa makes f {x} positive, and downward from left to right near a point for tohichf^pc) is negative. Jn other words, the function /(a;) increases with increasing X when its derivative /'(.c) is positive; it decreases with in- creasing X \\\\enf'(x) is negative. A maximum is a point on a curve which has a greater ordinate than any other point in its immediate neighbor- hood. It is, therefore, a point at which the curve changes from an upward to a downward course if we think of the curve as being described from left to right. (See the point marked A in Fig. 48.) Similarly at a minimum (such as ^, Fig. 48) the curve changes from a downward to an upward course. Consequently, if jo is a particular value of x which corre- sponds either to a maximum or minimum of the graph of y = f(x}, the derivative /'(a;) must change its sign as the variable x passes through the value jo. If/ (2;) is an integral rational function, /'(a;) is also an integral rational function and therefore a continuous function. (See Art. 96.) Con- sequently /'(a;) can change sign only by passing through zero. We draw the following conclusion : The abscissas of the maxima or minima of an integral rational function f{x) are included among those values of xivhich make the derivative f (x^ equal to zero. But not all roots of the ecjuation f'{x) = need to corre- spond to maxima or minima of /(a;). In fact the equation /'(a;) = really only means that the slope of the tangent is 174 INTEGRAL RATIONAL FUNCTIONS [Art. 107 equal to zero, that is, that the tangent at such a point is parallel to the a^-axis. And this may take place at a point, such as (7, Fig. 48, which is neither a maximum nor a minimum. In order to distinguish between these cases, we may pro- ceed as follows. Suppose that x = p is a root of the equation f '(x) = 0, which is obtained by equating to zero the derivative oi f(x). Let h be a very small positive number, so that —h is negative. We examine the signs of f'(p—1i) and f\p + ^)» ^^^ obtain the following criteria : If f'ip-Ji)>^, /'(f) = ^' /'(io + ^)<0, x = p gives a maximum of/(:r). If/'(^-A)<0, /'(jt>) = 0, /'(^ + /0>0, x=:p gives a minimum of /"(a;). If /'(^-A)>0, /'(jt>)=0, /'(jt> + 70>0, x = p gives neither a maximum nor minimum. If/'(^-A)<0, /'(p)=0, /'(;,+ A)<0, x = p gives neither a maximum nor minimum. The determination of the maxima and minima of a function /(a;) is often a matter of great practical importance. Most of the proljlems of Engineering are questions of tliis kind, since the engineer should attempt to make his constructions serve their purpose with a maximum of efficiency for a given outlay in money and time. From our present point of view, it is evident that a knowledge of the maxima and minima of a function is bound to be of great assistance in studying its graph. EXERCISE XLIV Discuss the functions given in Examples 1 to 5 for maxima and minima. 1. ?/ = x2 + 2 :c - 3. ■ 3. ?/ = 3 2-2 + 2 X - 1. 2. y/ = — 2 .r- + 7 X + 5. 4. y =: — 5 j:- -f 2 a; + 2. 5. y = ax'^ + hx + c. Compare your result as obtained by the use of the derivative with the result previously obtained in Art. 66. 6. Find the maxima and minima of ?/ = x^ — 3 x^ + 2 x. Plot the curve and find the roots of the equation x^ — 3 x^ + 2 x = 0. Art8. 108, 109] MULTIPLE ROOTS 176 7. Examine the function ,y = x^ — 10 a^ + 30 for maxima and minima and plot tlie curve. 8. Has the function y = (x — oy a maximum or minimum? Prove the correctness of your answer. 9. Examine the function y — x^ — (i x- + 10 for maxima and minima. 10. Examine the function y = x(x'- — 1) for maxima and minima. 11. A box open at the top is to be made from a square piece of tin, the length of one side of the square being a inches. It is proposed to do this by cutting equal squares out of the four corners and then bending up the tin so as to form tlie sides of the box. AVhat should be the size of the squares cut out of the corners so that the box may have the largest pos- sible volume ? 12. The strength of a beam is approximately proportional to its breadth and the square of its depth. What are the dimensions of the strongest beam that can be cut out of a circular log whose diameter is d inches? 108. Rolle's theorem. Let J.ifil/2^3^ (^'igs. 49 and 50) be the graph of an integral rational function 1/ = fipf)^ which crosses the rr-axis at the points A and B whose ab- scissas are equal to a and h respectively. Then we have /(a)=/(i) = 0. The figures show that there will be at least one maximum, or at least one minimum, between a and h. Since the deriva- tive f {x) will be equal to zero at such a maximum or mini- mum, these figures suggest the following theorem, which is known as Rolle's theorem. If an integral ratiowtl funetion f(^x) has the zeros x= a and X = by so that f{a~) =/(/>) = 0, then there exists at least one value of X, between a and b, for which the derivative f'(x) becoynes equal to zero. 109. Multiple roots. We shall not attempt to give a formal proof of Rolle's theorem, but proceed immediately to make an important application of it. Let us think of the 176 INTEGRAL RATIONAL FUNCTIONS [Art. 109 a:-axis in Fig. 51 as fixed, but let us think of the curve AMB as being gradually lowered from its original position, first to A'M'B', and then into the third posi- tion shown where the curve is tangent to the ic-axis at M". During this process the three points A, M^ B approach each other, and finally all three of these points coincide with each other at M" . We see therefore that a point of contact, such as M"^ may be regarded as arising from the union of two real points of intersection. If m" is the abscissa of M'\ the function /(a;), of which the lower curve of Fig. 51 is the graph, will of course have X — m" as a factor, since /(a:) becomes equal to zero for X = m". But f(x^ will actually contain {x — m")^ as a fac- tor, since each of the two factors x — a and x — b {a and b being the abscissas of A and B^ will tend toward x — m" as the curve AMB is lowered into its final position. We express this fact by saying that m" is a double root of the equation /(a-) = 0. According to Rolle's theorem there is a root of the equa- tion f'(x^ = between any two roots of /(.r) = 0. In our case tliis root of f (x) = clearly coincides with m" . This is also apparent from the fact that the x-axis is tangent to the graph at M" . The same conclusion follows if m" is a triple, quadruple, or multiple root of any degree of multiplicity. Consequently we obtain the following theorem: Every multiple root of f(x) = i% also a root off'{x) = 0*. By the factor theorem then, every multiple factor of ./"(a:) will also be a factor of /'(a;). We may therefore decide whether a given equation has multiple roots or itot as follows: We determine the highest common factor of f(^x) and f {x). If this highest common factor does not contain x, the equation *The converse of this theorem is not true. That is, not every root of /'(.'•) =0 is a multiple root off{x) = 0, nor indeed necessarily a root oi/{x) = at all. Art. no] RATIONAL KOOTS OF AN EQUATION 177 f(^z) = han no multiple root. If the highest common factor does contain a,", the multiple roots of /(a-) = will be those roots of the equation ivhich are obtained by equatim/ to zero the highest common factor of\f\u) andf'{x^. The process of finding the highest common factor of two polynomials, such as f(x) and /'(a;), is pi-obably familiar to the student from his first course in algebra. It is essentially the same as the process for finding the greatest common divisor of two integers (see Art. 5) and is justified by the same kind of reasoning. EXERCISE XLV Examine the following eiiuations for multiple roots. Determine the multiple roots if there are any ; and use these roots for the purpose of finding an equation of lower degree which the remaining simple roots will have to satisfy. Solve the etpiations completely. 1. ^-3 - 7 z- + 1<> X - 12 ^ 0. 3. .r* + 6 xH x^-2ix + 16= 0. 2. X* - 6 x2 - 8 2 - ;} = 0. 4. x" - x3 + 10 x^ - 8 = 0. 5. x^ - 15 x8 + 10 x2 + 60 X - 72 = 0. 6. x5 -3x*- o x3 + 13 x^ + 24 X + 10 = 0. 7. Show that x^ + 9 x'^ + 2 x — 48 = has no multiple root. 8. Show that x^ + :> ^x + r = has a multiple root if and only if 4 78 + r- = 0. 110. Rational roots of an equation with rational coefficients. If all of the coefficients of an equation of the form (1) Ax"" + Bx"-'^+ •■' + Lx + M=0 are rational numbers, we may at once reduce the equation to another one of the same form with integers as coefficients. To do this, it suffices to multiply both members of the equa- tion by the lowest common denominator of its various frac- tional coefficients. Let us assume therefore that all of the coefficients of (1) are integers. If we divide both members of (1) l>y A we obtain the equation (2) a:"-f ^r«-M- ••• +^^ + '^=0, ^ ^ A A A 178 INTEGRAL RATIONAL FUNCTIONS [Art. 110 iu which the coefficient of a;" is equal to unity, but in which the remaining coefficients will not, in general, be integers. But if we put (3) x = ^, 1/ = kx, y must satisfy the equation, y^ Btr^ ,,.LyM^^ k- Ak''~^ Ak A or (4) y- + !%«-! + ... + |f-V+ ^^" = 0, and the integer k may always be chosen in such a way that this equation for y shall have integral coefficients. (Com- pare with Art. 94.) In fact if we put k = A, the coefficients of (4) will certainly be integers, but often a value of k smaller than A will accomplish the same purpose. We have seeii that, by putting a; = |, y = kx, tvhere k is an integer, we can always transform the given equa- tion ivith rational coefficients into another one of the form (5) ?/" + 6t/"-i + c«/"-2 H \-ly + m — ^ which has the folloiving ttvo properties : (a") its coefficients are integers ; (5) the coefficient of the highest power of the unknown quantity is equal to unity. If the original equation has a rational root, (5) must also have a rational root, since every root of (5) is equal to k times a root of (1), and k is an integer. Conversely, to every rational root of (5) corresponds a rational root of (1) by means of (-3). Thus, the problem of finding the rational roots of (1) will be solved if we can find the rational roots of (5). Art. 110] RATIONAL ROOTS OF AN EQUATION 179 But the latter may be found with comparative ease, on account of the following theorem. Jf an equation has the properties (a) and (5), mentioned above, any rational roots which it may have must be integers. To prove this, let (5) be the given equation, having the properties (a) and (/>). If this equation has a rational root, let us denote this root by (6) y=^P 9 where p and q are integers without a common divisor, so that the fraction p/q is in its lowest terms. If (6) is a root of (5), we must have 9" \ ^"~ 9" q J or E. — — (bp''-'^ + cp^-^i + • • • + Ipq""'^ + mq"''^}. The right member of this equation is an integer. The left member is not an integer unless 5-= ± 1, in which case the root (6) is itself an integer, thus proving our theorem. Thus, it only remains to settle the question whether (5) has any integral roots. This can be done quite easily on account of the following theorem: Any integral root of an equation, of the form (5), tvith inte- gral coefficients, must be a divisor of the constant term m of the equation. In fact if an integer y satisfies equation (5), we have m = — y — by'*-^ — ly = — y{y''~^ -I- ^i/"~^ H f- showing that w is a product of two integers one of which is equal to y. Thus, if we wish to examine the given equation (1) for rational roots, we proceed as follows : 1. If the given equation in x does not have the properties (a) and (5), ive transform it into another equation in y which has 180 INTEGRAL RATIONAL FUNCTIONS [Akt. 110 these properties hy putting x = y/k^ using the smallest value of the integer k which ivill accomplish the purpose. 2. The resulting equation in y must have integral roots if it has any rational roots at all. These integral roots must he divisors of its constant term. Therefore, we test each of the divisors of the constant term of the equation, to see ivhether it is or is not a root of the equatiori in y. 3. From every integral root of the equation in y which is obtained in this way, we find a rational root of the original equation in x hy dividing it hy k. It may not be necessary to test all of the divisors of m as indicated in No. 2. If some of them lie beyond the upper limit for positive roots, or below the lower limit for negative roots (see Art. 105), they cannot be roots of the equation and need not be tested. Descartes's rule (Art. 106) may also frequently be used to reduce the number of trials. Moreover, if one rational root has been found, it will usually be advisable to make use of it to depress the given equation before proceeding farther. EXERCISE XLVI Examine the following equations for rational roots. 1. 108x3 - ,54x2 + 45 a; - 13 = 0. Solution. We first write this equation in the form (1) x^-\x^+j\x-^^\ = Q. If we put X = y/k, // will satisfy the equation The smallest value of k which will make the coefficients of this equation integers is A; = 6. Therefore we put x = y/6 and obtain the equation (2) / - 3 f + 15 y - 26 = for y. This equation has properties (a) and (b) ; any i"ational root which it may possess must therefore be an integer, and moreover a divisor of — 26. The only integral values of y that w^e need test there- fore are ±1, ±2, ±13, ± 26, since these are the only integral divisors of - 26. Art. Ill] SUMMARY OF OPERATIONS REQUIRED 181 But according to Descartes's rule, (2) has no negative roots. It suffices therefore to test + 1, + 2, + lo, + 26. We begin with + 1. Synthetic division gives 1 _ 3 +1.5 - 26 |_1 1 - 2 4- 13 - 2 +13 - 13 Therefore + 1 is not a rout of (2). We test + 2 next. 1 _ ;j +15 - 2() I +2 2 - 2 +26 1-1+13 Therefore + 2 is a root of (2). The depressed equation Z/-- 2/ + 13 = is a quadratic whose discriminant is equal to 1 — 4 • 13 = — 51, which is not a perfect square. Consequently this quadratic has no rational root, and therefore 3/ = + 2 is the only rational root of (2). Since x — y/Q, the only rational root of (1) is x — 1/3. 2. J-3 - 3 x'^ - 2 X + 6 = 0. 6. x< - 6 x8 + 6 x2 + 5 X + 12 = 0. 3. x3 - 8 x2 + 17 X - 10 = 0. 7. 2 x-» - x3 - 5 x2 + 7 X - 6 = 0. 4. x3 - 9 x2 + 23 X - 15 = 0. 8. x^ + -i/- -i'- - '/ ^ + V = 0. 5. 3x3 + 8x2 + X - 2 = 0. 9. 12x3 - 13x2 + fx - ^ = 0. 111. Summary of the operations required in solving an equation with given numerical coefficients. The application of iSewton's or Horner's method is likely to cause trouble when the root to be determined is a multiple root. It is advisable, therefore, to apply the method of Art. 109 in order to detect and determine any possible multiple roots before proceeding farther. It will then be easy to depress the equation (Art. 104). dividing by (^x — ay if a is a Ar-tuple root, by (x — by if b is an Staple root, and so on. The resulting equation will have only simple roots. If its coeffi- cients are rational numbers, it should be tested for rational roots (Art. 110). Descartes's rule (Art. 106) may then be used to find an upper bound for the number of positive and negative roots 182 INTEGRAL RATIONAL FUNCTIONS [Art. 112 of this equation. li the left member of the equation be called /(a;), we may compute the value of f(x), by sjm- thetic division (iVrt. 93), for various values of a;, obtain- ing incidentally an upper limit for positive roots (Art. 105), a lower limit for negative roots (Art. 105), and some indica- tion as to the location of the roots. If we wish, these calcu- lations may be used for plotting the graph of y =f{x). As soon as we have discovered in this way the approximate location of a root, we may determine its position as accu- rately as we please by means of Newton's or Horner's method (Arts. 100 and 101), or by the method of false position (Art. 99). While we have learned in Art. 109 how to avoid the trouble caused by multiple roots, there remains another case which requires a word of explanation. If two roots a and h are very close together without coinciding absolutely, essen- tially the same difficulties appear in applying Newton's or Horner's method as in the case of a double root, but we can- not use the same method for overcoming these difficulties. The methods best adapted for separating the roots in such cases are connected with a theorem of Sturm's and cannot be discussed here. See Dickson's Elementary Theory of Equations^ pag6 96. We have said nothing about any methods for calculating imaginary roots. Since a complex quantity contains two real numbers (its real and imaginary components), the prob- lem of calculating a complex root may be regarded as one involving two unknowns, and such problems are reserved for a later chapter. 112. Application of cubic equations to floating spheres. All calculations about floating bodies are based upon a funda- mental law usually called the principle of Archimedes after its discoverer. Archimedes (287-212 b.c), who is gen- erally regarded as the greatest mathematician of antiquity, lived in Syracuse, which was then a prosperous city of Sicily inhabited by colonists who had come from Greece. Accord- ing to a familiar story, Hiero, king of Syracuse, had given Art. n-2] APPLICATION OF CUBIC EQUATIONS 183 orders to a goldsmith to make a crown for him, and had given him the necessary amount of gold carefully weighed. When the crown was delivered, its weight was found to be correct, but the suspicion arose that the goldsmith had de- frauded the king by replacing some of the gold by an equal weight of silver. But how was this suspicion to be verified? Knowing the great reputation of Archimedes, the king laid the case before him, and Archimedes promised to make an attempt to solve the problem. A short time afterward, while in the public baths of Syracuse, he observed that the water seemed to exert an upward pressure upon his body, and that this pressure increased or decreased according as more or less of his body was immersed. Recognizing the bearing of this observation on the problem of Hiero's crown, he rushed out into the street shouting, " I have found it, I have found it." His solution of the problem was as follows. He weighed out a quantity of gold and an equal weight of silver, the weighing being performed in air as usual. He then attached these equal weights of silver and gold to the two ends of a bar with equal arms, which would therefore be in complete equilibrium. He then placed a vessel filled with water underneath the bar, so that both the gold and the silver were covered with water. The silver now seemed to weigh less than the gold. This being established, the problem of the crown could be solved easily. If on a balance the crown was measured against an equal weight of gold, and then the whole was immersed in water, the gold would outweigh the crown if the goldsmith had been dishonest. The fundamental principle of Hydrostatics discovered by Archimedes may be stated as follows. When a body is immersed in water, the water exerts an upward pressure upon it which, either partially or completely, counteracts the downward tendency due to gravity. Consequently a body will loeigh less when immersed in water than in air. This loss in weight is exactly equal to the weight of the water which the body displaces. 184 INTEGRAL RATIONAL FUNCTIONS [Art. 112 In the case of a floating boch% the upward pressure of the water just balances the downward effect of gravity. Therefore : The total iveight of a float iny body is ejjual to the weic/ht of the water which it displaces. Since the weight of a body varies as its mass (see Art. 80) we may, if we prefer, also state this law as follows. TJie total mass of a floatijig body is equal to the mass of the water ivhich it displaces. We proceed to make an application of this principle. Fig- ure 52 represents the cross section of a sphere of radius r float- ing in water and immersed to a depth AB = h. The volume of the sphere is The volume of the submerged por- tion of the sphere is of course equal to the volume of the water which has been displaced by tlie floating body. If we FiU. o2 call this volume -y', we shall have (2) v' =7rA2(>- 1 /O-t If we use the centimeter as unit of length and the gram as unit of mass, the density of water is equal to unity (see Art. 44), so that the mass (in grams) of the water displaced will be (3) m' = 'Kli\r -^h). If the sphere is composed of material of density p, the mass of the spliere will be (4) w = I 'TTT^P- * This is the formula for the volume of a sphere of radius /■. t This is the formula for the volume of a spherical segment in terms of the altitude h and the radius r of the sphere. AiiT. 113] APPLICATION OF CUBIC EQUATIONS 185 According to the principle of Archimedes, w' must be equal to m if tlie sphere floats, that is, 'rrh^(r — J A) = | Trr^p, whence ^^(3 r — k) = A r^p, or (5) A3 _ 3 rJfi + 4 rV = 0. Since the density p of u substance, when measured in terms of grams per cubic centimeter, is tlie same number as the specific gravity of that substance (Art. 44), we may state our result as follows: If a solid sphere of radius r, lohich is composed of material whose spenfic gravity is equal to p, floats upon water, the depth h to which it will sink into the water is a root of the cubic equation (-') ). Ill ;>|i|ilyini:;- (.')). it is not necessary to express )• ami li in centimeters; tlit'V may lie expressed in any conveiiieut unit, but of course the same unit must he used for both. Tlie reason for this is apparent from the equation (5) itself. Every term of (.5) has the dimension L^ since p, the specific gravity, is an abstract number. 113. Application of cubic equations in Trigonometry. From the addition formulas ^ sin (a + /3) = sin a cos (3 -f- cos a sin yS, cos (a + /S) = cos « cos /3 — sin « sin /3, we obtain the familiar formulas (2) sin 2^ = 2 sin 6 cos d, cos 2 (9 = os^ - sin2 by putting u = /3 = 6. If now we put in (1), a= '2 6. /3 = d and make use of (2), we easily find sin :5 6 ^ :] s'\n v.os^ 6 - sin3 6, cos 3 ^ = cos3 ^ — 3 sin^ 6 cos 6. Since we have sin^^ + cos^^ = 1, we may write instead of (3), , sin 3 ^ = 3 sin ^ - 4 sin^ 9, ^ ^ cos 3 ^ = 4 cos3 ^ - 3 cos 6. 186 INTEGRAL RATIONAL FUNCTIONS [Art. 113 If in these equations we regard sin 30 and cos 3 6 as known., the values of sin 6 aiid cos 6 may each be obtained by solving a cubic equation. This fact has an important bearing on the idiiwous problem of the trisec- tion of an angle. It can be shown that the solution of this problem is equivalent to that of finding a construction for the sine of one third of the given angle. It can further be shown that the cubic equation (4) cannot be solved by expressions involving only square roots. Finally it can be shown that a construction in the sense of elementary geometry, which only makes use of the ruler and compasses as instruments, can only be performed when the corresponding problem formulated algebraically is solvable by expressions involving only square roots. Consequently the problem of trisecting an angle of any size bij a ruler and compass construction is insolvable. This does not mean that an angle can not be trisected. It merely means that it cannot be trisected with the help of ruler and com- passes alone. The same remark applies to the problem of the dup>Ucation of the cube, the so-called Delian problem.* EXERCISE XLVII 1. How deep will a sphere of yellow pine one foot in diameter sink in water, if the specific gravity of yellow pine is 0.6.57? Compute to three significant figures. 2. How far will a cork sphere two feet in diameter sink in water, if the specific gravity of cork is 0.24? Compute to two significant figures. 3. Apply the principle of Archimedes to a floating cube, taking for granted that it will float with its upper face in a horizontal position. Find a formula for the depth to which it will sink. 4. Apply the principle of Archimedes to a rectangular parallelopiped floating in a horizontal position. Find a formula for the depth to which it will sink. 5. Show how to modify formula (.5) of Art. 112 if the sphere floats upon some fluid, not water, of specific gravity p'. Apply your result to a .sphere of iron of radius 6 inches, floating on mercury. The specific density of iron and of mercury are 7.2 and 13.6 respectively. 6. Given sin 30° = \. Make use of Art. 113 to calculate the sine of lO"" to three decimal places. Compare your result with that given in a table of natural sines. * For a detailed discussion of these and related questions consult Klein's Famous Problems of Elementary Geometry, translated by Beman and Smith, or the article by Dickson on Const nirtloits vlth Ruler and Compasses in the Mono- graphs on Topics of Modern Mathematics edited by J. W. A. Young. CHAPTER V INTEGRAL RATIONAL FUNCTIONS OF THE nth ORDER. THE PROBLEM OF THE ALGEBRAIC DETERMINATION OF THEIR ZEROS. AND THEIR GENERAL PROPERTIES 114. Distinction between the algebraic and numerical solu- tion of an equation. We have shown how the real roots of an equation of the form (1) Ax" + Bx''-'^ + '■■ +Lx + M=0 may be determined, if the coefficients A, B, •' L, Mare given numbers. But our solution of the problem was arithmetical rather than algebraic, since we did not find a general formula for the value of the roots of (1) in terms of the coefficients. In two cases, however, namely, when the given equation is of the first or second degree, we did find such general formulas. Thus we have actually accomplished the algebraic solution of equations of the first and second degree. (See Chapters II and III.) We shall now show that there are some other cases in which we are able to solve an equation algebraically . 115. The equation a:«-1 = 0. If ^ = 1, B=C= •• =X = 0, iHf = — 1, equation (1) of Art. 114 reduces to (1) a;" - 1 = or x" = 1. If n is an odd integer, the only real number which satisfies this equation is a;= 1. If n is even, the equation has two real roots, namel}^ x = + 1 and x = —\. The question now arises whether (1) may not also have some complex roots. This question may also be formulated thus. Can the wth power of a complex number be equal to unity ? 187 188 INTEGRAL RATIONAL FUNCTIONS [Art. 11(3 116. The TJth power of a complex number. In order to settle this question, we iimst first learn how to determine the nth power of a given complex number. The easiest way to accomplish this is to make use of the geometric representa- tion of a complex number as a vector, which was explained in Art. 24, and the definition of multiplication of two com- ])lex numbers as given in Theorem I, of Art. 31. Let r^ and r^ be the moduli of two complex numbers, and let 6^ and 6^ he their amplitudes. Then, according to the theorem just quoted, the product will be a complex quantity whose modu- lus is equal to r^r^ and whose amplitude is equal to ^1 + ^2- If rj and r^ are both equal to r, and 6-^ and 6^ are both equal to 6, the product will be the square of that complex number which has r as its modulus and 6 as its amplitude. There- fore the square of this complex number will have r • r = r^ for its modulus and 0-^6=20 as its amplitude. In the same way w^e see that the cube of the given complex number will have r^ for its modulus and 3 6 for its amplitude. In general, we obtain the following result. Jfr is the modulus and 6 the amplitude of the complex num- ber a + bi, the nth power of a + bi will have r" for its modulus and nd for its amplitude. The same result may be obtained by using the polar form of the com- plex number (Art. 30). If we write (1) X = a + hi = r(cos + i sin 6), we find first x^ = X- x = r- r[cos(^ + 6)+ /sin(^ + 6)}, (Equation (:3), Art. 31) or (2) x2 = r2 (cos 20 + 1 sin 2 6) . Again we have x^^ x-^x= r- ■ r [cos(L> $ + 0) + i sin(2 6 + 6)], or (3) .r8 = r3(cos :i6 + i sin 3 6) ; and continuing in this way, we finally obtain the formula (4) x" = r"(cosn^ + i sin nd). 1. r=\, 6= 9fr. 5. r=l, 6= 120°. 2. ;• ^ ^. 6 = 90^ 6. r = I, ^ = 120\ 3. r = i, ^ = 90 '. 7. r = 1, ^ = 240'^ Aim. 117] THE COMPLEX ROOTS OF UNITY 189 Coiiip.irisoii <>t'(l) ami (1) gives ri.se to an iiiiiiortant re.sult. If, in (4), we .sul»stitiit«' for x its value from (1). and then divide l)Oth meniber.s of the resulting- equation l>y '", we tind (5) (cos 6 + ; sin 6)" — cos nd + / sin nd, a remarkable equation usually known us De Moivre's formula. EXERCISE XLVIII Plot the vectors wliich corre.spond to the data given in the following examples and then plot their squares, cubes, and fourth power.s. 9. r=\, e = 240°. 10. r = \, 6 = 60^ 11. r = 1, ^ = 70°. Q«no 4. r=\, 6= 120-. Q. r=l, e = 2i(P. 12. r=l, 6 = '^". 117. The complex roots of unity. We are now prepared to answer the question raised in Art. 115, whether the wth root of a complex number may be equal to unity. In Fig. 53, the vector (9^j,one unit long in the direction of the positive aj-axis, represents tlie complex number 1=. 1 + ■ i. Let us draw a circle of unit radius with as center, so that A-^ will be upon its circumference. Let us divide the circumference into n equal part.s, Aj^ being one of the points of division, and the others being denoted by A^, A^, ••• -4„. (Li Fig. 53, we have chosen n = 8.) Then, each of the n vectors OA^ OA^. ^-^z' "' OAn represents a complex number ivhose nth power is equal to unity. PiiOfJF. Consider OA^ It represents the complex (piantity who.se modulus is 1 and whose amplitude is 0°. According to Art. 31, the nth power of this complex quantity Mill have as its modulus 1" = 1 and its amplitude w x 0° = 0°. Therefore the nth power of this complex quantity is 0(|ual to unity, as is also evident by calculation. Consider OA,^., which represents a complex quantity whose modulus is 1, and whose amplitude is 360'^//«. (In the figure this angle is equal to 45°.) According to Art. 31, the nth 190 INTEGRAL RATIONAL FUNCTIONS [Art. 117 power of this complex quantity will have as its modulus 1" = 1, and the value of its amplitude will be w(360°/w) = 360°. Therefore the nth power of the complex quantity represented by OA2 is the complex quantity represented by OA^ which is 1 + .^■ = 1. In the same way we can show that each of the n vectors mentioned represents a complex quantity whose nth power is equal to unity. It is easy to shotv further that the n complex quantities repre- sented by OA-^, OA^, • • • OAn are the only ones whose nth powers are equal to unity. For, if r is the modulus and 6 the amplitude of a complex quantity whose nth power is equal to 1 + • z, r" must be equal to the modulus of 1 + • z, and n9 may differ from the ampli- tude of 1 + ■ i only by an integral multiple of 360°. (See Art. 25, last statement.) Consequently we must have (1) r" = 1, nO = A" • 360°, where k may be zero or any positive or negative integer. Now r is the modulus of a complex number and is therefore positive, by definition. (See Art. 23.) Since r is positive, and since the onl}' positive number, whose nth power is equal to unity, is unity itself, we find from (1) (2) r=l, ^ = ^^^, /t = 0, ±1, ±2, ±3, .... n If in these equations we put in succession ^ = 0, 1, 2, • " n — 1, we obtain precisely the n vectors OA^, OA^, ••• OA^ of Fig. 53. If we give any other integral value to k, for instance a positive value greater than n — 1 or a negative value, we only obtain one of these same vectors over again. A number, real or complex, whose nth power is equal to unity is called an nth root of unity. We have shown that there exist exactly n distinct nth roots of unity. Tliey are the n complex 'numbers which are represented by the n vectors OA^, Art. 118] THE COMPLEX ROOTS OF UNITY 191 OA^^ ••• OAn of Fig. 53. Therefore the modulus of every nth root of unity is equal to 1. Their amplitudes are /Qx no 360° .-,360° ..360° ,360° . ...360° (3) 0°, ,2 ,3 ,---k , .-(m-I) n n n n n respectively. We may write out the value of each of these nth roots of unity in its polar form at once. A complex number of modulus /• and amplitude $ may be written in the form r(cos + i .sin 6). (See Art. 30) Consequently the n roots of unity, whose amplitudes are the angles listed in (3), may be expressed as follows : .r^ = cos 0'^ + i sin O^ = 1, 360^ , . . 360° Xj = cos 1- ism , (4) Xk — COS k ) + I sm .n \k—). X„-l (« - 1)360° , . . (n- 1)360° COS ^^ ^ h I sin '^ '- • 118. Numerical expressions for the complex roots of unity for n = 2, 3, 4. It is evident that the two square roots of unity are + 1 and — 1. In order to obtain expressions for the cube roots of unity, consider Fig. 54, where the vectors OA^, OA^, and OA^ represent these cube roots. Since the angle between OA^ and 0^12 is 120°, OB the bisector of this angle will make an angle of 60° with OAo and the triangle OA^B, isosceles by construction, will have to be equiangular and therefore equilat- eral. Moreover, each of its sides is of unit length. Con- sequently the length of A^C is equal to -|, and that of OC is equal to Vl -QY = V| = 1 V3. Since A^ is to the left of the y-axis, its abscissa is negative. Therefore the coordi- nates of A^ are a: = - .}, y = -h ^V3. If a vector is in its Fig. 54 f 192 INTEGRAL RATIONAL FUNCTIONS [Art. 118 standard position (see Art. 2-3) and the coordinates of its terminus are (a:, ?/), the vector represents the complex number x-{-yi. Consequently the complex number repre- sented by OA^ is i i • /- Similarly we find that OA^ represents the complex number -I- I iV8. Therefore, the three cube roots of unity are (1) 1' - i + i ^V8, -l-\ ^ V3. A figure may be constructed very easily to represent the four fourth roots of unity. Inspection of such a figure shows that the four fourth roots of unity are (2) 1, i - 1, - L These same results may also be obtained without any use of geometr}'. The cube roots of unity are those numbers which satisfy the equation (3) x^=l or x^ -1 = 0. This equation obviously has the root x = 1. Therefore .c — 1 is a factor of x^ — 1 and, in fact, we find .i-3_ 1 =(.f- l}{x^+ J+1). Thus the other roots of the equation (3), that is, those which are different from 1, must satisfy the quadratic equa- tion „ -, If we solve this quadratic, we find precisely the second and third of the expressions (1). Similarly the fourth roots of unity are the roots of the eti nation x^ = 1 or 2-4 — 1 = 0, which may be factored into (x^ - 1 )(.r2 +l^==(x-l)(x+ l)(x + 0(.r - i) = 0, showing that its four roots are precisely the four complex numbers (2). Art. 119] CONSTRUCTION OF REGULAR POLYGONS 193 We shall frequently denote the second of the three cube roots of unity^ namely^ — \ + \ iV8, by the letter (o. It follows either without calculation from geometry, or by direct multiplica- tion, that the third cube root of unity will then be equal to tu'-^. Thus we liave (4) w = - .} -f .] iV-), ^- = - 2 - 2 ''^•^' <«'^ = "1- We may find a numerical expression, at least approxi- mately, for any wth root of unity by drawing a figure, such as Fig. 53, accurately to scale and then measuring the co- ordinates of the points A^, A^^ ••• -4„. Or else we may use the trigonometric expressions (4) of Art. 117, making use of a table of natural sines and cosines for the purpose of 1 ^. . 360° 360° , evaluating sin , cos , etc. n n EXERCISE XLIX 1. Draw a figure representing the five fifth roots of unity and write each of them in the form x + yi, the vahies of x and // heing obtained to two decimal places by measurement. 2. Do the same thing for the seven seventh roots of unity. 3. Find the exact expressions for the six sixth roots and the eight eighth roots of unity. 4. Prove tliat each of the imaginary cnbe roots of unity is the square of the other. 5. Let Xj be that nt\\ root of unity wliose amplitude is equal to :}60Y«, and let Xf. be that ?jth root of unity whose amplitude is equal to k - :}60"/;*. Prove that x^. = xi*'. 6. Review the construction of a regular pentagon from ])lane geom- etry. By translating the steps of that construction into algebra show that one of the fifth roots of unity is equal to j( V5 - 1) + - VlO + 2x^5 . 4- 119. Construction of regular polygons. Our discussion of the equation .," - 1 — suffices to show how very closely its solution is connected with the problem of dividing the circumference of a circle into n equal parts or, what amounts to the same thing, with the problem of construct- ing a regular «-gon. On account of this connection the equations of the form x" — 1 = are frequently called cyclotomic equations. 194 INTEGRAL RATIOXAL FUNCTIONS [Akt. 120 It is a familiar fact of elementary geometry that it is possible to con- struct a regular polygon of 71 sides with the help of ruler and compasses if M is a power of 2, if n is equal to 3 or 5, or if n is a product of any two or three numbers of this kind. All of this was known to the ancients and was recorded by Euclid. No further progress in this direction was made for two thousand years, until Gauss in the early part of the nineteenth century proved that regular polygons of 17, 257, or 65,537 sides may also be constructed with ruler and compasses. More specifically the substance of Gauss's theorem is as follows : If n is a number which can be expressed as a product of any power of 2 and one or several other distinct factors, each (f which is a prime number of the form (1) 22* + 1, then and only then will it he possible to construct a regular polygon of n sides loith the help of ruler and compasses. The numbers of the form (1) which are obtained by putting h = 0, 1, 2, 3, 4 are in order 3, 5, 17, 257, 65,537 and all of these are actually prime numbers. The numbers (1) which correspond to h = 5, 6, 7, 8, 9 are known not to be primes. For a proof of the above theorem consult the articles quoted in the footnote on page 186. 120. The equation jr" — a = 0. If a is a positive number, this equation has one positive root which is usually denoted by (1) Xq = ^V + ^3. If we substitute these values of x^, x\ and x in (2), and ar- range the result according to descending powers of y. we find (3) y^ + (Bk + b)y'-+Qlk^+-2kb + c-)y + k^ + hk^ + ek + d = 0. We may now choose k in such a way as to make the coeffi- cient of y^ in (3) disappear, namely, by putting (4) k = -lb. U we substitute this value of k in ( o), this equation assumes the form (5) y^ + P^ + '1 = 0, where ( f) ) p = c - y)^ anil q = d — lhc+ .^^ h^. Thus, any cubic equation (2) may be reduced to the form (5) by making the transformation^ (7) •'^ = ^ - i f>. in other words (see Art. 02), by increasing the roots of (2) by ^b. Equation (5) is usually spoken of as the cubic in its reduced form, or as the reduced cubic. 198 INTEGRAL RATIONAL FUNCTIONS [Art. 121 The reduced cubic equation (5) may be solved as follows: We may introduce two unknowns, u and w, such that their sum shall be a root of the cubic equation ; that is, we put (8) y = u+ y, whence 7/3 = ^3 j^ 3 ^^2^, _j_ 3 ^^2 _|_ y3 — yZ j^ ^,3 ^ 2) Uv(u + w). Since ^ = m + y is to be a solution of (5) we must have ^3 j^ y3 _|_ 3 uiiQu _|_ 11^ -^ .pQn, -)_ ;) j +(2=0, or (9) V? + v^+ (3 uv +p')(u + v) = — q. Any two numbers, u and v, which satisfy this equation will have for their sum, w + v, a root of the cubic equation (5). And conversely, if we wish u + v to be a root of (5), then u and v must satisfy (9). But, since two unknowns, such as u and y, are not determined uniquely by means of a single equation, we have the right to impose the condition that u and v shall satisfy some other equation, besides satis- fying (9). This second equation ma}^ be chosen at our pleasure, provided it does not contradict (9). The infinitely many different choices which we can make of this second equation correspond to the infinitely many ways in which a given sum y may be split up into a sum of two terms u + v. In choosing a second equation for u and y, we are guided, of course, by the desire to make this second equation as simple as possible, and to have it aid us in simplifying our first equation, that is (9), at the same time. This is accom- plished by choosing (10) ?>uv+p = as the second equation. As a consequence of (10), equation (9) reduces to u^ -{• v^ =— q. We have obtained the following preliminary result. Ifu and V are two numbers which satisfy the two equations Aim. \-2\] THE CUBIC EQUATION 199 (11) ?f" -f v^ = — ' = y, will he x root of the cubic equation (5). Let us solve the second equation of (11) for v., and sub- stitute the resulting value in the first. We find 27 n^ ^ or (12) U^-\-qu^- .,\p^ = 0. But this is a quadratic equation for u^ and may be solved for u'^ by the formulae (5) of Art. 68. We find in this way (13) ,,3 =-iq± V:K, where R = J. P^ + I f- Let us choose the upper sign in (13) and let us denote by (14) 7/i=V- .] ry + VK one of the three cube roots of —\q-\- Vi2, the real one if — \ q + ^R is real. The other two cube roots of —\ q+VR will then be (15) i/g = (w?

. If the coefficients of (5) are real numbers and if B is positive, the cube roots «j and Vj may be selected as the real cube roots of — I q + ^B and — I q — ^ B respectively. Having selected our cube roots in this way, we know that (18) ij^ = n^ + i\ == + «- i(-(j)^==-0. >, = -2- i(-(!)6 + v^:(- (;)«=- 6. r = // + 2. From these values and (lo), Art. 121, we find (I) /,'= ,V(-(;)M- ](-fir= 1. The Cardano formulas (18), (Kl). (20) of Art. 121 now give us (o) (/j = >/T + v'2, I/., = (1) v4 + (o'-y/2, //g = a>'v^4 + ). 7. What value must k have in order that x^ + A:x + 9 may be identi- cally equal to (x + 3)^? 8. If the equation x'^ + px -[■ q = {x + r^ is an identity, what rela- tions must there be between/) and ?•, q and r, p and q'i If these con- ditions are satisfied, what relation is there between the two roots of the quadratic equation x'^+ px -\- q — i)"} 9. Let r^ and r„ be the roots of the equation x- + px + ^ = 0. If i\^ is just twice 1\, what relation must there be between p and ^y ? 10. If all three roots of the cubic equation x^ + px"^ + qx + r — Q are equal to each other, what relations are there between p, q, and r? 127. Application of the fundamental theorem to functions with real coeflBcients. If the coefficients, A^ -S, •••, M^ of an integral rational function (1) / {x) = Ax^ 4- Bx""-^ + ■■■ +Lx + M are real numbers, the values of the function which corre- spond to real values of x will be real. To an imaginary value of a:, such as (2) x=a + hi h ^ 0, tliere will correspond in general an imaginary value of the function, say (3) f{a + hi)==P+Qi, Akt. 1-27] FUNCTIONS WITH REAL COEFFICIENTS 213 which reduces to a real number, if and only if Q happens to be equal to zero. Let us now consider the value (4) X = a — hi, which is conjuf^ate to (2). (See Art. 34.) We shall find (5) f(^a-hi) = P-QU which is conjugate to (8). To prove this statement observe first that, since we are assuming the coefficients of (1) to be real, the symbol i enters into the expression (3) oi f (^a +hi) only because i occurs in x= a-{- hi. Since, in the second place, a — hi may be written a -(-^(— ^), the valueof /(« — ii) may be obtained from that oifQa + hi) if we replace i, when- ever it occurs in the latter expression, by — i. But this last statement is equivalent to (o). We may therefore state tlie following theorem : An integral rational function f {x) with real coefficients assumes conjugate complex values for conjugate complex values of the independent variable x. If rt + hi is a root of the equation /(.r) = 0, we have (6) f(a + hi) = P + Qi = 0, an equation which implies two others, namely (7) P = Q = 0. (See Art. 25.) But then we shall also have, on account of (5) and (7), (8) f(a-hi) = P- Qi = 0, so that a — hi is also a root of the equation / (a;) = 0. Thus, if an equation with real coefficients has as one of its roots an imaginarij numher a + hi, the conjugate of this number, namely a — hi, will also he a root of the same equation.* * III formulating: this theorem we have used the word iniaf/inanj rather than the word compk'.r, for the followiiis; reason. Accordintj to the hest usage, the term complex number is used for all numbers of the form a + hi and therefore in- cludes in particular the real numbers, namely, if ^ =. 0. We use the term imagi- nary number for those complex numbers which are not real, that is, for those for which It is not equal to zero. The above theorem is true for all complex numbers, inchidinji reals, but it is of no interest in the latter case since the conjugate of a real number a is that same real number. 214 INTEGRAL RATIONAL FUNCTIONS [Art. 127 This fact is often expressed as follows. Tlie imaginary roots of an equation with real coefficients occur in pairs. From this it follows at once, that an equation with real coefficients either has no imaginary roots at all or else an even number of them. It cannot have an odd number of imaginary roots. Consequently, an equation of odd degree., with real coeffi- cients, always has at least one real root. Let a + hi be an imaginary root of the equation with real coefficients, (9) f(^x) = Ax"" + 5a;"-i + • • • + ice + i¥ = 0. Then a — hi is also a root of (9). Consequently x— (^a-{-hi') and a^ — (a — 5^') are factors oif{x). (See Art. 84.) But we have [re — (a + hi)'] \^x — (a — hi')'] =■ (x — a — bi^Qx — a -\- bi^ = (^x— a)^+ b^. That is, these two conjugate complex linear factors of /(.r) combine into a single real quadratic factor. If we combine this result with the fundamental theorem, we obtain the fol- lowing theorem : An integral rational fu7iction with real coefficients may al- ways be expressed as a product of real linear and real quadratic factors. EXERCISE LIV 1. Given f(x) = 2 a:2 - 3 a; - 1. Compute /(I + i), /(I - i) ; f(2 + 3i),f(2-Si). 2. Given f{x) = x^-2x^ + 3x-7. Compute f(i), f(-i); /(-l + 20,/(-l-20. 3. State a reason for the fact that there exists no cubic function with real coefiicients which has x = 1, x = 1 + i, and x = 2 as its zeros. Find a cubic function with imaginary coefficients which has these three numbers as zeros. 4. A cubic equation, with real coefficients, has the roots x = I and X = 2 + 8 /. What must its third root be? Find such a cubic equation. 5. Find a cubic equation whose roots are 1, — I + ^ /Vy, and Arts. 128. 120] (JllAPH IN CASE OF DISTINCT FACTORS 215 6. Find a (luartic ecjuation "whose roots are 1, — 1, /, — i. 7. Prove that a cubic equation, with real coefficients, always has at least one real root. May it have two and only two real roots? (In an- swering this question remember that a double root counts for two roots. See Art. 126.) 8. How many imaginary roots may an equation of the fifth degree, with real coefficients, have ? 9. Discuss examples 1-10 of Exercise XLTII again, making use of what you have learned in the meantime. Obtain all of the information you can about the number of positive, negative, and imaginary roots of these equations. 10. Making use of the notions permanence and complete equation, ex- plained in Example 11 of Exercise XLIII, prove the following theorem : If a complete equation has all of its roots real, it will have as many posi- tive roots as variations, and as many negative I'oots as permanences. 128. Use of the factored form of /(jc) in plotting. If the linear factors of a function /(a;) are known, it is very easy to draw the graph of the function. Its intersections with the a;-axis are obtained by a mere inspection of the factors. Moreover, the sign of each factor, and consequently the sign of f(x), for a given value of a;, may also be obtained by inspection. The form of the graph, in a given instance, de- pends of course upon the nature of the factors of f(x)- Consequently we are led to distinguish the following three cases. Case I. All of the linear factors of /(a;) are real and distinct. Case II. All of the linear factors of /(a;) are real but they are not all distinct. Case III. Some of the linear factors of /(a,-) are imaginary. 129. Form of the graph in the case of real and distinct factors. Illustrative Example. Let/(jr) = (x - l)(x - 2)(x - 3). Then the graph of y - f{x) will intersect the a.-axis in the three points. A, B, and C, of Fig. 55, for which x has the values 1, 2, and 3 respectively. Let us divide the plane into four regions by drawing the lines AL, BM, and CN, parallel to the /y-axis through A, B, and C respectively. Fig. 55. INTEGRAL RATIONAL FUNCTIONS [Art. 129 Any point to the left of AL has an abscissa less than unity. For such a point we shall have x-KO, a;-2<0, x-3<0, and consequently y= (x - \)(x - 2){x -3)<0. Therefore all points of our graph to the left of AL have negative ordinates. In particular we find for X = 0, y = ( — 1) (— 2) { — -i) = — Q, giving the point D of Fig. 55. The abscissa x of any point between AL and BM satisfies the inequalities X - 1 > 0, X - 2 < 0, X - .} < 0. Consequently, we shall have for such a point y = (x - l)(x - 2)(x - 2) > 0. That is, all points of our graph which lie between AL and BM have positive ordinates. In particular, we find for x — 3/2, x=3/2, ^= i(- i)(_|)= + f, giving the point marked between A and B. The abscissa of any point between BM and CN satisfies the in- equalities X - 1 > 0, X - 2 > 0, X - 3 < 0, making ^ ^ = (x-l)(x-2)(x-3)<0. Therefore, all points of our graph between BM and CN have negative ordinates. In particular we find, for x = 5/2, ^ = l» y = i(i)(~ 2) = ~ f> giving the point marked between B and C. The abscissa of any point to the right of CN satisfies the inequalities X - 1 > 0, .r - 2 > 0, X - 3 > 0, making //=(x-l)(x-2)(x-3) positive. Therefore, all points of our graph to the right of CN have positive ordinates. The form of the graph, as indicated in Fig. 55, is now apparent. This form will not be altered very essentially if we replace (x-l)(x-2)Cx-3) by /l(x-l)(x-2)(x-3), where A is any positive or negative constant. Aim. loO] GRAPH IN THE CASE OF REPEATED FACTORS 217 EXERCISE LV Draw the graphs of the following functions. 1. 2. 3. 4. 9. 10. 11. 11= i(.r-l)(x-2)(x-:5). .y=-(.r-l)(.r-L>)(x-;}). ,,= (,, _ i)(,. _2)(x-3)(x- .y:^(..- \)(x~2){x-^){x- 5. y = x(.r- 1)(.?; - 2). 6. y={x+ \)x{x-\). 7. >,= (,-+l)(,,--l)(^-3). 8. y = {x- l)(x- :3)(x-o). 4). 4)(.i:-5). Describe the graph of /y = (x — «)(x — b){x — <•) if a, h, c are any three distinct real numbers arranged in ascending order of magnitude. 12. Describe the graph of i/ = (x— rt,)(x — ^2) "" (^ — ('u) if ('u 02 ••• a„ are n distinct real numbers arranged in ascending order of magnitude. 130. Form of the graph in the case of real factors some of which are repeated. Illustkative Example. Let f(x) = (x — l)'^(x — 2). Then the graph of the function >/ =/(x) has the points A and B of Fig. 56 in common with the x-axis, since ?/ = for x = 1 and for ^y j^ j,^ X = 2. But this time the curve does not cross the x-axis at A ; it merely touches it. We draw the lines AL and BM parallel to the ^-axis. For all points to the left of ^L we have a;< 1, X - 1 <0, X - 2<0 and therefore Fig. 56. (x - 1)2 > 0, X - 2 < 0, >j = {x - \y-{x - 2)< 0. Thus all points on our graph to the left of AL are below the x-axis. For every point between .4 L and BM, we have X - 1 > 0, X - 2 < 0. Since (x — 1)^ occurs as a factor of f(x) we find y <0 for x> 1 as well as for x,=(.r-\)(x-2)(x-Sy. 4. y = -2(x - iy^(x - 2). 10. y=(x- 2)^. 5. y= (x - l)(x - 2)2. 11. y = - (x - 2f. 6. ?/ = -(x- l)(z- 2)2. 12. ?/ = i(x- 2)3. 13. If Tj, ?•„, •••,r„ are distinct real numbers, show that the graph of y = A(x — r^y^(x — r.,)(x — r.,) ••• (x — r„) crosses the x-axisat the points for which x = r.„ r,, •••,;•„ and that it passes throughi the point of the a;-axis for which .c = ;-j, without crossing. Prove that the curve is tan- gent to the X-axis at this point. 14. Prove that the graph of y = A(x - ri)'5(x - ?-2)---(x - ;•„), where r^, r^--- i\ are distinct real niunbers is tangent to the x-axis at X = Tj and also crosses the x-axis at this point, as in the case illustrated in Fig. 57. 15. What can you say about the nature of the graph of y = A (x - rj)*(x - r^) ••• (x - r„) at the point of the x-axis for which x = ?-,? Distinguish between the cases when k is an even or an odd integer. Arts. l;il. 1:3-2] ROOTS AND COEFFICIENTS 219 131. Form of the graph when some of the linear factors are imaginary. If f{.r) is an integral rational function with real coeliicients, its imaginary factors occur in conjugate pairs, and each of tliese pairs gives rise to a real quadratic factor (Art. 127). Moreover the quadratic factors obtained in this way do not vanish for any real value of x. Conse- quently, if n is the degree of / {x) and if / (x-) has 2 k imaginary factors, the graph oif{x) will intersect the a;-axis only in n— 2 k points. EXERCISE LVII Draw the graphs of the following functions : 1. y = x^ + X -\- 1. 6. y = (x - l)(x2 + 1). 2. 7j = 3(x-^ + x+1). 7. y = (x + l)(x - l)(x2 + 1). 3. yz=- :3(x-^ 4 X + 1). 8. y = x« - 1. 4- i/ = K^' + ^ + !)• 9- y = (2 X + 5)(a;2 -x + 1). 5. y = (.r - 1)(.>;2 + X+1). 10. y = (x-iy\x^ -x+1). 132. Relations between the roots and the coefficients of an algebraic equation. We found it to be impossible (see Art. 125) to find a simple expression for the roots of an equation of the nth order in terms of its coefficients. But the inverse problem, to express the coeiHcients in terms of the roots, may be solved with ease. Let (1) «(,:?;" + a^a;"-! + a^x''-'^ + •• • + a^-iX + «„ = be any given equation. If we divide both members by Uq, we obtain the equivalent equation (2) . f(x') = x^ + p^x--'^ + />2.r"-2 + • • • + p„_ix + p, = 0, where we have put (3) ;., = ^, p2 = «2,...;,„_^ = ^, ^„ = £-". Let a:j, x^-, ■•■ Xn be the roots of this equation. According to the factor theorem x — x^, x — x^,--x — Xn will then be factors of /(a;), that is, of the left member of (2). (See Art. 84.) According to the fundamental theorem of Algebra 220 INTEGRAL RATIONAL FUNCTIONS [Art. 132 (Art. 126), f i^x) has exactly n linear factors. Therefore / (x) has no other factor depending upon x ; that is, the quotient (4) -^^ = A (X — X^^(X — Xc^^) ■■■ {x-x^} is independent of x, so that yl is a constant. The equation (4) or the equivalent equation (5) / (X) =X^+ p^X^ -1 + 7V~^ + • • ■ + Pn-l^ + Pn = A(^x - x^) {X - Xg) '■■ (x- .r„) must hold for all values of x, that is, it must be an identity. Therefore the following equation, obtained from (5) by multiplying together the factors x — x^, x — x^, x — x^, etc., of the right member, (6) a:" + |>i2-"'-i + ^^2:c"-2 + • • • + Pn-i-t' + Pn = A[x^-{x^ + x^-\- ■■■ +.r„).r"-i -f- (^XyV^ + X^X^ + • • •+ -^'w— i-*'n )*i' " • • ■ ± X^2'2 ■ ■ • Xj^j , must also be an identity. But according to Art. 126, Theorem F, an identity of this form can subsist only if the corresponding powers of x in the two members of the equation have the same coefficients. The coefficients of x" on the right and left members are A and 1 respectively. Therefore A must be equal to unity, and we shall have besides, comparing the coefficients of corresponding powers of x in the two members, i^i = ^^ = -(2-1 + ^2+ ••• +^n), P2 ^ ^^ "I" \-^\-^2 ' "^V^S ' ' ' ' I "^y^n ~i •^2'^S ' ' ' ' ' "^2*^" "r ■ ■ ■ "r '-^n—l-^nyt a O) PS=^=- C-^'rVs + •''r'V^4 + ■ • • + ^n-2«n-ia^n). a Aim. l:!:5] SYMMETRIC FUNCTIONS 221 where the + or — sign is to be used in the last equation ac- cording as n is even or odd. We have already obtained these relations in the special cases when w = 2 or 3 (Art. 68 and Art. 123). The gen- eral relations (7) were tirst discovered .by Girard. (See the historical note, Art. 123.) 133. Symmetric functions. The expressions which occur in the riglit members of (7 j, Art. 132, have important prop- erties. Except for sign, p^ is equal to the sum of all of the roots, and therefore jOj does not change its value if any two of these roots are interchanged. Again p^ is equal to a sum, each term of which is equal to a product of two of the roots, and all possible products of this sort occur in p^. Therefore p^ will not be changed if any two of the roots are inter- changed. The same thing is true of jOg, p^, •■•,Pn- We shall be able to express these facts very concisely with the aid of the following definition. An expression S(x-^, x.^, • • •, .r„), involving n letters x^, a^g, • • •» a;„ is said to be a symmetric function of x^, x^-, ■■■-, x^ if its value is left unaltered ivhen any two of these letters are interchanged. Thus, xi^ + x.^ + x^ is a symmetric function of x,, Xj, x^. But x-^ — 2 Xo^ + 3 x.i- is not symmetric. For if we interchange x■^^ and x^, this becomes x^^ — 2x^-+'^ x^-, which is not the same as the original expression. We may now express a part of what is involved in equa- tions (7) of Art. 132 as follows. The coefficients of an algebraic equation of the form X"" + p^X''-'^ -\- p^X""'"^ + ■•• +Pn-l^+Pn=^ are symmetric functions of its roots. The expressions Pj, p^^, ■•■ Pn ^re more specifically called the fundamental symmetric functions on account of the following important theorem due to Newton. Any integral rational function ofx-^^ x^^ ••o ^n which is sym- metric can be expressed as an integral rational function of the n fundamental symmetric functions p^, p-^ ■■■ Pn- 222 INTEGRAL RATIONAL FUNCTIONS [Art. 134 No proof of this theorem will be attempted in this book. It is clear from what we have said that the fundameiital symmetric functions of the roots of a given equation may be ob- tained from that equation by inspection^ although the indi- vidual roots themselves may be entirely unknown. This remark is very important in many applications. EXERCISE LVIII 1. Solve the equation x^ — Q x- + 26 a; — 24 = 0, making use of the in- formation that the three roots of the equation form an arithmetical pro- gression. 2. Find the roots of x^ — 8 a;^ + 5 a; + 14 = 0, making use of the fact that the sum of two of the roots is equal to 9. 3- Given a cubic equation x^ + p^x'^ + p.^x + p.. = 0. Find a formula for ./.'i'- + x/ + x^ in terms of the coefficients. 4. If 7^, = 0, what can we say about the roots of the corresponding equation ? 5. Solve x^ — 8 x^ + 5 a; + .50 = 0, being given that two of the roots are equal. 134. Vanishing and infinite roots. If the equation (1) /' (a;) = a^^x^ + a^x"-'^ + • • • + (tn-iX -^a^ = has one of its roots equal to zero,/(i') must reduce to zero when we put x = 0. Therefore «„ must be equal to zero. The same thing results from the factor theorem. For if a; = is a root of /(.r) = 0, x must be a factor of fii-c)- If x= is a multiple root^ taking the place of k simple roots^ x'' must be a factor off(x^; that is, the coefficients of the k terms of lotvest order in (1) must be eqtial to zero. Tlie same result might have been obtained from equations (7) of Art. 132. Instead of considering a single equation of form (1), let us now consider a whole chain of such equations. Let the coefhcients a^ a^--- a^ be the same for all of the equations of the chain, so that the individual equations will differ from each other only in regard to the value of «q. Let us assume further that a„ is different from zero, and that a^ approaches Art. l:3i] VANISHING AND INFINITE ROOTS 223 zero as we pass from the first equation to the second, from the second to the third, and so on. An example will make clear what these assumptions mean. Let the equations all be quadratics, and let 1) 1*5^-^4 5x -7 = 0, 2) T^ x--^ + 5 X - 7 = 0, 3) ToVff ^■'■^ + -"'a: - 7 = 0, k) _Lj;2+ 5x - 7 = 0, be the first k equations of the chain. For the A-th equation we have 1 - 7 The values of a^ and a^ are the same for all equations of the chain ; a„, in our case a„, is different from zero since a^ — — 7. Finally Oq is different from equation to equation and approaches zero as a limit as k- gi'ows be- yond bound. According to (7), Art. 132, we have if a^j, 2^2' ••■' ^n ai'e the roots of equation (1). As a^ approaches zero, the quotient aJaQ will grow beyond bound, since a^ is different from zero by hypothesis. Con- sequently at least one of the roots a^i.a^g, •••■tX^, whose prod- uct, according to (2), is equal to a„/«o, must grow beyond bound. So far in this discussion we have assumed a^^O. If in- stead a„ = 0, the equation (1) has at least one root equal to zero ; that is, f(x) has some power of x as a factor. If we divide f{x) by this power of .r and apply our argument to the resulting equation, which has no vanishing roots, we conclude that at least one of its roots, which is also a root of the original equation, must grow beyond all bound when a^ approaches zero as a limit. Therefore we have the following result. //' the coefficient a^, of the highest power of x in an equation of the form 224 INTEGRAL RATIONAL FUNCTIONS [Art. 134 (1) ((qX" + a^x"-'^ + •'• + a„_i^' + «„ = be regarded as a variable which approaches the limit zero, then at least one of the roots of the equation will grow beyond bound. This is sometimes expressed by saying that the equation has an infinite root. We may state the following more general and more pre- cise result. If the first k coefficients of (1) be regarded as variables which simidtaneously approach the limit zero, but if the (k + V)th coefficient of (1) remains finite., then precisely k of the roots of (1) ivill grow beyond bound. In this case the equation is said to have k infinite roots. To prove this last theorem we might proceed as in the case of one infinite root, making use of equations (7) of Art. 132. But it is easier to reduce the case of infinite roots to the case of vanishing roots by means of the transformation (3) a;=-, y = -. y ^ If we make this substitution in (1), we find, after clearing of fractions, (4) a^ + a^y + a^"^ + • • • + a„_^y«-'^ + a^y"" = 0. From (3) it is clear that the roots ?/p y^.. ••• y^ of (4) and the roots a^j, 2^21 ••• ^n of (1) ^^e so related that we may put 1 1 1 X-^ X,^ X^ Consequently, if k roots of (1), say x^, x^, ••• x,^, grow beyond bound, then k roots of (4), namely y^, y^, ■•■ 3/^, will approach the limit zero. But we have seen that (4) will have exactly k vanishing roots, if and only if the left member of (4) con- tains y''' as a factor, that is, if a^ = a^= ••• = a^._| = 0, %^ 0. Therefore we have proved our theorem about the infinite roots of (1). This tlieorem finds an important application in analytic geometry in the theory of asymptotes. Art. 134] VANISHING AND INFINITE ROOTS 225 EXERCISE LIX 1. By actually solving the quadratic equation iu the illustrative example of Art. IM, namely — x- + 5a;-7 = 0, 10* prove directly that one of its roots tends to become infinite with growing values of k, while the other one tends toward the value 7/5. 2. What relations must exist between a, h, k; and in, in order that the following equation (??i%'^ — b-)x- + 2 kiiKi'-j- + (k- + h-)n- = may have one infinite root? two infinite roots ? Discuss these relations under the assumption that a and h are different from zero, while k and m may be either zero or diffei'ent from zero. CHAPTER VI FRACTIONAL RATIONAL FUNCTIONS 135. Definition of a rational function. If we divide one integral rational function by another, two cases may present themselves ; the division may be exact or not. In the former case the quotient is again an integral rational func- tion, and may therefore be studied by the methods of Chapters IV and V. In the latter case, the quotient is a new kind of a function, known as a fractional rational function. The fractional and integral rational functions together constitute the class of rational functions. Every function which can he expressed either as an integral rational function, or else as a quotient of two integral rational functions, is called a rational function. For the sake of brevity we shall often speak of a fractional rational function as a rational fraction. The two integral rational functions, of which the rational fraction is the quotient, may be called its numerator and denominator. Thus ~ is a rational fraction. x^ — i x^ — 3 X + 2 is its numerator, and x^ — 4 is its denominator. 136. Proper and improper rational fractions. We shall say that a rational fraction is a proper fraction if its numerator is of lower degree than its denominator ; we shall call it an improper fraction if the degree of the numerator is as high as, or higher than, that of the denominator. We observe im- mediately that a rational function which is represented hy an improper fraction may he expressed as the sum of an integral rational function and a proper fraction. 226 Art. l:5(ij 1>R0PER AND IMPROPER FRACTIONS 227 Thus we have, for instance, X — 1 X— 1 X — 5 X — 5 The integral part of a rational fnnction is obtained as the quotient in the process of dividing the numerator by the denominator. The remainder obtained in this division, if there is a remainder, will be the numerator of the proper fraction which must be added to the integral part in order that the sum may be equal to the given improper fraction. If the remainder is zero, the given rational fraction is really an in- tegral rational function ; it is fractional only in appearance. The notions, proper and improper fraction, are very closely related to the corresponding notions in arithmetic. In arithmetic d/D is said to be a projier fraction if d is less than D, an improper fraction if d is not less than D. Thus the notion " lower degree " takes the place of the similar notion " less than," as we extend the terminology of arithmetic to the field of rational functions. In one very important respect the theorems about rational functions differ essentially from the corresponding theorems about the fractions of arithmetic. The sum of two proper arithmetical fractions may be an impi'oper fraction. Tims 1/2 + 3/4 = 5/4. But for rational functions we have the theorem: the sum of two rational functions^ each of which is a proper fraction, is always again a proper fraction. For let -p r ^ -f r \ be two proper rational fractions, so that /j(.?') and ./^^-^O ^^^ of lower degrees than ^/^(.r) and g^i-O i"cspectively. We This is again a rational fraction (compare the definition in Art. 135), and the numerator is of lower degree than the denominator. We shall soon n)ake an important application of this theorem. 228 FRACTIONAL RATIONAL FUNCTIONS [Art. 137 137. Reduction of a rational function to its lowest terms. Let /.^ X be a rational function of x, f{x) and ^(a;) being integral rational functions of degree m and n respectively. Accord- ing to the fundamental theorem (Art. 126), f^x) may be written as a product of the form /(a;) = A(x - aj^^)(x - a^) ... (x - a^), where ^ is a constant, and where a^, a^^ ••• a^ are m numbers whicli may be real or imaginary, all distinct, or some or all of them equal to each other. Similarly we shall have [/(x-) = B(x - h^X^ - h^y-.(x - 6„), and therefore From this expression we may conclude that R(x^ will become equal to zero when x assumes any one of the m values aj, ^2' ■■■' ^mi unless the same value of x should also cause the denominator of (2) to vanish. The fraction R(x) would not be defined for such a value of x, since it would then as- sume the form 0-^0, which is meaningless. (See Art. 21.) But the denominator of (2) can vanish only when x becomes equal to one of the values h^^ h^^ ■•■ h^. Consetiuently, the rational fraction R(x) can assume the indeterminate form. 0/0 only if its numerator and denominator, have a factor in common ivhich contains x. Thus, if J,,. ^ 2(a: + 5)(x + 3)(.r-l) ^ ^ (x + 7)(x + 3) we find r. ^ ^ We may divide both terms of a fraction by the same number (provided that the divisor is different from zero) without changing the value of the fraction. Now a; + 3 is different from zero whenever x is not e(]ual to — 3. Therefore, the above fraction /?(x), and the simpler function Art. 137] REDUCTION TO LOWEST TERMS 229 X + 7 are equal to each other for all values of x except for x = — 3. For X = — 3 these functions are not equal. In fact ii'(— 3) has no meaning, whereas R,(- 3) = ^'"^"^^ = - 4. 4 The illustration just given suggests the following defini- tion : //' the numerator and denominator of a rational fraction have a common factor containing x^ the rational function is not in its lowest terms. To reduce it to its lowest terms we divide both uuinerator arid denominator by their highest common factor. If the rational function Rix) is not in its lowest terras, and if R^Qc) is the function obtained from R{x) by reduc- ing to lowest terms, we shall have R^x) = R^(x) for all values of x excepting those for which R(x) is not defined at all ; namely, those which give rise to a meaning- less expression of the form 0/0 for RQx'). The reduction of a rational function to its lowest terms is easily accomplished when the linear factors of numerator and denominator are all known. But ordinarily this is not the case. However, it is not necessary to knoiv all of the linear factors of the numerator and denominator hi order to accomplish this reduction. It suffices to know their highest common factor, and this may ahvays be obtained by the method of successive division ivhich is essentially the same as the pro- cess for finding the greatest common divisor of two integers. (See Art. 5.) EXERCISE LX Reduce the following improper fractions to the form of an integral rational function plus a proper fraction : ^ x^-'x"^ + X -\ g x* - 1 X- -f a: + 1 X' - X + 1 X- + X + 1 x + 1 X* - 3 x3 + 2 x2 + X - 1 X8 + X2 + X + 1 230 FRACTIONAL RATIONAL FUNCTIONS [Arts. 138, 139 5. Find the highest common factor of the numerator and denominator of each of the fractions in Examples 1-4 and use the result to decide the question whether or not these fractions are in their lowest terms. Reduce the following fractions to their lowest terms : g (x~l)(x + 2) Q 3:2 - 5 X + 4 (j: + 3)(x-1)' ■ x-1 rj (x^ ^ X + l)(x- 1) 3 x^- 13r + 42 x*^ - I ' x2 - 7 X + 6 138. Zeros of a rational function. Let be a rational function in its lowest terms. If a^ a^, ■■• a^ are the roots of the equation /(.r) = 0, then R(x) will be equal to zero when x assumes one of these values. We shall therefore speak of these values a^, a^^ •■• a^ as the zeros of RQx). We should perhaps, more specifically, call them the finite zeros of R(x^t since 7t(.r), if it is a proper fraction, approaches zero as a limit when x grows beyond bound, as we shall see later. Of course some or all of tha zeros of RQc) inay be imaginary. Moreover, several of the roots ftj, ^21 •■■ *m o^ tliG equation f (x) = may coincide, that is, /(x) may have a repeated factor. If /(a;) has (x — rt^)** as a factor, a^ is called a multiple zero of f(x'), and the number r is called the multiplicity of this multiple zero. 139. Poles of a fractional rational function. If 5^, b^, ••• b„ are the roots of the equation g(^x) = 0, the function R{x^ is not defined for x = b^, x = b2', - • ■ x = h^. We should have, for instance, n^i ^ /-^i n where /(Jj) is different fi^om zero, since the fraction is sup- posed to be in its lowest terms. Thus the function ,,/ \ x + 4 X — 1 is not defined for x = 1. But it is defined for all other values of x. Let X approach 1 as a limit. Then the numerator will approach the limit 5, Art. UO] GRAPH OF A RATIONAL FUNCTION 231 and the denominator, x — 1, will approach the limit zero. As the denominator becomes smaller and smaller, the value of the fraction will grow beyond all bounds. We express this by saying that R(x) becomes infinite as x approaches 1 as a limit, and we say that x = 1 is a pole of the function R(x). If a fractional rational function is written in its lowest terms, the function c/roivs beyond hound, or (^as we sat/} becomes infinite, when x approaches as a limit one of the zeros of its denominator. These values of x are called the poles of the rational function. These poles will be real or imaginary, simple or multiple, according as the zeros of the denominator are real or imagi- nary, simple or multiple. 140. Graph of a fractional rational function. We may make a graph of a given fractional rational function RQc) as in the case of an integral rational function. We put y = RQx^), assume arbitrary values for x, compute the corresponding values of y, and plot the points whose coordinates are obtained in this way. But in doing this we should pay special attention to the poles of R(x}, if i2(a:) has any real poles. Thus, if we wish to make a graph of the rational function 1 2/ = -. X we observe at once that x = is a pole, the only one in this case. For X = 0, y is not defined. But we may compute values of y corresponding to values of x which are very close to zero. We find the values indicated in the following tables : ....1 BCD E- A' B' C D' E' - x-:5 2 1 I \ 1 _ i _l _o _3 ... >j.:\ \ 1 2 4 - -4 -2 -1 -i -y. The corresponding points are plotted in Fig. 58 which shows clearly the essential properties of this graph. As x approaches zero, decreasing toward zero from positive values, y becomes positively infinite. As x approaches zero through negative values, y becomes negatively infinite. 232 FRACTIONAL RATIONAL FUNCTIONS [Art. 140 Fig. 58 Thus, there is a discontinuity (break) in the vicinity of x = 0. The other jjoints of the grapli are easily supplied. They show that y approaches zero as x grows beyond bound, either through positive or negative values. The curve obtained in this example belongs to the class of curves called hyper- bolas. This particular hyperbola is, more specifically, called a rectangular hyperbola on account of the intimate relation which it has to two perpendicular lines, the X-axis and ^-axis, which are known as its asymptotes. EXERCISE LXI Draw the graphs of the following functions 1- y 2. J. y = X 3. 2 X 4. 1 y= , X — 1 5 o ■' x-1 6. - 1 y = ., -, X — 1 8. X — 5 y = — 7' x-1 9. y = 2 — -• X — 1 10. a X — D x-1 11. x-1 ^ (x--2)(x-3)- 12. X2 + X + 1 ^ ~ x(x - 1) m - is a curve of the same gen- 1 13. Show that the graph of y = h -V X — a eral character as that of y = 1/x, whatever the values of a, h, and m may be. That is, it is a rectangular hyperbola whose asymptotes are parallel to the X-axis and ?/-axis and which intersect at the point x = a, y = b. 14. Show that every function of the form y — ^^ + y ^ (jo, (/, r, s, being constants) rx -f- .s- which does not reduce to a constant or an integral linear function, may be rewritten in tlie form y = '' + > and that its graph thei-efore has the properties indicated in Ex. 13. Art. 141] FACTORED FORM OF A RATIONAL FUNCTION 233 141. General form of a rational function in terms of its zeros and poles. Let be a rational function in its lowest terms ; and let /(a:) and ^(a;) be of degrees m and w respectively. Then we may write f{x) = A{x~a^){x-a^) ■■• {x -a^), g(x) =B(x.-h^){x-K^ ... ix-h„), where A and B are constants, and consequently ^ ^ ^ ^ ix-b,)ix-b,)...(x-b^) where k = A/B is a constant. Formula (1) (/ives an explicit expression for the most (jeneral rational function which has the values a^, a^. ••• a^ as zeros, and the values b^, b,^, ••• b^ as poles. Since we have assumed R(x') to be in its lowest terms, each of the numbers a^, • ■ • a^ will be different from each of the quantities b^, b^ ■■• b^. If several of the a's are identical, say a-^ = a^ = ••■ — a^, then aj is called a zero of multiplicity r (Art. 138), and the numerator of R{x) has (x — a^y as a factor. If several of the 5's are identical, say b^ = b.^ = ■■■ = />„ then b^ is called pole of multiplicity s, and the denominator of i?(.?') has (a; — b^y as a factor. Thus, the must general rational fraction which has a^ as a zero of multiplicity r^ a^ as a zero of multiplicity rg, and so on, will be (o\ ji(^\ = k (.^- ^O^'C^ - ^i)"' • • • (^ - ^m)^» ^"^ ^ (x- b^y^(x - ^2)'^ ■■■ {x- b„yn When a rational function is expressed in tlie form (1) or (2), we shall say that it is written in its factored form. 234 FRACTIONAL RATIONAL FUNCTIONS [Art. 142 EXERCISE LXll Write down the most general rational function which has the follow- ing numbers as zeros and poles : 1. Simple zeros for x = 1, 2, 3 ; simple poles for x = 4, 5, 6. 2. Simple zeros for x= -1, 0, + 1 ; simple poles for x= - 2, +2, +3. 3. Simple zeros for a; = 1, a zero of multiplicity 3 for a; = 2, a pole of multiplicity 2 for x = 3, and a simple pole for x = 4. 142. Partial fractions. Let (1) ^(0 =4^ be a proper rational fraction in its lowest terms. Then /(a:) will be of lower degree than g^x}. If g(^x) is of degree n, let x — x^, X —x^, ■■■ X — Xr, be the n linear factors of g(x), and let us assume that all of these factors are distinct. Since R{x^ is in its lowest terms, none of these factors will be factors oif(x), and consequently, x-^, x^, ■■■ x^ will be poles of R(x). We may then write R{x~) as follows : (2) M(x) = -^ ^, ^ 7 T^' (x-Xi){x-x^) ■■■ {X-Xr,} where the numerator of i2(2;) is at most of degree w — 1, since we have assumed B(^x) to be a proper fraction. This expression contains precisely n constants a^, rtj, • ■ • a„_j which may have any value whatever, and it is the most general expression of its kind. Therefore, the most general proper rational fraction ivhich has the values x = x-^, x = x^, •■• x = .r„ as distinct poles^ contains n arbitrary coefficients a^, rt^, a^^ ■•• Let us now consider the sum (8) _^i]_ + ^2_+ ... 4--^, /y» /y /y ,y> nf» __^ nm where J.^, A.^,-' '" ^n are constants. This sum is a rational function of x ; its poles are the same as those of R(x)^ and since each term of (3) is a propef Art. 142] PARTIAL FRACTIONS FOR SIMPLE POLES 235 fraction, their sum is a proper fraction. (See Art. 136.) Therefore tliis sum may be rewritten in the form (2), by reducing the several fractions in (3) to the common denomi- nator (^x — x^(x — j'^) ••• {x — Xn) and adding. Moreover the'n coefficients A^ A^-, ••■ Anin (3) can be determined in such a way as to make the sum (3) exactly equal to any given expression of the form (2). The following example will illustrate this statement and show, at the same time, how these coefficients Ai ■•• .1,, may be determined. Let ^' ^' (x-l)(x-2)(x-3) This is a proper fraction, and it is in its lowest terms; for the numerator does not vanish when we put x equal to 1, 2, or 3. We therefore attempt to determine the coefficients A, B, C, in such a way as to make the sum (•'">) ^ + -^+-^,-^(^). X — 1 X — 2 X — i or, what amounts to the same thing, so as to make .Q. A (X - 2)(x - 3) + Bjx - l)(x - 3) + C(x - 1) (x - 2) _ p . . ^^ ■ (x-l)(x-2)(x-.3) -''^^^' or .-X A (x-^- .5 X + 6) + ^(x-^- 4 X + 3) + C{x^ - 3 x + 2) ^ „. . ^^ (x-l)(x-2)(x-3) ^^^- The denominator of -ff(x), as given by (4), and of the left member of (7) are exactly the same. Consequently the fractions (4) and (7) will be equal for all values of x (identically equal), if and only if their numerators are identically equal, that is, if and only if (.4 + B+ C)x-2+ (- 5^-4J5-3C)x+6^+35+2 C=2x2-r)x + 7. According to Art. 126, Theorem F, this will be so if and only if A + B + C^ 2, (8) -.5.4 -45-3 C==- 5, 6 /I + 3 5 + 2 C = 7. The solution of (8) gives A = 2, B = — 5, C = + 5. Consequently we have, from (4) and (5), the desired result, namely -.gx 2 X- - a X -I- 7 _ _2 5_ 5 ^^ (x- l)(x-2^(x-3) x-1 x-2 x-3" An easier way of obtaining the same result, avoiding the solution of system (8), is as follows. If (6) and (4) are identically equal, we must have 236 FRACTIONAL RATIONAL FUNCTIONS [Art. 142 (10) A(x - 2)(x - 8) + B{x - l){x - 3) + C(x - l)(x - 2) = 2 a:'-2 - 5 a; + 7 for all values of x. For x = I, (10) gives us A(- 1)(- 2) = 2 - 5 +7, whence A = 2. Similarly we find from (10) for a: = 2, 5( + 1) ( _ 1) = 2 • 4 - 5 . 2 + 7, whence B =-5; and for x = '■'> C(+ 2)(+ 1)= 2 • 9 - 5 . 3 + 7, whence C = + 5. The second method indicated in this example is especially convenient, and has the advantage of being explicitly appli- cable to the general case, thus enabling us to prove that the coefficients vlj, A^, ■•■ A^ in (3) can always be determined in such a way as to make the sum (3) equal to any given ex- pression of the form (2). We shall refrain from actually writing out the formulae, but we shall state the resulting theorem. Let Ii{x) he a proper rational fraction in its lowest terms, whose poles are all distinct, so that the denominator of R(x) has no repeated factor. Let x^, x^, ■■■ x^he the poles of R{x). Then R(x^ may he ivritten in the form (11) R{x~) = -^^ + -^a_ + . . . + -Ar^_. When a rational function is expressed in this way, it is said to be resolved into a sum of simple partial fractions. The metliod which we used for resolving RQr) into a sum of partial fractions is called the method of undetermined coefficients. It is cliaracteristic of this method that we assumed an expression with certain unknown coefficients (undetermined at the time), and that we found the values of these coefficients subsequently by comparing the resulting expressions with certain others which were known. In applying the above theorem, w% must not forget that i2(a;) is assumed to be a proper fraction. If R{x^ is not a proper fraction, we must first reduce it to a sum of an in- tegral rational function and a proper fraction. (See Art. Art. 14:3] PARTIAL FRACTIONS FOR MULTIPLE POLES 237 13G.) We may then apply the theorem to the hitter part of R(x). Again, we assumed also that R(^x) is in its lowest terms. If it is not, we must reduce it to its lowest terms before attempting to resolve it into partial fractions. Finally we assumed the poles of Rijc) to be simple poles, so that the linear factors of the denominator are all distinct. We did not assume these linear factors to be real. The tlieorem applies equally well to the case of real or complex linear factors, provided tliat no two of them are equal. In practice, however, the formula (11) is usually applied only to the case of real and distinct factors. EXERCISE LXIII Express the following rational functions as sums of simple partial fractions : 2. (x+l)(x-l)(x-2) x^-x 143. Resolution into partial fractions, when the poles are not simple. If R{x) is a proper rational fraction in its lowest terms, but if some of the factors of the denominator are repeated, the expression of R(x) as a sum of partial fractions will be somewhat different from that given in Art. 142. Let (1) R(^x-) = ^^^^ x2 + x - 3 ^ 1 - x + G x'- (•^•-l)(^-2)(x-3) X — x^ 2 a-2 - 3 X + 5 5 1 (.c + 2)(x-3)(x-6) x^ - a2 2 x2 - 7 .7: + 3 g 1 + X - 6 x2 {^x - a)\x — hy(x — cy be such a proper fraction, where the integers, r, 8, t, etc., in- dicate how many times x — a, x — b, etc., are repeated as fac- tors of the denominator. Tlien a is called a pole of R(x) of multiplicity r; b\s called a pole of multiplicity s; and so on. (Seo Art. 139.) If the degree of the denominator be still 238 FRACTIONAL RATIONAL FUNCTIONS [Art. 143 denoted by w, we shall have (2) r+s + t+ ■■■ =n, and the numerator must be of the form (3) / (x) = Oq + a^x + a^x^ + • ■ — t- a„_i2:"-i not involving x", 2'"'*'i, etc., since i?(.r) is a proper fraction. The most general proper fraction which has (a; — ay as its denominator is of the form ^ I X <^o ~r c-^x -}- C2X ~r • • • ~l~ c^_-^x ^^ ix-ay The numerator of this fraction, which is arranged according to ascending powers of x, may be rewritten as a polynomial arranged according to ascending powers of x — a. For we have X = a -{-{x — a), x^= a^ -\- 2 a(^x — a) + (2: — a)^, etc. etc. Consequently the numerator of (4) may also be written in the form A, + A,^^(x - a) + ^,_2(.r - a)^ + • • ■ + A^{x - ay-\ so that (4) becomes (J)\ ^r I Ai=i !-•••+ 2 |_ ^1 . (x — ay {x — ay~^ {x — ay X — a Thus, the most general proper fraction which has (^x — ay as its denominator may be replaced by a sum of the form (5), where the r coefficients A^, A^, ■■■ A^ are arbitrary numbers. Similarly the most general proper fraction which has (x — by as its denominator may be replaced by the sum (^x - by {x - by-^ (x-bf x-b' and so on. Art. 143] PARTIAL FRACTIONS FOR MULTIPLE POLES 239 We see that the sum Ar I ^r-i I ... I ^1 {x — ay (x — ay~^ X — a (6) + — ^ + ^'-1 — +•• -f-^^ + __ili + llAiJ + ... +_kl_ ^ (.i: _ e)' ^ (.r - c)'-l ^ ^^: - c + will be equal to a proper fraction, whose denominator (x — ay(x — hy(x — cy ■■■ is the same as tliat of i2(.r), and that the undetermined coefficients : A-^, A.^^ ■•■^ A^; B^, ■, B,; Cj, ■••, Of, ■■■ which occur in (6) are precisely r + s + r -I- • ■ • = n in number, on account of (2). But the numerator (3) of R{x) contains precisely n coefficients which we regard as given. We may therefore expect that it will be possible to determine the n coefficients: A^, •••,-4;.; jBj, , B^; (7j, •••, Ct, ■ •-, of (6) in such a way as to make the sum (6) exactly equal to R(pc). We shall not attempt to prove that this can actually be done, but leave it to the student to verify this fact in the following examples. EXERCISE LXIV O r -I- ,5 1. Resolve — ^^—^ into partial fractions. (,_1)3(.,._3) Hint. The theory of Art. 143 suggests that we shall put •2 X + 5 ' 1 ^ ^' , g , D ~~ + ~. ~rr^ T r H — (.f - l)3(x - 3) {X - \y (.1- - 1)- X -\ .1-3 Reduce the right member to a single fraction with the denominator (.r — l)3(.i- — 3) and compare numerators. Express the following functions as sums of simple partial fractions : 2 x + 1 4 6 x8 - 8 a:'' - 4 a: + 1 ■ {x-\y' ' x%x-\y ^ x+\ 5 .S x3 - f) x^ - 2 X - 1 x{x - \y^ (X + \)\x - l)2(x - 2) 240 FRACTIONAL RATIONAL FUNCTIONS [Art. 144 144. Modified form of the partial fractions in the case of im- aginary poles. Tlie developments of Arts. 142 and 148 are applicable whether the poles are real or imaginary. In the latter case, however, the partial fractions obtained in this way will also be imaginary, and it is usually convenient, al- though not strictly necessary, to avoid the introduction of imaginary elements when the function under discussion is a real function. Suppose then that R(x) is a real rational function, in its lowest terms, and a proper fraction. By this we mean that all of the coefficients of R(x) are real, so that R{x) assumes real values whenever x is real. The poles of Jl{.r)^ which are the zeros of the denominator of RQx). will then either be real or conjugate complex. (See Art. 34.) In other words, the denominator may be regarded as a product of real linear and real quadratic factors. (See Art. 127.) Let x^+jjx+q be one of these real quadratic factors which cannot be fac- tored into two real linear factors. The most general proper fraction which has x!^ -{- px -\- q as its denominator will have the form n\ Ax-\- B x^ + px -\- q and there must be a term of this form among the partial proper fractions whose sum is equal to R{x). This one real term takes the place of the two terms with conjugate imag- inary linear denominators, which would arise from tlie two imaginary linear factors of x'^ + px -f q if we were to apply the method of Art. 142. If the quadratic factor x"^ + px -\- q is repeated, the corre- sponding terms of the expression R(x) as a sum of partial fractions may be taken in th«! form (2) A,x + B, ^ A_yr + B,_ ^ ^ I --^r^ + ^i (x^ + px + qy (x^ + px -f- qy-"^ x^ -\- px + q The form (•_*) of the partial fraction development is suggested by the following argument. There must be in this development a proper frac- tion with (x'^ + px + qy as denominator. Since {x^ + jyx + (/)'• is of de- Art. 145] FRACTIONAL RATIONAL EQUATIONS 241 gree 2r in x, the most general numerator which a proper fraction with this denominator can have is of the form Hut it is not hard to see that such an expression as (3) may be rewritten in the form (4) (A,x + B,)(x' + px + f,y-^ + (A.,x + B„)(x^+px + «?)-2 + ... + (/l,.ix + Br_i)(x- + px + y) + A,x + B„ and if we divide (4) by (x- + px + (/)•■ we obtain (2). EXERCISE LXV 1. Resolve ; into a sum of simple partial fractious. (X - iy\x' +1) Hint. Here the factor (x — 1)- is a repeated real linear fat-tor, and X- + 1 is a quadratic factor whose linear factors are not real. Tiiis leads u«toput ^•2_4^^5 ^ A B Cx+D (X - 1)^(X2 + 1) {x - 1)2 X ~ 1 X^ + 1 Reduce the right member to a single fraction with the denominator (x — l)"^(.f- + 1) and compare numerators. Express the following functions as sums of partial fractions: 2 4 4 6 x3 + 2 x'^ + 2 X - 2 (x- l)(xH X + 1)' ■ x*-l . „ 9-2x - 5x2-4x+6 3. • •• 5.- (x + 2) (x2 - 2 X + 5) (x2- X + l)2(x - 3) 145. Fractional rational equations. An equation all of whose terms are rational functions of the unknown qnantiti/ is called a rational equation. If at least one of the terms of the equation is a fractional function, the equation is a fractional equation. If there are any terms in the right-hand member of the equation, these terms may be transposed. Consequently any rational equation may be written in tlie form (1) B^ix) + ll,(x) + ■■■ -\- E,(-i-) = 0, where R^, R^, ■■■ Rk denote rational functions of x. If all of these functions are integral rational functions, the left member of (1) is an integral rational function of .r, and we 242 FRACTIONAL RATIONAL FUNCTIONS [Anr. 145 have before us the problem discussed in Chapters IV and V. Let us assume, therefore, that at least one of the functions i?j, i^g, • •• -R/fc is a fraction. If one or several of these terms are fractions, let us assume that each one of them is written in its lowest terms. Let us assume further that no two of these fractions have the same denominator. If two such terms with a common denominator should occur, we could unite them into a single term. The sum of the rational functions R-^{x^, B^^x'), •••, M^^r^ is again a rational function which may be obtained by reduc- ing R-^(x), R^(x')^ •••,i?;t(^) to a common denominator and afterward adding the resulting numerators. The sum will not be in its lowest terms unless we use for this purpose the lowest common denominator, that is, the lowest common mul- tiple of the denominators of R^(x^, R^(x)^ '••,RkQc). But even if we do use the lowest common denominator, we can- not be certain that the sum is in its lowest terms. If we have, for instance, 1 2 ■^^^''^ = (:,-3)2(:.-2)' ^2^"^^ = (.,-3)2(.r-5)' the lowest common denominator is (.r — 8)^(2;— 2) (2;— 5). The sum, R^{x^ + R^(x^^ then becomes ^x- 9 ^_ Z(x- 3) (x - ^)\x _ 2)(a; - 5) - (:c - '6)\x - 2) (a; - 5) and is not in its lowest terms. If, however, we express every one of the rational fractions which occurs in (1) as a sum of simple partial fractions, then unite all of the partial fractions which have the same denomi- nator into a single one, and finally add these ^jartial fractio7is together, using as a common denominator the loivest common multiple of the denominators of the simple partial fractions, we may he sure that the sum R{x^ obtained in this way is in its lowest terms. Let R(x) be the sum obtained in this way. We may write Art. U5] FRACTIONAL RATIONAL EQUATIONS 243 where f(x) and g(x^ are integral rational functions which have no common factor involving- x, since R(^x) is in its lowest terms. The equation (1) will be replaced by (2) i2(^)=/M = o, and it will be satisfied by all of the roots of the integral rational equation (3) fix) = 0. Moreover, no other value of x will satisfy equation (1). Therefore, all of the finite roots of a fractional rational equation may he obtained by solving a certain integral rational equation. The process of deriving (3) from (1) is commonly described as " clearing of fractions.''^ But unless we clear of fractions by the process indicated above, first resolving each term into a sum of simple partial fractions, the resulting integral rational equation may not be correct. For, unless the sum B^ix) + B,{x) + ■■• + Rd-r) = Rix) ='t^ is in its lowest terms, we cannot conclude that all of the roots of (3) will also be roots of (1), since some of these roots would then also cause the denominator g(x) to vanish. If we are careful, however, while clearing of fractions not to divide by any integral rational expression involving x, we may be sure that the resulting integral rational equation will contain all of the roots of (1). But it may have other roots besides. We may formulate some of these results more briefly by introducing the following terms. A second equation, derived from a first equation, is said to be equivalent to the first, if it lias exactly the same roots 244 FRACTIONAL RATIONAL FUNCTIONS [Art. 146 as the first. It is said to be redundant if the roots of the first equation are included among its roots, and if it has other roots besides. The second equation is called defective if its roots do not include all of the roots of the original equation. We have seen that there always exists an integral rational equation equivalent to a given fractional rational equation. But we have also seen that the usual process of finding this equation hy the method of '•'- clearing of fractions'" is not always reliable. If., however., in clearing of fractions., we carefully refrain from dividing hy an integral rational function of x, we may he sure that the resulting equation loill not he defective. We can decide., after solving this equation., whether it is redun- dant or equivalent hy testing each of its roots to see whether all of them do or do not satisfy the original fractional equation. Special care is necessary to avoid dividing by a factor which contains X. Inexperienced people often conclude from such an equation as x(x - 5) = 0, that X = 5 is the only solution. But x = is also a solution of this equation. EXERCISE LXVI Reduce the following equations to integral rational equations. Discuss the question of equivalence, and solve. X 7 ^3,12 X + 60 3 x-5 X — 27 X = 13. 8x 6 = 20 x + 2 3x 146. Pressure exerted by gases. Let us suppose that the cylinder CC in Fig. 59 contains a certain volume, say Vq cubic feet, of air when the movable piston P occupies its highest position, and let us compress the air by i)ushing the piston down into the position shown in the figure. As we push tlie piston down farther and farther, we find that the resistance of the inclosed air becomes greater and greater. Art. 146] PRESSURE EXERTED BY GASES 245 Thus the inclosed air exerts a force tending to move tlie piston upward, and this force is overcome by the muscular force which we exert in pushing the piston ~'^'' . — 1 c down. ^ In Fig. 59 the air has been compressed to about one third of its original volume. If we take a second cylinder of the same height but of smaller cross section and compress the air in it to one third of its original volume, we notice that a smaller force will suffice for this purpose than in the case of the first cylinder. In fact, measurement of these forces will sliow that they are to each other as the cross sections of the two cylinders. Therefore, the force which the air exerts upon a unit of area will be the same in both cylinders. The force which any gas exerts upon a unit area of a containing vessel is called the pressure of the gas, and is usually denoted by p. The total force which a gas exerts upon an area of A square units will be (1) F=Ap if p denotes the pressure of the gas. Consequently we have (2) P = f. Since a unit of force has the dimensions ML/T^ (Art. 80) and a unit of area has the dimensions I? (Art. 41), a unit of pressure according to (2) has the dimensions (3) M/LT^. When gas is compressed in a cylinder, as shown in Fig. 59, the pressure increases, the volume decreases, and ordi- narily the temperature of the gas rises. If the cylinder is made of metal, we may wait until the original temperature, that of the room in which the experiment takes place, has reestablished itself. This may be done by inclosing a thermometer in G- which may be observed from the outside through a glass window. We are thus in a position to 246 FRACTIONAL RATIONAL FUNCTIONS [Art. 146 measure the various pressures which the gas exerts upon the piston at the various moments when its volume has dif- ferent values while its temperature remains the same. Ex- periments of this general character were first performed by Boyle in 1661, and by Mariotte in 1676. These experi- ments lead to what is usually known as Boyle's law. If the volume of a gas is changed isothermallg, that is, without chang- ing its temperature, its pressure ivill change in such a way that the product of pressure and volume remains constant. Thus the product pv can depend (for a given mass of a given kind of gas) only upon the temperature. The de- pendence of this product upon the temperature was investi- gated by Gay-Lussac in 1802. He found the equation (4) pv=RT,OTp = , V where T, the so-called absolute temperature of the gas, is (5) ^=273 + ^, 6 being the temperature expressed in Centigrade degrees, and where i2 is a constant which depends upon the amount of gas which is being used and upon the chemical constitu- tion of the gas. As a matter of fact, equation (4) is not perfectly exact. For high pressures it fails to agree completely with the ex- periments, and the following equation, due to Van der Waals, is more nearly in accordance with the facts : RT a V — v^ where a and h are small quantities which are practically un- noticeable except when v becomes very small. Observe that (6) reduces to (4) wlien a and b are equal to zero. If the temperature remains constant while the gas is changing its volume, eacli of the equations (4) and (6) de- fines p a,s a. fractional rational function of v. Art. 140] TRESSURE EXERTED BY GASES 247 EXERCISE LXVII 1. Use a unit on the x-axis to represent a unit of volume and a unit on the y-axis to represent a unit of pressure. For a given temperature represent the relation expressed by the combined law of Boyle and Gay- Lussac graphically. Are negative values of p and v admissible? How will the curve change if the temperature changes? 2. Investigate the form of the curve given by Van der Waal's equa- tion. If appropriate units are used, we have II = 0.003G9, a = 0.00874, b = 0.0023 when the gas is carbon dioxide. Trace the curves which correspond to 7' = 250°, 300°, 350°. I CHAPTER VII IRRATIONAL FUNCTIONS 147. Existence of irrational functions. All of the functions which we have studied so far were rational functions. But there exist many important functions which are not rational. The function Va; is such a function. However, the fact that Va; is not a rational function of x requires proof. It does not follow from the mere presence of a radical sign. Let a; be a variable capable of assuming all possible real and complex values. Every number x has two square roots each of which is determined by the value of x and is there- fore a function of x. Let V^ denote either of these square roots. We observe first that Va: will he finite whenever x is finite. To prove this statement, let x assume any finite com- plex value whose modulus is r and whose amplitude is 6. (See Art. 30.) Then r is a finite positive number. Both square roots of x will have the positive finite number Vr as their common modulus, and their amplitudes will be 6/2 and ^/2-h180° respectively. (Art. 120.) In other words, if x is finite, both square roots of x are also finite. We shall now show that 'Vx is not a rational function by the method of reductio ad absurdum. Suppose it were a rational function. We should then have, for all values of a:. (1) Vi = /(^) where / (a;) and g(x~) are integral rational functions, which we may assume to have no common factor involving x. (Definition of a rational function in its lowest terms. Arts. 135 and 137.) According to the fundamental theorem of Algebra (Art. 126), there exist finite values of x (real or 248 Am. U7] EXISTENCE OF IRKATIOXAL FUNCTIONS 249 imagiiKii'}') wliicli satisfy the equation g(x^=0, if the func- tion g(^x) involves x at all. Such values of x would cause the right member of (1) to become infinite. But we have just seen that the left member of (1) remains finite for all finite values of x. Consequently, equation (1) gives rise to a con- tradiction unless (/{x^ does not involve x at all, that is, unless ff(x) reduces to a constant. In that case, the quotient f (x^lg{pc) would reduce to an integral rational function of a", and we should have, for all vahies of x, (2) V~v = a^-\- a^x + a^x9- + • ■ • + a^x"" or, squaring both members, (3) x = a^^^l a^a-^x + (a^^ + 2 a^a^^x^ 4- • • • + «„V". But an identical equation of form (3) can hold only if the coefficients of like powers of x in tlie two members are equal to each otiier. (See Art. 126, Theorem F.) Therefore (3) can be true for all values of x^ only if = Uq^, 1 = 2 a^a^, = a^"^ -{^ 2 a^a^., etc. But the first and second of these equations contradict each other. Therefore equation (2) is impossible and Va; is not a rational function of x. We could prove, in similar fashion, that Vx^ Va^, ••• '\/ .r are not rational functions of a:, and more generally that the function Vx"" is not a rational function of x uidess m happens to be an integral multiple of n. We shall speak of all of these functions as irrational functions, in accordance with the following definition. A function which cannot he expressed either as an integral rational fwiet ion or as a quotient of tivo integral rational func- tions^ is called an irrational function. Tt would be entirely erroneous to conclude that the jDresence of a radical sign in the expression which defines a function is either necessary or sufficient to make the function irrational. Thus \/4 .r^ involves a radical sign, but this function is equal to ± 2 x and is therefore rational. 250 IRRATIONAL FUNCTIONS [Art. 148 On the other hand the trigonometric functions, sin x, cos x, etc., are irra- tional although they involve no radical signs. In applying the above definition we are not concerned with the form in which the function happens to be written. The real question is whether it is or is not possible to rewrite it in such a way as to make it assume the form of a rational function. 148. The function y=\/x and its principal value. If a; is a given number, and if t/ is another number sucli that (1) ?/" = X, y is said to be an wth root of x and we write (2) y =)" = (a")"" = a'"", where m and n are positive integers.* Consequently, the one-valued function (1) satisfies the equation (2) ^" = a:"*. But this equation, which is of the nth order in y, has n roots (Art. 120) each of which is called an wth root of x"". A consistent use of the symbols would lead us to denote any one of these roots by Vx"'. But our desire to have this symbol defined unambiguously leads us to the following more specific definition. The symbol ^/x"" shall be used to stayid for that particular one of the nth roots of x"" ichich is equal to (\/:c)"', tvhere the symbol Vx is defined as a one-valued function of x by the agreements laid down in Art. 149. Thus the principal value of \'x"' is associated with the principal value of Vx in such a way that we shall have (8) V^={^xy. * The proof of these formulas may he reviewed by referring to Art. 157. 254 IRRATIONAL FUNCTIONS [Art. 151 The choice which we have made of the principal value of Vx^ may also be expressed as follows. Whenever there exists only one real nth root of x^ {x itself being real), this real root is the principal value. Whenever there are two real nth roots of a;'", one will he positive and one will be negative. In that case the positive nth root of x^ is the principal value. EXERCISE LXIX Draw graphs of the following functions : 1. y = yf^. 5. J/ = 2 + Vz3. 2. y -- Vx^. 6. ^/ = 2 - yJ~^. 3. ij = vx2. 7. ?/ = - 2 + v;^. 4. y = - \/x^. 8. ?/ = - 2 - Vx3. Making use of the definition of the jirincipal values of y/x and of yjx'^ as given in Arts. 148 and 150, prove the statements in Exs. 9-12. 9. ^x"' is defined as a real one-valued function of x for all real values of x, if n and m are both odd. This function is positive or nega- tive according .as x is positive or negative. 10. \/x"» is defined as a real one- valued function of x for all real values of x, if n is odd and m is even. It is positive for all values of x. 11. Vx"' is defined as a real one-valued function of x for all real values of x, if n and m are both even. It is positive for all values of x. 12. Vx"* is defined as a real one-valued function of x only for values of x which are positive or zero, if n is even and m is odd. For all such values of x the function is positive. 13. For what values of x will the function V(2 x — 7)^ be defined as a real one-valued function ? 14. For what values of x will the function V(7 - 2 x)^ be defined as a real one-valued function ? 15. Formulate the four statements which correspond to Exs. 9, 10, 11, 12 for the function \/— x'". 151. Properties of radicals. The following formulas are used very frequently in calculations involving radicals : (A) -v/^ = (-v^^)- (C) -^^='V^, (B) VV^ = "'-v/^, (D) V^Vx^=V^, Art. 151] PROPERTIES OF RADICALS 255 The first of these equations has already been proved. It is identical with equation (3) of Art. 150. Let us prove equation (B). The right member of (B) is the principal value of the mnth root of x, and is. therefore, a positive number whenever x is positive. The left member of (B) is also an wmtli root of x. For it is easy to verify that the mnth power of v Va: is equal to a;. If a: is positive the left member of (B) will also be positive, each of the radical signs being used to designate the principal value of the corresponding root. Thus each of the two members of (B) is a positive mni\i root of the positive number x, con- sequently these two members must be equal. For, a positive number x has one and only one positive 7nnt\\ root, namely, the principal value of that root. (See Art. 120.) We have now proved the correctness of equation (B) for all positive values of x. If X is negative, both members of (B) will be imaginary unless m and n are both odd. In the latter case the prin- cipal values of both members of (B) will be negative. Thus, both members of (B) are mnth roots of the negative number x^ and both of them are negative. Therefore, they are equal. For, a negative number has only a single real mnth root and this is negative. (See Art. 120.) The remaining case when a- = requires no discussion. Equation (B) is obviously true \i x=0. Thus we see that equation (B) is actually true for all real values of x in all cases in which both members of (B) are defined as real functions. To prove formulas (C), (D), and (E) we may proceed as in proving (B). The details of the proof are left as an exercise for the student. The student may iw.ssibly have some doubt in his mind as to the necessity of proving these equations at all. The following- example will help to clarify the situation. Unless we specify that the radical signs 256 IRRATIONAL FUNCTIONS [Art. 152 are to be used for the principal values of the indicated roots, we should have the alternatives V4 = ± 2, V9 = ± 3, Vm =±Q and we could not affirm that the equation ViV9 =V36 is necessarily true ; it might just as well be ViVO = - VM. In all of the equations (A) . . . (E) both members are easily seen to be like roots of one and the same number. Since we are discussing only the case when the quantities concerned are real, it is evident at once that the two members of the equation are either equal, or else numeri- cally equal and opposite in sign. The essential result which we have obtained is this: both members of each of the equations (A) . . . (E) will actually be equal not only in magnitude but also in sign, whenever the functions concerned have real values, provided that every radical sign which occurs is used for the principal value of the corresponding root. EXERCISE LXX Simplify the following expressions : 1. Vl8. 4. V- \/-27^- 5. 'f'^'<^^' 3. 9. v'lOOOO. 6. V25 v/24 + V54 - V6. . as/jiopi 15. 16. 17. 18. 19. 20. 10. 11. 12. ^ />3 + \ /,,/■> \ 5e2 v^a y/b'y/c. (3+V5)(2-V5). (9 -7Vi3)(5 - evTa). 13. 14. Va^c + a^d. Vv'ti4. ^b ^h 152- The square root of a rational function. When we iiLtcmpt to extract the square root of a ratioual function R{x)., two essentially different cases may present themselves. The square root may again be a rational function, or else it may not. In the former case we may say that i?(.r) is a perfect square. Akt. 152] SQUARE ROOT OF RATIONAL FUNCTION 257 Thus the rational function J. (x-iy(Sx + 5r ^ ^ (x - 3)4(x - 7)6 is a perfect square. One of its square roots is the rational function .0) (X -])(:} x + 5)^ ^'^ (X - 3)^(x - 7)3 The other square root is this same function multiplied by — 1. This example shows quite clearly why the square roots of (1) are rational functions, namely, because all of the ex- ponents which occur in (1) are even numbers. Now an^ rational function may be written in what we have called its factored form (Art. 141), and it will obviously be a perfect square if all of the exponents which occur in it are even numbers. We may express this result as follows : I. If, in the factored form of a rational function, every linear factor of its numerator and of its denominator appears raised to an even power.- the function is a perfect square; that is, both of its square roots will be rational functions. Let R{x) be a rational function. Let us assume that RQx^ has been written in its factored form, and that not all of the linear factors of R(^x) which appear in this factored form are raised to even powers. Then each of the square roots of i?(a;) will be an irrational function. For let C'3) i/=VR(x), be one of these square roots, so that (4) f=R(ix^- Equation (4) shows that y cannot be a rational function. For if it were, i/^ in its factored form would contain only even exponents, contrary to the assumption that 7?(.r), in its factored form, contains some odd exponents. We have obtained the following theorem: IL If, in the factored form of a rational function, any of the factors appear raised to odd powers, both of the square roots of the function will be irrational. 258 IRRATIONAL FUNCTIONS [Art. 15.} Tlius, the function is not a perfect square because not all of the exponents are even. The two square roots of R(x) may be expressed as follows, (6) v7fw = ±ii^iKM?Vj5l. This example illustrates the following general theorem. III. If a rational function is not a perfect square, its square root may he expressed as a product, one of whose factors is rational, while the other factor is a square root of a rational function with simple zeros and poles. The first of these factors ma}% of course, reduce to a mere constant, and this constant may be equal to unity. This is the case in the follow- ing example, X ~1 \x —i The expression for the square root of a rational function may be simplified still more. In the expression (6), let us multiply both terms of the fraction which occurs under the radical sign by x — 7. We find (x - ly where we now' have an integral rational function with distinct linear factors under the radical sign. The corresponding general theorem is as follows : IV. If a rational function is not a perfect square, its square root may he expressed as a product, one of tvhose factors is rational, while the other is a square root of an integral rational function without repeated factors. 153. Functions which involve the square root of a rational function and no other irrationality. Let R(x) he a rational function which is not a perfect square, so that -\/ R(x) is Art. 153] SQUARE ROOT AS ONLY IRRATIONALITY 259 irrational. Consider a function / {x} which is made up of a sum of terms of the form (1) A,Cx}, A,ix)^E{x), ^2(a.)(Vi2(a;))2, ^3(x)(V;ft(^)3, etc., where ^o(^)' -^i(-'')^ ^2(-*')' ^^^•' ^^® rational functions of x. Since (V7^0r))2 = 7e(:r), (Vi2(a;))3 = i2(2;)Vi2(a:), etc., each of these terms will be either a rational function, or else a product of a rational function times ^I{{x). Conse- quently the sum of all of these terms will be an expression of the form (2) rational function -f another rational funetion x V^(2;}. If R(x^ is not an integral rational function without repeated factors, we may, as in Art. 152, express ^R(x^ in the form of a rational function times ^r{x) where r(x') is an integral rational function without repeated factors, so that Vr(2;) is irrational. Thus, the sum of any number of terms of the form (1) may finally be written in the form (3) /(x) = A(x) + B(^x)Vr(ix% where A{x') and B(^x') are rational functions of x, and where r{x) is an integral rational function without repeated factors. Suppose we have a second function ,g{x^ of the same kind as /(a;). It may then be written in the form (4) g{x) = Cix) + I)(x) Vr(2:), where C(x^ and -Z)(x) are rational functions. The quotient of two such functions ma}^ again be reduced to the same form. To prove this we write A + B^/r A + B^r C-BVr (5) C+BVr C+D^r C - D^r ^ AC- BDr + (BC- AD^^r c^-mr 260 IRRATIONAL FUNCTIONS [Art. 15i where C^ — I>h' will not be identically equal to zero. For, if it were, we should have _ (72 contrary to our assumption that Vr(a^) is irrational. We may rewrite (5) in the form where L(x) and M(x^ are rational functions. Thus, ever// function of x, which depends upon the square root of a rational function in the way indicated, may he re- tvritten in the form (6) L (a-) + M(x:) Vr{x) where L{x^ and M(pc) are rational functions arid where r(a-) is an integral rational function with no repeated factors. We may speak of (6) as the normal form of such a func- tion. The process used in (5) for reducing an irrational function of this kind to the normal form is frequently known as rationalizing the denominator. If a function contains more than one independent square root, the normal form may be obtained by rationalizing first with respect to one of the square roots, then with respect to the second, and so on. EXERCISE LXXI Reduce the following functions to the normal form : .. x^-hxy/x 4 3a:- 4V.r2 - 9 3. 2Vx 3x + 4v'x2-9 .5 + 6 x Vx c a + ftVx^ — m^ _ X — 2Vx a — hy/x"^ — m'^ 3+4 \\ - x2 - 7 a; + 5 + ^'x^ - x^ + 1 ,3 _ 4 Vl - x2 3 x + 2 + 4Vx8 - x^ + 1' 154. Irrational equations of the simplest type. Let f(x) be an irrational function of x. Tiiis function may assume the value zero for one or several particular values of x. Akt. 1:)4] simplest IRRATIONAL EQUATIONS 261 These values of .r, if they exist, are called the zeros of f(j-), or the roots of the irrational equation (1) /(•r) = 0. Let us confine our attention to the case where the function /■(.r) contains as its only irrationality the square root of a single rational function. We have just shown (in Art. 153) that we may then write /(i-) in the form /(.r) = i(r) + i>f(.r)Vr(^.i') wheic L(x) and 3I(x) are rational functions, and where r(a-) is an integral rational function with no repeated factor. Consequently equation (1) will assume the form (2) L(^x)+ M(x)Vr{x) = or (3) L(x)=- 71if(x)Vr(a-). If i- is a root of (3), it will also satisfy the equation (4) [X(.0]^ = [i»f(.r)]V(.r), obtained from (3) by sc^uaring both members, and (4) is a rational equation which may be solved by tlie methods of Art. 145. If we remember what those metliods were, we shall recognize the truth of the following statement : The problem of solving an irrational equation of form (2) 77iai/ he reduced to the problem of solving a certain integral rational equation in .r. But Avhile it is certain that all solutions of (2) will also be solutions of tlie rational equation (4), it is not at all certain that every solution of (4) will be a solution of (2). //■ thr si/i)i/iol \/r(.r) stands for the principal value 71 ). Formulas I, II, III are usually known as the index laws. The definition (1) of x"" is entirely devoid of meaning 265 266 THE GEXERAL POWER FUNCTION [Akt. 157 unless m is a positive integer. For, in this definition, m stands for the number of factors in a certain product, and it is absurd to speak of a product composed of a fractional num- ber of factors, a negative number of factors, or no factor at all. Consequently the definition (1) cannot be used in any case in which m is not a positive integer. Thus, if we wish to give a meaning to the symbol a;'" when m is not a positive integer, we must seek a new delinition essentially different from (1). Now, as a mere matter of logic, rue have the right to define the symbols of algebra i)t any way we may desire. Thus we might, for instance, define the symbol x'^ in such a way as to make it stand for x/2, for 2/x, for Va;, or for anything else we please, and nobody would have a right to object to any of these definitions on the score of logic. But one might very properly object to some of these definitions on account of their inconvenience. In order to obtain a definition for x"^ which shall not only be logically admissible, but which shall also be convenient and useful, we argue as follows. Nothing compels us to introduce such a symbol as a;^ at all. The very fact that this symbol resembles the familiar symbols a-^, x^, x^, etc., so much, makes it inconvenient to introduce it at all unless it can be done in such a way as to enable us to perform calcu- lations with this new symbol according to the same rules (the index laws) which hold for x^, x^, a:^ etc. Thus, our desire for a convenient definition leads to the following ques- tion : Is it possible to define the symbol x^ in such a way that the index laws /, /i, and III may he used m all of the calcula- tions in which this new symbol occurs? This question is easy to answer. If it is possible to define x^ in such a way, we may apply index law II to it, for the purpose of computing its square, giving (a;2-^2 _ ^2 _ ^^ Therefore, the symbol x'^ must be taken to mean one of the two square roots of x.* If x is positive, one of its two square * Since its square is equal to x. Ain. 157] THE INDEX LAWS 2G7 roots is positive and the otlier is negative. We therefore define x^ as the positive square root of x, whenever a: is a positive number. If x is negative, the two square roots of X are both imaginary and either one of them may be identified with the symbol x-. 1 In the same way we are led to define a" bi/ means of the equation 1 _ (2) x^ = ^/x, the principal value of the nth root of x being meant when x is real. (See Art. 148 for definition of principal value.) According to the second index law, we shall have 1 m (^xy = (x"y' = x\ m thus leading to a definition for x'\ namely., m _ (3) x"=(Vxy=Vx'\ where the second and tliird members are equal on account of (3), Art. 150. We observe that the third index law (4) — = a;--" ^ x^ was subject to the restriction m > n. If ni is less than n or equal to w, the exponent in the right member becomes nega- tive or zero. If we wish to introduce negative or vanishing exponents in such a way as to preserve the validity of the index laws, we must therefore do so in accordance with what equation (4) tells us. But if we put w = w in (4) we find the following definition of .r^ : (5) ^ = ?- = l. 268 THE GENERAL POWER FUNCTION [Aim. 1.> If m is less than /«, let it be equal to n—k. If we put m = n — k in (4) we find X X -k ^n—k — n /y» — k But the left member of this equation is actually equal to l/a:^ thus leading us to write (6) x-^ = \ x" as the definition of x~^. Combining (3) and (6), we have -™ 1 1 JU — ,,„ x^ We have now defined the s3'mbol x^ in all cases in which r is either a positive or negative integer or zero, or a positive or negative rational fraction. We know that no other defi- nitions than those actually adopted were possible, if the in- dex laws were to be satisfied. But we are not yet absolutely certain that all of the index laws will actually be satisfied by these symbols in all cases. For we have only used one of these laws for the purpose of guiding us toward the appro- priate definition in each case. It is possible, however, to verify, by actual test, that the powers with fractional, 7iegative, and vanishing exponents defined in this wa// actually obey all of the index laws. We may therefore operate with them ac- cording to the same rules which hold for positive integral ex- ponents. It is this fact which makes tliese definitions, not merely logically admissible, but extremely convenient and useful. 158. The principle of permanence. The j)oint of view which guided us in fornndating these definitions is known as the priyiciple of the permanence of theforvnal latvs of algebra. This same principle aided us in Chapter I, when we were engaged in enlarging the number system of algebra, ])y add- ing to the system of all positive integers the negative, frac- tional, irrational, and complex numbers. The principle of Art. 158] THE PRINCIPLE OF PERMANENCE 269 permanence is not a principle of logic. It is a heuristic principle which leads us to new definitions and notions, enabling us to obtain results which are, not merely logically admissible, but simple and convenient. Its main object is to make a few formal laws (formulas) suffice where other- wise there would be many. It accomplishes this purpose by making a formula which is already in existence do more work than was originally intended for it, thus preventing the introduction of a new formula. Thus, in the present instance, the index hiws were originally intended to apply only to powers with positive integral exponents. The principle of permanence has led us to adopt such definitions as to make the index laws accomplish very much raoi-e. They now apply to powers with negative and fractional exponents as well. EXERCISE LXXIIi Find the value of the following numbers : 1. 8^ 2. 3-^ 3. 16"l 4. (VO^- 5- 25"^. 6. (fl)"^: Obtain equal expressions free from negative and fractional exponents : 7. 3a~\ 8. Sx--b-^c^. 9. (a-hc)\ 10. a'ph'^c'l- Perform the operations indicated and simplify : 11. (a^y*. 13. "_) • 15. ^""^ ^ 12. (ahK^^y. 14. (^\\ 16. ^f^" y Va Perform the following multiplications: 21. (a^+b^-){a' -b-^). 22. (a^ + //^)(a- + b^). b')(J - aW + 6*). 29. a — b by a^ + b'-. 1 1 30. a — b hy a- — b^. 31. a-b by a^ + aV + OK 32. a + b by J - aM +b^ . 3 5 17. aia^. m p 19. «"«?. 18. a'h-^ah'c. _m p 20. a 'HiQ. 23. {a^-b^)(J + a^b^ + b'^). 24. \ Divide : m p 25. o» by as. m p 27. a « by ««• m _p 26. a" by a «. _m _p 28. a " by a i. 270 THE GENERAL POWER FUNCTION [Art. 159 159. The case of an irrational exponent. If the exponent k is an irrational number, the symbol x'' is as yet undefined. To define x'^ in this case also, we proceed as follows. We know that any irrational number k can be expressed with any desired degree of approximation by a decimal. (See Art. 77.) Let k^ be a number written in the decimal notation, with n figures to the right of the decimal point, such that kn is less than k but differs from k by less than one unit of the nt\\ decimal place. Let kj be the number ob- tained from kn by increasing the digit in the nth. decimal place by one unit. Then As n (the number of decimal places) grows beyond bound, the rational numbers k^ and kj both approach k as a limit. It can be shown that the numbers /j-JCl /yiA2 /y"3 , , /y»n'l /yt^^'l ry^.i JU * • ••• be the corresponding values of y. Then we shall have Vx ^^ ax-^ , 7/2 ^ ax2 , y^ = ax^ •••, whence (2) ^2 = -V = (-hy^ !h ^ ffsY, etc. If the values x^, x^-, 3*3, ••• form a geometric progression whose ratio is r, we hiive = r and therefore, according to (2). ^ = ^= ...^rK AuT. 160] TUE POWER FUNCTION 273 In other words, ?/j, i/^^ 3/3, ••• will also form a geometric progression, whose constant ratio r* is, of course, in general different from r. We obtain the following theorem which expresses the essential property common to all power functions : Let y he a pou'er function of a-, and let us take any number of values of x which form a geometric progression. The cor- responding values of y ivill then also form a geometric pro- gression. This may also be expressed as follows : In a power func- tion., to values of the independent variable which form a geometric progression, there correspond values of the function which likewise form a geometric progreij = x^, and y = ar-i = - will illustrate this state- X ment. 274 THE GENERAL POWER FUNCTION [Akts. 161, 162 161. The exponential function. Up to the present moment we have always thought of the exponent in the equation (1) y = x>' as a constant, and the base as variable. We now propose to think of the base as a constant and the exponent as variable; that is, we propose to study the expression (1) as a function of the exponent. We shall indicate tliis new point of view by a change of notation, writing a for the fixed base, and x for the variable exponent, so that (1) becomes (2) y = a-. After the base a has been chosen, the value of a^ will depend only upon x, so that a"" is a function of x. It is called an exponential function. We shall eo7isider only the exponential functions whose bases are positive numbers different from unity. The reasons for these restrictions are fairly obvious. If the base a were negative, a^ would not be real if x were equal to 1/2, 1/4, 1/8, or any fraction with an even denominator. If the base were equal to unity, a^ would be equal to 1^ and this would be equal to 1 for all values of x. We agree., moreover., when a is positive and when the exponent x is a fraction m/n, that a^ shall always be defined as the prin- cijyal value of a'' = Va™, quite in accordance with what ivas said in Art. 157. In most applications, the base a is, moreover, taken to be greater than unity. However, this is not at all essential. 162. Graphs of exponential functions. The method of constructing the graph of an exponential function needs no detailed explanation. It is precisely the same as for all of the other functions which we have studied so far. Art. 16:]] THE EXPONENTIAL FUNCTION 275 EXERCISE LXXIV 1. Construct the graph of l?^. Solution. We compute the table of values on the right- hand margin of this page, by putting 1/ = 2" and computing the values of // which correspond to the values x = — 4j - 3, - 2, - 1, 0, + 1, + 2, + :}, + 4. We plot the corresponding points in Fig. 62 and connect them by a smooth curve. This curve is the required graph. Construct the graphs of the following functions : 2. 3"^. 5. (V2)'^. 3. 4*. 6. 2-'. Fig. 02. X -4 -3 -2 - 1 + 1 + 2 + 3 + 4 i 1 2 4 8 16 4. 5^. 7. 3-». 8. 9. 10. ay 163. Properties of a'^. The graphs obtained in the pre- ceding exercise indicate several properties of the exponen- tial function by mere inspection. In the first place, all of these graphs are continuous, unbroken curves. To this fact corresponds the following theorem. I. The exponential function is continuous for all fiiiite values of x. AVe shall not attempt to give a formal proof of this theorem. In the second place, we observe that all of the exponen- tial curves are situated entirely ahove the .r-axis. This is a consequence of the following theorem, which follows at once from the definition of a^. II. The function a^ loith a positive base a, is positive for all values of x. 3Iore specifically ; if a > 1, then a"" > 1 for x > 0, and rt^ < 1 for X <0; if a < 1, then a-" < 1 for x > 0, a7id a'^ > 1 or X < 0. Let us draw a line parallel to the ^--axis and above it. The graphs, which we have constructed, indicate that such a line will intersect the curve in one and only one point. 276 THE GENERAL POWER FUNCTION [Art. 163 This remark suggests the following theorem which we shall not attempt to prove except in this intuitive fashion. III. If y is any positive number^ there exists one and only one real number x^ such that a^ = y. We observe further, if we draw all of the graphs on the same sheet and referred to the same axes, that they all pass through the point a;= 0, ^ = 1. This is due to the follow- ing fact (see Art. 157) : IV. For any base «, we have a^ = 1. It is clear, botli from the graphs and otherwise, that a' grows beyond bound when x grows beyond bound in the positive direction ; but that a^ approaches zero as a limit when X grows beyond bound in the negative direction, pro- vided that the base a is greater than unity. These facts may be summarized as follows : V. If a y 1, a^ increases as x increases^ and lim a^= +od. VI. If a > 1, a"" decreases as x decreases^ and lim a^= 0. The following properties are not quite so evident from the graph, but follow immediately from the definition of a^ and the index laws, VII. a^ • ay = a""*"^, the addition formula for the exponential function. VIII. — = «^~'', the subtraction formula for the exponential a" function. IX. (a^)" = a^^, the multiplication formula for the exponen- tial fuyiction. The division formula (a-^)i''^ = a-^''-" may be regarded as being contained in IX, since y in IX may be a fraction, and need not be listed separately. Similarly VIII may be regarded as a consequence of VII, since y in VII may be positive or negative. However, VIII has so many important applications as to justify an explicit state- ment. Art. 164] DEFINITION OF LOGARITHM 277 X. If the numbers^ .r^, x,^^ x^, and so on, form an arithmetic progression, the corresponding exponentials a^\ a""", a'3, and so on, form a geometric progression. This follows from VII and the definitions of arithmetic and geometric progressions (Arts. 56 and 59). As a consequence of X, the theory of geometric progressions may be connected with the following question : what values does an exponential function ar' assume when x assumes in succession the values 0, 1, 2, 3, and so on? In just this way the theory of arithmetic progressions was connected with the question : what are the values of the linear function clx + a for X = 0, 1, 2, 8, and so on? (See Art. 56.) EXERCISE LXXV Simplify the following expressions : 1. 10^ . 1..- .. 3. (-_^^j . 5. (4^ . s^y. 7. [-^-^) . 2_ io--io"-\ 4. (■■^±3:^Y. 6. (4x' . 8x')i.' 8. :^E2/^. 10-' \ 9-3^ / ^ ^ 25^ 164. Definition of logarithm. If rt^ is a positive base differ- ent from unity wq know, according to Theorem III, Art. 163, that there exists a real exponent x such that (1) a-=y, where y is any given positive number. This exponent x is called the logarithm of y with respect to the base a, a relation which is expressed in symbols as follows : (2) .r = log„y. Thus, the logarithm of a positive ^lumber y, ivith respect to a given base a, is the exponent of the poiver to -which the base a must be raised in order to obtain the number y. It should be noted that we have defined the logarithms of posifive numbers only. The question whether negative numbers have any loga- rithms need not be discussed here. In the theory of functions, however. logaritlimsof negative numbers are actually defined ; but these logarithms of negative numbers are imaginary. 278 THE GENERAL POWER FUNCTION [Art. 165 EXERCISE LXXVI 1. Express the contents of the equation 5^ = 125 in the language of logarithms. Solution. This equation states that the base 5 must be raised to the 3d power in order to produce 125. According to the definition of a log- arithm, we have therefore log, 125 = 3. 2. What are the logarithms of 2, 4, 8, 16, 32, 64, 128 with respect to the base 2 ? Write out each of these results in symbols ; thus log2 4 = 2. 3. What are the logarithms, 3, 9, 27, 81, 243 with respect to the base 3? 4. What are the logarithms of 10, 100, 1000, 10,000, with respect to the base 10 ? 5. What are the logarithms of 3, 9, 27, 81, 243 with respect to the base 27? 6. What are the logarithms of 1, \, \, ^V' sT) 2*3 with respect to the base 3 ? 7. What are the values of 2^, 3*, 4*, 10^ when x is equal to zero? What, then, is the logarithm of 1 with respect to each of the bases 2, 3, 4, 10? 8. What is the logarithm of 1 with respect to any base a ? 9. What is the logarithm of any number with respect to itself as base? 10. Find, approximately, to two decimal places, the number whose logarithm, with respect to the base 2, is equal to 1.5. 165. Graph of a logarithmic function. The functions (1) y = a^ and x = \og^y represent the same relation between x and y, merely written in a different form, just as is the case with the relations y =z aP' and X = ± V,y- In other words, the two functions «=" and log„?/, which are inverses of each other (see Art. 149), have the same graph. If, however, we prefer to write (2) y = loga X Art. lOf)] PROPERTIES OF LOOARTTHMS 279 SO that the independent variable is denoted by x, as we are in the habit of doing, the graph of (2) will be the same as that of the equivalent relation (3) X = a^. But this latter graph may be obtained from the graph of //= a"" by the process of reflection described in Art. 149. EXERCISE LXXVII Draw the graphs of the following functions: 1. ?/ = logo X. 5. 1/ = logio X. 2. y = logg X. e. >j = log^- X. 3- y = log4 ^- 7. y = log^ X. 4- y = logs X- 8. y = logi X. 3 166. Properties of logarithms. The properties of loga- rithms follow at once from those of the exponential function, and from inspection of the graphs. The most important ones are as follows: I. If a is a jjositive number different from unity, the function log^x is defined for all 2)ositive values of x, butnotfor x = nor for negative values of x. It is a continuous function for all positive values ofx. II. If the base a is greater than unity, we have log^x < for < a; < 1, log^x > for x > 1. III. The logarithm of unity with respect to any base is zero^ that is log^ 1 = 0, since a^ =1. IV. The logarithm of any number with respect to itself as base is unity, that is, log a a = 1, since a^ = a. V. Jf a > 1, then lim log^x = + oo . 280 THE GENERAL POWER FUNCTION [Art. 166 This is meielj' a re-statement of V, Art. 16o. For if y = logaX, we have a« = x and, according to V, Art. 163, x will become infinite as y grows beyond bound. This property is sometimes expressed by the symbolic equation logo CO = CO . VI. 7^ a > 1, then Urn logaX= — co . Thus log^x is not continuous in the neighborhood of .r = 0, This is merely a re-statement of VI, Art. 163. For, if // = logo a;, we have ay = X and, according to YI, Art. 163, x will approach zero as its limit when y grows beyond bound through a sequence of negative num- bers. This property is sometimes expressed symbolically as follows : logo = - CO . VII. The logarithm of a product is equal to the sum of the logarithms of the factors. Proof. Let M and N be two jjositive numbers, and let x and y be their logarithms, so that X = logo M, y = loga N, or M = a^, N = nv. We then have MN = (fay = a^+y, (Theorem VII, Art. 163) or, making use of the definition of logarithms, logo {MN) = z + ?/ = logo M + logo iV, and this equation proves the theorem. VIII. The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithyn of the divisor. Proof. Using the same notations as in the proof of VII, we find and therefore ^ = c^-y, N logo^ =x- y = logo M - log N, which was to be proved. IX. The logarithm of the jo"' power of a number M is ob- tained by mnltiplging the logarithm of M by p. Art. 107] COMMON LOGARITHMS 281 Proof. If x = logo M, we liave M = a", and Mp = (a^)p = a^* accord- ing to IX, Art. IGo. Therefore loga .1/'' = px = p logo J/. X. The logarithm of the n"" root of a number Mis obtained hi/ dividing the logarithm of M by n. Proof. Tliis theorem follows from IX by putting/) — -• n XI. If the numbers :?-j, .z^, .jg, etc^ are in geometric progres- sion^ their logarithms will be in aritlimetric progression. Pi:ooK. This follows at once from VII, if we make use of the defini- tion of an arithmetic and a geometric progression. Compare this theorem with theorem X of Art. 163 and with the fundamental property given in Art. 160 of the power function. EXERCISE LXXVIII 1. If logio^ = 0.3010, log,o3 = 0.4771, and log,o 5 = 0.6990; find log,o 12, logio(i), logio(V-), log,o and log^r/, the following quantities : l0g„(;/Y), l0ga(^^'^), l0g„^/^J, log^aV^T- 3. Prove the equation X + y/x^ - 1 loga ^ " = 2 log„ (X + Vx-i - 1). 4. Prove the following statement : In order to be able to calculate the logarithm of any integer, it suffices to know the logarithms of all prime numbers. 5. What functions are those which have the following property? If the argument x takes on a sequence of values which are in arithmetic [>rogres8ion, the corre.sponding values of the function will also be in arithmetic progression. 167. Common logarithms. With scarcely an exception, the civilized nations ol' all times have made use of the deci- mal system for expressing numbers, both in the spoken and in the written language.* For this reason, the number 10 is especially well adapted to serve as base for a system of loga- * It is usually admitted that the prednminance of the decimal system over all others is due to the fact that the normal hmnaii btiiii; has ten finders. Tliis opinion has certainly been generally held since the time of Aristotle. 282 THE GENERAL POWER FUNCTION [Art. 168, 169 rithms. Logarithms with respect to the base 10 are usually known as common logarithms ; they are also sometimes called Briggsian logarithms, in honor of Henry Briggs* (1556- 1630), who constructed the first table of common logarithms. For purposes of numerical calculation, common logarithms are by far the most convenient. We reproduce in tlie ap- pendix a four-place table of common logarithms, whose use we shall now explain. Articles 168-177 may be omitted by students who have studied trigo- nometry, or postponed until they take up ti'igonometry. 168. Characteristic and mantissa. The positive integral powers of 10, sucii as 10, 100, 1000, etc., the negative integral jaowers of 10, such as 0.1, 0.01, 0, 0.001, etc., and the zero power of 10, which is equal to 1, are the only numbers whose common logarithms are integers. The loga- rithms of all other numbers have an integral and a fractional part. The fractional part of the logarithm is called the mantissa, ivhile the integral pa7-t of the logarithm is known as its characteristic. 169. Properties of the mantissa. We consider the mantissa and the characteristic separately because, in practice, the method of finding the characteristic of a logarithm is entirely different from that employed for finding its mantissa. The reason for this will appear from the following discussion. Let us grant that we have found out in some way (1) log 1.7783 = 0.2500. From the theorem about the logarithm of a product, we conclude log 17.783 = log (1.7783 x 10)= log 1.7783 + log 10 = 0.2500 + 1 = 1.2500, log 177.83 = log (1.7783 x 100) = log 1.7783 + log 100 = 0.2500 + 2 = 2.2500, We observe that the numbers 1.7783, 17.783, 177.83, etc., contain the same succession of digits, and differ from each other only in the position of the decimal point. Their logarithms, on the other hand, whose values we have just calculated, differ from each other only in the value of the characteristic. * Briggs was the first Savilian Professor of geometry at Oxford. According to Ball (see Ball's Primer of the Historij of Mathematics). Briggs was also the first to make systematic use of the decimal notation iu working with fractions. Art. 170] DETERMINATION OF THE CHARACTERISTIC 283 Again, if we make use of the theorem about the logarithm of a quo- tient, we find from (1) log 0.17783 = logi—^ = 0.2500 - 1, log 0.017783 = log ^-^^ = 0-2500 - 2, Now, the negative quantities, whicli appear in the right members of these equations, are not written in the form wliich we ordinarily use for negative quantities. Thus, for instance, we have found the value of log 0.017783 to be 0.2500 — 2, a result which we should ordinai-ily write in the form — 1.7500 to which it is obviously equal. If we agree to write every negative logarithm in this unusual form, as a difference between a positive proper fraction and an integer, thus making its frac- tional part positive, we gain the advantage that the mantissa will be the same for any two numbers which contain the same succession of digits, even if none of these digits appear to the left of the decimal point. We avoid, in this way, the necessity of using two different tables of mantissas, one for numbers greater than unity and one for num- bers less than unity. Let us recapitulate the result of our discussion in two formal statements. I. We agree to express the logarithm of any positive number N in such a form that its mantissa shall be positive. This can be done w'hether log N is positive or negative, that is, whether N be greater or less than unity. In the latter case, the nega- tiveness of log iV is brought about entirely by means of the negative characteristic. As a consequence of this agreement, the following statement will be true in all cases. II. If two numbers contain the same succession of digits, that is, if they differ only in the position of the decimal point, their logarithms will hare the same mantissa and icill differ only in the value of the characteristic. It is for this reason that the tables give only the mantissas of the logarithms and that, in looking up the mantissas, we pay no attention to the position of the decimal point in the given number. 170. Determination of the characteristic. The characteristic of a logarithm is easily determined by inspection. Its value depends merely on the position of the decimal point. Since we have 10"= 1, 101 _ 10, 102 ^ 100, 103 = 1000, etc., or log 1=0, log 10 = 1, log 100 = 2, log 1000 = 3, etc., we draw the followino- conclusions : 284 THE GENERAL POWER FUNCTION [Akt. 170 If 1 < .V< 10, then < log iV< 1. .-. log N has the characteristic 0. If 10riomikeh, the seven-place tables of Vega, the eight-place tables of Bauschingek. 290 THE GENERAL POWER FUNCTION [Art. 177 Fig. G3. We mark the points on our line-segment whose distances from the left- hand end, measured in terms of the whole line as unit, are in order equal to log 2, log 3, log 4, ••• log 9, and label them 2, o, 4, ••• 9, respec- tively. If the whole line-seg- 1_ I ? f ? ^ ] ? ? y ment is 10 centimeters long, these points will, on account of (1), be at distances 30, 48, 60, 70, 78, 85, 90, 95 millimeters, respectively, from the left-hand end of the line-segment (see Fig. 63). A scale constructed in this way is called a logarithmic scale, and its usefulness for purposes of calulation was first pointed out by Edmund GuNTER* in 1620. It enables us to find a line-segment equal in length to the logarithm of any number between 1 and 10. It is easy to see how, by means of such a scale and a pair of dividers, multiplication and division may be reduced to the simple graphical processes of adding and subtracting line-segments. 177. The slide rule. Some years before 1630, William OuGHTRED f noticed that the use of .the dividers might be avoided by constructing two equal logarithmic scales (Scales A and B of Fig. 64) capable of sliding by each other, as indicated in the figure.^ The use of this simple bit of apparatus for the purpose of multiplica- tion and division will be apparent from the following examples : To multiply 2 by 3. Place scale B in such a way that its left-hand index (i.e. the division marked 1) falls directly under the division marked 2 on scale A. Directly above the division marked 3 on scale B, we shall find, on scale A , the product which (of course) is 6. To justify this pi-ocess it suffices to note that it is equivalent to adding the loga- rithm of 3 to that of 2. A 1 1 r •i 4 1 1 5 1 6 1 7 8 1 9 10 1 1 1 1 1 1 3 4 5 6 7 8 1 1 9 10 B Fig. (54. Figure 64 shows scales A and B in the proper position for the pur- poses of this example. To divide 6 by 3. Under the division 6 of scale A, place division 3 of * Professor of a.stronomy in Gresham College, London (1581-1626). t OuGHTRED (1,")7.5-1()()0) was a fellow of King's College, Cambridge. t Oughtrcd's instruments were described iu publications of William Foster, one of his pupils, in 1632 and 1633. Art. 177] THE SLIDE RULE 291 scale B. .Over tlie division 1 of scale ]j we shall find the quotient (|= 2) on scale .1 (cf. Fig. (!l). The instrument actually in use, the Mannheim slide rule, is a slight amplification of the one just described (cf. Fig. 65). It has four scales, usually denoted by A,B, C, D, respectively, the scales A and D being on the rule, and B and C on the slide. Fk;. (W. The scale A is composed of two logarithmic scales such as that of Fig. 63, so that its right-hand end might be labeled 100, since log 100 = 2. On most slide rules, however, the first principal division on scale A after 9 is not labeled 10, as in Fig. 63, but 1, the next one is not labeled 20, but 2, and so on to the last one, which is again labeled 1 instead of 100 or 10. Thus, the two halves of scale A are exact copies of each other. This is done for precisely the same reason that the mantissas only are printed in our tables of logarithms. The slide rule also makes use of the mantissas only. The characteristics, or what amounts to the same thing, the position of the decimal point in the result, must be obtained by inspection or by special rules. Scale B is on the upper edge of the slide, in direct contact with scale A on the rule, and is an exact copy of scale A. These two scales together may be used for multiplication and division as explained above. Scale D is on the lower part of the rule. It is a single logarithmic scale, from 1 to 10, of the same length as the combined two scales of A. The logarithm of any number is therefore represented, on scale D, by a distance twice as great as that which represents the logarithm of the same number on scale A. It follows from this that the number which is found on scale A, vertically above any number of scale D, is the square of the latter. Any number on scale D, on the other hand, is the square root of the number vertically above it on scale A . Scale C is on the lower edge of the slide, in direct contact with slide D on the rule. It is an exact copy of scale D. These two scales to- gether may be used for multiplication and division, according to the same rules which hold for scales A and B. Besides these four scales, the slide rule is sxipplied with a runner (cf. Fig. 65), which is useful in performing compound operations, and also in comparing two scales (such as A and D), which are not in direct 292 THE GENERAL POWER FUNCTION [Art. 178 contact with each other. The runner was made a permanent feature of the slide rule by Mannheim in 1851.* It often happens, in manipulating the slide rule, that the result is to be sought opposite a number of the slide which falls outside of the scale on the rule. In such cases, we may shift the slide, bringing the right- hand index to the place which the left-hand index occupied previously, and read off the result as before. For such a shift has no influence on the mantissa, since it merely amounts to dividing the result by 10. On the Mannheim rule, this shifting of the slide may be avoided by work- ing with scales A and B rather tlian with C and D. Scales C and D, however, have the advantage of greater accuracy. If the slide be withdrawn entirely, it will be found to have three other scales on its reverse side, two of which are labelled S and T. These are scales of logarithmic sines and tangents, respectively, and may be used for calculating such products as c sin A , c tan A . The middle scale on tlie reverse side is used for finding the value of the logarithm of a number, and is important if we wish to compute a power of a number with a complicated fractional exponent. For more complete information concerning the slide rule, we must refer to the manuals which are usually presented to the purchaser of such an instrument.! Cheap slide rules, especially constructed for the beginnei-, may now be obtained of all dealers under the name Student's or College Slide Rule. Engineers and coniputers use the slide rule so extensively that the student will find it advisable to make himself familiar with the instrument by actual use. The Mannheim slide rule, which we have described, admits of three- figure accuracy. In some (exceptional) cases, results correct to four decimal places may be obtained by its use. The Thacher and Fuller slide rules, more complicated instruments, but constructed on essentially the same principles, admit of far greater accui-acy. 178. The general notion of a scale. The logarithmic scale, which is used in the slide rule, and the familiar scale of inches on a yard- stick are two special instances of the general notion of a scale. In both of these cases the scales are straight. One of them, the scale of inches, is also uniform, that is, the divisions of the scale are numbered in such a way that the points labeled 1, 2, 3, 4, etc., are at equal distances from each other. The logarithmic scale is straight but not uniform. The * Amkdke Mannheim (ISiU-lPOf)), a distinguished geometer of recent times. The nnuier \v,u\ Iiowever been used occiisionally, long before Manuheim, by a iiumbei' of English mathematicians. t See also Raymond's Plane Surveying. Art. 179] LOGARITHMS OF DIFFP^RENT SYSTEMS 293 scale of degrees on a graduated circle is iiiiifonn but not straight. The scale of hours on a sun-dial is neither uniform nor straight. These illustrations will suffice to explain, even without a formal def- inition, what is meant by a scale in general. The essential cliaracterisdc of a scale is that it establishes a one-to-one correspondence between the points of a straight or curved line on the one hand and the numbers of a certain set on the other hand. Moreover, this correspondence is such that if a point describes the line continuously in a certain direction w ith- out ever going backward, the corresponding numbers will grow continu- ously from the lowest number of the set to the highest. Such scales, both straight and curved, both uniform and non-uniform, are exceedingly useful for purposes of measurement and graphical com- putation and are used extensively in practice. 179. Relation between the logarithms of two different sys- tems. Let a; be a positive number. If we choose a positive number a as base, let (1) \o^^x = p. If we clioose a second positive number h as base, let (2) log6 x=q. We wish to investigate the relation which exists between the logarithms of the same number x taken with respect to the two different bases, a and h. From (1) and (2) we have, by the definition of logarithms, (3) a; = aP, x = b'' and therefore which, on account of the division formula for the exponen- tial function (see IX, Art. 163), may be written in either of the two forms a'' = h or a = b^. By the definition of logarithms we have therefore logg b = ' and logj, a = i, q ' p 294 THE GENERAL POWER FUNCTION [Art. 179 whence, by multiplication, (4) logj, a . log„ b = l. This result may be expressed as follows. Theorem I. If a and h are two positive numbers, the loga- rithm of a un'th respect to the hase h is the reciprocal of the logarithm of h with respect to the hase a. Let us now take the logarithm, with respect to the base J, of both members of the first equation of (3). We find (see Art. 166, Theorem IX) \og^x=p\ogi,a. But, according to (1), p is equal to log„a^. Consequently, we find (5) logft X = log„ X • logs a. On account of (4) this may be written, more conveniently, as follows : Theorem II. Equation (6) enables us to compute the logarithm of x with respect to the neiv base b, when the loga- rithm of X and the logarithm of b with respect to the old base a are known. Suppose we actually have at our disposal a table of com- mon logarithms. Then we know log^Q x for all values of rr, and we may also find from the table the value of logjg 5, where b is any positive number. Theorem III. Consequently the formula enables us to construct a table of logarithms with respect to any base b. For this purpose it is only necessary to divide all of * This means los^a x divided by loga b, and not loga \,\< which would be equal to logo a; — logo &. Art. 180] STANDARD LOGARITHMIC CURVE 295 the logarithms of the common system hy one and the same divi- sor, namely by the common logarithm of the new base. More generally, formula (6) tells us the following : Theorem IV. If the logarithms of all numbers are known with respect to some particular base a, the logarithms of all nu7nbers ivith respect to any other base b may be found by di- viding all of the logarithms of the first system by one and the same divisor, namely by the logarithm of the second base tvith respect to the first. It is clear, then, that the knowledge of one system of logarithms is sufficient to give us complete information about all other systems of logarithms. This becomes even more evident if we express our last theorem geometrically, as follows : Theorem V. If the graph of the function log^x has been constructed, the graph of any other logarithmic function, log^ x, may be obtained from it, by either diminishing or increasing all of the ordinates of the first graph in the same ratio. This is so, because division by log^ b will decrease all ordi- nates in the same ratio if log,, Z> > 1. It will increase all ordinates of the graph in a fixed ratio if log„ ^' < 1 . EXERCISE LXXX 1. Making use of the table in the Appendix, compute the logarithms of 2, 3, 4, 5, 6, 7, 8, 9, 10 with respect to the base 5. How would you proceed if you wished to construct a four-place table of logarithms with respect to the base 5 ? 2. Draw the graph of y = log.,x. What nuist you do to this graph iu order to obtain the graphs oi y = log^ x, ij = logg x, y = log^- x? 3. Construct a scale for tlie function x\ for values of x between and 1, making use of the computed values of x- for x — 0.0, 0.1, 0.2, ••• 0.9, 1.0. 180. Selection of a standard logarithmic curve. Whatever may be the base a, the curve obtained as a graph of y = loga X 296 THE GENERAL POWER FUNCTION [Art. 181 will always pass through the point x = 1, ?/ = 0, since log„l = for all bases. (See Fig. Q6.^ But although these curves all have this point in common, each of them will have a different tangent at this point. We shall select as a stand- ard logarithmic curve that one whose tangent at the common point makes an angle of 45° with tlie a;-axis, on account Fig. 60. ^ ' of tlie central cliaracter of. this curve as compared with all of the others. The question which we sliall have to settle is this : what is the base of that par- ticular system of logarithms whose graph is the standard logarithmic curve ? We shall denote tliis base by e and speak of the corresponding logarithms as natural logarithms. 181. The derivative of the logarithmic function. In order to be able to answer the question raised in Art. 180, we must first show how to compute the slope of the tangent at any point of a logarithmic curve. The same argument which was used in Art. 87, and which was there applied only to integral rational functions, gives us the following result : Let f (x^ he a continuous function, and let us construct its graph hy putti7ig y=f(^)- If this curve has a tangent at that one of its points whose coordinates are x and y, the slope of the tangent will he ohtaitted hy evaluating the limit : (1) /' (,0 = lim /(^ + /0-/(x) , A->o h This limit is called the derivative off(x'). In our case we have /(a;)= log^z, and therefore f(x-\-h)=]oQ;,(x+h). Art. 181] DERIVATIVE OF LOGARITHM 297 Consequently ^2) f(^ + h~)-fix) ^ \og„(ix + h)- log, a; ^ 1 ,^^^,^ f x + h \ h h h '"""V X J where the last two members are equal ou account of VIII, Art. 166. We wish to evaluate the limit which (2) ap[)roaches when X has any definite fixed value while h approaches zero in any manner whatever. Since the function log^x is not con- tinuous for x = (see Art. 166, Theorem VI), we shall, on this account, assume that the fixed v^lue assigned to x is not zero. We first rewrite (2) as follows : (3) /(2L±^l^££) = l,og,(i+^^). Since, in evaluating this limit, x is to be I'egarded as a fixed number different from zero, we may introduce (4) ^ = t X as a new variable in place of h. As 7i approaches zero, t will also approach zero. But from (4) we find h = xt and (3) now becomes h X t ' or, on account of IX, Art. 166, /(.+A)-/(.) ^i ^^,)j. k X If now we make use of (1), we find the expression (5) /(a:) = ^limlog„(l + 0' X /->o for the derivative of the function log^ x. (6) J\x') = -\og^e X 298 THE GENERAL POWER FUNCTION [Art. 181 1 Let us assume provisionally that (1 + ^ ' actually ap- proaches a definite finite limit, different from zero, when t approaches zero in any manner whatever, and let us denote this limit by the letter e. Since the function log^a; is con- tinuous in the neighborhood of any finite positive value x =p (see Art. 166, Theorem I), we may write lim logo a; = logojt). (See Art. 96.) We shall^ therefore, Jifid from (5) the following expression (6) for the derivative of or, what amounts to the same thing, for the slope of the tangent of the curve y — log„ x at that one of its points which has the abscissa x. Let us apply this theorem to that point of the curve for which x=l and therefore y = 0.* We find the following result : The slope of the line which is tangent to the graph of logo a; *^ ^^^ point a; = 1, y = 0, is equal to log^ e where 1 (7) e = lim (1 + 0^ In order that this tangent may make an angle of 45° with the a;-axis, its slope must be equal to unity. Therefore in this case log„e must be equal to 1, and this is so if and only if a = e. Consequently, the standard logarithmic curve, ivhose tangent at the point a; = 1, ?/ = makes an angle of 45° with the x-axis, corresponds to that system of logarithms whose base is the number 1 e = lim (1 + ty. The number e is called the natural or Napierian base, and the corresponding system of logarithms is called the system * This is the point common to all logarithmic curves. (See Art. 180.) Art. 182] THE NUMERICAL VALUE OF e 299 of natural logarithms. The notation In x is sometimes used for the natural logarithm of x. 182. The numerical value of e. It is a rather difficult matter, and quite beyond the scope of this book, to prove that 1 (1 + ty actually approaches a definite finite limit when t ap- proaches zero. To prove this it is necessary to show that 1 (1 + t)'- will approach the same limit when t approaches zero throuoli a sequence of positive values as when t approaches zero through negative values, that the limit will be the same whether t approaches zero continuously or by a series of jumps. If, however, we grant the existence of the limit we can easily gain a very fair idea as to what the value of it will be, by allowing t to approach zero in some particularly convenient fashion. Let us then think of t as assuming in succession the values .111 1 ' 2' 8' 4' "■ n "" We shall then have (1) e = lim (1 + 0' = lim f 1 + - Y • We actually find (1 + 1)1 = 2, (1 + . 1)2 =2.25, etc. By using a five-place table of logarithms we easily find (to four significant figures) (1 + ^)10 = 2.594, (l + 3io)ioo = 2.704. The true value of e to eight decimal places is (2) g = 2.71828123. We may also obtain this value as follows. Let us expand f 1 + - j by means of the binomial theorem. (See Art. 88.) We find 300 THE GENERAL POWER FUNCTION [Art. 182 /-J , 1Y = 1 I y^l ■ n{n-l) 1 n{n-l)(n — 2} 1 \ 7iJ In 1 • 2 w^ 1 • 2 • 3 n^ n(n-l)(n-2)(n-d) 1 1.2.3.4 71^ 1-1 (i-^Yi-?) = 1 + 1+1.-' ■ l.:i.8 l_lYl-2\,i_ + rnnri — -+ 12 3 As w grows beyond bound, -, -, -, etc. all approach zero, n n n and it seems })lausible that we should find limfl+-Y = l + l+ — + ~ + + ••• n-^K nj 1.21.2.31.2.3.4 where the law, according to which the terms on the right- hand side are formed, is evident. Thus we find the follow- ing formula for e, (3) e=l + l + -l^ + 1 + 1 — ^+ .... ^^ 11 .2 1.2.3 1 .2-3.4 We have not actually proved this formula. For as w 12 8 grows beyond bound some of the fractions -, — , -, etc. will 71 71 n have numerators which also grow beyond bound, and the number of factors which occur in sucli i)roducts as (1 )(1 — -)(1 — -] will also grow beyond bound. Con- V nj\ nj\ n) _ ^ / IV sequently it is not at all certain tliat ( 1 H — j actually ap- proaches the right member of (3) as a limit. It is possible to prove, however, that this is actually the case. Art. is;}] EXPONENTIAL EQUATIONS 301 Formula (3) enables us to compute the value of e with great rapidity to as many decimal places as may be desired. Thus, we have (to five decimal places) 1 + 1^ = 2.00000 ^^=0.50000 ^ — 1 fififiT 1.2.3~ ^^ ^ _ oilfiY 1 .2.3.4- "-"^^^^ 0099^ __. . ._ — W.uUOOO 1.2-3.4.5 1 _ A AA1 on r .2.3.4.5.6-"-"^^'^^ 1 _ ooo^n 1-2 .3.4.5.0.7 ^ — nono'^ 1.2.3 .4.5. 6.7. s-"-"^^^"^ 2.71829 a result which agrees with (2) to within one unit of the fifth decimal place. 183. Exponential equations. An equation, some or all of whose terms are exponential functions of the unknown quan- tity, is called an exponential equation. We have no general method for solving exponential equations, but the method of trial and error explained in Art. 99 will usually enable us to find approximate values for the unknown quantity. Tliere are two cases, however, in which the problem may be reduced to that of algebraic equations. Case I. The equation is of the form where a, 5, c, .••, /, m, n, ••• are constants, and where the ex- ponents /(x), g{x), 7i(2-), ..-, /r(.r) logiy m -h x(x) It'Sio « + ••• 302 THE GENERAL POWER FUNCTION [Art. 183 and this is a rational equation, which may be solved by the methods of Chapter VI, Art. 145. This method consists in reducing (2) to an equivalent integral rational equation, which may then be solved by the methods of Chapters II to V. Thus, the equation 23x521-1 _ 45131+1 becomes, if we take the logarithms of both members, 3 x log 2 + (2 X - 1) log5 -Tjx log 4 + (x + 1) log 3 where common logarithms are meant. This may be written (3 log 2 + 2 log 5 — 5 log 4 — log 3)x — log 3 + log 5, whence ^ ^ log 3 + log 5 ^ 3 log 2 + 2 log 5 — 5 log 4 — log 3 ' or X log3 + log5 ^ 0.4771 + 0.6990 ^ _ ^ gg^g 2 log5 - 7 log 2 - log 3 1.3980 - 2.1070 - 0.4771 Case II. The equation is of the form (3) Aa"-f''-^ + ^a'"-i'-^^-^> + (7a("-2'/('> + ••• + La-^^-''^ + M= 0. where f(^x^ is a rational function of x. In this case we put (4) «/(-) = ^, thus reducing (3) to an algebraic equation for ^, namely (5) A2/" + Btr-' + -' + Li/ + M=0. Lety^ be one of the n roots of (5). Then we find from (4), taking logarithms of both members, /(a:) log a = log .Vi- and this is an algebraic equation for x. Thus, the equation, in which c denotes a given number, (6) 1(6=^ + e-»^) = c, may be reduced to the form (3) by multiplying both members by 2 e". It then becomes e2i + 1 = 2 e'c or ga» _ 2 e*c + 1 = 0. Art. 184] CALCULATION OF LOGARITHMS 803 If we put (7) e* = /y or x= logey, we find a quadratic equation for ij, namely which gives, when solved for y, y = c ± Vc^ - 1, and therefore, on account of (7), X = log, ((• ± \/c2 - 1). EXERCISE LXXXI Solve the following equations. Give the numerical results correct to four significant figures. 1. 2' = 64. 3. 4»='-2* = 64. 5. S'^-^^+i = \. 2. 2'-* = 5. 4. .5»'-8* = 2^. 6. 2^ = 14. 7. l;r^+5 = 14^+7. 10. a 3'^ + /; 3^ + c = 0. 8. log Va; — 21 + ^ log x = 1. 11. a ;> -\- J> b-^ -{■ c — 0. 9. K2^ - 2-*)= f- 12. Aa^ + Ba' + C = 0. 184. The calculation of a table of logarithms. We ap- proached the theory of logarithms by way of the index laws. But this was not the path pursued by the first inventors of logarithms, John Napier (1550-1617) and Jobst Burgi (1552-1632).* In fact, the notation x" was not in use in their time and consequently the index laws were not avail- able to them, although in a certain sense they were probably well known even then.f Both Napier and Biirgi observed that the numerical opera- tions involved in multiplication are much more burdensome than those required in addition. They, therefore, sought a * Napier was of Scotch and Biirgi of Swiss nationality. Biirsi's discovery of logarithms was unquestionably independent of Napier's and was made at about the same time. But Napier's book Miriftci Lof/aritkmonmi canonis descriptio, containing an account of his method, was published in 1614, six years earlier than Biirgi's Arithmetisdte and Grometrische Pror/ress-Tahulen. For an account of the history of logarithms consult Cajori in the Aiaerican Mathematical Monthly, Vol. 20 (19i;i). t David Eugknk Smith, Tht: Law of Exponents in the Works of the Sixteenth Centunj. Napier Tercentenary Memorial Volume. Royal Society of Edinburgh. 1915. 304 THE GENERAL POWER FUNCTION [Art. 184 method of reducing multiplication to addition, and both of them accomplished this purpose independently by the follow- ing scheme : Let \i-) d-^i 0-c^i (l^>, d^i •••, (In-, •" be a sequence of numbers in geometric progression, and let (2) 5^, 62, ^3, ?>4, ..., J„, ... be a second sequence of numbers which are in arithmetic progression. Moreover, let us think of these two sequences as being in correspondence, so that to a^ corresponds 6„. Then to the product of two numbers «i and a;^ of the first sequence will correspond the sum of the corresponding two numbers hi and b/^ of the second sequence. Consequently, if we actually have two such sequences worked out we may multiply a^ by % as follows. Find the numbers h^ and 5^ of the second sequence which correspond to a,- and % respec- tively, add bi and J^., and then find the number of the first sequence which corresponds to b^ -\- b/.. This will be the product of tti and a^.-. Of course this scheme is for us merely an application of Theorem XI of Art. 166. In fact, the numbers Jj, b^-, ••• will be the logaritlims of tfj, a^-, "• with respect to some base. But for Napier and Biirgi, tliis correspondence between the terms of a geometric and an arithmetic progression was not a theorem, but served as a definition for logarithms. It is easy enough to find sucli correspondences. Thus, the numbers (3) 1, 10, 100, 1000, 10,000, ... form a geometric progression, anil the numbers (4) 0,1,2,3,4,... form an arithmetic progression of the kind described in (1) and (2). But in order that this correspondence may actually be useful for purj)oses of calculation, the terms of each of the two progressions should be much closer together Akt. 184] CALCULATION OF LOGARITHMS 305 than they are in the two progressions (3) and (4). We therefore insert any convenient number of geometric means between any two terms of (3), and just as many arithmetic means between the corresponding two terms of (4). The easiest way to do this is to insert one mean of each kind at a time, since tliis may be accomplisluMl l)y merely extracting a square root. Thus, tlie geometric mean between tlie first two terms of (3) is V10= 3.1623. (See Art. 61.) The arithmetic mean between the first two terms of (4) is 1.5. (See Art. 57.) Our two progressions now read as follows : (5) 1, 3.1623, 10, 31.623, 100, 316.23, 1000, (^!) 0, 0.5, 1, 1.5, 2 2.5, 3, In order to have the terms of the progression still closer to each other, we use the geometric mean between the first two terms of (5) namely V3.162o = 1.7783, and the arith- metic mean between the first two terms of (6) namely 0.25. Our two progressions now become (7) 1, 1.7783, 3.1623, 5.6234, 10, ... and (8) 0, 0.2500, 0.5000, 0.7500, 1, .... In our notation this signifies that log 1.7783 = 0.2500, log 3.1623 = 0.5000, etc. It is clear how, by continuing this process, a table of logarithms will result. To be sure, this will not yet be in convenient form, since the numbers are arranged in geomet- ric progression instead of being spaced equally. But from the logarithms obtained in this way, the logarithms of inter- mediate numbers may afterward be obtained by interpola- tion, thus finally enabling us to find the logarithms of the equally spaced numbers 1.0, 1.1, 1.2, 1.3, etc. There are other, far more convenient, methods for calculat- ing a table of logarithms, and some of these will be explained 306 THE GENERAL POWER FUNCTION [Arts. 185-187 later on, although a complete proof of these other methods is beyond the scope of this book. 185. Applications of logarithms. In all extensive numeri- cal calculations which involve multiplication or division, the introduction of logarithms is advisable. Trigonometry is full of illustrations of this remark. Another field in which logarithmic calculation is almost indispensable is offered by the problems of compound interest, some of which we shall now discuss. 186. Simple interest. When a capitalist lends out money, he usuall}^ charges the borrower a fee which is called interest. The sum loaned is called the principal ; the princi- pal plus the interest accrued at the end of any period is called the amount due at that time. The rate is said to be i2% annually if interest is charged at the rate of R cents per year for every dollar of the principal. If f P is the principal, and R the rate, the interest at the PR end of one year will be ^- At the same rate, the interest nPR at the end of n years will be $ , and the amount due at ^ 100 the end of n years will be (1) ^=^-+^=^(1 + In what follows we shall use the letter r to stand for ^/lOO. Then (1) assumes a simpler form, namely (2) A^ = P(\ + nr). 187. Compound interest. If the borrower pays his interest annually, the formula just derived gives a correct result for the total amount which he should return to the lender. But if he wishes to pay nothing until the n years have passed, this formula should be modified. For, by retaining the various installments of interest as they become due from year to year, he is depriving the lender not only of the use Art. 187] COMPOUND INTP:REST 307 of the principal, but also of the interest which each installment might have earned for the lender in the meantime. Account is taken of this circumstance in computing compound interest. The interest at 7v% annually on a jirincipal of 8P is PR — — -• Thus the amount at the end of the first year is 100 -^ where again we put ^ R ** lOO' During the second year we regard JLj as the principal. The amount at the end of the second year will be A^ = A^-\- A^r = A^Q. + r). If we substitute for A^ the value just found, this becomes Similarly, during the third year we regard A^ as principal, and Und A,^F(l + ry as the amount due at the end of the third year. Finally we find the formula (1) A^ = P(l + ry. for the amount due at the end of n years on a principal of $P at the rate of R% a year compound i7itere8t, if the interest is compounded annually. Formula ("2) of Art. 18G shows that the amounts due at the end of one, two, three, etc., years form an arithmetic progression at simple interest. Formula (1) of the present article shows that they form a geometric progression when compound interest is charged. Formula (1) may also be used to solve the following prob- lem. What sum must be invested now, allowing compound interest at the rate of R(fo annually, so that the amount at the end of n years shall be a given sum ? Let P be the unknown sum to be invested, and let A^ be 308 THE GENERAL POWER FUNCTION [Art. 188 its amount at the end of n years. Then A^ and P are con- nected by equation (1) and A^ is to be regarded as a given quantity ; consequently we tind (2) P=^„(l+rr" by solving (1) for P ; the value of P is called the present value of a sum A^ which is to be paid at the end of n years, allowing compound interest ?it R %. 188. Annuity. A fixed sum paid annually is called an annuity. Let us compute the amount A of an annuity of •fa per year which is allowed to accumulate for n years with compound interest at 72 % . This problem is important in such cases as the following. A corpora- tion (perhaps a life insurance company) makes a contract with a person X, promising to pay him a certain sum of money at the end of twenty years, in return for certain fixed sums called premiums which are to be paid by X to the corporation annually. The corporation invests its money at R per cent. In order to be able to fulfill its contract, the company must know how to calculate the sum of all the payments made by A' plus the interest on these payments. If the first annual installment a is due at the end of the first year, it earns interest during w — 1 years ; therefore its amount will be a(\ -\- r)""^. The second installment earns interest during w— 2 years; its amount will therefore be a(l -I- r)"~^. The last installment will earn no interest. Thus the total amount will be A = a(l-j-r)"-i-fa(l-|-r)"-2 4. . . . j^ a{\ ■\- r') ^ a^ or if we write the terms in inverse order, J. = a + a(l + r) -F a(l -f- r)2 -t- • • • -f a(l -}- ry-^. The terms in the right member form a geometric progres- sion of n terms, the common ratio being 1 -|- r. Therefore, the sum is (Art. 60) (1) A^a^-^Sl^^a\^(l±ir = 'L^a + rr-^. 1 — (1 + r) —r r akt. i«!)] compound interest 309 Formula (1) enables us to compute the amount A of an annuity of ^ a per year which is allowed to accumulate for n years at R per cent compound interest. It is often necessary to compute the present value of an annuity of '^a ?i year to continue for 71 years, allowing com- pound interest at the rate of R per cent, Tims, a person wishes to pay a corporation noiv a lump sum to be returned to him as an annuity of f 500 per year for ten years. The corporation must know how to compute the lump sum which is adequate for this purpose. The present value of ^a due one year hence is a(\ + r)"^. (See equation (2), Art. 187.) The present value of $a due two years hence is •a(l + r)"^, etc. The present value of $a due 71 years hence is a(l + r)~". Therefore, the total present value of this annuity is P= a{\ +r)-i + a(l +r)-^+ ••• + a(l +r)-". This is a geometric progression of n terms, with the common ratio (1 + r)~i. Tlierefore, we have -(1+^- P = a(l+r)-i[i^ (1 + 0-M 1 — 1/(1 4-r) or 1 + r r/(l + r) r If n grows beyond bound, (1 + r)~" tends towards zero, since 1 + r is greater than unity. Therefore, the present value of a perpetual annuity (called a perpetuity^ of -fa per year is a/r(\ — 0) = a/r. 189. Interest compounded more than once annually. We found in Art. 187 the formula A = P(\ + r)", where r = - — -, ^ lUO for the amount due at the end of n years on a principal P at the rate of R per cent a year, if the interest is compounded annually. If the interest is compounded more than once a year, this formula must be modified. 310 THP: general power function [Art. 189 Let us suppose that the interest is to be compounded t times a year, and let us speak of the f"" part of a year as a period. The interest at the end of the lirst period will be P-; therefore the amount at the end of the first period is (1) A = ^ + i'^ = ^(i + 0. During the second period we regard A^ as the principal. The interest on this principal earned during the second period will be A-^- so that the amount at the end of the second period will be A, = A, + A/- = A,(l + '^ which, on account of (1), gives Since n years contain nt periods, we find finally the formula (2) ^ = p(l + 0" where r=^, for the amount due at the end of n years on a principal P at the rate of R per cent annually., if the interest is compounded t times a year. In this formula we may think of t as growing beyond bound. As a consequence the period will approach zero as a limit, and we find the following result. The formula (3) A = P lim f 1 + ^ will give the amount due at the end of n years on a principal P at the rate of R per cent annually, if tlie interest is com- pounded instantaneously or continuously. But we are in a position to actually evaluate this. limit. Let us put t = kr. Since r is a fixed number, k will then Art. 100] THE COMPOUND INTEREST LAW 311 grow beyond hound as t becomes infinite. We may there- fore write in place of (3), ■\\knr (4) A = P lim 1 + But we have found (see Art. 182), lim('l + ^Y=., x->i' \ kJ so that (4) becomes (5) A = Pe-\ Till s i8 the final formula for the amount due at the end of n years on a principal P at the rate of R per cent annually, if the interest is compounded continuously. We observe that the amount, as given by (5), is an ex- ponential function whose base is e, the exponent being the product of the annual rate and the number of years. EXERCISE LXXXII * 1. Find the amount of f 157.38 for 7 years at '^>\ ^c compound interest. 2. How much money must I put into the bank at 3 % compound in- terest, so that the amount may be $ 500 at the end of five years? 3. What will be the amount of f 10,000 after ten years, at 4^0 com- pound interest, if the interest is compounded annually? if the interest is compounded semiannually? quarterly? 4. What is the value of an annuity of !f 1000 for a term of twenty years, if money is worth 3 % per annum ? 5. How long will it take a sum of money to double itself at 5 9? com- pound interest per year ? 190. The compound interest law. Formula (5) of Art. 189 has many applications besides the one already noted. The essential thing about this formula is that it represents the * In many of these examples four-place tables of logarithms are not sufficient to give results accurate to the nearest cent. In solving such examples, the student should l)e satisfied with an approximate result if he has no other tables at his disposal tlian those given in the Appendix of this book. Otlierwise he should use some of the more extensive tables mentioned in the footnote on page 289. 312 THE GENERAL POWER FUNCTION [Art. 191 amount ^ as a function of w, the number of years (P and r being regarded as constants), and that this function has the special property that its rate of growth at any instant is pro- portional to its own magnitude. For that is the way in which the amount due on a loan grows when the interest is compounded instantaneously. Whenever we have a relation of this kind between two variables x and ?/, such that the rate of change of y is proportional to y itself, then y will be connected with x by means of an equation of the form (1) y = ae'"' where a and b are constants ; that is, y will be an exponential function of x. On account of this connection, the relation between two variables expressed by an equation of the form (1) is frequently called the compound interest Imv^ a name in- troduced by Lord Kelvin (1824-1907), a famous British physicist. The importance of this law lies in the fact that it occurs so frequently in the applications, some of which we shall now explain. In making these applications it should be remembered, however, that 5, the coefficient of x in (1), may be either positive or negative. If a and h are both positive, as in the case of Art. 189, y is an increasing function of X and the constant increase in log y which corresponds to an increase of one unit in x is called the logarithmic increment. If a is positive while h is negative, ?/ is a decreasing function of a;, and the decrease in log y which is caused by a unit in- crease of x is called the logarithmic decrement. 191. Dampened vibrations. If a weight is attached to one end of a string, whose other end is attached to a fixed sup- port, an impetus given to the weight will cause it to oscillate about its position of equilibrium. This simple piece of ap- paratus is called a pendulum. Let the motion of the pendu- lum take place in a vertical plane. The displacement of the pendulum from its position of equilibrium at any instant may be measured by the angle which the string makes at that moment with a vertical line. The larsfest value of this Art. 192] PRESSURE IX THE ATMOSPHERE 313 angle during one comjjlete oscillation is called the amplitude of the oscillation. Owing to friction and resistance of the air, the amplitude of the vibration gradually decreases. Ob- servation shows that the amplitudes of successive vibrations of the pendulum form (very approximately) a decreasing geometric progression. We may therefore write k-l (1) A = A,e- where t represents the time, expressed in seconds or some other unit of time, which has elapsed from the beginning of the motion, where Aq represents the amplitude (expre.ssed in radians or degrees) of the vibration at the beginning of the motion, and where A represents the amplitude after t time units have passed. The coefficient of t in the exponent has been written in the form — k^ to emphasize the fact that it is negative. This must be so to make each amplitude smaller than the preced- ing one. The actual value of this coefficient depends upon the amount of friction and resistance. If k is large, for in- stance if the pendulum is swinging in water, the vibrations will cease very soon. If k is small, it will take a long time before the pendulum comes to rest. Strictly speaking, if formula (1) be regarded as absolutely true, it will always take an infinite time to produce absolute rest. For, accord- ing to (1), A cannot become equal to zero for any finite value of t. But when the continually increasing quantity k-t l)ecoines large enough, the corresponding value of A will become so small as to become immeasur- able. When this condition has been reachetl, the pendulum may be re- garded as being at rest, for practical purposes, since its motion has become imperceptible. Formula (1) is applicable to other cases of dampened vi- bration. The case of a pendulum swinging in air is merely a particular case of dampened vibrations. 192. Variation of density and pressure in the atmosphere. It is a familiar fact that the air is less dense on the top of a mountain than at sea level. It is easy to see why this should be so, since the air at higher levels, by its weight, helps to 314 THE GENERAL POWER FUNCTION [Art. 193 compress the air which is below it. For the same reason the pressure (as measured by a barometer) is less at greater alti- tudes than at sea level. The following formula due to Halley (1656-1742), a famous English astronomer, very nearly represents the facts as obtained by observations at various altitudes. Let p^ be the density of air at sea level (compare Art. 44 for definition of density), and let p be the density of air at an altitude of h meters above sea-level. According to Halley's formula we shall have (1) p = PqB 8000 The pressure of the atmosphere at any height may be measured in pounds per square inch, or may be expressed by means of the barometer in terms of millimeters of mercury. Since the pressure is proportional to the density, we may also write (2) P=Poe~^. where Pq is the pressure at sea level (15 pounds per square inch, corresponding to a height of the barometer of 760 millimeters), and where p is the pressure at a height of h meters above sea level. 193. Transmission of light by imperfectly transparent me- dia. When light passes through a medium like air, water, or glass, some of the light is absorbed, although most of it is transmitted. The amount of light absorbed depends upon the nature of the medium and its thickness. Even glass ceases to be transparent if it is thick enough. Let L be the intensity of the light transmitted by a sheet of glass X millimeters thick. We shall have (1) L = L,e-^'^ where i^ is the intensity of the incident light, that is, the intensity of the light before any absorption has taken place, and where k"^ is a coefficient whose value depends upon the quality of the glass. Art. 194] COOLING BODIES 315 Let us take the intensity of the incident light as unit of intensity. Suppose that a pane of ghiss ;} millimeters thick absorbs 2% of the inci- dent light. Then we shall have Z^, = 1, and L — 0.98 for x = 3, so that 0.98 = e-3*'. Consequently we have - 3 ^-2 log e = log 0.98, whence ^, ^ _ logos ^ _ 9.9912 - 10 ^ 0,0088 ^ 31oge 1.3029 1.3029 so that we find for this quality of glass the formula for the percentage of light transmitted by a pane x millimeters thick. 194. Cooling bodies. If a warm body is placed in a niedium whose temperature is kept fixed and which is cooler than the body, the latter will cool off at a rate proportional to the difference between its temperature and that of the sur- rounding medium. This law of cooling, due to Newton, may be expressed by the formula = 0^ + (0^-e,)e-''\ where Oq is the temperature of the medium, ^j is the original temperature of the body, 6 its temperature after t time units have passed, and F is a coefficient whose value depends upon the material of wliich the body is composed and upon the nature of the surrounding medium. EXERCISE LXXXIII 1. Assuming Halley's formula, at what height will the pressure of the atmosphere be just one half of what it is at sea level ? 2. A body of temperature 55" C. is cooling off, surrounded by air whose temperature is 15° C. After eleven minutes the temperature of the body was found to be 25'' C. Find the value of k- and write out Newton's formula for this body. 3. According to the formula found in Example 2, what time is re- quired for the body to cool off 5° ? -4. A dampened vibration begins with an amplitude of 10 centimeters. After nine minutes the amplitude- is found to be only one centimeter. Express by a formula the change which takes place in the amplitude in the course of time. 316 THE (GENERAL POWER FUNCTION [Akt. 195 195. Semi-logarithmic paper. Wlienever we have two variables X- and y connected by an exponential equation, (1) y = ae'"^, as in Arts. 190-19-1, we may construct the graph of this equation as in Art. 162. The resulting graph is, of course, an exponential curve. We may, however, convert the graph into a straight line by the following device : Let us take the logarithms of l)oth memliers of (1). We find (2) log y = log a + log(e''^) = log a + hx log e. If now we put (3) A' = X. Y = log y, h log e = M, log a = B, equation (2) may be written (4) Y = MX + B. If we regard X and Y as the coordinates of a point in the plane, the locus of equation (4) is a straight line of slope M which intersects the I'-axis in the point (0, B). (See Art. 54.) Thus, the transformation which consists in putting (5) A' = X, Y =\ogy transforms the exponential curve which is the locus of equation (1) into a straight line. In order to actually carry out this transformation we might proceed as follows: If (x, y) are two numbers which satisfy equation (1), we put X equal to x and compute Y = log y. Then we plot the point whose coordinates are A' and F instead of the point (.c, y). But an easier way of accomplishing the same result is to provide the 3^-axis with a logarithmic scale such as is used on the slide rule. (See Arts. 170 and 177.) The point of the //-axis which corre- sponds to any number y on this scale will then be at a distance from the origin which is equal to log y. If the a:-axis is pro- vided with an ordinary uniform scale, we can lay off our x-co6r- ^ '^ 1 s 'J 10 dinates as usual. Ruled paper Fig. 67. on which the rulings are ;ir- Art. 196] LOGARITHMIC PAPER 317 riuiged in this way is called se mi-hxiarithnic papfr. and is particularly well adapted for the grapliic representation of exponential eciuations of tlie form (1). On semi-logarithmic paper, tlie graph of such a relation is a straight line. Figure 07 represents a sheet of semi-logarithmic paper. 196. Logarithmic paper. If both axes are provided with loga- rithmic scales, instead of one of the axes only, and if the paper is ruled in accordance with these scales, we obtain a sheet of logarithmic paper. See Fig. 08. This is particularly adapted for the graphic representation of relations such as (1) 7/ = flX*, in which a power function occurs. For, if we take the logarithms of both members of (1), we find (2) log y = log a + log a* = log a + I: log x. Consequently, if we put (3) X = log X, Y = log y, B = log a, M = k, eipuition (2) becomes (-1) Y = MX + B, the graph of which is a straight line. Thus, on lor/nrithmic paper wJiere both sets (if rulinr/s are made on a logarithmic scale, the graph of any power function is a straight line. If, as a result of a series of observations, pairs of corre- sponding values of x and ^ are given, and if a preliminary in- spection of these values sug- gests that the relation between X and y may be of the foi'm of a power function, y = ax^, it will- be advisable to plot the points on logarithmic paper. If the i-esulting graph is a straight line, we can easily find the values of M and B and afterward, from (8), the values of a and k. If the relation seems to be of the form of an exponential function, it will instead be advisable to use semi-logarithmic paper. ■) 6 1) 1 Fiu. (jH. 318 THE GENERAL POWER FUNCTION [Art. 196 EXERCISE LXXXIV On semi-logarithmic paper plot the following relations : 1. y = 10^ 4. y := 10-^. 7. ?/ = 2^ 10. y = 2"'. 2. y = 102^ 5. y = 10-2^. 8. y = 3\ 11. y = 3"*. 3. y = 108^. 6. 7/ = 10-3'. 9. y = 5^ 12. y = 5"^ 13. Plot Halley's formula (Art. 192) for the density on semi-loga- rithmic paper, h being expressed in kilometers and putting p^ = 1, that is, using the density at sea level as unit of density. 14. Plot the relations obtained in Examples 2 and 4 of Exercise LXXXIII on semi-logarithmic paper. 1 _1 2 _2 15. Plot the relations y = x-, y — x ^, y = x^, y = x ^ on logarithmic paper. CHAPTER IX LINEAR FUNCTIONS OF MORE THAN ONE VARIABLE. LINEAR EQUATIONS AND DETERMINANTS OF THE SECOND AND THIRD ORDER 197. Functions of two variables. A variable z is said to be a function of two independent variables^ x and y, if to defi- nitely assigned tmlues of x and y there correspond definite values of z. Such relations are usually indicated by equa- tions of the form (1) z=f{x,y'), where / may be replaced by other letters sucli as JP, ^, i/r, and so on. The equation (1) may be read z is equal to the /-function of x and y. The area, .1, of a rectangle, whose base is h units long and whose alti- tude contains h units of length, is bh square units. Thus A = M is a function of b and h. EXERCISE LXXXV 1. Tf /(x, y)^ x + y +\, find the values of /(O, 0), /(I, 0)./(0, 1), /(I, 1),/(1, ^-2). 2. If F (x, //) = li - 3 X + 7 y + xy, find the values of F (0, 0), F {\, 1), F{% -1), F(_:5,2). 3. If (x, y) becomes equal to zero. 4. If f(r, y) =x^ + xy + y\ prove that /(- x, - y) =f(x, y), that f(x, y) =f(y, .r), and that f(hx, ky) = k-f(x, y) where k denotes any number whatever. 5. If f(x, fj) = x^ + y^, what will be the value of /(- x, - y), of /(y. x), and oi /(kx, ky)? 6. Express the area of a triangle as a function of its base and alti- tude. 319 320 LINEAR EQUATIONS AND DETERMINANTS [Art. 108 7. Express the volume of a rectangular parallelepiped as a function of the length of three mutually perpendicular edges. 8. Express the volume and the total surface of a right circular cylin- der as functions of the altitude and the radius of the base. 198. Linear functions of two variables. Ami function of the two variables x and y/, which cayi he expressed m the form ax + by + t/ -\- c. to assume a given value k are on a sfraic/ht line. Thus there is one such straight line for everg value of k, and the straight lines wli/rh correspond to any two dttferent values of k are parallel. EXERCISE LXXXVI 1. Plot the straight line upon whicli the function f{x, 1/)= x + 1/ + I assumes the value — ^3 ; also jilot the lines upon which f{x,y) = — 2 f{x, //)--!, /(x, y) =0, f{x, y) = + 1, /(x, ^) = + 12, /(x, y)= + -d. Observe that the straight line upon which f{x, y) = divides tlie plane into two portions such that ./'(x, y) is negative on one side of this line and positive on the other. 2. Study each of the following functions by the nietiiod outlined in Example 1. (a) 2x-^ + 3. (c) 3a:-4.y+5. (e) 2x + ^y-l. \b) x + 2y + 3. {/ = 0. I 3 X — I // = 5, 6 / 3 X = I/, \l/ -6x + 2 = 0. J j X + my — 1, \y = mx. ^^ + y 8. 1, f - ^ = 1 I ft a ^ + f = l, a Hint. In plotting, [4 x + o y = 5. regard in as a positive J X + .y — 3 = 0, number, and use a line- 1 -^ ~ "*"^' \x — y + i = 0. segment of convenient length to represent m. f X + fj — 'd = Similarly for a, b, and m in Examples 8 and 9. \y :^7 X. 201. General formulas for the solution of two simultaneous linear equations with two unknowns. Let us transpose the terms c^ and c^ of equations (1) and (2) of Art. 200, and let us call — , a., and let us define this symbol by saying that it shall be equal to aj)., — ac,b^^ tlie common denominator of the two fractions (5), Art. 201. We have then by detinition (1) = aj>., — a^by This quantity, a siiiirlc number fui'nied fi-oin the f(jur num- bers flj, ^p (^2' ^^T '^^ indicated in (1), is called a determinant of the second order. Tims, we hiive o -1 3 2 _ 1 . ;i = 10 _ 12 We now observe tliat the nuniei'ators of the expressions (o) of Art. 201 may also be written as determinants, namely: Ir I /rj* CLyfCty CLk^fC't "1 a., 326 LINEAR EQUATIONS AND DETERMINANTS [Art. 203 We have therefore the following new form for the solution of equations (1) and (2) of Art. 201 : (2) h h «i ^1 ^2 h ' I/ = «2 h «1 h «1 h «2 K ^2 h Observe that the denominator of both expressions is the same determinant, namely, that one which is formed from the coefficients of x and i/ in the equations a^x + h^y = k^. Observe further that the numerator of a:: is a determinant ob- tained from the denominator determinant by replacing in it the coefficients of x (a^ and a^) by the right members (^k^^ and k^'). Observe finally that the expression in the nu- merator of 9/ is a determinant obtained from the denominator determinant by replacing in it the coefficients of t/ (b-^ and Jj) by the right members {k^ and ^2)- EXERCISE LXXXIX Find the values of the foUowino- determinants: 1. 2. I 2 3 4 II 4 2 5 3. 1 - 6 2-3 -26 2 5 a b c d -d h c — a 7-15. Solve the equations given in Examples 1-9 of Exercise LXXXVIII by using determinants. 203. Homogeneous linear equations with two unknowns. If the right members, k^ and k^^ are both equal to zero, the ec^uations are said to be homogeneous. Tlius, the two equations (1) (2) a^x -f h^y = 0, a^x + h^y = 0. Ai:t. 2o:5] HOMOGENEOUS LINEAR EQUATIONS 827 form a Jioniogeneous system. If the determinant is different from zero, equations (2) of Art. 202 show that the only solution of (1) and (2) is .'r = ?/ = 0, since both of the numerators which occur in (2), Art. 202, will vanish when Tc^ and h.^ are equal to zero. The solution ./• = y = of the equations (1) and (2) is often called an obvious or trivial solution, because it is evident that every system of this form has this solution. But if the determinant D is equal to zero, it does not follow that the system (1) (2) has x=i/ = as its only solution, since the expressions (2) of Art. 202 become useless in this case, on account of their indeterminateness. We can easily show that in this case (when D = 0), there do exist other solutions of (1) and (2), besides the obvious one x= 7/ = 0. In fact, if we put in (1 ) (4) X = kh^, y = — ka^, where Jc is any number whatever, equation (1) will be satis- fied siuce, for these values of x and ij, we find a^r -\- h-^ij = a^kh^ + /)^( — ka^) = ki^a^h^— a^h^ = 0. But these same values of x and y will also satisfy (2); for we have a,2^- + h^jj = a.Jch^ -\- h.^( — ka^') = — k(a^b^ — a^h^) = 0, since, in our case, D = a-jb., — a.yb^ is equal to zero. Thus all of the pairs of numbers, which can be obtained from (4) by giving different values to k, will be solutions of both (1) and (2). Moreover, all of these solutions will not coincide with the obvious solution x= y = unless a^ and 5j are both equal to zero. Let us exclude this case and also the analogous case in which a^ and b^ are both equal to zero, since in either of these cases we should really be dealing with only a single equation ; one at least of the equations (1) or (2) might then be resrarded as absent. 328 LINEAR EQUATIONS AND DETERMINANTS [Art. 2():J We may summarize our results as follows: If the determinant D of the homogeneous equations (1) and ( 2 ) ?'.s' not equal to zero, these equations have only a single solu- tion, namely, the obvious one, x = y = Q. If T) is equal to zero, the two equations have infinitely many solutions. More specifi- cally we can say that if both equations are actually present, that is, if neither of them has both of its coefficients equal to zero, then if D is equal to zero, the two equations are equivalent ; that is, every solutio7i of one equation is also a solution of the other. Although the equations (1) and (2) are satisfied by infi- nitely many pairs of values, x and y, \i D = 0, the ratio of x to y will nevertheless be determined uniquely. In fact, this ratio may be computed from each of the two equations, and the resulting two values will be found equal on account of the relation 2) = 0. Thus, the homogeneous equations (1) and (2) forx and y may be regarded as two no7i-homogeneous equations for a single un- known, the ratio of x to y ; the condition that these two equations 7nay be consistent, that is, that they shall furnish the same value for this ratio, is again D = 0. To clarify still more the significance of the condition i>= 0, Ave add the following remarks. If a^ and b^ are common multiples of a^ and b-^, that is, if a^= ma-^, b^ = mb^, the determinant D = a^b^ — agi^ is equal to zero. Conversely, if i> = and if a^ and ^j are not equal to zero, we have a A - ^A = 0, or h = ^. If we denote by m tlie common value of tliese two fractions, we find «2 = ma-y, ho = mh-^, so that (5) a^x + />2,y = m(ayr + ftj?/). Art. 204] DISCUSSION OF GENERAL CASE 329 Therefore, if the determinant D of (1) and {2) is equal to zero, and if not both of the coefficients of either equation are equal to zero, then the coefficients of one of the equations will he proportional to those of the other ; in other ivords, the left mem- ber of one of these equations will be a mere multiple of the left member of the other. Ill our proof we have assumed that Wj and i, are liotli different from zero. The student should complete the proof by considering separately the cases a-y4^(), 6, = 0, and r/j = 0, b^ =^ 0, the further case rr, = />, = being excluded by our liypothesis that not both of the cofficientsof either ecpiation shall vanish. The student may easily decide, however, what would happen in this case also. The theorems of tliis article are easily explained graphi- cally. The graplis of the two e(iiuiti()ns (1) and (2) are straight lines through the origin of coordinates. If the deter- minant D is not equal to zero, these lines are distinct and their only common point is the origin, x= y = 0. lfi) = 0, the two lines coincide, and have all of their points in common. 204. Discussion of the solutions of two linear equations with two unknowns. We have seen tliat the equations (1) a^T + b^i/ =k^, (2) a.^.r + k,!j = k^^, have a single solution if i>=^0. (See Arts. 201 and 202.) If 2>= 0, we tind from (3) and (4) of Art. 201, (3) k^b., - kj^^ = 0, (4) a^k^ — ajc^ = 0, so that k^ and k.y must satisfy these conditions if (1) and (2) are to have any solutions at all. Bnt the left members of (3) and (4) are the determinants which occur in the numer- ators of the expressions (2) of Art. 202 for x and /y. Thei'efore, the si/stem of equations (1) and (2) has no solu- tion at all if D = U. unless at the same time both of the determi- nants 330 LINEAR EQUATIONS AND DETERMINANTS [Art. 204 (5) 1 '*i and l«2 k, k., are equal to zero. If (6) i>=0, = 0, and k, k^ = 0, then (1) awe? (2) have i7ifimtel'y many solutions. Unless one of the two equations has all of its coefficients equal to zero., each of the two equations may he obtained from the other by multiply- ing both of its members by an appropriately chosen number., and the tivo equations are equivalent, that is., they have exactly the same solutions. In fact the three equations (6) are equivalent to the con- tinued proportion a-^ : b^ : k^ = a^ '• bo : ko- If we remember that each of the equations (1) and (2) has for its graph a straight line, we see at once that the corre- sponding geometrical situation may be described as follows : The graphs of equations (1) arid (2) are distinct intersect- ing lines if D 4^ 0. They are distinct parallel lines if D = and if not both of the determinants (5) are equal to zero. The two graphs coincide throughout if conditions (6) are satisfied. EXERCISE XC First discuss the following pairs of equations by the methods of Arts. 202 and 204. Then find all of their solutions. 1. / 2 X + .V = 1, '■^y / X + ?/ = 5, \2x + 2y = 1. j X + y = o, \2x + 2y=10. 4 f:3.r--ry-.5 = 0. ■ \ 2 X + 3 y + 1 = 0. 5 {3,r-4.v-7 = 0. ■ \l8x-24?/ = 38. 6 f2x+3?/-7 = 0. ■ \8a; + 12y = 28. 7_ / y = mx + h, \y = m'x + h', m ^ m'. Q \ y = mx + h, \y = mx + b', h ^ h'. a 1, y = h--x. a Art. 205] DETERMINANTS OF THE SECOND ORDER 331 10. ^Vl^at value or values must /: have in order that the equations 3 X + 4 y = 0, 1-2 X + khj = 0, may have other solutions besides x = ?/ = 0? Find these solutions when thev exist. 205. Properties of determinants of the second order. The following theorems are important for what follows : 1. A determinant of the second order does not change its imlue if its columns {vertical lines^ are converted into rows {horizontal lines^^ that is. a. a b^ h Proof. Both members are equal to aih^ — a-ihi. 2. A second order determinant is equal to zero if both ele- ments of any one of its rows or columns are equal to zero. For instance (I., b.j 0. if Wj = a„ = 0. 3. A determinant of the second order chanc/es its si — uki-^, and then ai In a.2 />■ = (iih-2 — (t-Jn = m(^a\(i-2 — ) = 0. 6. If evert/ eleynent of any column (or roiv^ is expressed as a sum of two terms, the determinant may he rewritten as a sum of two determinants as in the folloiving cases : (1) «1 + ^1, h «i ^1 ^1 h ^h + ^2' h «2 h + ^2 h a^. h + d. «1 h + «1 d. ^imilarly, for (2). 7. The value of a second order determinant is not altered if to the elements of any colunm (or row^ he added common multi- ples of rorrcsponding elements of a parallel column (or roiv'). Proof. According to theorem (J of this article, we liave Oi + mbi, bi n.s (1) (2) (3) a^x + b^y + c^z = r^, a^x + h^y + c^z = /"a, a^x + h^f/ + c^z = rg, for X, y, and z. We proceed just as thongli the coefficients were specilic given numbers, and eliminate y and z, so as to Art. 206] DETERMINANTS OF THE TIIIKl) OllDER 333 obtain an equation invulving- x alone. To accomplish this we rewrite (2) and (3) as follows: (4) hi/ + ''2~ = ''2 - "2^' ^3^ + ^'3- = >'-6 - «3^' and solve (4) for ?/ and z. This gives, according to (2) Art. 202, and theorem 6 of Art. 205, (5) y = z = a^x, c^ _ '"2 e^ |_ ^2 ^2 (f^X, 6'3 '•3 ^3! I«3 ^3 ^2 - «2^ ^2 ''2 ^2 ^2 ^3 - «3^' h ^'3 *3 «3 X, where we have not divided by the coefficient of ^ and z, thus avoiding cumbersome fractions. Let us substitute the values of y and z from (5) into (1). But again, so as to avoid fractions, let us first multiply both members of (1) by f>2 ^2 We shall then find, after substituting and uniting all of the terms which contain a: as a factor, ^'2 ^'2 ^'3 ^3 + h 2 + C, h ^'3 = r '\h <'. or (making use of Theorem 3 of Art. 205) (6) ^2 ^^2 I ^2 ^2 1 *3 H -h. -h. «2 H / 1} Oct + ^1 + ^1 «2 f'2 ^2 ^2! The coefficient of x, in this equation, depends upon the nine coefficients aj, 6j, e^, a^^ \^ c^, a^, 63, Cg, of the three un- 334 LINEAR EQUATIONS AND DETERMINANTS [Art. 206 ^1 ^2 = «1 ^2 ^3 «3 ^2 ^3 + ^1 «2 ^2 ^'3 ^3 kiiowiis in the three equations (1), (2), (3). It does not depend upon the values of r^, r^^ r^ We shall henceforth speak of this coefficient as the determinant of these nine quantities, and use the symbol h to represent it. Thus we are led to the following definition : When nine Clumbers are writteji in the form of a square array, the value of their determinant, represented hy the symbol (7), is hy definition equal to the coefficient of x in (6). We have therefore the following defining equation for a determinant of the third order : (8) D = This formula, whicli we have used as a definition for the symbol in the left member, is easy to remember. Each of the three terms of the right member is a product of two factors ; one of the two factors of each of these terms is one of the elements a^, 6j, c^ of the first row of the determinant ; the other factor is that second order determinant which we obtain from the tliird order determinant (8) by crossing out the row and the column in which stands that one of the elements rtp h^, c^, wliich is tlie first factor of the term in question. The signs of the three terms are in order +, — •> +• Let us substitute, in (8), for the second order determi- nants their values and multiply out. We find in this way B = a^(h^c^ - h^c^) - ^'lO'gC'g - a^c^ + c^^a^h^ - a^h^), which may be rewritten as follows ; and this expression might also be used as a definition for the determinant D. We see that a determinant of the third Art. 206] DETERMINANTS OF THE THIRL) ORDER 335 order consists of 6 terms, each of wliich is a product of three of the nine numbers a^, ■•• Cg, three of tliese products being preceded by the plus sign, and the other three b}^ the minus sign. We find again the expression (8) for D from (9), if we unite tlie two terms of (!)) which contain a,, the two terms which contain ftj, and tlie two terms which contain Cy Let us instead unite the terms which contain a^, b^, cv- Then we may write D as follows: (10) D aj'^ ^1 + h «i ^1 — C.y "\ h '\h ^3 ^'3 '•3 «3 h a new expression for D similar to (8). There are again three terms. But this time the first factor of each term is an element of the second row cr„, h„, c„. The second factor of each term is again obtained, as in (8), as a second order determinant by suppressing that row and column of D to which the first factor of the term belongs. In order to express this hiw of formation more compactly we introduce a new word by the following definition. The minor of a particular one of the nine eleynents of the third order determinant , , D = h. is that second order determinant which is ohtained from D if the row and column he erased to which that particular elernent of D belongs. Thus the minors of n^, h^, c^ are respectively *2 ^2 ) «1 ^1 "1 h h <^3 "3 <^3 a„ b.^ We can now say that the two expressions. (8) and (10), for D have this in common. Each term of either expression is the product of an element of D by the minor of that element. The two expressions differ in so far as the elements which are used in (8) are those of tlie first row, while those used in (10) belong to the second row. Moreover, the terms 336 LINEAR EQUATIONS AND DETERMINANTS [Art. 206 are preceded by the signs +, — , + in (8), and by the signs — , +, — in (10). In order to get to the bottom of this matter, we now arrange the expression (9) for D according to the elements a^, b.^, c^ of the third row. We find (11) D = cu -K a„ c, + c^ "i h a„ h. Tlie resnlts (8), (10), and (11) may be summed up as follows : We may expand a determinant D of the third order accord- in;/ to the elements of the kth row^ where k may he equal to 1, 2, or 3. This expansion vnll present D as a sum of three prod- ucts of the form 1st element of the kth row times its minor, 2d element of the kth roiv times its minor, ?>d element of the kth row times its minor. Each of these products is preceded by a plus or nmius sign, which is determined in accordance with the following diagram of signs : (12) ^= _ + - + - + Tliree further expressions for I) may he obtained by expand- ing with respect to the elements of any column according to the same rule, using the columns in the diagram of signs. To prove tlie last statement, it suffices to arrange D, as given by (9), with respect to the elements a^, a^, a^ of the' first column, or with respect to b^ b^, b^, or c^ c^, Cg. EXERCISE XCI Compute the values of the following determinants. 1 1 1 11 1 1 ;3 4 2 - 6 - 1 2. -6 - 1 3. 7 :} 3 1 2 4 2 :5 .5 :i 2 :} 2 1 1 1 - 1 - 1 - :} 5. ■1 - (i 2 6. 3 2 4 - 1 1 1 1 - .0 207] COFACTORS « a h g a h g b 8. h b / 9. a k g 0c 9 f c 9 f <^ 337 7. 207. Cofactors. The six expansions of a tliiid order de- terminant wliit'h we have found in Art. 20(3 assume a some- what simpler form if v,e introduce tlie notion of cofactor in place of the notion minor. Every element of the determi- nant D has associated with it, as its minor, a certain detei- minant of the second order. The expansion of D with respect to the elements of one of its rows or columns is com- plicated by the fact that some of the terms of such an expansion are preceded by the minus sign. We may get rid of this complication by uniting the minus sign with the corresponding minor, in accordance with the following definition. Consider the third order determinant D and the diagram of signs S, defined by D K S= + — + — + — + - + £g the cofactor of a given element of Z>, we mean the minor of that element preceded by the plus or minus sign, according as the place in the diagram S ivhich corresponds to the given element of D is occupied by a plus or minus sign. Thus, the cofactor and the minor of a given element of D differ at most in sign. The cofactor of c, is If the cofactors of a^, ftj, Cj, etc., be denoted by A^, B^, C\, etc., tve may noiv write Din any one of the following sir forms. • D = a^A^ 4- b^B^ + c^ C^, D = a^A^ -f a^A^_^ 4- 'a^Ay. (1 ) B = a^A^ -f- ^2^2 + ^I^V ^ = ^^1 + ^2^2 + ^'3^3' B = a^A^ + b^B^ + t-gCg, B = c^C^ + c.,t\ + c^C^ 338 LINEAR EQUATIONS AND DETERMINANTS [Art. 208 EXERCISE XCII Write down and evaluate the cofactors of each of the elements of the following determinants. 1 2 3 1. 4 5 6 7 8 9 1 1 1 2. a h c a'^ b-' j2 1—1 1 1 3. n h c a* b* c* a h g 4. h b f 9 f c 208. The principal properties of determinants of the third order. It is now easy to prove tlie following theorems, which are quite analogous to the Corresponding properties of second order determinants. (Compare Art. 2U5.) 1. A determinant of the third order does 7iot change its value if its elements are transposed, that is, if its roivs he converted into columns, and its columns into rows, the relative order of rotvs arid columns 7iot being changed. To prove this, let i) = If we expand D according to the elements of its first row, and D' according to the elements of its first column, we ob- tain identical results, thus proving the theorem. 2. A third order determinant is equal to zero if all of the elements of any one of its roivs or columns are equal to zero. This follows directly if we expand the determinant with respect to the elements of such a row or column. 3. A determinant of the third order changes its sign if any two of its columns or any two of its roivs are interchanged. For «1 h ^1 «i a.2 a.. h ^2 , D' = h h a., h C;, Ci Co instance c, «i ^2 Cj b, Co as may be shown by expanding both members of this equation and com- paring the two results. Art. 208] PROPERTIES OF DETERMINANTS 339 4. If all of the elements of a roiv (or column') are multiplied hy the same number m, the whole determinant is multiplied by m. Proof. Let D ma^ iii//^ inc^ «2 h '"2 "3 h <^3 If we expand D according to the elements of the first row, we have (see Art. 207, equations (1)) : SO that ma^ mb^ mc^ (1.2 f>., Co "s h c. — in \a.2 It., cJ, a„ b., c. thus proving the theorem for the case where the elements of the Jirst row are multiplied bj^ a common factor. The proof for any other row or column may be carried out in the same way. 5. A third order determinant is equal to zero if two of its parallel lines (rows or columns) read alike, that is, if all pairs of corresponding elements in two parallel lines are equal to each other. This follows at once from Theorem 2. For, let us denote by D the de- terminant under consideration. If we interchange the two lines which read alike we should find, according to Theorem 2, a new determinant D' such that (1) D' =- D. But since the interchanged elements are equal in value, we must also have (2) D' = D. But from (1) and (2) we conclude by addition* D' = D = Q). 6. If every element of any rotv or column is expressed as a sum of tii'o terms, the determinant may be expressed as a sum of two determinants, thns Oo + m., h„ cj = a, + ni, bo Co I «1 ''. '"i a.. />., c, + «3 fh <^3 ««j b^ cA m., b.-, cJ »n„ bo cJ 340 LINEAR EQUATIONS AND DETERMINANTS [Art. 208 To prove this, expand the determinant on the left according to the elements of the first column. A similar proof will establish the theorem for any column or row. 7. If to the elements of any row or column we add the cor- respondin(/ elements of a parallel rotv or column^ multiplied hy one and the same factor, the value of the determinant is not changed. For, by Theorems 4 and 6 we may write r/j + m/^j. ^ Cj «i ^^1 ^1 h h r/o + mh.^, h ^2 = a.. h ^2 + ??? h h 'I3 + mbg, ''3 (^3 ";! h ^3 ih h and the second determinant of the right member is equal to zero on ac- count of Theorem .3. This actually proves Theorem 7 for the first and second columns. All other cases may be proved in the same fashion. 8. If the elements of any rotv or column he multiplied hy their respective cof actors, the sum of the products ohtained in this ivay is equal to the value of the determinant. This is merely a restatement of equations (1) of Art. 207. 9. If the elements of any roiv or column he multiplied hy the cofactors of the corresponding elements of a parallel row or column, the sum of these products is equal to zero. Proof. We know, for instance, according to Theorem 8, that (3) a^A^ + a^A^ + a^A^ = Now b^A^ + h^A„ + ^g.lg is obtained from tlie left member of (3) by re- placing rtj, flo, f'3, by It^, l>2,.l>s, respectively. Therefore, /\A^+ l>^A^ + ^s^^a must be equal to the determinant ''2 which is obtained from the right member of (8) when we replace Oj, Oj, Og by 61, 62, 63. Rut this determinant is equal to zero on account of Theorem 5. Therefore hiAi + biAi + hAs-O. All other cases of Theorem 9 may be proved in the same way. Art, 208] PROPERTIES OF DETERMINANTS 341 The theorems which we have developed have many impor- tant applications, some of which we shall explain in Art. 209. They may frec^uently be used to simplify the work involved in calculating the value of a determinant, as illustrated in the following Exercise. (1) We may write 4 7 7 5 - 4 2 -2 5 1 EXERCISE XCIII 4-2.5, 7-2(-4), 5 -4 -2 5 15 31 -4 2 5 1 siuce the second determinant in (1) may be obtained from the first by multiplying the elements of the second row by - 2 and adding to the corresponding elements of the first row (Theorem 7, Art. 208). But the third determinant in (1) is equal to zero, since the elements in its first row are exactly three times as great as the corresponding elements of its third row (Theorems 4 and 5, Art. 208). Therefore 4, 7, 7 5, - 4, 2=0. -2, 5, 1 Show that the following determinants are equal to zero : 1 3 5 15 7 9 a + d b + e c +f 2. 2 6 10 . 3. 1 o 3 4. a b c 8 9 11 4 5 6 d e f 5. Prove that 1 1 1 a b c = (a-b)(b-c)(c-a). a"- h-^ 6-2 Proof. The determinant will vanish if a = ft, or if ft = c, or if c = a (Theorem 5, Art. 208). Therefore it has a — b, h — c, and c — a as factors. (See Factor Theorem, Art. 84.) But the product (a — b)(b — c)(c — n) and the expanded determinant are integral rational functions of a, ft, r, of the same (the third) degree. Therefore the determinant can differ from this product merely by a numerical factor L independent of a, ft, and c. Thus, we must have (1) 1 1 1 n b r cfi b'^ c'^ L(u ~b)(b-c)(r — a). 342 LINEAR EQUATIONS AND DETERMINANTS [Art. 209 If the determinant be expanded it will be found to contain the term hc^. The corresponding term in tlie expanded right member of (1) will be Lbc'^. Therefore L must be equal to 1, which proves the desired result. 6. Prove that 1 a a^ 1 h h^ ■1 1 c ,.3 {a - h)(h - c)(c - a){a + h + c). Prove that oi + kao + In^ 02 + ma^ (iz a, ^2 h + ^'bz + Ih 5 2 + mh^ bz br b.-. Cl + kC2 + ICi Co + mc3 cz ^1 C, Find the value of a b a2 h^ c a3 b^ c^ 209. Solution of a system of three simultaneous linear equations with three unknowns. We now return to the problem of Art. 20G, to solve the equations (1) (i^x + h-^y + c^z = rp (2) a^x + Ky + c.^ = r^, (3) a^x + h^y + c^z = rg, for x^ ?/, and z. It was this problem which first suggested to us the introduction of determinants of the third order (see Art. 206), and we shall now see how easily the solution of such a system of equations may be effected by means of determinants. Let (4) D^ «1 ^ Oo b^ a.. b. be the determinant of the coefficients of x, y, and z, in the equations (1), (2), and (.3), and let ^i, Bu Ci, etc., be the cofactors of n\, by, c^, etc., in the determinant D. (See Art. 207 for definition of cofactors.) Let us multiply both members of (1) by Ai, both members of (2) by A2, both members of (3) by .I3, and add. AVe shall find, after collecting terms, (ai.li + u.Ai + a^A3)x+ (biAi + M2 + bzAa)// + (r^Ai + C2A2 + ^3^3)2 = riAi + /•2.42 + rs^s- Art. 209] THREE SIMULTANEOUS EQUATIONS 343 The coeHicieiit of ,/• in this equation is equal to D (Theorem 8, Art. 208), wliile the coeHicieiits of y and z are both equal to zero (Theorem 9, Art. 208). Tiierefore, we find (5) Dx = r\Ai + r-zAo + nAz. If we multiply (1), (2), (;J) by Bx, Bo, B^ respectively and add, we find in the same way (6) nil = nBi + r.B^ + r^B^, and by a similar process we find (7) Dz = riCi+r.>C; + rzCz. The right members of (.5), (6), and (7) may also be written as determinants of the third order. In fact, if in the equation, D ax hx C\ a-2 ho C2 as bz C3 axAx + a2A2 + a3A3, we replace ai, 02, 03, by ri, r-2, rs, we find rx />x Cx r-2 hi Co = rxAx + r2A2 + rsAs, rs hs C3 which is the right member of (.3). Similarly we conclude that Ol ri Cl ('2 1-2 <"2 «3 ^3 (•3 riBi + 7-2^2 + rsBi, «! bx rx ao bo »'2 f3 b3 n rxCx + /•2C2 + raCs. (8) We may therefore write equations (.5), (0), and (7) as follows: n bi Cx rj &2 Co rz b?, C2 ai rx cx\ ax bi Ci Co ro Co\, ao bo C2 f'3 ''3 ^'a! «3 bz Cz rtl bi ''1 Oo b2 C2 X = 03 bz Cz ai 61 Cl ao bo C2 y^ «3 bz C3 ai bi n «2 bo r-2 as bz rz If i) is different from zero, we may divide both members of each of these equations by D, and thus obtain the solution of (1), (2), and (3) in the form of three fractions. These fractions have as their common denominator the determinant 344 LINEAR EQUATIONS AND DETERMINANTS [Art. 210 D. The numerator of the fraction for x is obtained from D by replacing in it the coefficients a^, a^, ag, of x by the right members, r^, r^, rg, of equations (1), (2), and (3). The. numerator of tlie fraction for y is obtained from D by replac- ing in it the coefficients, hy, h^^ 5g, of y by r^, r^, r^. The numerator of the fraction for z is obtained from D by replac- ing the coefficients, fj, e^, r/g, of z by r^, 7'^, r^. EXERCISE XCIV Solve the following systems of equations by determinants 3. 4. 5. f 2 X + ^ + c = -i, \x + 1/ + 2z = i. i X + ij + z = SO, 3 x + 4 y + 2 ^ = 50, [27 x+ 9i/ + '3z = 64. [ 18x - Ty - 57 = 11, Hi y - f -^ + ^ = 1^8, \^z + 2y+ |x= 80. 7x — 5z = y + x — 86, ix + ^y + lz = 5S. X + y = \0, x + z=19, y + z = 23. 6. 7. y -{- z = a, z + x = h, x + y = c. ~ + ~ - a, y ~ Z X 1 J 1 - H- - = c. X y 8. [■ fl.r + hy = c, ,lx 4- ey =/, ^ //// + hz — I. \ ax + hy + cz = d, I n'~.r + li-y + r~z = ^/'^, I r<3.i; + h^y + c^^ = d. Hint for Ex. 9. Make use of Examples 5 and 8, Exercise XCIII. 210. Homogeneous equations. If the equations (1), (2), (3) of Art. 20*J are homogeneous, that is, if r ^ = rg = rg = 0, and if their determinant D is different from zero, equations (8) of Art. 209 and Theorem 2 of Art. 208 show that the only solution of (1), (2), (3) is the obvious one x = y = z = 0. It would be quite improper to draw the same conclusion in the case where D is equal to zero, since equations (8) of Art. 209 do not admit of division by D in this case. (See Art. 21.) What actually takes place in this case is ex- pressed by the latter part of the following theorem, which covers both the case when i) ^ 0, and i> = 0. Akt. 210] HOMOGENEOUS EQUATIONS 345 The Jiomogeneous equations (1) a^j- + />!// + e^z - 0, (2) a^j' + ^.3// + ^2.' = 0, (3) a^.r + b^>/ + CgS = 0, Aaug 72 ^2 b., C3 does not vanish. We may then solve equations (2) and (3) for y and z in terms of r, as in Art. 206, thus obtaining equations (5) of Art. 20G, where, however, we now have to put ?-j = r., = 0. If we take account of this, and make use of the notation introduced in Art. 207 for the co- factors of the various elements of D, we find that these equations (the equations (")) of Art. 206) reduce to A^y = B^r and A f = C\x, whence (4) 2/ = ^-^-^ ^ = ^^- .4, A, These values of 1/ and z will satisfy equations (2) and (3), no matter what value be given to x. But they will also satisfy (1). For if we substitute the values (4) of // and ~ into the left nieni])er of (1), we find a^x + h^ii + c^z — a^x + h^^x + r, ^ ' j; - ' 1 • M = — {a\A 1 -1- bilii + <\C\)x - — .r, -'1 '1 and this is equal to zero, no matter what value x may have, since D is equal to zero by hypothesis. Since equations (4) give a solution of (1), (2), (3), no matter what value be given to x, we see that there exist in this case infinitely many solutions of the tlii-ee given equations, as our theorem asserts. Our proof was based 011 the assumption that A^ is not equal to zero. It is easy to see how the proof should be 346 LINEAR EQUATIONS AND DETERMINANTS [Art. 211 modified if A^ is equal to zero, provided that there exists some otlier second order minor of D which is not equal to zero. We may, in fact, state our theorem a little more precisely as follows: If the determinant D of the homogeneous equations (1), (2), (3) is equal to zero, hut if there exists at least one second order minor in D which does not vanish, theji there exist infinitely many solutions of (1), (2), (3), hut the ratios of the three un- knowns, X, y, z, will he determined uniquely hy these equations. For instance, if ^4^ ^ 0, equations (4) show that these ratios are y -.x - B^.A^, z: X - C-^-.yi-^, x:y:z = A^:B,:C\. If i) = 0, and if, besides, all of the second order. minors of D are equal to zero, it may be shown that two of the three unknowns, x, y, z, may be chosen arbitrarily. In this case the ratios x : y : z are not determined uniquely. Our method of proof showed us that, in the case D = 0, any solution of two of the three equations would also satisfy the third. We may express this by saying that the three equations are not independerit ivhen D is equal to zero. In the case when J) = 0, and when, besides, all of the second order minors of D are equal to zero, any solution of one of the equations satisfies both of the others. In this case, then, two of the three equations (1), (2), (3) are mere multiples of the third, and the three equations taken together convey no more information about the unknown quantities than a single equation. 211. An application of linear equations in chemistry. There are certain substances like iron, silver, lead, sulphur, hydro.- gen, oxygen, chlorine, etc., which chemists have not been, able to separate into other constituents, and which they call elements. Very frequently several elements occur in a mix- ture which may contain its various constituent parts in any proportion. Thus pulverized iron and pulverized sulphur Art. 211] LINEAR EQUATIONS IN CHEMISTRY 347 may be mixed in aii}' [H'oportion, but the various particles are still recognizable as iron or sulphur particles. If heat be applied, some of the iron and sulphur particles will combine to form a new substance which is essentiall}^ different in kind from both of its component elements, and which is said to be a chemical combination or compound of iron and sulphur. It is a fundamental fact of chemistry that the elements enter into chemical combinations only in certain fixed proportions, while in a mixture the proportions may be any whatever. For instance, one gram of hydrogen (chemical symbol H) will combine with 35.4 grams of chlorine (chemical symbol CI) to form 86.4 grams of hydrochloric acid (HCl). If the vessel in which the reaction takes place should contain just one gram of hydrogen and more than the proper amount of chlorine, only 35.4 grams of the chlorine will be used up. The rest will remain unchanged. This reaction is repre- sented by the chemical equation H + CI = HCl. To account for the law of fixed proportions, the English physicist and chemist Dalton (1766-1844) proposed the following theory. A gram of hydrogen contains a certain very large number of exceedingly minute particles called atoms. Similarly, the chlorine in the vessel is composed of atoms of chlorine. Each atom of hj-drogen combines with one atom of chlorine to form a smallest particle, called a molecule, of the compound. If all of the material is used up, there is one atom of hydrogen combined with every atom of chlorine, and the law of fixed proportion will be accounted for if we assume that all hydrogen atoms have the same weight, and that every chlorine atom weighs 35.4 as much as a hydrogen atom. Every chemical reaction is capable of a similar interpreta- tion, and as a result of a systematic study of all the elements and their compounds, chemists have come to attribute to every element a number, called its atomic weight. The atomic weight of an element tells* us how the weight of an 348 LINEAR EQUATIONS AND DETERMINANTS [Ain. 211 atom of that element compares with that of a hydrogen atom. Thus the atomic weight of chlorine is 35.4, as determined by the reaction which was described above. These atomic weights are sometimes called relative atomic weights. They do not teach us how heavy any atom really is. When we say that an element has the relative atomic weight jw, Ave only mean that one of its atoms weighs p times as much as a hydrogen atom. If the absolute weight of a hydrogen atom (expressed in grams) is w grams, the absolute weight of an atom of this other kind will then be pw grams. Let us now consider three elements A^ B, and P, whose relative atomic weights are a, b, and p respectively, and let AP and BP be two chemical combinations the molecule of each of which contains only one atom of each constituent. Suppose that we have a mixture of the two compounds AP and BP, and that we know the total weight g (in grams) of the mixture, and also the weight h of the element P which is present in the mixture. We can then calculate the amount present in the mixture of each of the two compounds AP and BP. The solution of this problem is known as indirect analysis of the mixture, and is often of practical importance. For it is easy to hud g the total weight of the mixture, and in many- cases it is easy to lind the amount h of the element P, while it may be very difficult to separate the elements A and B by chemical means and to determine their weights directly. To solve the problem we proceed as follows : Let us denote by x and y the number of grams present in the mix- ture of the compounds AP and BP respectively. Then of course (1) ^ + y = 9^ since g represents the total weight of the mixture. Let w be the weight (in grams) of a hydrogen atom. Then an atom of A weighs aw grams, one of B weighs hw grams, and one of P weighs pw grams. Consequently a molecule of AP weighs at<;+jow or (a + jt?)M'grams, and there Art. 211] LINEAR EQUATIONS IN CHEMISTRY 349 are x I {a + p^iv molecules of AP and 7/ / {b + p)w molecules of BP in the mixture. Again, since there are h grams of the element P in the mixture and since each atom of P weighs pw grams, there are h/pw atoms of P in the mixture. But there is one and only one of these atoms in every molecule of AP and BP. Therefore the number of molecules of ^P and BP taken together must be equal to the number of atoms of P ; that is X _,_ y ^ A {a-\-p^w {b + p)w pw^ whence, multiplying both members by w, (2) ^ 1- y =^. a+p b+p p Thus, we have found two equations, (1) and (2), for our two unknowns. We may write them as follows, (3) I ^ + y = 9^ \p(b +p)x-\-p(a +p)i/=(a+p^(b +p}h. The determinant of the coefficients of x and 1/ is 1 1 p{h+p'). lK(^+p) Consequently this determinant will be different from zero whenever a is different from J, and it will actually be possi- ble to solve (3) for the two unknowns. The result is p a— b p a — b Illustrative example. A mixture composed of sodium chloride (NaCl) and potassium chloride (KCl) was found to weigh 3 grams. The com- pounds were then decomposed and tlie amount of chlorine was found to be 1.7 grams. Find the amount of NaCl and of KCl present in the original mixture. The atomic weights in this case are a =39.1 for potassium (K), i = 23 for sodium (Na), and p = 35.4 for chlorine (CI). We liave also g — S, h = 1.7. Substitution of these numbers in formulas (i) gives (to the nearest tenth of a gram) X = 0.9 gram, // = 2.1 grams, so that there were 0.9 gram of KCl and 2.1 grams of NaCl in the mixture. --p(ia+p) — p{b+p) = p(a- 5). 350 LINEAR EQUATIONS AND DETERMINANTS [Art. 212 212. Generalization to systems of n linear equations with n unknowns. The attempt to generalize the methods of this chapter to the case of n equations with n unknowns, leads to the introduction of determinants of the nth order. This might be done by the method of mathematical induction. (See Arts. 84 and 88.) But a more elegant treatment of this subject may be based on the theory of permutations, which is also important from many other points of view. We shall therefore reserve the discussion of w linear equa- tions with n unknowns for Chapter XII, devoting the next two chapters to some of the numerous questions concerned with permutations and combinations and the theory of probability. EXERCISE XCV 1. Find two numbers whose sum is equal to 60 and -whose difference is equal to 16. 2. If the first of two numbers is multiplied by 2, the second by .5, the sum of the products is 31 ; if the first be multiplied by 7, the second by 4, the sum of the products is equal to 68. Find the two numbers. 3. A father was three and one-third times as old as his son six years ago. Three years from now his age will be two and one-sixth times the age of his sou. What are the present ages of father and son? 4. A father was m times as old as his son six years ago. Three years from now his age will be n times the age of his son. "What are the present ages of father and son ? Shall we obtain a reasonable answer no matter what the values of m and n may be ? 5. A tank containing 21,000 liters may be filled by means of two pipes. If the first pipe is opened for 4 hours and the second for 5 hours, 9000 liters of water are obtained. If the first pipe is o]ien for 7 hours and the second for ^ hours, 12,600 liters are obtained. What is the flow, in liters per hour, of each pipe, and how long will it take to fill the tank if both pipes are opened at the same time ? 6. It is found that 21 kilograms of silver weigh 2 kilograms less in water than in air, and that kilograms of copper lose 1 kilogram when weighed in water. An alloy of silver and copper weighing 118 kilo- grams in air is found to weigh 14| kilograms less in water. How much silver and copper are there in the alloy? (See Art. 112.) Art. 212] GENERALIZATION TO :M0RE UNKNOWNS 351 7. Generalization of Ex. 6. p kilograms of metal yl lose n kilograms in water, p kilograms of metal B lose b kilograms in water. An alloy of the two metals, A and B, weighing p' kilograms in air, loses c kilo- grams in water. How much of each element is there in the alloy? 8. According to the story told by Vitruvius (see Art. 112), the crown of King Hiero, composed of gold and silver, weighing twenty pounds in air, lost 1.25 pounds when weiglied in water. liut 1!).(J1 jJOiinds of gold and 10.') pounds of silver each lose one pound in water, AVhat was the composition of the crown? 9. Two trackmen are practising on a circular track 126 yards in circumference. When running in opposite directions, they meet every 13 seconds. Running in tiie same direction, the faster passes the slower every 126 seconds. How many minutes does it take each of the men to run a mile ? 10. The planet Mercury makes a circuit about the sun in 3 months. Venus makes a circuit in 1\ months. Find the number of months be- tween two successive times when Mercury is between Venus and the sun. 11. Find three numbers whose sums, taking two of them at a time, are equal to a, h, c respectively. 12. Three brick masons. A, B, and C, are building a wall. A and B alone would require 12 days to complete the job ; B and C would require 20 days; and A and C would require 15 days. How much time would be required for each working alone, and how long will it take them to build the wall if all of them work together? 13. Generalize Example 12 by substituting a for 12, h for 15, c for 20. 14. A cistern is filled by three pipes. A, B, and C. A and B together will fill it in 70 minutes, A and C in 84 minutes, B and C in 140 minutes. How long will it take each pipe separately to fill the cistern ? How long when all of them are running at once? 15. A first mass of alloy contains 5 oz. of gold, 15 oz. of silver, and 30 oz. of copper. A second mass contains 20 oz. gold. 28 oz. silver, 48 oz. copper. A third alloy is composed of 12 oz. gold, 39 oz. silver, 24 oz. copper. How much must we take of each of these alloys in order to obtain a fourth alloy composed of 10 oz. gold, 25 oz. silver, and 26 oz. copper? 16. A certain number contains three digits in arithmetical progres- sion. If the number he divided by the sum of its digits, the quotient is 48. If 108 be subtracted from the number, the remainder will have the same digits as the original number, but arranged in the opposite order. Find the number. CHAPTER X PERMUTATIONS AND COMBINATIONS 213. The notion of order. If we have n elements of any kind such as numbers, letters, chairs, tables, animals, or per- sons, we may think of these n elements as being arranged along a straight line. In any such arrangement we shall speak of one of the elements as the first, another as the second, and so on. Any arrangement of this sort differs from any other arrangement of the same elements merely in the order in which the various elements are thought of, not in the total number of elements included in the arrangement. 214. Permutations. Each of the various ordered arrange- ments tvhich can be made of n elements is called a permutation of these elements. The principal problem which we shall have to solve is this : how many permutations are there of n elements, if each of the elements occurs in each permutation ? Clearly a single element can be arranged in one way only. If we have two elements, let us represent them by the letters a and h.* There are clearly two arrangeinents and only two in this case; namely a first one with a in the first place and h in the second, and a second arrange- ment with h in the first place and a in the second. We may represent these two arrangements symbolically as follows : ab and ha. In the case of three elements, a, b, and c, there are Two arrangements, ahc and ach, in which a occupies the first place. Two arrangements, bac and bco, in which b occupies the first place. Two arrangements, cab and dm, in which c occupies the first place, or 3 . 2 = 6 arrangements altogether. Thus there are six permutations of three elements. * This is somewhat of a departure from our general practice. So far we have used the letters a and b only to stand for numbers. In this connection, however, a and b may represent two different persons, two animals, or two elements of any kind. 352 Art. 214] PERMUTATIONS OF n ELEMENTS 353 Let us consider the case of four elements, a, b, c, d. By our previous argument there are six permutations in which a occupies the first place. For, after a has been put into tlie first place there remain three other elements, 6, c, r/, which may be permuted among the remaining places. Similarly there are six permutations witli h in the first place, six with c in the first place, and six with tl in the first place, or 4 x 6 = 24 permu- tations in all. Let us now pass to the general case. We denote the n elements by (1) a^, a^. «3, .•• a„, and we use the symbol P„ to denote the number of permutations of these n elements. If we omit any one of the elements (1) there are n — 1 left, and the number of permutations of these n— 1 elements will be represented by the symbol P„_j. Now all of the permutations of a^, a^. •••, a^ may be divided into n classses according as a^, or «2i ^^i' «3' ••••> or a„, occupies the first place. The number of permutations in any one of these classes is represented by tlie symbol Pn-\- I' or, the first element having been fixed, there are onl}^ n — 1 elements left to be arranged. Since there are n such classes, there will be n times Pn-i permutations in all. But we have denoted the total number of permutations of n elements by P„. There- fore we must have (2) P,, = nP^_y We have found already Pj = l, P2 = l-2, ^3= 1.2. 3 = 6, P4 = 1.2.3.4 = 24. By using (2) we now find P^ = .5P4 = 1 • 2 . 3 . 4 . 5 = 120, Pg = 1.2. 3. 4. 5. 6 = 720, suggesting the general formula (3) P„ = l .2.3.4. ..w. The product of all of the integers from 1 to w, which makes its appearance here, is called factorial n, and is usually de- 354 PERMUTATIONS AND COMBINATIONS [Art. 215 noted by one of the two symbols n ! or \n. Thus (3) be- comes (4) Pn = n ! (read P sub n is equal to factorial ti). To complete the proof of this formula, we use the method of mathematical induction. We know that (4) is correct for w = 1, 2, 3, 4, 5, 6. We can prove that if (4) is correct for n = k^ then it ivill also be correct for n= k+ 1. In fact, if P, = ;^! = 1.2.3...yt, then, according to (2), P,^^=(k+l)P, = kl(k+l)=1.2.S:.k.k+l={k + l^U tlius proving our assertion, and consequently the validity of (4) for all values of w. 215. The number of permutations of n elements taken k at a time. Suppose again that we have w elements a^, ag^ •*• ^w and let us examine in how many ways these may be arranged in groups of k elements each, where k S w? attention being given to the order of the arrangement in each of the groups of k elements. The following is a concrete problem of this sort. There are 25 base- ball players in a college (n = 25). Each of them is willing to take any position on the college team composed of nine players {k = 9). In how many ways can the team be formed? Each of the required permutations is composed of k ele- ments. The first place may be filled in n ways, since it may be occupied by any one of the n elements. After the first place has been filled there are only n — 1 ways of filling the second place. Thus the first two places may be filled in w(w — 1) different ways. There are 7i — 2 elements still available for filling the third place, so that the first three places may be filled in n{n — 1) (w — 2) ways. If we continue this argument we see that the k places may be filled in n(n — l)(w — 2) ••• (ri — A; + 1) dif- ferent ways. Art. 215] n ELEMENTS k AT A TIME 355 Let us now use the symbol „Pt to,denote the number of per- mutations of n elements taken k at a time. We have found (1) „P, = n(w - l)(w _ 2) ... (n - y^ + 1), the right member being a product of k factors, namely of n — 0, n — 1, n — 2, ••• n — (k — 1). Of course for k = n, we find, as in Art. 214, Since we may write nl=n(n-l}(H-2^ ... (n-k + l)(n-^^(n-k-l) ...2.1, and since the product of the first k factors in the right member is „Pfc, while the product of the remaining n — k factors is Qn — ^) !, we have n\ = ^P,(n-ky., whence n ! (2) . „P. = (n-k)\ EXERCISE XCVI 1. In how many ways can eight soldiers be arranged in a row ? 2. How many permutations of the letters a h c d e f g are there if each of the letters occurs in each permutation ? 3. Of the permutations mentioned in Ex. 2, how many begin with a? How many begin with aft? with abc'i with ahcd'i 4. How many of the permutations mentioned in Ex. 2 contain the letters ahcd consecutively and in this specific order? 5. How many different permutations are there of the letters of the word " stone " when tliree are taken at a time? 6. There are 15 baseball players in a college, each of whom is will- ing to take any position on the college team. In how many ways can a team be formed ? 7. How many of the niimbers between 10 and 100 contain two dis- tinct digits, not counting zero as a digit? 8. How many numbers of tin-ee different digits can be formed from the seven digits 1, 2, •••, 7? 356 PERMUTATIONS AND COMBINATIONS [Art. 216 9. With eight flags of different color, how many signals can be formed by displaying four of them at a time? 10. How many permutations are there of n things taken r or fewer than r at a time, that is, if it be admitted that we may select only one, or two, or three, •••, or as many as r of these things? 11. How many signals, composed of one, two, or three flags can be formed from five different flags ? 216. Circular arrangements. The notion of order which we introduced in Art. 213 may be called more specifically linear order, since it is suggested by the arrangement of n objects placed in a row, or on a straight line. If the n objects are instead placed on the circumference of a circle, or on any other simple closed curve, we may adopt the point of view that, in such a circular arrangement, it is unnatural to fix upon any one of the elements, rather than upon any other one, as being the first. Thus, in Fig. 69, we may consider that the three points A, B, C have the same circular order whether we start from A and then proceed in the order ABC, or wliether we start from B and proceed in order to and A, or finally whether we start from O and proceed in the order CAB. Thus, although there are six permutations of these three elements, there are only two cir- cular permutations. In one of these permu- tations we go around the circle in clockwise fasliion ; and the other order may be described as a counter-clockwise arrangement. In general, if we wish to arrange n elements in circular order, we may place any one of the elements in a fixed position and leave it there. The remaining elements can then be arranged in (rt— 1)! different ways. Therefore, there are (w — 1) ! circular permutations of n elements. In some cases there are pairs of clockwise and counter-clockwise arrangements which are indistinguishable; namely, whenever it is admissible to turn the circle around a diameter through an angle of 180°. In such cases the number of arrangements reduces to ^(w— 1) I. Art. 217] ELEMENTS NOT ALL DISTINCT 357 EXERCISE XCVll 1. lu placing a party of people at a round table, two arrangements are regarded as equivalent which give each person the same left and right hand neighbors. How many different ways are there of seating 8 people at a round table ? 2. Seven beads of different colors are to be strung on a closed wire. In how many ways may this be done ? 3. Six distinct points have been selected on a closed curve (an ellipse for instance). How many different hexagons can be formed with these six points as vertices? (Hexagons whose perimeters intersect themselves are admissible.) 4. In how many orders can a liost ami seven guests sit at a round table so that the host may have the guest of highest rank upon his riglit, and the next in rank on his left? 5. If we have « beads of different colors to form a bracelet, how many distinguishable arrangements are possible? 217. Permutations when all of the elements are not distinct. It happens quite frequently that the elements which are to be arranged are not distinct. Such is the case, for instance, if we wish to answer the following question. How many distinct words of eight letters each can be formed from the letters i, Z, ?, i, n, o, ^, s ? There will not be as many as 8 !, because several of the 8 I permutations of these eight letters will give rise to the same word, since there are three I's and two Vs. The general question is easily answered. Among the n symbols which are to be arranged, let us suppose that a occurs r times, b occurs s times, c occurs t times, and so on. We find it convenient, for the purpose of our proof, to write aj, rTg, ^3, •••, a^ for the r symbols a ; b-^, b^. •••, b^ for the s sym- bols b ; and so on. Thus we have r symbols a^, a^, •••, a^, each of which means the same as a, (1)8 symbols b^, b^, •••, 6,, each of which means the same as b, t symbols c^, c^^ •••, c^, each of which means the same as c, etc. etc. Since there are w symbols all together (counting repetitions), we have n = r + s + t -\- ■■■.^ 358 PERMUTATIONS AND COMBINATIONS [Art. 217 and there are n ! permutations of w symbols. But not all of the n ! permutations will be distinct. Let X be the number of these permutations which are distinct, and let us imagine that all of these X permutations have been written down. From each of these X distinct permutations we can obtain r ! permutations by interchanging the r symbols a, leaving the 5's, c's, etc., fixed. We obtain in this way X • r\ arrange- ments. From each of these we can obtain s ! arrangements by permuting the 6's, leaving the «'s, c's, etc., fixed. This gives rise to X ■ r\ s\ arrangements. Proceeding in this way, we find that the total number of arrangements will be Xrlsltl-.' . But, since there are n symbols all together, the total number of arrangements is also equal to m !. Therefore X 'r\s\t\ -" =n\, or (2) X = r\s\t\ ..• Consequently, (2) gives the number of distinct permutations of n elements, r of which are a's, s of which are b's, t of which are c''s, and so on. EXERCISE XCVIII 1. How many permutations can be made of the letters of the word Illinois ? 2. How many permvitations can be made of the letters of the word Mississippi V 3. A desk has r pigeonholes; n documents are to be filed in these pigeonholes so that a of them sliall go into the first, jS into the second, y into the third, and so on. In how many ways may this be done? Remark. The pigeonholes are distinguished from each other as first, second, and so on. But it is regarded as indifferent in what order the documents are arranged inside of the holes. 4. How many different numbers of seven digits each can be formed by permuting the figures 1112225? Aim. 218] TWO CLASSES OF PERMUTATIONS 359 5. Piove the following theorem. // there are n distinct elements Ou «2> •••) o„, which are to be arranged in sets of r elements at a time, and if it be permitted that each element be repeated as often as r times, then there are n'' permutations. 218. Two classes of permutations. It happens frequently that some particular arrangement of 7i elements is regarded as more important or more natural than any other. Thus, if the elements under consideration are the n numbers 1, 2, 3, •••, w, we naturally think of the smallest number first and then arrange the others in the order of increasing magni- tude. If the elements are letters a, b, c, d, ••-, we naturally tliink of their alphabetic order as^being the most important. Whenever, for ayiy reason, one of the n ! permutations of n symbols is to be regarded as more important than any other, we call it the principal permutation. Let X, c, m, y, b, ••• be tlie principal permutation of a sys- tem of n symbols. We may rename x and call it aj, rename c and call it a^, and so on. Thus, by changing the names of the elements, if necessary, we cati always fix our notation in such a ruay that the principal permutation will assume the form rtj a^a^ ••• rt„ in ivhich the subscripts 1, 2, 3, •••, n appear in their natural order. In every other permutation of w elements a^, •••, a„, some of the lower subscripts will be preceded by a higher one. Every instance of this kind is called an inversion. By the number of inversions in a permutation we mean the total number of instances in ndiich a loiver subscript is preceded, in that per- mutation, by a higher one. Thus, in the permutation 23514G7 of the numbers from 1 to 7, there are four inversions. 1 is preceded by 2, 3, 5 (3 inversions). Neither 2 nor 3 is preceded by a higher number. 4 is preceded by 5 (1 inversion). 5, 6, 7 are not preceded by higher numbers. Thus there are 3 + 1 = 4 inversions. A permutation is called even or odd according as it contains an even or an odd number of inversions. 360 PERMUTATIONS AND COMBINATIONS [Art. 218 The principal permutation contains no inversions and is regarded as an even permutation, thus leading ns to classify zero as an even number. Theorem 1. If any two elements of a permutation are in- terchanged^ the class of the pefmutation tvill he changed from even to odd, or vice versa. Proof. Let us begin with the case in wliich a pair of adjacent elements are intei-changed. Let these adjacent elements be called a^ and a^ and suppose that /i< ^•. We may represent by a single letter .1 the collec- tion of all of the elements of the permutation which precede the pair QhOki and by B the collection of all those elements which follow Offlif. If the original permutation was (1) Aa^akB, h < k, the second one, after the interchange of o^ and a^., will be (2) AakOhB, h < A-. In both of these permutations every element of A, and also every ele- ment of B, is preceded by exactly the same elements, and therefore by the same number of elements with higher subscripts. But in (2) a^ is l^receded by a/c (/; < k). Therefore (2) contains one more inversion than (1). If, instead, Aaua^B was the oi'iginal permutation, and Aa^akB the second one, we see by the same argument that the second permutation contains one inversion less than the original one. In either case, the number of inversions is changed by one, and therefore the class of the permutation is changed from even to odd or vice versa. If the two elements, o^ and Uk, which axe to be interchanged are not adja- cent, we may proceed as follows: Let there be m elements, ci, C2, ■■•, r„„ between a^ and a^, so that the original permutation may be represented by (;3) Aa!,CiC2 ••• CmOkB, where A represents the collection of elements which precede a^, and B the collection of elements which follow aa.-. We can accomplish the interchange of «/, and rtt by a series of operations each of which con- sists in merely interchanging adjacent elements. Thus, we obtain, in order, the following permutations : AayfivCiCz ••• c^-iCmakB, the original permutation, Acia^^C2C:i ••• Cm-iCmakB, result of 1st operation, AcidtthCs •■• Cm-iCmCikB, result of 2d operation. AciC2CsCi ••• ('m-i'^'hCm.(ikB, result of m — 1th operation, Aciczc^d ••• c,n-\c„/iiflkB, result of mth operation. Acic-zCzCi ••• Cm-ic^QkahB, result of m + 1th operation. Art. lMO] COMBINATIONS 361 After these m + 1 operations, Uh actually occupies the place originally occupied by a,;. But a^t is not yet in the place originally occupied by o^. To get it into that place, we change the last of the above permutations by interchanging a^ and c„, giving the new permutation Ac^c.fiC^ •■• QkC^ahB. As a result of m operations of this kind, we finally obtain the desired permutation (4) AokC^c.^ ••• c„ahB. Thus, the interchange of o/, and Uk is equivalent to ( »?+ \) + m — 1 m + 1 interchanges of adjacent elements. Since each of these interchanges changes the class of the permutation from even to odd, or vice versa, and since 2 m + 1 is an odd integer, the total effect of interchanging a^ and flfc will be to change the class of the permutation, and we have proved our theorem. We may also state this same theorem as follows: Theorem 2. The interchange of any two elements of a per- mutation changes the number of inversions by an odd integer. Let us think of a list composed of all of the n ! permutations of n elements. If we intei'change any two of the elements, say the first and second, in every one of these permutations, the list, after this change has been made, will still include all of the permutations but not in the origi- nal order. For two permutations different from each other before the change will also be different from each other after the change. But by this change every even permutation is converted into an odd one and vice versa. Hence the number of even permutations must be equal to the number of odd permutations. This gives the following theorem. Theorem 3. Of the total number (n!) of permutations of n elements, one half are even and the other half are odd. These theorems are of great importance in the theory of determinants of the ni\\ order. (See Chapter XII.) EXERCISE XCIX Count the number of inversions in each of the following permuta- tions of the natural numbers. 1. 43-21. 2. 1324. 3. .54321. 4. 13524. 219. Combinations. If n elements are given, we may select k of these elements in viirious ways. Erery set, of k elements each, which can be obtained from n elements, no atten- 362 PERMUTATIONS AND COMBINATIONS [Art. 219 tion being given to the order in which they may be arranged, is called a combination of the n elements. Thus out of three elements, a, h, c, we can form three combinations if we take two at a time ; namely be, ca, and ab. The combinations be and cb are the same, since no attention is to be paid to the order or arrange- ment of the elements, although the permutations be and eb are different. We have denoted by ^Pk the number of permutations of n elements takmg k of them at a time. Let us denote simi- larly by J^C^. the number of combinations of n elements taken A; at a time. From each of these combinations we can form h ! permutations by writing the k elements of the combina- tion in their k ! different orders. Consequently we have SO that we find (1) nOk = '^= ''• k\ kl{n-k)l' if we make use of formula (2) of Art. 215. This formula may also be written as follows ,„. /7 _ yt(n-l)(n-2) ... (n-k + 1) C^; n^k- ^^ ^, on account of (1), Art. 215. Whenever we select k elements out of a total number of n elements, there are n — k left. Consequently, to every combination of n elements taken ^ at a time, there corre- sponds just one combination of the n elements taken n — k at a time. There must exist, therefore, just as many com- binations taken ^ at a time, as there are combinations taking n — k at a time, that is, it must be so that C") n^k=^n^n~kl a formula which may also be verified directly by means of (1) or (2). Of course, we have in particular (4) nC-'i=„C„_i = 7l, and (5) „(7. = 1, Art. 220] INDEPENDENT COMBINATIONS 363 since there is only one combination of n things taken all together. The application of formula (3) in this case would give and we shall occasionally, as a mere matter of convenience, make use of this formula although, from our original point of view, combinations which contain no element were not contemplated. In deducing formula (1) we assumed that the elements which were to be combined were all different from each other. The formula, of course, undergoes a modification if some of the elements are alike. 220. Independent combinations. The following problem in combinations arises frequently. There are « classes of elements of which the 1st class contains a elements, the 2d class contains h elements, the 3d class contains c elements, etc. etc. etc. We are to form combinations of s elements by picking out one element and only one out of each class. In how many ways may this be done? We select any one of the elements of the first class, any one of the second class, any one of the third class, and so on. Clearly there are i different ways of doing this. In particular, if each of the « classes contains the same number a of elements, the number of combinations is a'. EXERCISE C 1. Write out all of the combinations which contain two of the five symbols a, b, c, d, e. 2. Write out all of the combinations which contain three of the five symbols a, h, c, d. e, and state a reason why the number of these combi- nations is the same as of those found in Ex. 1. 364 PERMUTATIONS AND COMBINATIONS [Art. 221 3. If four points in the same plane, no three of which ai'e on the same straight line, are joined in pairs by straight lines, how many of these lines will there be ? Note. The configuration obtained in this way is called a complete quadrangle. 4. In how many points will four straight lines of the same plane intersect? Note. Four lines of the same plane together with all of their points of intersection are said to form a complete quadrilateral. 5. How many lines are obtained by joining n points of the same plane by straight lines in all possible ways? 6. If we have a system of n points in space, such that no four of these points lie in the same plane, how many planes do we obtain by passing a plane through each set of three of the n points? 7. Prove „C;, = „C„_i by using (1) or (2) of Art. 219. 8. Find the values of ^^C^^ and looQs- 9. How m.any committees of 9 can be selected from a group of 12 men ? 10. A committee of six is to be selected from seven Englishmen and four Americans. The committee is to contain at least two Americans. In how many ways may the committee be chosen ? 221. The binomial theorem. The foinuila (2) of Art. 219 for „(7^ is precisely the same as the expression (2) of Art. 88 for tlie coefficient of rc"~V' in the expansion of (x + a)". Of course this agreement is not a mere accident. In fact, we shall now give a new proof of the binomial theorem based on the theory of combinations. Let us consider n binomials .r + f7i, x + a-y, ■■•. x -\- a„ and let us form their product, (1) (.r + (t\){x+ n.^ ••• (.r + «„)• If this product is multiplied out we shall, of course, obtain an integral rational function of x, of degree n. The complete product will be the sum of all of the partial products which can be obtained by selecting one and only one term from each of the binomial factors of (1), and nniltiplying these together. In order that such a partial product may contain exactly the (n — ^•)th power of x (no higher and no lower power) as a factor, n — k of the factors of the partial product must be x's, and the other k factors must be a's. Consequently, there will be as many partial Art. 222] TOTAL NUMBER OF COMBINATIONS 365 products of tlii.s kind (containing x"~'-" as a factor) as there are ways of selecting L- of the as from ai, a^, ••-,«„• Tiierefore, there will be „Ca partial products iu the expansion of (1) which contain x''~* as a factor. Each of these partial products will reduce to x"~*"-i 4- „C,a;"-- + ••• + „tV~*a* + ••• which is precisely tlie binomial formula, if we remember that (A) c - ^(" - 1) ••• (» - A-+ 1) 222. Total number of combinations. If we have n objects, we may form combinations of them, taken one at a time, two at a time, ••• , ^ at a time, •••, and finally all of them together. The total number of combinations will be But if, in equation (2) of Art. 221, we put 2; = a = 1, we finrl (l + l)n=l + „C\ + „C2+ ••• +na, SO that the total number of combinations is equal to (1) nO,+ nC,+ •■■ +„C„ = 2"-1. EXERCISE CI 1. How many different sums of money may be formed with a penny, a nickel, a dime, a quarter, a half dollar, and a dollar? 2. A merchant has a set of 12 different weights. IIow many differ- ent weights can be obtained from these by combination? 3. In how many ways can two letters be filed in five pigeonholes? 4. In how many ways may 52 cards be dealt to four persons, if each person plays for liimself ? (Only those ways are to be regarded as differ- ent which have some influence on the game.) CHAPTER XI PROBABILITY 223. Definition of Probability. As our knowledge of Natui-e becomes deeper and more extensive, we become more and more convinced that tlie laws of Nature are permanent and universal and that, in the strictest sense of the word, there is no such thing as chance. Nevertheless there are many phenomena which may be studied from the point of view of chance. Thus, if we toss a coin, it is quite evident that the peculiar way in which we toss it will cause it to land either head or tail and that the result, whether head or tail, is really determined by our method of tossing and the laws of mechanics. But these laws are difficult to apply to any specific instance of this sort, and a very slight impercep- tible change in our method of tossing would change the result. We therefore profess complete ignorance of the causes which govern the process of tossing coins, and say that the two possible results, head or tail, are equally likely or equally probable, and that the probability of tossing a head is equal to |. Again, if a bag contains three white and seven black balls, and if we draw out a ball at random, we say that the probability of drawing a white ball is -j^^, since there are three chances out of ten for this. In all such cases, to use general language, we estimate the probability of an event. In the above illustrations, the event is either the appearance of a head after the coin has been tossed, or the appearance of a white ball as a result of the drawing. By a trial we mean any operation which gives the event an opportunity to happen. In our illustrations the tossing of the coin, or the act of drawing a ball from the bag, are trials. 366 Art. 223] DEFINITION OF PROBABILITY 367 In estimating the probability/ of an event, it is very important, first of all, to decide which ones of the various possibilities are to be regarded as equally probable. Thus in tossing a coin tliere is no reason to suppose that one side will turn up rather than the other. Consequently we say that the probability of a toss resulting head is ^, and the probability for tail is also |. But suppose we aie tossing two coins. Either face of either coin may turn up, giving 2 x '2 = i possibilities. We might be led to conclude that a double head would be just as likely as a combination of head on one coin and tail on the other. But this would be erroneous. For a double head is only one of four possible cases, so that its probability is |. But there are two chances out of four for a combination head-tail, since we obtain such a result in both of the following cases : (1) the first coin lands head and the second tail, (2) the second coin lands head and the first lands tail. Thus, the probability of tossing head-tail with two coins is I or |. Suppose we have made a list of all of the results which mag possibly apjjear when we make a trial, and that each of these results is as likely as any other. Let us call each of these possibilities a case, and let there be m + n cases all together. Suppose that m of these cases are favorable to the event under consideration ; that is, let ^ls suppose that each of the m cases makes it certain that the event will happen. Let us suppose further that each of the remaining n cases is unfavorable, that is, makes it certain that the event will not happen. Then ice say that the probability of the event is .^ . m the ratio of the favorable to all possible eases. The probability that the event will not happen is (2) p' =^^1— m + n so that (3) p + p'=l, (•i) p'=l-p. 368 PROBABILITY [Art. 224 Therefore, the sum of the probability that an event will happen and the probability that it will fail is equal to unity. If all possible cases are favorable, then w = so that, according to (1), we have p = 1. If all possible cases are unfavorable, then ^ = 0. In all other cases p is a positive proper fraction. Consequently the probability of an event is always a positive proper fraction unless we are certain either that the event will happen or else that it will fail. Tlie prob- ability of an event which is certain to happen is equal to unity; that of an event which is sure to fail is equal to zero. EXERCISE CM 1. A die has 2 white and 4 black sides. AVhat is the probability that a white side will turn up ? 2. Five coius are tossed. What is the probability that the result will be five heads ? 3. What is the probability of tossing 4 heads at one throw with 5 coins? 4. What is the probability of tossing 3 heads at one throw with 5 coins ? 5. Show that if there are n coins, the probability of /• heads and n — r tails at one throw is nCr/'^'^- 6. An ordinary die has six faces marked 1. 2, .'}, 4, .3, 6 respectively. What is the probability of throwing two sixes with two dice? 7. What is the probability of a score of 11 with two dice? 8. If three dice are thrown, what are the probabilities of throwing three sixes? two sixes and a five? a six, a five, and a four? 224. Compound events. It is frequently convenient to think of an event as being composed of several simpler events. Thus, if we are tossing two coins, we may i^refer to think of what hap- l)ens (head or tail) when we toss each of the coins separately, and speak of the result of tossing the two coins together as a compound event. But whenever we decompose any event whose probability is to be computed into simpler events, it is very important to know wliether the component simpler events are independ- ent, dependent, or exclusive. Akt. 2-H] COMPOUND EVENTS 369 Two or more events are said to be independent or dependent, aecording as the occurrence of one of them at a given trial does not or does affect the prohahility of the others. Suppose we are drawiug two balls out of a bag containing three white and seven black balls. We may decompose this process into two separate drawings of one ball at a time. The probability of a white ball at the first drawing is y*j. But if we have actually drawn a white ball, the probability of a white ball at the second drawing will be |. If the first ball had been black, the probability of a white ball at the second drawing would be |. Thus the probability of having the second drawing result in a white ball is aifected by the result of the first drawing. If instead tlie balls are returned to the bag after each drawing, the probability of a white ball will be the same at each trial. In this case, the event of drawing a second white ball would be independent of the event of drawing a first white ball. Two or more events are said to he exclusive if only one of them can happen. For instance the two events of a single coin landing head and landing tail at a single toss are exclusive. Either may happen, but not both. If |)j, p.^, ••'-, Pn <^>'^ the probabilities of n independent events, the prohahility that all of these events will happen together, at a given trial, is (1) P=PlP'i-' Pn^ the product of their separate probabilities. Proof. Let us suppose that there are two events only. Let there be rtj cases favorable to the first event and />j which are unfavorable. Then the probability of the first event is Pi = ij_ Oj + i, Similarly we shall have ^2 = — fr if a„ cases are favorable to the second event, and ^2 are unfavorable. Since the events are independent, both of them will happen together in a.^a.2^ cases out of a total number of (ai + h\)(^a-i + b-i) cases, thus giving , n^a„ 370 PROBABILITY [Art. 224 as the probability that both may happen together. We may now regard the combination of these two events as a single event of a probability p'. The probability that a third independent event of probability /jg may also happen will therefore be P' Pz = PiP-.Pz- If we continue to reason in this way, we finally obtain (1) which was to be proved. We have a similar method for calculating the probability of a com- pound event which is composed of several dependent events. This method may be expressed by means of the same formula (1), but with a somewhat different meaning attached to the symbols pi, p.j, •••, jo„, as ex- plained in the following theorem. Let p^ be the prohahility of a first event ; let p^ he the proba- bility of a second event after the first has happeiied ; let p^ be the probability of the third event after the first two have happened ; and so on. Then the probability that all of these events will occur together is (2) ^ = PiP2Ps---Pn- Proof. Let us suppose first that there are two events only. If there are a^ cases favorable and b^ cases unfavorable to the first event, we have «i pi- «i + ^1 Now both events cannot happen together unless the first event happens. Consequently the cases which favor both events can be found only among the Oj cases which favor the first. If 62 of these cases favor the second event, the probability that both events will occur is P' = Oj + 61 If the first event has actually happened, one of the a^ cases favorable to it must have occurred. Since b^ of these a^ cases are favorable to the second event, the probability of the latter, after the first has occurred, will be But we have I _ ^2 — ^1 ^2 _ thus proving our theorem in the case of two events. The extension to n events is made as in the proof of the preceding theorem. Art. 224] COMPOUND EVENTS 371 The following theorem is concerned with mutually exclu- sive events. Let pi^ p^, ••■,Pn ^^ i^^ probabilities of n mutually exclusive events. The jj^'obabiHt// that one of these events (we care not which} will occur, is equal to the sum of their separate probabilities. Proof. Let there be m equally probable cases. Let ai of these be favorable to the first event, a., to the second event, and so on. Then a, a„ (/„ ■^' to' to to Since the events are mutually exclusive, the «, cases which favor the first event, the a.^ cases which favor the second event, and so on, must be entirely different from each other. Consequently there are just «i + flo + ••• + On cases among the m eqiuiUy jii-obable cases which are fa- vorable to one of these events, and the probability of one of these events is a-, + a.y + ■■■ + a,, ' ' -^= P1 + P2+ - +Pn, 711 as was to be proved. EXERCISE cm 1. A traveler has five connections to make in order that he may reach his final destination on time. He estimates that, for each of these con- nections, the chances are two to one in his favor. What is the probability of his getting through on time? 2. A traveler argues as follows before starting on a sea voyage. It is an even chance that the ship will encounter a storm. The probability that she will spring a leak in tlie storm is ^V- If ^ l*^^k occurs the chances are 9 to 10 that the engine will pump her out. Tf they fail, the chances are 3 to 4 that the compartments will keep the ship afloat. If she sinks, the chances are even for him to be saved by a boat. What is the prob- ability of his being lost at sea? 3. The probability that A will live ten years is |, and the probability that B will live ten years is ^. Wliat is the probability that both of them will be alive after ten years? 4. A bag contains two white, three black, and four red balls. If a ball be drawn from the bag at random, what is the probability that it will be either white or red ? 5. What is the prol)al)ility of throwing either an ace or a deuce in a single throw with one die ? 372 PROBABILITY [Art. 225 225. Repeated trials. If p denotes the prohahilitij that an event will happen on a single trials the prohahility that the event will happen exactly r times in n trials is equal to (1) ^CrpX^-py-'-. Proof. We separate the proof into three cases ; r = 0, r = 1, and the general case when r has any value. I. It r — 0, the event does not happen at all ; it must fail at each of the n trials. If p is the probability of the happening of the event at any trial, 1—p is the probability of its failing to happen (Art. 223). Therefore, since the n results which correspond to the n trials are inde- pendent, the probability of failure in all n trials is (1 — p)" (Art. 224). II. Let r = 1. If the event is to happen once and only once, it must happen at one trial and fail at the other n — L The probability of its happening at a specific one of these trials, say the first, and failing at the other n — 1, is jt»(l — p)""^. (See Art. 224.) But since there is an equal opportunity for it to happen at each of the n trials, we find the value np(l — p)"~^ for the probability that the event will happen just once in n trials. III. The general case. In order that the event may happen exactly r times, it must happen on some combination of r trials and fail on the remaining n — r trials. The probability that the event may happen on any specific combination of r trials and fail on the remaining n — r is /)-(! - p)"--. (See Art. 224.) Since there are „C, combinations of r trials out of n, the probability that the event shall happen exactly r times in n trials is as stated in the theorem. Of course, on account of (3) of Art. 219, this is also equal to „C„_j;-(l -/))"-. In many cases we are interested in the question whether the event occurs at least r times in n trials rather than whether it occurs exartly r times in n trials. Such is the case, for instance, if a chess phiyer under- takes to win three games out of four. He will make good his intention if he wins either three or four games. The solution of the question is immediate. The event will happen at least r times, if it happens exactly n times, or n — 1 times, •■• , or r + 1 times, or r times. Therefore (see Art. 224) the prnfxiJiillti/ Ihal nn ereiil of prohahilitii p will happen (it least r times in n trials Is (2) p'^ + r,c..^p--\\ -p) +/Voi»"-'(i -py+ - + uC\p\i-py. Art. 226] APPLICATION TO LIFE INSURANCE 373 EXERCISE CIV 1. Prove by the methods of Article 225 the result of Example 5, Exercise OIL 2. What is the probability that in six throws of a coin, at least three will result in heads ? 3. A is playing chess with B. Ilis chances of winning any one game from B are 2 to 1. What is the probability that A will win three games out of four? 226. Application to life insurance. When a company insures a man's life it makes a contract with him according to which the company agrees to pay his heirs, after proof of death has been offered, a certain sum of money, in return for payments made by him to the company at certain stated intervals during his lifetime. These payments, called premiums, may be continued during the whole life of the insured person, or else for a stated number of years. The company undertakes a risk in each individual case, since the insured may die long before his premiums are sufficient to cover the sum for which the company is liable to his heirs. To put the insurance business on a sound basis it is necessary, therefore, to know what is the probability that a person of a certain age shall live a certain number of years. In order to be able to do this, actuaries have constructed so-called Tables of Mortality which are based on statistics. Suppose we make a record of 100,000 people at the age of 25 and find that, at the end of a year 99,194 of them are still living. We should conclude from these data that the proba- bility of surviving one year, for a person 25 years of age, would be 99,194/100,000 = 0.992, The results obtained in this way become more reliable the greater the numlier of cases which have been taken into account. The Appendix contains such a table of mortality. The general i)rinciple, upon which this method of esti- mating probabilities is founded, may be formulated as follows : 374 PROBABILITY [Art. 226 If n is a very large number and if observations show that a certain event has happened m times in n possible cases, the probability of this event is equated to m/n. In other words, the assumption is made that at the next opportunity the event will again happe7i m times out of a possible n. To find the probability p that a person of age a will live to age 5, we take from the table the number of persons living at age b and divide it by the number living at age a. If the insured person of age a promises to pay $ m per year until he is b years old, the value of this promise to the company may be calculated as follows : His first payment is certain, since the policy will not be in force until the first payment is made. The second payment is contingent upon his living at least one year. Let p^ be the probability for this, as obtained from the table of mortality. Then the value of the promise to pay I m at the end of the first year, or the value of the company's expectation is estimated to be $7wpj. Similarly the value of the company's expectation for the third premium of I m is f mp^i if P2 ^^ ^^^® probability that the insured will survive two years. These estimates are in accordance with the principle that the value of a prob- able payment (or the expectation) is equal to the sum to be paid midtiplied by the probability that it will be paid. Of course, in the case of life insurance the actual value, to the company, of each payment is really greater than in- dicated above. For the money is invested by the company and produces income. Therefore any complete solution of the question of the value of these payments to the company must take into account the interest earned by each payment. The company must also make the proper deduction for the expenses connected with the conduct of its business and for a reasonable profit.* Only by investigations of the sort here indicated can the proper rates and the proper methods of conducting the busi- * For a more detailed discussion of life insurance and other applications of algebra to commercial questions, see E. B. Skinner, T/>e Mathematical Theory 0/ Investment, Giun & Co., l'J13. Art. 227] OTHER APPLICATIONS 375 ness of life insurance be found. Similar questions arise, of course, in accident and fire insurance. 227. Other applications of the theory of probability. There are many other applications of the theory of probability. One of the most notable of these is the theory of errors of observation.'"' Every measurement made by man is subject to inaccuracies due in part to unavoidable defects in the in- struments, and in part to physiological causes. Thus, if the same quantity is measured independently several times in succession, the results obtained will usually differ to some extent. The questions, how to obtain the most probable value from these discordant results, and how to find the probable error of this most probable result, are of great importance in physics and astronomy. * Consult a very readable presentation of this important subject by L. D. Weld, Theory of Errors and Least Squares, New York, The MacmillauCo., 191G. CHAPTER XII DETERMINANTS OF THE »iti> ORDER AND SYSTEMS OF LINEAR EQUATIONS "WITH n UNKNOWNS 228. Definition of a determinant of the nth order. We studied determinants of the second and third order in Cliapter IX. They were defined by the equations = a^ - a,^h^ (Art. 202). (1) and «1 «2 ^1 «i ^1 H (2) «2 h H «3 *3 H = «1^2^3 + «2*3^1 + ^3^1^2 - «1^3^2 " ^2*1^3 " ^3*2^1 (Art. 206). We shall now make use of these expressions to suggest by analogy the definition for a determinant of the nWi order. In both (1) and (2) we have a square array of numbers. The first term in the right member is, in both cases, the product of all of the elements which occur in that diagonal of the square array which passes from its upper left-hand to its lower right-hand corner. All of the other terms in the right member are obtained from this first or principal term by making a permutation of the subscripts while the letters are left in their alphabetic order. Those terms are preceded by a 'plus sign whose subscripts are so arranged as to form an even permutation, the principal permutation being 1, 2, 8. (See Art. 218.) The other terms are preceded by a minus sign. Thus 13 2 has one inversion (3 before 2). It is therefore an odd permutation and the corresponding term a\h;\c-i of (2) is preceded by a minus sign. Again 3 1 2 has two inversions (3 before 1, and 3 before 2). The corresponding term a-^\C.^ is preceded by a plus sign. 370 Art. 228] DEFINITION OK A DETKllMIXANT 377 We now proceed to dt'liiic a dL-tenninant of the nth order. 1. Consider a >(quare arrai/ of n^ numbers^ each of ivhich is called an element of (he determinant ; this array may he. writ- ten as follows : (3) ^1' 1"' ^r "■■' 1' ag, Oj' ^2'' ■"' ^2'' = We wish to show that ai h Cl • • ^1 ao h Co ■ • k. a„ ^n r„ • ■ K D' Oi 02 as ■ • «n ^'l h-2 h ■ • f'n ^-1 h h ■ ■ K Z>= D'. We observe in the first place that, when we apply the definition of Art. 228 to each of these determinants, exactly the same terms will appear in both, since these terms are formed by computing all possible products of n elements obtained by taking one and only one element from each row, and one and only one element from each column. ]Moreover, the corresponding terms of the two determinants will have the same sign prefixed to them, on accoimt of the final theorem of Art. 229. Therefore the two determinants are equal. 2. If all of the elements of a row or column of a determi- nant are equal to zero, the value of the determinant is zero. For every term of the determinant will contain as one of its factors an ele- ment of the row or column in question, and will therefore be equal to zero. Art. 230] PROPERTIES OF DETERMINANTS 381 3. If two roivs or columns of a determinant are interchanged^ the relative order of the elements in such rows or columns not being altered^ the determinant merely changes its sign. Proof. Change of two rows is equivalent to an interchange of two of the subscripts (row marks) in every term of the determinant. But such an interchange increases or diminishes the number of inversions among the subscripts by an odd number (Theorem 2, Art. 218), and therefore clianges the sign of tlie corresponding term of the determinant. (No. 4, Art. 228.) Similarly for interchange of two columns according to tlie final theorem of Art. 229. 4. If all of the elements of a row (or column) are multiplied by the same factor w, the determinant is multiplied by m. Proof. Of the elements which are multiplied by m, one and only one occurs in every term of the determinant. 5. A determinant of the nth order is equal to zero if two of its parallel lines (rows or columns) read alike, that is, if all pairs of corresponding elements in two parallel lines are equal to each other. The proof is exactly the same as for determinants of the third order. (See Theorem 5, Art. 208.) 6. If every element of any row or column is expressed as a sum of two terms, the determinant may be expressed as a sum of two others. For instance, "l + '"l» h, .. ki (lu bi, ■■ h mi, bu •■ h do + 1112, h, ■ ■ hi = (!■>, b,, .. h + '«2, b,, ■■ 1-2 «„ + m,., K, ■■ f^-n ««. bn, ■■ K m„. />„, • K To prove this theorem observe that, as a result of the hypothesis, any term of the original determinant may be expressed as a sum of two terms, each of which is one of the terms occurring in the expansion of another determinant. 7. If to the elements of any row or column we add the corre- .^ponding elements of a parallel row or column, multiplied by one and the same factor, the value of the determinant is not changed. 382 DETERMINANTS OF THE n**^ ORDER [Art. 281 Proof as in the case of third order determinants. (See Theorem 7, Art. 208.) Before we can generalize the remaining theorems of Art. 208, we must discuss the notions of minors and cofactors of a determinant of the wth order. 231. Minors. Consider a determinant of the nth order and suppress both the row and the column in which any particular element lies. The determinant of the {n — 1)^^ order formed from, the remaining elements, ivithout disturbing their relative position, is called the minor of the element in question. For examples of such minors see Art. 206. Let us use the notation Dg for tlie minor of any element of D which is named e. Thus, the minor of «j will be called Da^i that of Cg will be i><,^, and so on. We then find the fol- lowing theorem. 1. If the determinant D is expanded, the sum of all of those terms which involve the element a^ (^which stands in the upper left-hand corner of D) is a^D^^. Proof. According to the definition of a determinant of the nth order^ Art. 228, every term of D which contains a^ as a factor is formed by multiplying a^ by a product of n — 1 elements chosen from the n — 1 rows and columns of D which are different from the first row and the first column, and chosen in such a way that one and only one element is selected from each of these rows and columns. Consequently every term of this kind is obtained by multiplying a^ by a term of D^j. Moreover, the sign which precedes such a term of D will be the same as that which precedes the corresponding terms of D^^, since the number of inversions among the subscripts in any term of D^^, such as '2 '3 '/l' will not be changed if we write Oj in front of it. 2. If we denote by e the element in the ith roiv and the kth column of D, the sum of all of the terms of I), lohich have e as a factor, is ^_ -^y+,^j)^^ To prove this theorem, we observe that e may be brought into the position originally occupied by a^ without disturbing the relative posi- Art. 231] MINORS 383 tion of any of the elements of D excepting those which lie in the row and column in w^hich e stands. This may be done by first interchanging the row in which e stands with the preceding row and repeating this operation until e stands in the first row. Since e was originally in the ith row, i — \ such operations will accomplish this result. We may then, by interchanging columns in the same way, bring e finally into the posi- tion originally occupied by Oj. This will be accomplished after k — \ further operations, making i-\ + k-\ = i+k-'2 operations in all. Each of these operations changes tlie sign of the determinant (Theorem 3, Art. 230). Consequently we shall have if we denote by D' the new determinant obtained from D by these i -\- k — 2 operations. The minor, D'^ of e in Z>', is the same as the minor, D^ of e in D, since the elements in both determinants are the same, and since the rows and columns of the two determinants, excepting only the row and the column which contains e, are arranged in the same rela- tive order. Hence D^ = D\. By Theorem 1 the sura of all of those terms of D' which contain e as a factor is eD'^ = eD^' Therefore the sum of the corresponding terms in D is as we wish to prove. If this proof does not seem quite clear to the student, let him actually trace the changes of sign which take place as some element, say d^, of the determinant of the fourth order a^ a^ ag a^ &i &2 ^3 ^'4 c-y Co <'^ c^ f/j d.^ d^ d^ is gradually brought into the upper left-hand corner. As a result of Theorem 2 we obtain at once the following theorem. 3. Let us fix our attention upon any particular row or column of a determinant. Multiply every element of such a row or column by its minor and prefix the plus or minus sign 384 DETERMINANTS OF THE n^^ ORDER [Art. 2:52 to this pi'oduct according as we find a plus or minus sign in the place which corresponds to that element in the diagram of signs + - + - • - + - + • + - + - • which contains n^ signs arranged in a square array ^ alternating in checkerboard fashion^ the signs along the principal diagonal all being -\- . The algebraic sum of all such products, with the proper sign prefixed^ will be equal to the value of the deter- 7ninant. This follows from Theorem 2 because (— 1)»+* is equal to + 1 or — 1 according to the indications given by the diagram of signs. This method of expressing a determinant is called expand- ing with respect to the elements of a definite row or column. Since the minors are determinants of a lower order, to which the same method of expansion may be applied, this method is often convenient when we wish to calculate the value of a determinant. (Compare Art. 206 for the special case of determinants of the third order.) 232. Cofactors. If e denotes again that element of D which stands in the ith row and the kth column, the quantitg which differs from the minor Dg at most in sign^ is called the cofactor of e. Let us denote by Ai, Bi, •••, A2, Bo, •••, the cofactors of di, /y,, ■••, 112, h-i, • •• respectively, in tlie deterniiiiaiit D. Then we may write Z>= r/,.li 4 (uAi + ••• + a„A, D^h.B^ + b.B. + ••• + ^/J„ D= f.Jx\ + k-.K; + ••• + KK,, also CI) D^h.B, + b.B. + ••• + hjl,. and also D = niAy + h,Bi + ••• + k\Ku D = a.A. + h,Bn + •■• + l.J\2, D =a„.4„ + IkB,^ + ••• + A„/v„. Akt. 232] COFACTORS 385 To prove, for instance, the first equation of (1) it suffices to remark that flj^i represents the aggregate of all of those terms of D which con- tain Oj, that a.,.!, is equal to the sum of all of those terms of D which contain (z.^, and so on. But equations (1) and (2) are equivalent to Theorem 8 of Art. 208, which was there proved only for determinants of the third order. Theorem 9 of Art. 208 may be generalized to determinants of the rjth order by an argument whi(Oi involves merely a repetition of the essential features of the proof given in Art. 208 for third order determinants. Thus we have the further result: Theorems 8 and 9 of Art. 208 are true of determinants of any order. EXERCISE CV Compute the value of the following determinants by expanding with respect to the elements of some row or column : 1 2 3 4 1 1 2 1 1 1 o 1 3 _ X 3 o 1 1 - 1 1 1 2 1 2 1 3 3 1 4 4 4 3 2 5 5 2 1 3. 1 -1 2 2 1 3 1 By applying some of the theorems about determinants evaluate the following : a - a - a - a 1 1 1 a 1 1 1 a 1 1 1 b c h d h c 1 1 1 1 o rt2 1 h 6^ 1 c C2 1 d rf2 68 Hint. Compare Ex. 5 in Exer- cise xcni. la + mp, 11) + Hi 7, Ic + mr, Id + ms 8. Slaow that a h f f / c d fl '> a b J ^^ J '^- « / I m I m 386 DETERMINANTS OF THE 71^^ ORDER [Art. 233 233. Solution of a system of n linear equations for n un- knowns. It is now an easy matter to solve n linear equa- tions, such as a^i + ^i-J'a + ••• + ^"i^\ = ^p ag^i -h b^x^ + •• • + Vn = ^2' for the w unknown quantities x^, .j-g, ••• a;„. Let us denote bv D the determinant (2) £> «1 ^ • ■ '^i Qg h ■ •• ^2 a„ b„ ■ •• l^-n of the coefficients of x^, Xo, •••, x„ in the ?; equations (1), and let us denote by capital letters Ai, •••, Bi, ■■■ the cofactors of a^ ••-, i^ ••• in D. If we multiply both members of the equations (1) in order by A^, A.-,, •••, An, and add, we find (a^A^ + a.-.A. + ■•■ + a„An)xy + (b^A^ + KA„ + — + bnA„)x.;, + ■•■ + (k.^^A^+ k„A.^ + ••• + k-nA„)x„ = 7\A^ + r.-,A., + ••• + r„vl„. According to the final remark of Art. 232, the coefficient of xi is equal to D, and the coefficients of x„, x.,, •■■ x„ are equal to zero. The right member is obviously equal to an nth oi'd'er determinant which may be obtained from D by writing i\, r.,, •••, r„ in place of a^, a,, •••, «„• Thus we find (3) Dx, ^1 h Similarly we find (4) Dxo = cin r„ k, Dx„ «•> h In the case n = 3 these formulas reduce to equations (8) of Art. 209, if we put x^ = X, X2 = y, Xg = z. From (3) and (4) we obtain the values of the unknowns x,, Xo, •■■ , Xn if we divide by D, provided of course that D is not equal to zero. Art. 234] HOMOGENEOUS EQUATIONS 387 Just as in the case n = o we obtain tlie following theorem: A Si/stem of n linear equations ivith n unknowns has a single solution composed of the n numbers i'j, .rg, ■■•, t„, obtainable fronn (3) and (4) by dividing by />, provided that the determinant D of the coefficients of the unknoivns is not equal to zero. If D is equal to zero and if even a single one of the deter- minants occurring in the right members of (3^ and (4) is dif- ferent from zero^ the system (1) lias no solution, that is, the n equations (1) are not consistent. The following supplementary theorem we shall state with- out proof. If D = 0, and if besides all of the nth order determinants which are written in the right members of (3) and (4) are also equal to zero, then system (1) has infinitely many solutions. More specifically let us assume that among the minors of D there exists at least one determinant of the (n— l)th order which is not equal to zero. Then a single one of the n unknowns may be left arbitrary and all of the others may be expressed in terms of that one. 234. Homogeneous equations. If rj = 7*2 = ••• = r„ = 0, equations (1) of Art. 233 assume the form a-^^x^ + ^^3-2 + ■ • • + ^,.r„ = 0, n\ «2^i + V2 + •• •+ h^n= 0, «na^i + br^x^ + • • • + k^x^ = 0. Such a system is said to be homogeneous. The discussion of Art. 210 may be generalized to the case of n unknowns without any trouble, giving rise to the following theorem. The homogeneous equations (1) have no solution except the obvious one x-^^ = x^ = ■•• = Xn= 0, if their determinant D is different from zero. If I> = these equations have infinitely many solutions besides the obvious one. If in this case when D = 0, among the minors of D, there exists at least one deter- minant of the (jn — l)th order which is not equal to zero, then the 388 DETERMINANTS OF THE n^^ ORDER [Arts. 235-237 ratios of the n unknowns^ x-^: x^: •■■ : 2;„, ivill he determined unique! 1/ b>/ the given equations. If all of these minors vanish, even these ratios luill not be determined uniquely. 235. Systems of linear equations with more equations than unknowns. Obviously such systems will ordinarily not be consistent. We might solve n of these equations for the n unknowns. The resulting values will ordinarily not satisfy the remaining equations which were not used in the solution. The conditions for consistency are obtained by demanding that the values of the unknowns obtained from w of the equations shall also satisfy the remaining equations. These conditions can always be expressed most conveniently by ec^uating to zero certain systems of determinants. 236. Systems of linear equations with fewer equations than unknowns. Let there be m equations and m -{■ n unknowns. We can ordinarily solve these equations for m of the un- knowns in terms of the remaining n whose values remain perfectly arbitrary. Such systems therefore usually have infinitely many solutions. Thus the system (1) x + y - z=\, X- y + -2z = 2, may be solved for x and ij in terms of ~, giving (2) :,^__i(3 + ~), y = 1(3 2-1). The values (2) will satisfy (1) no matter what the value of z may be. 237. Application of determinants to the theory of elimina- tion. A quadratic equation (1) ax'^ + bx + c = has two roots x^ and x^. Similarly a second quadratic equation (2) a'x"^ + b'x-\- c' = also has two roots which may be called x^ and x^. Ordi- narily neither of the two roots of (2) will also be a root of Art. 237] ELIMINATION 389 (1), but in special cases it may happen that this is the case. What condition must the coefficients of the two quadratics satisfy in order that they may have a root in common ? We may find this condition as folh)\vs. If tlie equations (1) and (2) have a root in common, let it be called x. Then X will satisfy the following four equations : + ax^ + l>.r + c = 0, .3. + a'x^ + h'x + c' = 0, ax^ + bx^ + C.V + (J = 0, a'x^-hh'x^ +c's +0 =0. But we may regard (3) as a system of four homogeneous equations for the determination of the four (quantities Not all of these quantities can be zero, since x^ is equal to unity, even if x^, x^^ and x^ should happen to be equal to zero. Therefore according to the tlieorcm given in Art. 234, the determinant E = must he equal to zero if the equations (1) and (2) have a root in common. Our argument does not suffice to prove the converse; that is, if E=0, then the two (quadratics actually have a root in common. Su(!h, however, is actually the case, a statement which we shall leave without proof. U is called the eliminant or resultant of the two quadratics (1) and (2). EXERCISE CVI 1. Solve by deteriniiiants :;./■ -f -J// + 1 ^ — w = 18, .') ./■ + 1/ - z +'2 ir = 0, 2 .r + 3 // - 7 ;: + 3 w - 14, 4 r - 1 // + ;]-- .5 w = 4. a h c a' h' a a h c a' h' c' 390 DETERMINANTS OF THE n^^ ORDER [Art. 237 Discuss consistency or inconsistency of tlie following systems and obtain the most general solution in those cases in which the systems are con- sistent. 2. 3. 4. 2x + 7j + 3z = 1, 4:x+2y- z = -S, 2x + y + 3z -1, 4:x + 2y — z — 3, 2x + y — 42 = 4. x + y + -dz=:(), x + 2y + 2z = 0, x + 5y— z — 6. Qx + 4:y + 3z - Siio = 0, x + 2y + 3z-^8 10 = 0, x-2 y + z -12w = 0, 4:X + iy-z-2'i:U' = 0. 2x - y + 3z -2w = 0, X + 7 y + z - IV = 0, 3x + oy — 5z + 3 to = 0, ^x-3y + 2z-w=^0. CHAPTER Xril QUADRATIC FUNCTIONS OF TV70 INDEPENDENT VARI- ABLES AND SIMULTANEOUS QUADRATIC EQUATIONS 238. Integral rational functions of two independent vari- ables. A function of the two independent variables x and y is said to he an integral rational function, if it can be expressed as a sum of a finite number of terms of the form (1) Ax'>/\ where A is a constant coefficient (that is, a quantity not de- pending for its value on x or y^ and ivhere the exponents, r and s, are positive integers. Tlie sum of the exponents, r and s, is called the degree, order, or dimension of tlie terra (1). An integral rational function is said to be of degree, order, or dimension w if there occurs in it at least one term of order n and no term of degree greater than n. According to this rnle xr-, xy, and y"^ are terras of the sec- ond order in x and y. Some students find it difficult to understand why the terra xy should be classified as being of the second order. To clear up- this difficulty, consider x and y as the numerical measures of line-segments (ex- pressed in inches) ; then each of the three terms x"^, xy, and y"^ may be regarded as the numerical measure of an area (expressed in square inches). EXERCISE evil State the degree of each of the following functions of x and y : 1. 3x + 4y-5. 4. xy + X -\- y. 7. xy'^. 10. x'^y- -\- 3*. 2. x'^-y. 5. x^ - 7/. 8. xhj + y'^ -?,,x. 11. x - xhj. 391 392 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 239 Express the following as functions of two independent variables: 13. The volume and surface of a rectangular parallelopiped of dimen- sions X, X, y. 14. The volume and total surface of a right circular cylinder of radius r, and altitude /(. 15. The volume and total surface of a right circular cone of altitude 11, if the radius of its circular base is r. 239. Quadratic function of x and y. A?! integral rational function of the seco7id order is called a quadratic function. Thus, a quadratic function can contain only terms involv- ing a:^, xy., y^^ x, ?/, and a term independent of x and y. The most general quadratic function of x and y may be written as follows (1) fi^x, y) =Ax^ + 2 Hxy + By'' ^ -1 Gx + 2 Fy + O, where the constant coefficients A, B, (7, F, (x, H may have any values whatever. We have used the notations 2 F., 2 6r, and 1 H to represent the coefficients of ^, a;, and a;y, in this expression, rather than F, G-, and H., because the for- mulas assume a somewhat simpler aspect when we use this notation. (Compare the corresponding remark in Art. 71.) EXERCISE CVIII In each of the following examples, determine the values of A, B, C, F, G, and H. 1. f(x, .'/) = B 3-2 + 4 xi/ + 7 //2 - 4 X + 8 ?/ - 10. 2. /■(.!■. //) = 7 x^ - ") .r// + 4 If- + -ix -2y + G. 3. f(x, y) = - x- - 2 x + \. 4. /(.r, y) = X- + ?/2 — 4. 240. Composite and non-composite quadratic functions. Clearly, the product of two linear functions of x and y is a quadratic function. Therefore, some quadratic functions can be expressed as products of two linear functions. Whenever this is the case we shall say that the quadratic function is composite or factorable. We shall now show how Art. •J40] COMPOSITE QUADRATIC FUNXTIOXS 393 the question may be decided whether a given quadratic function, (1) fix, y) = Ax-^ +-2EXI/ + Bf + 2 Gx + -2 Ft/ + C, is composite or not. Let us le-write (1), arrano'ing the terms according to descending powers of ?/. \Vc fnul (2) f(x, y ) = ^//2 +2(^Hx + F)y + Aj^ + 2 Gx + O. Let us assume B=^0. Then we may write (3) f(x,y}=B o , .,Rx + F , Ax^ + 2ax+ >r + ^ ]■ We may, for a moment, think only of the way in which this expression depends upon y, ignoring the fact that it de- pends upon X also, hi order to do this it will help us to write where we have put (•^) ^Hx+F A2'^+2ax+C P = '—B-^ ^ = ^ Tlie quantity in the square bracket of (4) may be resolved into two factors of the first degree in y by the method of Art. 68, giving first and then (ti) f(x,y) = B .E where, on account of (5), p''-Aq= —MHx + Fy - B(Ar^ +2ax+ (7)] ^2 or. (7) pi-Aq=^[(E^-AB^x^ + 2(Fff-BG')x + F^-BC]. B^ 394 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 240 Since the factors of /(a;, ^) as exliibited in (6) contain the square root of jo^ — 4 q, these factors, although rational and of the first degree in y, will in general be irrational func- tions of X. As shown by (6) and (7), the only irrationality which these factors contain is the square root of the integral quadratic function (8) Q {x) = (ir2 - AB)x'' + %FH- B a)x + F'^ - BO, and the square root of Q{x} will be a rational function of x if and only if the two linear factors of Q (a;) are the same (see Art. 152), or, as we may express it briefly, if Q{x^ is a perfect square. We may rewrite Q{x} as follows: (9) Q(^x) = ax^-\-bx-{-c, if we put (10) a = H^-AB,b= 2(Fff- BG), e=F^- BO. The two linear factors of Q{x^ will be the same if and only if (11) b^-4:ac=0. (See Arts. 68 to 70.) If we introduce the values (10) of a, 5, c into this condition (11), we find , 4(iFIf - Bay - 4(H^ - AB)(F^ - BC} = 0, or, dividing both members by 4 and performing the indicated operations, F^IT^-2BFaH+ B'^G^ - (i^2^2 _ ^^^2 _ ^CH^ + AB^C) = 0, which reduces to (12) - 2 BFQH^ B^G-'^ + ABF'^ + BCm - AB^O^ 0. Every term of this expression contains B as a factor. Since we are just now treating the case B ^0, (12) can be satisfied only if Art. 1240] DISCRIMINANT OF A QUADRATIC FUNCTION 395 -2FaH+BG^ + AF^+ CH^-ABC=0, or, rearranging the terms and changing all of the signs, if (13) ABC+IFGR-AF^-BG^- (77/2 = 0. Thus, in order that the given quadratic function may be a product of two linear functions, its coefficients, A, B, C, F, G, £[, must satisfy the condition (13), at least if B is different from zero. Moreover, if B^O, and if A, J5, C, F, G, H do satisfy this condition (13), the given quadratic function can actually be expressed as a product of two linear functions. The factors are given by (6) where, if (13) is satisfied, the square root which occurs will reduce to a linear function of X. The quantity which occurs in the left member of (13) is usually denoted by A (pronounced delta) and is called the discriminant of the quadratic function (1) of x and y. Our argument only covers the case B ^ 0. If i? = we have, instead of (1), the simpler expression f(x, y) = Ax^ + 2 Hxy + 2 Gx + 2 Fy + C. Let us assume that A is not also equal to zero and let us arrange /(r, y) according to descending powers of x. We may now repeat our argu- ment essentially as above merely interchanging the part played by the two variables x and y. The resulting condition for factorability will turn out to be (14) 2FGn- AF^ - CH^^O. But this is precisely w-hat (13) would reduce to in the case 5 = 0. We conclude that the condition (lo) may be used even it B = 0, although our original method of finding this condition was not applicable to this case. Our argument now covers all cases in which at least one of the num- bers A and B is not equal to zero. It remains to consider the case ^ = 5 = 0. The function (1) reduces to (15) f(x, y)=2 II xy + 2 Gx + 2 Fy + C, where we may assume /f ^ 0, since otherwise the function would reduce to a linear function. If (15) is a product of two linear functions only one of the factors can contain x. For otherwise the product would contain a term involv- ing X-. Similarly, only one of the factors can contain y. Therefore, if 896 QUADRATIC FUNCTIONS OF TWO VARIABLES [Aur. 240 (15) is a product of linear factors at ail, it must be equal to a product of the form (16) {px -\- q){r7j -{- s). Tims, in this case, (15) must l)e equal to (16), that is, we must have • 2 Ilxy + 2 Gx + 2 Fi/ + C = prxy + psx + qry + qs for all values of x and y. But this is possible only if (17) 2II^pr, 2G^ps, 2 F = qr, C = qs. Since, in the case under consideration, we also have A = B = 0, the left member of (1:5) will assume the value 'qr\( P><\f pf\ p-r'^ _1 „ ., 1 '(f^jlf )(?■) ~ 9^^ = |PV'« - ^P'V 0. Consequently condition (13) is satisfied by the coefficients of a composite quadratic function even ii A = P> — 0. Conversely, if the coefficients of a quadratic function satisfy the conditions (13) and A = B = 0, we have (18) ^ = jB = 0, 2 FGH - CH^ = 0, H =^0, and we can actually find four numbers p, q, r, and s which satisfy equa- tions (17). It suffices for this purpose to put p = 2II, q = 2F, r^l, s = ^ = -^ f ' / H 2F where the two values obtained for s are equal on account of (18), and where F is assumed to be different from zero. If F should be equal to zero, we should, according to (18), find C = also, since H ^ 0, and (15) would reduce to f(x, y) = 2 Ilxy + 2 Gx = 2 x (Hy + G^ the composite nature of which is evident by insjiection. We liave proved the following important theorem: An integral rational quadratic function ( 1 9) Ax"^ + 2 Hxy + Bf + 2 G.v + 2 Ft/ + may he expressed as a product of two integral rational functions of the first degree., if and only if the coefficients satisfy the con- dition that the discriminant (20) A ^ ABC +2 Faff- AF'^ -BG"^- Gff'^ is equal to zero. It is easy to apply this criterion to any given quadratic function. It is also easy to actually find the factors if the Art. 240] COMPOSITE QUADRATIC FUNCTIONS 397 condition is satisfied. It is sufficient for tliis purpose to apply the general methods of this article. If tlie function (10) is homogeneous, that is, if it contains no terms of degree lower than the second, we have (7=0, F= 0, (r = 0, and therefore, according to (20), A will also be equal to zero. Therefore, every homof/eneous quadratic function may be resolved into integral rational factors of (he first degree. The factors of (19) need not be real. Thus the factors of the homo- geneous function x- + y'^ are .r + /// and :r — i;/. EXERCISE CIX 1. Is 2 x^ — xy — 3 _?/■- +9 X- + 4^ + 7 a composite function ? If so, find its factors. Solution. We have .4=2, 73 = - 3, C = 7, F = 2, G = f, H = - j, and, therefore, find A = 0. Therefore the function is composite. To find the factors follow the general method of Art. 240. The given function is equal to ^ .r/^ l-'--^)-' (•f-4)-^ + 12(2x'^ + 9x + 7) -| __ .^ r/ 2- + y - 4 \^ 25 x^ + 100 X + lOO l ~ ' Lv 6 / 36 J = - u(^ + Qi/- -^r - (5 X + 10)2] = - 1^2 [x + 6 .y - 4 + 5 X + 10] [x 4- 6 J/ - 4 - (.5 x + 10)] = -J,(Gx+()^ + G)(-4x + Gy-14)=-(.t-f//+l)(-2x + 3^-7) = (x + i/ + l)(2x-3y + 7). Thus, 2 a;2 - x/y - 3 y- + 9 X + 4 y + 7 = (x + y + 1) (2 X - 3 3/ + 7). The result may be checked by multiplying out the product in the right member. In each of the following examples state whether the given function is composite or not. Find the factors if they exist. 2. - x^ + 4:Xij - ij^ -4:^/2 X + 2^2 y -11. 3. y- — xy — Q x^ + y — 'i X. 4. 4 x^ — 4 X2/ + ^2 + 4 X — 2 y. 898 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 241 241. The values of a quadratic function. In order to obtain some idea of tlie values wliich a quadratic function takes on for various values of x and ?/, we formulate tlie question : what are all of the values of x and y which cause the function to assume the given value h ? That is, what are the values of x and y for which (1) ^z2 + 2 Hxy + By'^+2ax^2Fy+0=V'^ These values are obviously the same as those which cause the new quadratic function (2) Ax'^ + IHxy + By'^^-^ax+IFy+O-h to assume the particular value zero. Since (2) is a function of the same kind as the left member of (1), our question may be regarded as answered if we can only solve the fol- lowing, somewhat simpler, problem. To find all of those pairs of values for x and y which will satisfy any given equa- tion of the form (3) Ax"^ + IHxy + By^ -^ 2 ax + ^Fy + C = 0. Every pair of numbers which, when substituted for x and y, will satisfy equation (3) is called a solution of the equa- tion. The equation is called a quadratic equation. EXERCISE CX In each of the following examples test the given pair of numbers, to see whether or not they constitute a solution of the given equation. Give the details of your work. 1. X- — 4 z^ = ; X = 2, ?/ = 1. 2. x2 + 2 x?/ + ?/2 = 5 ; X = 1, 2/ = 1. 3. x^ — ?/2 _ ; X = any number, y = x. 4. x2 + »/2 :^ ; X = 1, ?/ = «■ = V"^r. 5. x2 — xij + ?/2 = 7 ; X = 1, ?/ = 0. 6. (X - ?/)2 + 2x - 3/ = 2; X = 0, 2/ = 2. Art. 242] EXISTENCE OF SOLUTIONS 399 242. The existence of solutions of a quadratic equation. Every quadratic equation (1) A.i- + 2 Hxy + %2 + ^ax+ IFij + 6'= Aas infinitely many solutions. For we may substitute for x any number we please (real or complex), and then obtain, if B =^ 0, two corresponding values for y by solving the equation (1), which then becomes a quadratic equation with a single unknown, for y. Thus (1) may be tliought of as defining y as a two-valued function of x, provided that B is not equal to zero. If B = and if at least one of the coef- ticients ^and F is not equal to zero, we find that (1) defines 3/ as a one-valued function of x. Finally if B, F, and II hre all equal to zero, the equation (1) does not contain any y ; it will be satisfied by the two values of x which are roots of the equation Ax^ -\- 2 G-x-\- C= 0; with each of these values of X may be associated arbitrary values of y. Although, as we have just seen, every equation of the form (1) has infinitely many solutions, it does not follow that these solutions are real. There actually exist equations of this sort which have no real solutions at all. Thus the equation x- + //- + 4 = has no real solutions. For, if x and y are real numbers, x^ and y'^ cannot be negative, and therefore a;2 + ^- + 4 cannot assume any value less than 4 for real A'alues of x and y. The equation x^ + ^^ _ o ^as one and only one real solution, namely X = ^ = 0. We have shown in Art. 240 how the left member of (1) may be resolved into two factors of the first degree in y if B ^ 0. In order that the equation (1) ma}^ be satisfied, one of these factors at least must vanish. Therefore, if we use the notations of Art. 240, we find the following explicit expression for ?/ as a function of x : (2) ^^ -p±y-4y ^ where (.3) p = hnx+F~), q='^(iAx^ + 2ax+C\ B=^0. B B 400 QUADRATIC FUNCTIONS OF TWO A^ARIABLES [Art. 242 These formulas show clearly the two- valued character of this function, since we may use either the plus or minus sign in (2). We may also, and in many respects this is desirable, regard (2) as defining two one-valued functions rather than a single two-valued function. The resulting tivo functions are rational or irrational according as p^ — 4q is a perfect square or not, that is, according as the discriminarit A o/" (1) is or is not equal to zero. The case B =Q is easily settled. We shall leave the dis- cussion of this case as an exercise for the student. The formulas (2) and (3) give us the general solution of equation (1) in the case B ^ 0. If we substitute in them for X any number whatever, and then compute the corre- sponding values of ?/, all pairs of numbers, x and i/, obtained in this way are solutions of (1). EXERCISE CXI Solve each of the following equations for ?/ as a function of x and dis- cuss the following questions. Is the resulting function one-valued or two-valued ? If it is two-valued in general, are there any particular values of x for which the two values of y coincide? For what real values of X will the resulting values of y also be real? 1. X- + _y2 = 4. 8. X- + if- = a^. 2. X- - y- = 4. 3. xy — 5. 4. 4x- -\- y'^ = 10. 5. a,-2 + 4 7/2= 16. ^^- ~ ^ 11. 7. x^ - 4y- = 16. 18. (// - ^•)'- = 4;>(.r- A). 19. {x - hy^ = ip(y - k). 20. (x - 'Ay + (y - iy = 5": ^^ (x - /,y _ (y - /.)- ^ ^ 21. (x - hy + (y - l-y = «2, • " a-2 b- 24. _I:^^0_%I .'/-/-V^ = l. (r h- 25. - x2 -H 4 xy - ?/2 - 4 \/2 x + 2 V2 ^ - 11 = 0. 26. y^ - x^ - 6 X- 4- y - 3 X = 0. Akt. 21:3] GRAPH OF QUADRATIC EQUATION 401 243. Graph of a function defined by a quadratic equation in X and y. Whenever a function, detinecl by means of a quad- ratic equation in x and y, is real, that is, if tlie quadratic equation has real solutions, these may be plotted as points in accordance witli the metliod which we have used so fre- quently. The general question as to the nature of the graphs obtained in tins way, while not very dilTicult, is usually reserved for the course in analytic geometry. There are, however, several special cases in which it is quite easy to draw the graphs. We shall now discuss some of these cases, a few of which have appeared already in this book in a different connection. Case 1. The graph of a quadratic equation in x and y con- sists of a pair of straight lines if the discriminant A is equal to zero. For we have seen in Art. 240 that, in this case, the quad- ratic function is a product of two linear functions, say a^T + b^g + c^ and a^x+b^g + c^ ; and we have also shown how these linear factors may be found. The values of x and y, which cause the quadratic function to vanish, must make at least one of the factors equal to zero. But the locus of all of those points, whose coordinates cause a linear function of X and g to vanish, is a straight line. Therefore, the graph of a quadratic equation with a vanishing discriminant consists of the two straight lines a^x + b^g -(- (?! = 0, a^r -f- b.^r/ + c.^ = 0, whose equations are obtained by equating to zero the linear factors of the quadratic function. Case 2. If B = 0, g becomes a rational function of x. If A = 0, X becomes a rational function of g. \\\ either case, the graph may be constructed by the methods of Art. 140, special attention being devoted to the poles of these rational functions whenever they have any poles. (See Art. 139.) For illustrations see Art. 140. 402 QUADRATIC FUNCTIOXS OF TWO VARIABLES [Art. 243 Case 3. If F = G- = H = 0, the locus of the equation is symmetric with respect to both the x-axis and y-axis. In this case the origin is called the center of the locus. Proof. The equation, in this case, reduces to (1) Ax'^ + %2 ^ c=0. If this equation is satisfied by a pair of numbers (a;, j/), it will also be satisfied by the pair (a;, — y'), since only the second power of y occurs in the equation and since (~~ I/)^ = (+ ^)^- ^^t ^li6 ^^^o points (2;, ^) and (.t;, — y} are symmetrically situated with respect to the a:-axis. A similar argument shows that for any point (x, y^ which is on the graph of (1), there exists a second point (— a;,["?/) which is also on the graph and which is symmetric to the first point with respect to the y-axis. It is easy to see what the graph of (1) will look like in the various cases. If B = 0, (1) does not contain any y term at all, and solution of the equation for x shows that the graph consists of a pair of lines parallel to the y-axis. These lines may be real and distinct, coincident, or imaginary. Of course in this case A is equal to zero. li B =^ 0, we may rewrite (1) as follows y^ = Ax^-\- A ^ B = B^- «/2 = mx'^ + w, ' B' or (2) if we use the abbreviations, m and n, for — A/B and — C/B respectively. The appearance of the graph depends essen- tially on the values of m and w, and more especially upon the signs of these numbers. Case 3 a. If m = 0, the graph of (2) reduces to a pair of straight lines, parallel to the a;-axis, namely the lines for which y — ± Vw. Art. 243] GRAPH OF QUADRATIC EQUATION 403 These lines are real and distinct, coincident, or imaginary, according as n is positive, zero, or negative. Case ?> h. \i )i = 0, the graph of (2) reduces to a pair of straight lines throngli the origin, namely y = ± Vma\ these lines being real and distinct only if ni is positive. Case 3 c. If m and n are both negative, the right mem- ber of (2) will be negative for all real values of x. Conse- quently, in this case, no real values of i/ will correspond to real values of x; the equation has no real solutions and it has no real graph. There remain three cases, namely Case 3 d when w < 0, % > 0, Case 3 e when m > 0, w < 0, and Case 3/ when m > 0, n>0. Case Z d. w < 0, w > 0. We may write, in this case, (3) m = — k\ n = k^a^, where a and k may be regarded as positive numbers whose values may be computed from (3) in any case where m and n are given, m as a negative and w as a positive number. Then (2) becomes y = — k^x"^ + k'^aP- = k'^(^a^ — a;^), whence (4) y = ± kVa^ — x\ If x^ > a^ a^ — x^ will be negative and therefore y will be imaginary, as shown by (4). Therefore, there are no points on the graph whose distances from the ?/-axis are greater than a. Let us mark the points A and A' on the a;-axis (see Fig. 70) whose abscissas are equal to + a and — a respectively. If we draw lines parallel to the y-axis through these points, we know then that no point of the graph can be outside of the strip inclosed by these two lines. The points A and A' themselves are points of the graph. For if we put a; = ± a in (4), we find y = 0. 404 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 243 For X = 0, equation (4) gives y = ± ka. Let us construct the points B and B' on the y-axis (Fig. 70) whose ordi- nates are equal to + ka and — ka respectively. Both of these points are on the graph of equation (4). For any value of X which lies between — a and + a, we find from (4) two values of y, of the same absolute value but of oppo- site sigh, thus proving again that the graph is sjanmetric with respect to the a^-axis. Moreover as x increases from to a, the numerical value of y decreases from ka to 0. If ^ < 1, the case assumed in constructing Fig. 70, OB is less than OA. If ^ > 1, we should find OB > OA. If ^ = 1, we have OB = OA^ and it looks as though the curve would be a circle with as its center. This is actually the case. For if ^ = 1, (4) becomes whence (5) ?/ = ±Vt y^ = a^ — 3? .r2 or Fig. 71 rjfi J^ y^ — (jfi^ But, if X and y are the coordinates of a point P (see Fig. 71), x^ -\- y'^ will be the square of the distance from the origin to the point P. If the point P moves in such a way that x^^f- = OP always remains equal to the same number a^^ it must remain on the circumference of a circle whose center is and whose radius is equal to a. Therefore, the graph of (5) is indeed a circle whose center is at the origin and whose radius is equal to a. When k is not equal to unity, the graph of (4) is called ill! ellipse. The line-segments AA' and BB' are called the principal axes of the ellipse, the larger one being the major axis and the smaller the minor axis. Case 3 e. m > 0, w < 0. In this case we put m = k^, n = — k'^a^, Art. 243] GRAPH OV QUADRATIC EQUATION 406 so that (2) becomes whence ?/2 = Hi:2 _ ^2^2 _ ^2^ 2;2 — a^), (6) y = ± k^aP- i'lG. 7'^ The discussion of this case is ^y quite similar to that of Case 3 d. But this time the values of x which give rise to imaginary values of y are the small values, namely those for which a? < a^. The vertical strip of width 2 a, between A and A\ instead of in- cluding all of the points of the curve as it does in Case 3 d, does not include any point of the curve. The curve has two infinite diverging branches, as illustrated in Fig. 72, and is called a hyperbola. Case 3/ also leads to a hyperbola, but located in a different way. It may be obtained by rotating Fig. 72 through an angle of 90°. Case 4. If H= 0, if A and B are both different from zero, while at least one of the numbers F and (7 is not equal to zero, the (jraph will have the same general characteristics as in Case 3, but the tivo axes of symmetry will not both coincide with the coordinate axes. They will merely he parallel to the coordinate axes. The following illustrative example will show how the axes of symmetry may be obtained, by the process of com- pleting the squares, and how the locus of such an equation may be plottc^l. Example. Discuss 4 a;^ + sy^ _ g x — 4 y + -4 — 0. Solution. We collect the terms involving x and complete tbe square, and proceed similarly for the terms involving //. This gives (7) 4 x^ - 8 .r + 4 + if' _ ] ,/ + 4 = 4 + 4 - 4 = 4. (8) 4(x - 1)^ + 0/ - -ly = 4. 406 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 243 +y +!/' B r \ P 1 A ,"' / A M' V y B' il I Fig. 73 and therefore Let us mark the point O' whose co- ordinates are x = 1, y = 2 (see Fig. 73) and ciroose a new coordinate sys- tem with O' as origin, the new (x' and y') axes being parallel to the old (x and y) axes. Let P be a point whose coordinates referred to the old coordinate system are x and y, and let its new coordinates be called x' and y'. Then we have (see Fig. 73) X = OM, y= MP, x' = O'M', y' = jWP, OB' = 1, B'O' = 2, so that (8) x= 0M== OB' + B'M = 0B'+ O'M' = 1 + ar', y = MP = MM' + M'P = B'0' + AFP =2 + y', x- 1 ^, y -' = ?/• Let us substitute these values in (8). AVe find 4 x'- + ?/'2 = 4 which is an equation of the form discussed under Case 3. Thus the x' axis and y' axis are the axes of symmetry of this curve, and the point 0' is its center. We may continue the discussion as in Case 3 and verify that the loans is the ellipse ABA'B' shown in Fig. 73. Case 5. If F = G- = Q^ that is, if the equation contains no terms of the first degree, the origin is the center of the locus. Such an equation lias the form (10) Ax"^ + 2 m-g + Bg^ +(7=0. If X and g constitute a pair of numbers which satisfy this equation, the numbers (—.?-, — ^) will also form such a pair. But the origin is always halfway between two points whose coordinates are (x, g^ and ( — a;, — ?/) . Consequently the locus of any equation of the form (10) has the property that, to every point P which is on the locus, there corre- sponds another point P' also on the locus, sucli that the segment PP' is bisected at the origin. This is what we mean by saying that the origin is the center of the curve. Art. 244] LINEAR AND QUADRATIC EQUATIONS 407 Case 6. This is the general case which we shall not dis- cuss in detail. It may be stated, however, without proof, that the graph will always be an ellipse, a parabola, a hyper- bola, or a pair of straight lines. The distinction between this case and the others which we have discussed lies merely in the fact that the axes of symmetry of the curve will not be related to the .r-axis and y-axis in such a simple manner. All of the curves, which may be obtained as loci of quad- ratic equations in x and y, are known by the collective name conies. EXERCISE CXIl Plot the loci of the equations given in Exercise CXI. 244. Solution of a system of simultaneous equations one of which is linear and one of which is quadratic. All of the points whose coordinates satisfy an equation of the first degree (1) Ir + 7ni/ + 71 = are on a certain straight line. All of those whose coordi- nates satisfy an equation of the second degree (2) Ax^ + 2 my +Bif + 2 a.r + '2Fij+C=0 are on the conic (parabola, ellipse, hyperbola, or a pair of straight lines) which is the locus of (2). The points whose coordinates satisfy both equations, if there are any, will be the common points, or points of intersection, of the straight line and the conic. Therefore we may find the real solutions which the two equations (1) and (2) have in common, hy draw- ing the line and the conic represented hy these two equations individually, and then measuring the coordinates of their points of intersection. The common solutions of the two equations may also be found quite easily by algebra. We solve (1) for y, and sub- stitute the resulting value of y in (2), thus obtaining a quad- ratic equation for x alone. If Xj^ and x^ are the two roots of this 408 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 244 quadratic equation, there will correspond, by means of (1), to each of these values of a; a value of y. If y^ and y^ are these values of ?/, the common solutions of (1) and (2) will be (Xy, y-[) and (x^, y,^. This process would fail only if m were equal to zero. In that case we solve (1) for x^ and substitute the resulting value in (2), obtaining a quadratic for y alone. We always obtain two solutions (a^j, y-^ and (x^-, y^) which may, however, as in the simple case of Art. 70, be real and distinct, coincident,' or imaginary. The corresponding loci actually intersect in distinct points only when the solutions are real and distinct. If the two solutions coincide, the straight line is tangent to the conic. If the solutions are imaginary, the straight line and the conic do not intersect at all. EXERCISE CXlll Solve the following systems of simultaneous equations, and verify your results geometrically by graphs : ' \xrj = 187. ' [xy = 45. ^ |x-// = G, g |a;2 + ?/2-8.r-4f/-5 = 0, [ xy = 91. ' [ 3 X + 4 ?/ = 5. ^ |.r+ y= 11, ^ ^.^ + j/ = 4, I X- + ^2 = 73. [ X2 + 7/2 _ 1(3. |.i:-7/=4, \x + y = a, ■ I X' + y'= 208. [ x'2 + y/2 = «2. 9. What must be the value of h in terms of m and r, in order that the two solutions of the system y = mx + b, a;2 + //2 = r2 Tuay be identical? Hint. The quadratic equation for x, obtained by eliminating y, must have equal roots. (See Art. 70 and Exercise XXIV.) 10. What must be tiie value of r in terms of a, b, and m, in order that the two solutions of the system may coincide? 3:- , y- 1 y = mx + c, - + ^ = 1 Arts. 245, 246] SIMULTANEOUS QUADRATICS 409 245. Simultaneous quadratics. The problem of finding those pairs of numbers, x and //, which satisfy eacli of two quadratic equations (1) Ax^^1Hxij^Bif^2ax-\-1Fy-^C=^, (2) Ax^ + 2 E'xy + B' if +'2a'x + 2 F y + C' = is equivalent graphically to that of finding the coordinates of those points in which the two conies intersect which are obtained by making the graphs of (1) and (2). Conse- quently we have a graphic solution of the problem immedi- ately available. We need merely draw the graphs of the two equations, determine their points of intersections, and measure the coordinates of these points. However, this graphical method will only furnish the real solutions of the problem, and these only approximately. We might think of attacking the algebraic sohition of the problem by means of the following direct but rather clumsy method. Arrange (1) according to descending powers of ij, and solve the resulting equa- tion, which is a quadratic in y, for y as a function of x. The resulting expression for y is given by ("i) and (3) of Art. 242. If we substitute this value of y in equation (2), we obtain in general an irrational equation for x. If this be rationalized (see Art. 154), the resulting equation for x turns out to be, in general, an equation of the fourth de- gree. Since such an equation has four roots (Art. 126), the solution of this equation will furnish four values of x. In similar fashion, elimination of x between the equatinns (1) and (2) gives rise to an equation of the fourth degree for y, the oolution of which furnishes four values of y. The four values of x and the four values of //, obtained in this way, can be combined into jiairs in sixteen ways. But not all of these pairs are solutions of (1) and (2). It remains, therefore, to ascertain, by actual substitution in (1) and (2), which of these sixteen combinations are actually solutions of the simultaneous system (1) and (2). This rather laborious process may he simplified very consid- erably by making use of the results of the following article. 246. Equivalent systems of simultaneous equations. Let us denote by /(a;, ^) and f\x^ y) the two quadratic functions 410 QUADKATIC FUNCTIONS OF TWO VARIABLES [Art. 246 (1) f(x, y) = Ax'^+2 Hxij + By"^ + 2ax+ 2 Fy + C, (2) f'(x, y) = A'x-2 + 2 H'xy + B'y^ +2a'x + 2 F'y + C", and let us put (3) g(x,y) = a.f(x,y:, + b.f(x,y\ (4) g' (X, y) = a' ■ /(.r, y} + h' -fix, y), where a, J, a', and 6' are four constants whose values may be chosen arbitrarily. Then g(x^ y/) and g' (x, y') will also be quadratic functions of x and y, although in some special cases ^(a:, ?/) or ^'(.r, ^) may reduce to a linear function, ^(.r, y~) and ^(.r, ^) are said to be linear combinations of fix, y) and/'(.r, ^). If we multiply both members of (3) by h\ those of (4) by — 6, add, and then interchange members, we find {ah' - a'b}f{x, y) = b' ■ g{x, y) - b ■ g' (x, y}. Simihirly, let us multiply both members of (3) by — a', those of (4) by + a, and add. We find (ab' - a'b^f'(x, y) = - a' ■ g{x, y') + a - g'Qx, y). If ab' — a'b is different from zero, we may write the last two equations as follows ; (5) fix, ?/) = -^^ ^/, r/-^^' ab ~ a'b ^^\ /C-^'^)- ab'-a'b In other words, if a b (7) .' A' = ab' — a'b^O, not only will g(x, y^ and g' (x, ?/) be linear combinations of f(x, ?/) and f'(x, y}-, but conversely f(x, y') and f'(x, y^ will also be linear combinations of g(x, ?/) and g' {x, ?/). Now let (x, y) be any solution of the simultaneous quad- ratic equations (8) /(^, J/) = 0, f'ix,y-) = 0. Art. 247] NORMALIZATION 411 According to (3) and (4), (a;, y/) will also be a solution of the simultaneous quadratics (9) gix,n-) = % g'{x,y)==Q. Conversely, according to (5) and (6), any solution of (9) will also be a solution of (8). Consequently the two sys- tems (8) and (9) have exactly the same solutions, that is, every solution of one system is also a solution of the other. In other words, the two systems are equivalent. We have proved the following important theorem : All of the solutions of a system of two simultaneous quadratics (10) /(.r,2/) = 0, /'(.r,i/) = are also solutions of the system (11) a .f(x, y)+h ./(.T, y)=0, a' .f(x, y}+b' .f'(x, y) = 0, whose left members are said to he linear combinations with con- stant coefficients a, b, a', ?/, of the left members of (10). More- over, the tivo systems (10) and (11) are equivalent, if the constant coefficients a, b, a\ b' are such as to make (12) a b a' b' = ab' - a'b =^ 0. 247. Normalization. We shall now make use of this theorem to simplify the solution of a system of simultaneous quadratic equations (1) f(x, y) = Ax^ +2Hxy + By'^ + 2ax+2Fy + C=0, (2) f'{x, y-) = A'x^ + 2H'xy + B' y'^+2a'x+2 F'y + C =Q. Let us make the following linear combinations of f(x, y') aud/(.c, ^): (3) gir, y^ = B' .fix, y)-B -fix, y), g'Qc, y) = -A' -fQx, y^) + A -fix, y). 412 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 247 In the notation of Art. 246, this amonnts to putting a=B',h = -B,a' = -A',h' = + A, so that we have, in our case, ah' -a'h = AB' -A'B. According to the final theorem of Art. 246, the equations g(x, 3/) = and g' (x^ ^) = will therefore form a system equivalent to (1) and (2) if (4) AB'-A'B^O. But we find, from (1), (2), and (3), that these equations become (5) c/{x, g) = (AB'-A'B).i^-\-2(HB'-H'B)xi/+^ + - = 0, and (6) g'{x,y)=^-{-2(AII'-A'Hyxi/+(AB'-A'B}f-h"- = 0, where we have merely indicated the terms of lower degree than the second by dots, and where the asterisk >|< indicates that the corresponding term has not been omitted by mis- take, but that the coefficient of that term is actually equal to zero. We may rewrite equations (5) and (6) as follows : "^ ^ ^ + 2H^xy + B^y'^+2a^x+2F^ij+ C^ = 0, if we put A^ = AB' - A'B, H^ = HB' - H'B, etc. Thus, we may replace system (1), (2), by a system equiva- lent to it of the simpler form (7) provided that AB' — A'B is not equal to zero. Let us consider now the case, hitherto excluded, that (8) AB' - A'B = 0. Then equation (6) contains at most one second degree term, namely (9) 2(iAH' - A'H^xy = 2 H^xy, Art. 247] NORiMALIZATION 413 siuce the coefficient of a-^ is equal to zero, on account of (8). It may happen tliat the term (9) is also absent, namely if H^ = AH' — A'R= 0. In that case the equation g'(x, «/) = becomes linear,* and our system of equations may be solved by the method of Art. 244. If, however, H^ =^ 0, we may replace the two equations (1) and (2) by tlie equivalent system ^2^+2 Hxy + By^+'iax + 2Fy+ O' = * + 2 K^y + * +2 (^..r + 2 F^_y + C^ = 0, and, by means of the second equation, we can eliminate the xy term from the first, giving rise to a new equivalent sys- tem of the form ^.. A.x'' + * + By- + 2 a,x + 2F^y+ C\ = 0, '^ ^ * + 2 ff^xy + * + 2 (7,.r + 2 F,y + (7. = 0. Thus, by the process of linear combinations, we may always reduce a system of two independent simultayieous quadratics tvith two unknowns either to the form (7), or to the form (10), or else to a system in which one of the equations is of the first degree only. When a system of simultaneous quadratics has been re- duced to one of these forms, we shall say that the S3'stem has been normalized. One great advantage of this normalization consists in the greater facility which it gives us in making a graphic solution. The loci of both of the equations of the normal- ized system (7) are easily plotted because one of these equations gives y as a rational function of x, while the other gives X as n, rational function of y. These graphs fall under the case 2 of Art. 243. If the normalized system is of form (10), the second equation will again give y as a rational * It might even happen that all of the coefticiciirs of this equation are equal to zero. In that case we say that theoriijinal two equations are not independent; one of them is a mere multiple of the other, the graphs of the two equations coincide, and every solution of one equation satisfies them hoth. We are assum- ing tacitly that the two given equations are independent, so that this case may be I'egarded as excluded. 414 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 248 function of x. The graph of the first equation will also be obtained easily, although it does not determine ^ as a rational function of x, because this graph comes under Case 4 of Art. 243. The student should not make the mistake of thinking that the graphs of the two original equations are the same as those of the two equations obtained from them by normaliza- tion. They are, in general, quite different, and it is pre- cisely this difference which accounts for the fact that the graphs are easier to construct for the normalized system than for the original. However, the points of intersection of the two pairs of graphs are the same, and that is the only thing we care about just now. EXERCISE CXIV Normalize the following systems of quadratics : 1. X- + xy + if- - X - .?/+ 2 = 0, 3. 2 x- + Z xy-\-if-\-1x- 5 ,y+ 1 = 0, .r^ — 3 zy — _y- + x + // — 3 = 0. 6 x^ + 9 x^ + 3 y/^ + x -f ?/ + 1 = 0. 2 x2 + XT/ + ?/'^ + 3x -4 ?/ + 7 = 0, 4. 2x- + 4xir-3 ?/2 + 2x-5?/4-l = 2 x^ - xy + 2 y'^ + X + y - 2 - 0. o x'^-oxy + 2y^-x+ 7 ?/+ 2 = 0. 248. Existence of four solutions. Let us assume that AB' — A'B is different from zero, and let us normalize our system. We may then consider the normalized system (7) of Art. 247 instead of the original two equations. PVom the second of these equations we find, solving for a:, so that a; is a ratiotial function of y. If we substitute this value of X in the first of the equations (7), Art. 247, we find the following fractional equation for y : , f ^^2 + 2 F, ^ + C,r- jj- B,f^ + ^. F4, + (7, _ ^^^2^!+lZ^±^ + 2 #,^ + (7, = 0, or, clearing of fractions. Art. 2i8] EXISTENCE OF FOUR SOLUTIONS 415 - 4 H,y~. F h K (3) a,- = ± 3, y = ± 4, giving the four solutions, (4) (4-3, +4), (-3, +4), (-3, -4), (+3, -4).- The graj)!! of the first ecjuation of (1) is the ellipse ABA'B', that of the second equation (1) is tlie circle CDEF oi Fig. 74. These intersect in the four points Q, R, S, T whose coordinates are given by the four solutions (4). But observe how much easier it is to draw the graphs of the equations (2). The graph of the first equation (2) consists of the two lines IIK and H' K', that of the second equation (2) consists of LM and L'M'. These two pairs of lines intersect in the same four points Q, R, S, T as the ellipse and the circle, as they should according to the general theory, since systems (1) and (2) are equivalent. Solve the following systems algebraically and graphically. In Ex- amples 6-9 discuss also the conditions under which the solutions obtained will be real and distinct, coincident, or imaginary. 2. X- + //2 = 113, ^ £: !_ .?^ - 1 X2 _ ,y2 =15. ■ a-2 fy^ 3. 9 x2 + 2.5 f = 225, 2,-2 + v/2 ^ r^ x2 + 3/2 = 16. 4. 9 x2 + 25 2/2 = 225, X- + ?/2 = 9. 5. 9x2 + 25 7/2 = 225, x2 + f = 4. 6. ./■2 + //2 = k; x-2 - y/2 =. I. Fig. 74. 8. a2 - i2 1, x2 + //- = ,.2 9. X- + />2 1, X-2 «2 - /;2 1. 251. Case II. F=F'=G=G' = 0. This case includes the preceding one and is more general. Neither equation con- tains any first degree terms, but either or both equations may contain an xy term. The equations are of the form ; 418 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 251 ,. . f A.r^ 4- 2 Ht?/ + Bf+C=0, ^ ^ \a'x'^+ 2 R't>/ + B'f + C = 0. By linear combination we can obtain from these, the equation C QAx^ + 2 Rri/ + %2 + C-) - C(A 'x^ +2 R'xy-\-B'f+ C') = or (2) (iAC -A' C)x^- + 2(HC' - IT' C)xy + {BC'~B' 0)^=0, which is homogeneous in x and y and whose graph therefore consists of a pair of straight lines through the origin. (See Art. 240.) The system composed of (2) and either one of the original equations is equivalent to (1). But (2) may be decomposed into two separate equations of the form (3) ax + hy=0, {■\~) a' x + h' y = 0, if ax + hy and a' x + h'y are the factors of the left member of (2). Hence by solving each of the first degree equations (3) and (4) simultaneously with one of the equations (1) Ave obtain the four solutions of (1). Illustrative Example. Solve the following system of equations: (4) a;- + 3 xy - 28 = 0. (5) x2 + f- 20 = 0. Solution. To form the homogeneous equation (2) in this case, we multiply both members of (4) by — .5, those of (5) by 7 and add. This gives 2x-- loa-// + 7//2 = 0, or, factoring, (6) (2x-y){x-ly)=0. The system composed of (5) and (6) is equivalent to the original system (4), (5). But according to (6) we have either (7) y = 2 X, or (8) y = ^ X. Substituting the value (7) in (5) gives ^2 + 4 x2 = 20, 5 .r2 = 20, .r2 = 4, x=± 2, and therefore from (7), y =±i. Thus (+2, +4) and (-2, - 4) are two solutions. Substituting the value (8) in (.5) gives the other two solutions (+|\/10, + ^VlO) and (-^VIO, -|VlO). Art. 251] NO FIRST DEGREE TERM 419 A second solution. The following modified form of the solution is convenient. Since we know that by finding the factors of the homo- geneous function which occurs on the left member of (2), and equating one of these factors to zero, we shall obtain an equation of the first degree of the form y = 7nx (see (7) and (8)), we substitute // = mx in the given equations, and regard m as an unknown quantity. Thus (4) and (.5) become a;2 + 3 nix^ = 28, x- + m-x- = 20, Solving each of these equations for x- gives 28 20 1 + 3 7« 1 + m^ whence a quadratic equation for //(, namely 7 7n2 -lo?H + 2=0, m = 2 or +. ■whose roots are Since we had piit // = ?nx, we now know that either ?/ = 2 .r or ?/ = f X. These are the same equations as (7) and (8) and now we proceed as in the first solution. Figure 75 shows the graph- ical solution. The graph of (4) is the hyperbola of Fig. 75 and the graph of (5) is the circle shown in the same fig- ure. The coordinates of the +y ^ / / r 7 Y I _, A^ / / / \^ s ^ / / ^ n\ _ _- N / _— - -m t;. — - '70 ^ V. c\ s / 1 ■•^ \ iy / / A/ OjV ■^ / ' \ / 1 / — 7^ — - — points of intersection ^1, B, C, D are the solutions of our sys- tem. The graph of (G) is composed of the two straight lines AC and BD which pass through the origin 0. We might find the solutions graph- ically by drawing the circle (the graph of (5)), and these two straight lines, and then measuring the coordinates of the four points of intersection. Our algebraic process is exactly equivalent to this. Fig. 75 EXERCISE CXVI Solve the following systems algebraically, using the simplest graphs which you can find to accomplish the purpose. This means, for instance, to use the circle and the two straight lines of Fig: 75, rather than the cii'cle and the hyperbola. 420 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 252 1. a;'^ 4- xy — y- = 5, 6. 2 //'^ — 4 ^7/ + 3 X- = 17, 2x^ - oxy +'2y'^ = 14. y/2-.r-2='l6. 2. x2 + yi = 10, xy = 3. 7. X2 + ?/2 _ c[2^ xy ^ k. 3. x^ + y^ = 20, xy = 8. 8. Z2 , .y^ _ J a2 i 4. x^ + 10 //2 = 44, xy = ^•. 5. xy + 1 = 0. •r^ + 3 xy = 28. iy^ + xy = 8. 9. X2 .?/2 _ xy = A:. 252. Case III. Both equations contain x and y in sym- metric fashion, so that each equation is left unaltered if x and y are interchanged. The equations are of the form (1) A(x^ + f)+2Hxi/-h2a(x+2/}+ (7=0, (2) A' (x^ + j/2) + 2 H'x^ + 2 a' ix + y) + C" = 0. In this case it is advisable to introduce the fundamental symmetric functions of x and y (see Art. 133) as new un- knowns. We put therefore (3) x + y = u, xy = v. Since a;2 j^y'i-^(x^yy'—2 xy, we shall have (4) a;2 + ^2 ^ w2 _ 2 y, so that the equations (1) and (2) assume the form (5) aw^ + 5m H- 6'v + c? = 0, (6) a!iC"^Vu^ c^v + S! = ^. From these two equations we can always eliminate ti? by linear combination, so as to obtain an equation of the first degree between u and o. If this first degree equation be solved simultaneously with either (5) or (6) (see Art. 244), Art. 252] BOTH EQUATIONS SYMMETRIC 421 we obtain two sets of values (Wj, Vj) and {u^, v^) ^^^' ^ ^^^^^ ^• It now remains only to solve the systems and in each of which one of the equations is of the first degree. ExAMPLK. Solve the system X- + y^ = 25, xy = 12, by this method. Solution. Putting x + /y = u, xy = v, we have u-2 - 2 (' = 25, V = 12. Therefore u- = 25 + 2 y = 49, w = ± 7. It remains to solve the two systems •^■ + .^ = ^' and l^+U = -7, xy = 12, [ xy = 12. The solutions of the first system are (3, 4) and (4, 3). Those of the second system are (—3, — 4) and (— 4, — 3), A second method applicable to this case is to substitute (7) X = m + n, y = m — n, so that (8) X + ^ = 2 ?n, xy = m? — n-, x- + y'-^ = 2(7n- + n-). In the above example this would give us 2(m2 + n^) = 25, m- - n^ = 12, whence 4 m2 = 49, 4 n2 = 1 TO = ± I, n =± |. and tlierefore X = I + ^ = 4, or I - 1 = 3, or - I + 1 = - 3, or - I - ^ = - 4, to be combined respectively with y=i-i = -i, or I-(-D = 4, or - | _ i = - 4, or _|-(-'i) = -3, giving the same four solutions as before. Geometrically, the symmetry of the equations (1) and (2) has the follow! no- signiticance. If x = a, y = h is a point on the graph of (1), then the point x=h, // = a is also on the 422 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 252 graph of (1). But (see Art. 149) this means that the locus of (1) is symmetrical with respect to the line y = x, which bisects the angle between the positive a:-axis and y-axis. If this is true of both equations, then the solutions of the system must correspond to each other in pairs in such a way that the two members of a pair belong to points which are symmetric with resj^ect to this angle bisector. Observe that this is actually so in case of the system which we have just solved. For the graphic solution of this case it is advisable to replace (1) and (2) by an equivalent system composed of two equations one of which, at least, shall contain no xy. For the graph of such an equation will be a circle whose center lies on the bisector of the angle between the coordi- nate axes. Sometimes (if ]I= W = 0) both graphs are of this kind. In all other cases the second equation may be chosen so as to con- tain no x^ + y^ term. Its graph will be a hyperbola with a vertical asymptote (see Art. 140) unless it happens to degenerate into a pair of lines, one parallel to the rr-axis and one parallel to the y-axis. Figure 76 represents these graphs for the system x^ -\- y^ — 25, xy = 12 whose algebraic solution we have just given. Fig. 76 EXERCISE CXVII Solve the following systems algebraically and graphically, using the simplest graphs which you can find to accomplish the purpose : 1. x2 + 7/2 ^ 13^ 3_ ^i + y = Q. Now (4) is a cubic equation for k. If the k which occurs in (3) is one of the roots of this cubic equation, the left member of (3) will be a product of two linear factors, and the system composed of (1) and (3), which is equivalent to the original system, may be solved by the method of Art. 253. This method is applicable in all cases, but requires the solution of the cubic equation (4) for k. Thus we see that the general problem of solving any system of simtdtaneous quadratics may he reduced to that of solving a certain auxiliary cubic equation with one unknown. Tliis method is somewhat simpler than that of Art. 248, where we required the solu- tion of an equation of the fourth degree. However, the two methods are closely related. For, as we saw in Art. 124, the solution of an equation of the fourth degree requires as a preliminary the solution of an auxiliary cubic equation. The method outlined in this article becomes immediately available in particular, whenever one of the roots of the cubic equation (4) happens to be evident by inspection. Akt. 2r)o] THE METHOD OF SMALL CORRECTIONS 425 Cases II and IV (Arts. 251 and 253) are really illustrations of this general method. EXERCISE CXIX 1. Set up the cubic equation for k in llie case where the two given equations are of the form Ax^ + 2 Hxi/ + Bf + r* = 0, A'y- + 2 //'./■// + />"//- + <"' = 0, and show how the method of tliis article applies to Case II. 255. The method of small corrections. It often happens that the graphic solution gives only approximate values of x and y and that none of the algebraic methods which we have discussed will be convenient for actual numerical work. In such cases it is usually desirable to employ the method illus- trated in the following example. * Find a solution of the simultaneous (juadratics (1) 4x-' + !)y^=l (2) {x-\y+{ii-\Y = i correct to six decimal places. The graph of (1) is the elliiise ABA'B' (Fig. 77) which has the X-axis and ?/-axis as axes of symmetry and for which OA — \, and OB = \. The graph of (2) is a circle of radius unity, whose center C has the coordinates x = 1, y = \. Figure 77 shows that one point of intersection 5 is given approximately by x = 0, y = \, and the other point of intersection T has coordinates which are approximately x = \, y = — \. We propose to find more exact values for the co- ordinates of T. We regard the values X = 0.33, */=— 0.2.5 as a first approximation. Let (3) X = 0..33 + h, y=- 0.25 + k * This example has been taken from Ruxge's Praxis der Gleichnncjen, Leipzig, 1900, p. 73. +y +1 / ^ / B s •(1,^4) A O \> +1 / B' ^-^ - — -^ -1 Fig. 77 426 QUADRATIC FUNCTIONS OF TWO VARIABLES [Art. 255 be the true values of these coordinates. Then h and k, the corrections, may be regarded as small quantities whose squares and products may be neglected. If we substitute the values (3) in (1) and (2), neglecting h', hk, and ^-^, we find the equations of the first degree ,,. 2.64 A - 4.5 A: = 0.0019, W 1.34 A + 1.50 yL- = 0.0114 for the corrections. Let us solve these equations of the first degree for Ti and k, neglecting all figures beyond the 5th decimal place. We find (5) /« = + 0.00541, k = + 0.00275. Substituting these values in (3)- gives the values (6) X = 0.33541, y^- 0.24725 which we regard as a second approximation to the desired coordinates of T. We now use these new values of x and ij and put X = 0.33541 + h^, y^- 0.24725 + k^, where /;, and k^ are the further corrections. We obtain the following equations for Aj and Atj, 2.68 \ - 4.45 k^ = 0.000192, 1.33^1 + 1.49 yti = 0.000062, giving hi = - 0.000001, ^■l = + 0.000042. The corrected values of x and y will be (to six decimal places) X = 0.335409, 1/ = - 0.247208. As a check we substitute these values in our original equations. We find 4x^ + 9 >f - 1 = 0.00000283, (x - 1)2 + (/y — *)•■=- 1 = 0.00000097. A third approximation would not alter our results by as much as a single unit of the sixth decimal place. Therefore the values obtained are correct to six decimal places. This method, known as the method of small corrections, is applicable not only to simultaneous quadratics with two un- knowns but to equations of higher degree involving more than two unknowns. Newton's method (see Art. 97) is really a special case of the method of small corrections. Art. 256] APPLICATIONS 427 EXERCISE CXX 1. Find to six decimal places that solution of X-! - 5 x^-;/* + 1506 = 0, y5 _ ;3 -giy _ 103 = 0, which is given approximately hy x = 2, y = 3. 256. Applications which involve simultaneous quadratics. The following exercise contains concrete pi'oblenis which lead to simultaneous quadratics. They have all been formu- lated in terms of general numbers. If the student wishes to obtain specific numerical applications, he may of course substitute particular numbers for the quantities a, 6, c, etc., which are to be regarded as known in each of these examples. But the practice of setting up the equations and solving them by general formulas is more important, at this stage, than that of obtaining numerical results, since the student has had ample practice in obtaining such results in the earlier parts of this chapter. EXERCISE CXXI 1. The Slim of two numbers is «, and their product is b. Find the numbers. 2. The area ^ of a right triangle and its hypotenuse c are given. Find formulas for the lengths of the other two sides, a and b. 3. The width of a circular track is 1/n of its inside diameter. The area of the track is A square feet. Find the dimensions of the track. 4. Two circles, whose centers are on the same diameter of a third circle, are tangent to each other externally and touch the third circle internally. (See Fig. 78.) If r denotes the radius of the outer circle what must be the railii, Tj and 7-„, of the smaller circles in order that the sum of their areas may be just half the area of the outer circle? What values must r^ and Tj have, in order that the sum of the areas of the small circles may be equal to any specified fractional part m/n of the large circle? Wiiat is the smallest value which the sum of the areas of the two small circles can ever have ? CHAPTER XTV SEQUENCES AND SERIES WITH A FINITE NUMBER OF TERMS 257. Continuous and discontinuous variation. All of the questions which have occupied our attention so far were in- timately connected with the notion of a function^ and we discussed, in order, tlie principal properties of linear and quadratic functions, rational functions both integral and fractional, algeln'aic irrational functions, exponentials and logarithms. Finally, in the last two chapters we studied linear and quadratic functions of two independent variables. In some cases the function f(x)^ which was under con- sideration, was defined as a real function for all real values of X. In other cases the function was defined only for a certain range of values. For instance, the functions log^a; and -\/x were defined, as real functions of x, only for posi- tive values of x^ and the function Vl — a^ is defined as a real function only for values of x between — 1 and + 1. But in all cases we admitted that x might assume any one of the values of the range for which the function was defined. We always thought of x as changing its value gradually^ from its initial to its final value, without omitting any of the intermediate values. In other words ive have so far^ almost exdusively^ thought of x as varying continuously. We now propose to think of x as varying discontinuously. More specifically, we shall allow x to assume only integral values. Moreover, we shall usually confine ourselves to positive integral values of x. Although we are only now beginning to enter upon a systematic study of functions of x when x is confined to integral values, we have considered a few especially important cases of this kind earlier in our course. (Com- pare Art. .'36 on arithmetical progression, also Art. 163, Theorem X.) 428 Art. 2r,8] DKFIXmoX OK A SKQUENCE 429 258. Definition of a sequence. Let us consider the values which a function /'(•'■) iissumes when x assumes in succession tlie vahies 1, 2, 3, 4, ■•■, n. These values will he ecjual to /(I), ./•(•2), ./•(;{), ..., /•(//). respectively. Let us call these values Wp w.^, u^ •■•, ?/.„, so that (1) Wi =/(-!), W2=/(2), ..., 7^.=/(/0- We naturally think of the numbers ?/p u^^ ••• u„ obtained in this way as being arranged in order ; Wj first, u^ second, Wg third, •••, iin in the ?ith place, and we shall say that they form a sequence, in accordance with the following definition : The 91 umbers of a set are said to form a sequence if the in- divii/ual numhers are regarded as standing in a definite rela- tion of order with respect to each other^ so that we may in a perfectly definite manner speak of the first, the second, •••, the nth number of the set. The kth number of the set (1), or the kth term of the sequence (1), is (2) %=.f(^). Since k may assume any one of the values 1, 2, 3, •••, n, the expression (2) may be regarded as representing any one of the terms of the sequence. For this reason we also speak of u,, as the general term of the sequence (1). In the sequence 1, 2, o, ••• n, we have w, = 1, »o = 2, W3 = 8, •■■,Uk = k, •••, w„ = 71. In this case/(x-) = x. In the sequence 1/2, 1/4, 1/8, ••■, 1/2", we have Mi = l, M„ = — , M„ = — , •■•, «i= — » •••» "»= ?r ' '^'^d /( j) = — • 2 2^ 2* 2* 2" .' V ' 2x We cannot always write down an explicit formula of the form (2) for the ^th terra of a sequence, even when it is clear that the sequence is perfectly well defined. 480 SEQUENCES AND SERIES [Art. 259 Thus, let ^^ ^ j^^ ^^^ ^ J ^j^ ^^ ^ j_^^4^ g^^^ and for any value of k let Hk be the largest decimal fraction with k digits to the right of the decimal point whose square is less than 2. In other words, let ui, mo, us, ••■, U). be that sequence of decimal fractions which appears when we seek the approximate value of ^2 to 1, 2, 3, ••• k deci- mal places. Although u^ is perfectly well defined by this statement, we cannot write down a general explicit formula for m^- We shall, nevertheless, be able to say in all cases, that the kth term of a sequence is a function of k, and ivrite Uk=f(k). This equation merely means that the ^th term of the se- quence depends upon k for its value, whether we can find an explicit formula for it or not. We shall confine most of our discussion, however, to the cases where such an explicit formula is given, and we miglit begin with the simplest case when the function f (x^ is an integral rational function of the first order. However, this case has been treated already and leads to the theory of arithmetic progressions. (See Art. 56.) 259. Higher progressions. Let us study the case of those sequences which naturally come next, that is, the sequences which are obtained from a quadratic function ax^ -f hx + c when we substitute for x in order the values 1, 2, 3, ••• n. The kth. term of such a progression will be (1) % = aP + hk + c, and w^_^ = a(k - 1)2 + h(k -!)+(? will be the (k — l)th term. If we denote by A'^_j the dif- ference between these two terms, we find A'^._j = %. - Uk-^ = aB + hk+c- la{B-'2 /c + 1) -f- h(k- 1) -f e] = ak'^-\-hk + c—[^ak'^+{h— 2 d)k -\- a -/> + e], Art. 259] HIGHER PROGRESSIONS 431 whence (2) A',_i = u, - n,_, = 2ak-a + h. Since the right member of (2) is of the £rst degree in k, the differences ^ , , , , , A A A ... A' form an arithmetical progression (See end of Art. 258), and the differences of these differences (called second differences^ are all equal to each other. In fact, if we denote these second differences by A";i_j, so that AVi=A',-AVi, we find, from (2), (3) A",_i = -2a{k + 1^ - a +h ~ \^ak - a + h^=2 a, and the third differences are all equal to zero. There is no difficulty in generalizing this interesting result. If the kth term of a sequence is expressible as an integral rational function of k, and if this function is of the nth degree^ so that u^ = ak"" + hk"-'^ + r-A;"-2_|_ ... 4. /^ _|_ ,^^ the terms of the sequence which is formed by the nth differences tvill all be equal to each other. In other ivords, the nth differ- ences are constant^ that is, their value does not depend upon k. TJie (n + V)th and all differences of a higher order will be equal to zero. This theorem has a very important application. When the numerical values of a function f(x) have been tabulated for certain equidistant values of x, as in the case of a table of logarithms, the values of /(.f) which correspond to inter- mediate values of x must be obtained by interpolation. (See Art. 171.) In the ordinary logarithmic tables the values of a;, for wliicli the function is tabulated, are so close to each other that the ordinary method of interpolation, in which only first differences are used, is satisfactory. In fact the second differences are either zero, or else their values do not exceed a single unit of the last decimal place which is used 432 SEQUENCES AND SERIES [Art. 260 in the calculation. But in many cases it is impracticable to make the tables so extensive. The Nautical Almanac, for instance, gives the declination of the sun, that is, the dis- tance in degrees of its center from the celestial equator, for noon of every day of the year. If we wish to find the declination of the sun at 6 A.M., we must interpolate and, in this case, the first differences are not approximately constant, so that the ordinary method of inter[)olation does not suffice. The method of interpolation by higher differences rests upon the fact that the converse of tlie above theorem can be established. That is, if the nth differences of a function are constant, the function may he regarded as an integral rational function of the nth order. For a proof of this theorem, and for further details in the theory of interpolation, we must refer to other sources. 260. Geometric progressions. Another important sequence is obtained by considering the values which an exponential function ar"" assumes for a: = 0, 1, 2, 3, ••• w. These values, namely, a, ar, ar"^, ar^^ •••, «r", form a geometric progression, and we have already had tlie opportunity of becoming ac- quainted with the principal properties of such progressions. (See Arts. 59-63.) EXERCISE CXXII Write down the first four terms (corresponding to x = 0, 1, 2, 3) and the general term of the sequences defined by each of the following func- tions : 1. /^+ 1. 5. — i— . 9. 2*. - ' + 2. 6. -^ . 10. 2 3. x2 + 3 X 1 + 1 x^ X- ' - X X- — 1 X- + 1 1 11. 3''+''5^-i. 4. J—. 8. -J—. 12. log (3-). x + 1 .r^' + 1 13. Write down eight terms of the sequence defined by the function X- + 3 .r — 1, and the first and second differences of this sequence. 14. Prove that the sequence defined by the function x^ has the prop- arty that its fovn-th differences are all equal to zero. Akts. 261, 262] SUMMATION OF SERIES 433 261. Series. Whenever we have a sequence Mj, u^, W3, ••• m„, the sum of all of the terms of this sequence, S„=Ui+ u^+ ■■■ + u„, is called a series. Since we may think of ;t, the number of terms, as being large or small, and since the sum of n terms of a sequence will depend upon the value of /i, we may say that jSn is a function of n, and this fact is indicated in the notation. Tn the case of arithmetic and geometric progressions we have been able to find simple formulas for S^ (see Art. 56 and Art. 60), which make it possible to compute the value of *S'„ without actually adding up all of the individual terms. Whenever such a formula has been found, which gives explicitly the value of >S'„ as a function of ?i, we say that the series u^ + U2+ — h «„ has been summed: the prob- lem of finding such a formula is known as the problem of summation of series. 262. Summation of series by mathematical induction. In very many cases we can obtain a formula for the sum of a series by the method of mathematical induction. In fact the summation of series by this method is one of the best means for becoming thoroughly familiar with the method of mathe- matical induction, which is one of the most important forms of mathematical reasoning, and Avhich we have used several times in this book (See Arts. 84 and 88.) The following example will illustrate the method. Illustrativk Exampi,k. We observe that (1) 1+3 = 4 = 22, 1 + 3 + .■) = = 3'^ 1 + 3 + 5 + 7 = 16 = 42. These equations suggest that the following law may be true ; the sum of the n first odd integers is equal to ur. In fact, equations (1) prove that tlie law is true in the three special cases when n — 2. 3, or 4. To prove that the law actually is true for all values of n. we first prove the following leinina, or auxiliary theorem. Lkmm A. If the sum of the first k odd integers is equal to k-, then the sum of the first k + I odd integers will be equal to (k + l)^. 434 SEQUENCES AND SERIES [Art. 203 Proof of the lemma. The kt\\ odd integer is 2 A — 1, since the odd integers form an arithmetic progression whose first term is 1 and wliose constant difference is 2. Therefore, //'the sum of the first k odd integers is equal to k'^, we have (2) 1 + 3 + 5+ ... +(2A--l)=A---2. The sum of the first k + 1 odd integers will be obtained by adding the {k + l)th odd integer (which is 2 k -\- 1) to the sum of the first k odd integers. If the latter is given by (2), we shall therefore have (3) 1 + .3 + .0+ ... +(2/.-l)+(2/.+ 1) = B + 2k + l = {k+ \y. In other words : if (2) is true, then (8) must also be true, thus proving the lemma. But we know from (1) tliat the sum of the fii-st four odd integers is 42. Apply the lemma for the case k — 4. The lemma tells us that the sum of the first five odd integers is 5-. Apply the lemma to the case k = 5. It tells us that the sum of the first six odd integers is 6^. Pro- ceeding in this way we finally conclude that the sum of the first n odd integers actually is equal to n^. 263. General characteristics of the method of mathematical induction. The theorems which are capable of proof by mathematical induction have the following characteristics : (a) The theorem asserts that a certain property or relation is valid in all of a certain well-defined set of cases. (6) The cases with which the theorem is concerned can be arranged in a definite order, so that there is a first, a second, • ••, a kih. case. (c) The total number of cases may be large or small. But the most useful applications of the method of mathematical induction to theorems of this sort are those in which the number of cases is large. The proof of a theorem of this kind by mathematical induction always consists of two parts. The first part of the proof consists in verifying that the theorem states the truth in the first few cases. We shall call this part of the proof, the verification. For the purposes of the proof by mathematical induction, it really suffices to verify the correctness of the theorem for the very first case. But frequently, as in the illustrative example of Art. 262, the statement Art. 263] MATHEMATICAL INDUCTION 435 whose general validity is to be proved is suggested in the first place by inspection of what actually happens in the first /ew cases. It is for this reason that the word induction is used in this connection. The second part of the proof consists in proving the lemma : if the theorem is true in the kth case, it will also be true in the (^k + l^fh case. This usually constitutes the more difficult part of the proof. If we have made the verification and proved the lemma, tiie proof of the theorem is complete. For the lemma enables us to conclude, from the fact that the theorem is true in case one, that it is true in case two. From this the lemma enables us to assert the truth of the theorem in case three, and so on for all cases. The lemma may or may not be true, but until we prove it to be true we have not proved the general theorem. In a proof by mathematical induction both parts of the proof are equally important. The following example, which is of considerable interest historically, will illustrate the fact that the first part of the proof alone does not suffice to jjrove the theorem. The numbers 2 + 1 = 3, 2-^ + 1 = 5, 24+ 1 = 17, 28 + 1 = 257, 2i6 + 1 = 65537 are all prime numbers (See Art. 3) and they are all expressible in the form .,.^fc ^ J Thus the numbers of the form 2-* + 1 are certainly prime numbers for k = 0, 1, 2, 3, 4. An incomplete induction would therefore make it seem likely that all numbers of this form are prime numbers. In fact, Fermat (1601-1665) thought that this was the case. But Euler showed later (in 1732) that this was not so by proving that the number given by 2-^ + 1 = 232 + 1 = 4,294,967,297 is divisible by 641. Another illustration of this sort, due to Euler, is given by the expression n'i—n + 41. If we compute the values of this function for n = 0, 1, 2, 3, ..., 40, we find that all of the resulting values are prime numbers. The evidence seems to be very strong in favor of the assertion that ifl — n + ^\ is always a prime number. But it is very easy to see that this is not so. For n = 41 we find n2 _ „ + 41 = 412, and this is obviously not a prime number. Two similar expressions due to Legendre (1752-1833) are ,j-2 + n + 17, which represents prime numbers for all integral values of n less than 17, and 2/t- + 28, which represents prime numbers for all values of n less than 28. All of the examples given so far emphasize the importance of the 436 SEQUENCES AND SERIES [Art. 263 second part of the proof by mathematical induction. We shall now give an example to illustrate the fact that the second part, when not accom- panied by the first part, does not constitute a complete proof. The veri- fication, or first part, is also essential. If we were to omit the verification we might easily prove the follow- ing proposition, which is an obvious contradiction to the illustrative example of Art. 262 : the sum of the first n odd integers is equal to 2.5 + n2. In fact, the proof of the lemma is easy. If (1) 1-^3+5+ ... +(2k- 1)= 2.5 + k% then (2) 1 + 3 + 5+ ...-f(2^:-l) + (2A: + l)=25 + A:H2^ + l=25+(A,- + l)2. From this it would follow that 1 + 3 + 5 + ... + C2 n - 1) = 25 + n^ in all cases, if this equation were true for 7i — 1, which, however, is not the case. The process of mathematical induction may be compared to the process of climbing up a ladder. The first part of the proof (preliminary verification for the first case) assures us that we can put our foot on the ladder ; the ladder actu- ally touches the ground and the first round is within reach. The second part of the proof assures us that there is no round missing ; we can actually pass from the ^th to the (k + l)th round. EXERCISE CXXIII Prove the following formulas and theorems by mathematical in- duction : 1. 1 + 2 + 3 + 4 + ... + n = "('^ + ^^ . 2. 1 • 2 + 2 . 3 + 3 . I + ... + n(n + 1) = i n(n + l)(ri + 2). 3. 1 .2.3 + 2.3-4+ •• +,,(« + l)(n + 2)=l«(n+l)(n + 2)(n + 3). 4. 12 + 22 + :',2 + ... + „■! = 1 n(n + l)(2n + 1). 5. P + 23 + 33 + ... + „3 = (1 + 2 + 3 + ... + n)2 = {n2(», -|- 1)2. - 1 + 1 , ... + 1 1-2 2-3 n{n + 1) n + 1 7. 2 . 4 + 4 . (5 + 6 . 8 + ... + 2 /((2 /i + 2) = i «(2 n + 2) (2 n + 4). Art. -264] THE SUMMATION SIGN 437 8. x" — y" is divisible by x — // if n is any positive integer. (See Art. 84.) 9. ./:" + .'/" is divisible by x + y if n is any odd integer. 10. .(•" — //" is divisible hy x + y ii n is any even integer. 264. The summation sign. When dealing with a series (1) u^ + u^+ ••• + M„, whose terms are formed according to some complicated law, it becomes very burdensome to write out all, or even many of the terms. If % is the ^th term, we may replace (1) by the symbol (2) X ^^^' which is read sum of such terms as % from k = 1 to k = n. The 2 which appears in this symbol is the Greek capital S and is called sigma. Thns we may write n 1 . o + 2 . ;5 + ... + ^•(^■ + 1) + - + n{n + 1) = ^J ^'^^ + 1)' *=i n 1 . 2 . 3 + 2 . :5 . 4 + - + n(» 4- 1)(h + 2) = J /'(/•• + l)^- + 2), 1 ,,^,^,...,^.X- 1+12 1 + 02 1 + :}2 1 + ,j2 ^ 1 + ^■2 EXERCISE CXXIV 1. Use the summation sign to represent each of the series which apjiears in Examples 1-7 of Exercise CXXIII. Write out. the first four tiMins of cacli of the following series: ^1 + ^-2 A 1 + 3fc ^ /•! 3. yi. 5. yl. ■ 7. y '"-' k=\ k=\ k=\ ^ ' 438 SEQUENCES AND SERIES [Art. 265 265. Summation of a series whose /tth term is an integral rational function of k. We kno^v from the theory of arith- metic progressions that (1) 2*=^^^^. ft=i -^ and we have seen in Exercise CXXIII, Examples 2 and 3, that (2) yt(j + i) = 'i£^+ll(!i+2i, (3) ^yfc(/^ + i)(^ + 2)=<^ + ^><^^+^><^^'+^). These results suggest that the following formula may be true (4) ^^(1-Hl)...(^+^_1) /t=i _ n{n->rl). ' -{n + l—l^jn+l) l + l where I is any positive integer, and where every term of the sum in the left member contains I factors, while the nu- merator of the right member is a product of / + 1 factors. Formula (4) is certainly true for Z = 1, 2, 3, for in these cases it reduces to equations (1), (2), (3) respectively. To prove the validity of (4) in general, we proceed as follovps. The ^th terra of the series vrhich occurs in (4) is (5) n, = ^-(^^ + l)(^- + 2)-(^+/-l). It is a product of I factors, and may be expi'essed simply in terms of two similar expressions, each of which contains I + 1 factors. In fact, if we put (6) Vk = k(k +l)(Ic + 2)...(k + l- IXk + 0, and therefore (7) r,_, =(k - l)k(k + 1) ... (k + l- 2) (A- + 1-1), we observe that the first / factors of r/, are the same as the last / factors of J'i-i, so that Vk - i'A-i = K'' + 1)(^- + 2)...(k + l-l)lk + l- (k - 1)], Art. 265] SUMMATION OF A CLASS OF SERIES 439 or, (8) V, - v,.t = k(k + l)(^• + 2) ... (k + l- !)(/ + 1). But the product of the first / factors of the right member of (8) is equal to «fc, so that we find (9) v^-r,_i=(!+l)H,. Let us write down the particular equations which follow from (9) if we put in order ^ = 2, 3, 4, .•■ n. We find ^'2 - '"1 =('' + 1)«2- ^IQ. '-4 -'-3 =(l+l)>^. ?;„_!- i'„_2=^(/ + 1)«„_„ . r„-r„_i =(/+!)«„. The sum of all of the left members reduces to r„ — I'l, since each of the other terms i'2. is •■• i'„_i occurs twice, once with a plus and once with a minus sign. Therefore we find, by addition, (11) I'u - '-i =(/ + l)('/2 + "3 + »4 + ••• + "„)• Moreover we have, acccording to (6), i-i = 1. 2. :3.- /(/ + !), and according to (5) ui = 1 -2 •:}.••/, so that (12) a=(/ + l)"i. If we substitute this value of I'l in (11) and transpose it, we find (•„ =(/ + l)(Ui + Mo + ... + u„), whence .-..^^ , , , v„ n(n + l)(n + 2)— (n +1) (10) III + Uo + ••• + U„ — ^!— = —5^ ^-^ ■ i- '> <-, ^ ^ l+\ l+\ which is merely another way of writing the equation (4) which was to to be proved. Formula (4) is important, because it enables us to find the sum of n terms of any series whose kth term is an integral rational function of k, the degree of this function and its coefficients being the same for all values of k. For, let the A^th term of a series be given by an expression of the form (14) ut = ak-^ + hk'-^ + •.. + mk + n, where a, h, ■•• nj, n, and I are independent of k. By the method of unde- termined coefficients we can rewrite u^ in the form 440 SEQUENCES AND SERIES [Art. 265 (15) ui, = Al-(k- + 1) ••■ (/.• + /-!)+ Bk(k + 1) ■■■ (1- + I - 2) + - + Mk+ N, so that the sum ui + m2 • • • + M,i will be equal to ,g. , «(n + l) ••• (n + I) ^n (n + 1) ••• (» + Z - 1) ^ ^ ' / + 1 / Illustrative Example. Find the sum of n terms of the series whose ^th term is (17) Mt = 3A--!-4A- + 2. Solution. We wish to write iik in tlie form (18) w, = Ak{k +\) + Bk + C. If we expand the products indicated in (18) we find (19) Uk = A (t^ + k) + Bk + C = Ak-^ + (A + B)k + C. In order that this expression may be identical with (17) for all values of k, we must have A =Z,A + B = -^, C = 2, whence A =%,B = -'t,C = 2, so that we have found tik = -ik{k^ I) ~1 k + 2. Consequently n n V Y^u,= -6"2^k{k+l)-l^k + 2n, k=\ fc=l A=l or, if we substitute for the sums which appear in the right member their values, (20) 2 C^' ^■' - * '^^ + -) = 3 "<^" + V^" + -> - 7^ <^" + ^^ + 2 n. EXERCISE CXXV By til!' method of Art. 26.") prove the following formulas : 1- V X-2 = y n{n + 1)(2 // + ]). 2. y F = {.' n{n + \)f. 3. y 2 k(2 A: + 2) = \ n{2 n + 2) (2 n + 4). Art. 2G6] SUMMATIOX OF OTHER SIMPLE SERIES 441 266. Summation of some other simple series. We may find the sum of n tei-ins of a series, wliose ^th term is the reciprocal of the ^th term of series (4) of Art. 265, by a very similar method. Let (1) Uk= ^ , kxk + i)(A + -_>) - (^■ + /-l) and let us put (2) '•* = ^ , -l(/--t-l)(/.-fi) - {k + l--2y the inuuber of factors in the denoniinator of '•/.. being one less than in t/*. From ("2) we find 1 '■'•+' ~ {k + l)(^• + -1) :■ {k + i-2){k + i-\y and therefore I'k - '•<+! = —1 ri- {k + \){k + 2) ... (^• + / - L>) L^- k + I 1 A- 4- Z - 1 - /; '-] (k + 1) (^• + -2) ... (k + l- 2) k(k + 1-1) ^ l-l k(k + l)(k + 2) ... (k + l- 2)(k + / - 1)' whence, making use of (1), (■i) Vk - Vk+\ = (I - l)«f Thus, we have in particular '•i - "2 = G - l)"i. f2 - rg = (/ - 1)^2, (4) '•u-l - '■„ = (' - l)«n-l, '"n - ''h + I = G — !)««• From the.se ecjuatious we find by addition, '•i - 'n+i = (I - l)("i + 11.2 + ■■■ + n„), 2"* = ^(''i- '•,.+!). if/ 9^1. If finally we substitute for v^ and (•„+! their values from (2), we find (5) y ^ =-J-r-^^ 1 . 1 ;_ lL(/- 1)! (n + \)(n +2) ••. („ + / _ 1)J ' a formula which is valid for all positive integral values of t except / = 1. 442 SEQUENCES AND SERIES [Art. 266 In particular for I = 2, 3, 4, we find V 1 -1 L_ ^^^ ^ k(k + l)ik + 2) " 2 12! ~ (n + l)(n + 2) J ' • V 1 =iri-. 1 1. n k(k + l)(k + 2)(^- + 3) 3 L 8 ! (n + l)(n + 2)(n + 3)J These formulas, and the method used in deducing them, may be applied to the summation of many other series. CHAPTER XV LIMITS 267. Limits suggested by series. Let us consider again the geometrical progression m 1 1 1 1 ... -i- ... ^^) ' 2' 4' 8' 2"-!' and let us denote by S^ the sum of its first k terms. Then S! — 14.1— 3_9_1 AJ2 — ^'2 — 2 — 2' ^3=l + 2^+i = l = 2-i, (2) ^^=i + |+i + i=JJi = 2-i, ,V— I4-I4-I4-...4- ^ =2 — and we observe that *S'„ approaches the limit 2 as w grows larger and larger. Figure 79 illustrates the same situation graphically. s. s,s,s, The notion of limits also presented itself to 1 2 our attention in Arts. 87 and 89, when we ^^^' '^^ were concerned with the determination of the tangent at a given point of a curve. The theory of limits is of importance also in discussing the so-called incommensurable cases of elementary geoyietry and in the closely allied questions con- cerning irrational numbers. It also comes up in the men- suration of the circle. There are many other subjects in pure and applied mathematics in which the notion of limits is indispensable. The theory of infinite series, of which the geometric progression (1) is an illustration, is one of these. Since we wish to give a brief treatment of infinite series in this book, we must first discuss some of the more important questions which are connected with the notion of a limit. 443 444 LIMITS ^ . [AuT. 268 268. Definition of a limit. The notion of a variable which approaches a limit is a fairly familiar one. Thus the sum *S'„ of n terms of the geometrical progression (1), Art. 267, approaches the limit 2. If P„ denotes the perimeter of a regular polygon of n sides circumscribed about a circle whose circumference is (7, P„ approaches Q as a limit. At the same time the area A^ of the polygon approaches the area A of the circle as a limit, and the same statements are true of the perimeters and areas of the regular inscribed polygons. When an automobile slows up and finally stops, it approaches its stopping place as a limit. If we prefer to think of the variable as a number^ we may, in this last'illus- tration, say that the distance from the starting place of the automobile approaches as a limit the distance from that point to the place where it stops. A pendulum which swings in a resisting medium like air approaches the vertical position as a limit. In Arts. 87 and 89 we developed the notion that a straight line which joins a fixed point of a curve to a mov- ing point of the same curve, approaches a limiting position, called a tangent of the curve, when the moving point ap- proaches the fixed point as a limit. In order that we may be able to reason logically about limits in general, we must think about these various instances, discover the essential feature which they all have in common, and then formulate a definition which shall cover them all. If we do this we are led to the following definition. A variable x is said to approach the constant a as a. limits if the law ivhich describes the variation of x is such, that the numerical value of the difference between a and x ivill ultimately become and remain smaller than any positive number which may have been assigned in advance. B}^ the numerical value of the difference between a and x we mean, as usual, the magnitude of this difference, no atten- tion being given to its si n. We shall denote the numerical value of this difference by the symbol \x — a|, as in Art. 17. Art. 268] DEFINITIOX OF A LIMIT 445 The positive number assigned in advance, which is men- tioned in the definition, may be called 8. If we use these notations, we may reformulate our definition as follows: A variable x is said to approach the constant a as a limit, if the law according to ivhidi the. variable x changes is such as to insure that uldmately \x — a\ will become and remain less than ang positive number h which has been previously selected, and which mag be chosen as S)nall as we please. Thus, to test whether a variable x approaches a certain number a as a limit, we may proceed as follows : Step 1. Choose a positive number 8. It is understood that this number may, in particular, be chosen arbitrarily small. Step 2, Examine whether the law according to which the variation of x takes place will permit |a: — a| to become less than 8. Step 3. Examine the variation of x after this stage has been reached, to see whether ja; — a| will not merely become, but ever afterward remain, less than 8. Step 3 is very essential. Thus in the geometric progression 1, ^, \, \, ••• of Art. 2(57, the numerical value of S„ — 1-5/8 does become less than any positive number 8 for n = 4. In fact we have St — 15/8 = 0. But S^ does not approach 15/8 as a limit, since j S,, — 15/8 | does not remain less than 8 after n has increased beyond the value n = 4. In accordance with our definition x may approach the limit a from above, so that x — a is always positive, or from below, so that X — ah always negative. Or else x — a may be posi- tive during some of the stages of the approach and negative during others. Since the definition merely speaks about the numerical value of x—a, all of these cases are equally admissible. The geometric progression of Art. 267 is an instance of a variable which approaches its limits from below. The area ^„ of a regular cir- cum.scribed polygon aj^proaclies the area of the circle from above. If we consider a sequence of regular polygons alternately inscribed and cir- cumscribed about a circle, the area of a polygon of this sequence also 446 LIMITS [Art. 268 approaches the circle, but alternately from below and above. The same thing is trne of a geometric progression whose common ratio is negative but numerically less than unity. Again, ovir definition says nothing about whether the variable reaches its limit or whether it does not. The variable S^ of Art. 267, , . 'S'n = l+f,+ ••■ + — 9 — 1 9n-l On-1 never reaches its limit 2 for any value of n. Neither do the variables P„ or A^-, if P„ and J.„ denote the perimeter and area of the regular circumscribed n-gon. But an automobile not only approaches its stopping point as a limit, but reaches it. Therefore, the variable distance from a fixed starting point to an automobile approaches a7id reaches a limit when the automobile stops. Thus, a variable which approaches a limit, may or may not reach its limit.* It is true, however, that in many problems it is desirable, or even necessary, to think of the variable x as approaching its limit a without reaching it. But such a restriction, if desired, we shall add explicitly. It is not included auto- matically when we say that x approaches a as a limit. We use the symbols lim X = a (read the limit of x is eqnal to a), or X — ^a (read x approaches a as a litnit^, whenever we ivish to say that a variable x approaches a constant a as a limit. The definition of a limit leads at once to the following property, which is used very frequently. If trvo variables., x and y., simultaneously pass through the same values, so that x and y are equal to each other at each and * This remark is valid on the basis of the definition of limit as we have formu- lated it in tliis bool^. Some authors define a limit differently, including in their definition explicitly a statement that the variable shall not reach its limit. For some purposes this definition is preferable to ours. But it leads to some compli- cations which we prefer to avoid. Art. 269] IXFINITY 447 every stage of their variation, then if one of these variables approaches a limit, so does the other, and their limits are equal. In symbols; if x = y, and if liin .v = «, theyi Vww y = a. 269. Infinity. The simplest sequence of numbers is that of tlie positive integers (1) 1, 2, 3, ..., k, :.. According to our fundamental assumptions (see Arts. 1 and 2), if k is a positive integer, no matter how large, there always exists another one still larger, namely ^ + 1. Con- sequently the sequence (1) has no last or largest number; it has no limit, it is unbounded. A variable which assumes such values is said to grow beyond bound, or to become infinite. j\Iore generally, if the law, according to ivhich a variable x changes, is such, that the numerical values of x ultimately he- come and remain greater than any positive number M chosen in advance, x is said to become infinite. This is often expressed by the symbols (2) lima:=x or x — >-xi. These symbolic statements are in common use, and we, therefore, find it necessary to become acquainted with tliem. It should be remembered, however, that, strictly speaking, the use of the symbol lim in this case is inappropriate. For when we write (2) we mean to indicate that x does not approach a limit, but grows numerically beyond bound. Tu fact, the word infinite means just this; unbounded or unlimited. In accordance with this dcluiition a variable x may become infinite by passing through a sequence of values such as (1), all of which are positive, or by passing through a sequence of values all of which are negative, or else by passing through values some of which are positive and some of which are negative. If we restrict the variation of x, by compelling x to assume only positive values while it is grow- ing beyond bound, we write lim X = -|- GO. 448 . LIMITS [Arts. 270, 271 Similarly, the symbolic equation lim X = — CO means that x grows beyond bound, exclusively through nega- tive values. This distinction is frequently very important. 270. Infinitesimals. If a variable approaches zero for its limit, if. is called an infinitesimal. The following statements are immediate consequences of this definition. 1. If X approaches the limit a then x — a is an infinitesimal. 2. If X becomes infinite, the reciprocal of x is an infinitesimal. 3. Ifx is an infinitesimal, the reciprocal of x becomes infinite. The last two statements are sometimes expressed symbolically by writing j ^ — = 0, - = CO. CO Literally these equations have no iiieauing, since division by zero is ex- cluded from algebra (Art. 21) and since the symbol -jo does not stand for a number. 271. Variables which remain finite. If a variable x be- comes infinite it has no limit although we write symbolically lim X = cc . But a variable may remain finite and neverthe- less not approach a limit. Thus, if x assumes in succession the values +1, — 1, -f 1, — 1, and so on, it remains finite and nevertheless it does not approach a limit. A variable x, whether it approaches a limit or not, is s'- n- 464 LIMITS [Art. 275 the sequence of values for ^hj \A 'ill be X _ 1, , 2, 3, -.jn, . y so that, in this case, 1 im - = GO . y Finally, if we should liave 1 - 1 + 3' 4' ±1 the quotient would assume the values ^=+1, -1, +1, -1,-, so that xl]i remains finite but does not approach a limit. Thus, if both x and y approach the limit zero, the quotient xjy may approach the limit zero, it may become infinite, it may approach a finite limit, or finally it may remain finite without approaching a limit. To decide what becomes of the quotient x/y^ if both x and y approach zero as a limit, requires special investigation in every particular case. EXERCISE CXXVI Investigate whether the limits indicated in the following examples exist or not. Find the limit when it exists, quoting the theorems needed in your argument at every step. 1. lim f2 + 1 2. lini n — ^-00 lim r\ + (-l)n2l. 3. li m (3 X - - 5). 5. \im(x+7)(x-S) x->4 4. lim (2 X - !)• 6. lim2- + l. x_>i X + 3 8. 2-1 n lllll 9. lim 5 n-, "-^3+- >i— >-« 7. n > °o 275. Limit of the nth power of a positive number as n grows beyond bound. Let r be a positive number and let us con- sider the sequence of numbers r^, r^, r^, •••, r", ••• for all positive integral values of w. Art. 270] CONTINUITY OF A FUNCTION 455 If r = 1, we have r" = 1" = 1, and therefore (1) lim 1" = 1. If ?• > 1 let us put r=l + h wliere /; is a positive number. By the binomial theorem we have r" =(1 + h)" = 1 + )ih + positive terms (See Art. SS). so that r" > 1 + 7ih. Clearly we may choose n so great that 1 + //// will become and remain greater than any jiositive number ^f, no matter how large M may be. It suffices for this purpose to choose ^^- 1 for then we shall have "> /, 1 +nJi>l +'^^^^h = M. h In other words; if r is a positive number greater than unity, the ex- ponent n may be chosen so great as to cause r" to become and remain larger than any positive number ^f. That is. r" becomes infinite, or in symbols (2) lim r" = X if ?>1. If ;• < 1, we may put r — — , r' > 1, and r" = We .shall have lim (;•')" = oo. since ?•'> 1. Therefore (see Art. 270), (3) lim ;•" = if r 1, lim|r"l = l if |r| = l, lim|r«| = if \r\ < 1. 276. Continuity of a function. Let a and h, where b > a, be two real numbers. By the interval (a ••• b) we mean the assemblage of all real numbers between a and i, includ- ing the numbers a and b themselves. Any such number, 456 LIMITS [Art. 276 excepting a and b, is said to be m the interior of the interval (a ••• b). Let JO represent any number in the interval (« ••• b^. We say that afunctio7i of x is defined for x= p, if it is clear from the definition of tlie function, as expressed by a formula or by some other description of the function, what tlie value of the function will be for re = p. The function is delined as a real oyie-valued, function for x = p if there is only one value prescribed for x = p and if this one value is a real number. Let y=f{x) be defined as a real one- valued function of x for all values of x in the interval (a ••• 6), and let us think of Fig. 80 as representing the graph of such rx a function, where we have made 6)^ = a, OB = b, OX=x. and where the values of f(a), /(^), and f(x) are repre- sented by the ordinates AA', BB\ and XX' respectively, so that ^^, ^^^^^^ BB'^fib), XX' = fix). Let lis now pick out any particular point P of the interval AB, whose abscissa is p, so that a■ A P X li Fiu. 81 Art. 27G] CONTINUITY OF A FUNCTION 457 (as is permissible according to tlie definition of a limit), XX' will oscillate between values nearly equal to PP' and PP" respectively, and not approach any limit. We see that the possibility of such a break in the continuity of the graph is excluded if equation (1) is satisfied. This remark leads to the following defini- tion of continuity. A real one-valued fnnction f(x} is said to be continuous in the vicinity of a particular value x = p ii the following conditions are satisfied. 1. The function f(x') is defined for t = p, that is, the defini- tion of the function assigns a unique definite finite value to the sijmhol fi^p}- This symbol f(^p) then represents a definite finite real number. 2. TJie function f(^x) is defined for all values of x in the neighborhood of x = p, in such a tvay that when x approaches p as a limit, the function f{x) tvill approach one and the same definite finite number as a limit, 7io matter according to what particular laiv x may approach p. This is expressed by saying that the limit lim f(^x) exists. 3. Finally, if the function is to be continuous, the condition lim f(x)=f{p^ must be satisfied. This last statement is read as follows; the limit of f(.r), as x aj> proaches p, is equal to f( p). If the function f(x') is defined only over a finite interval (a ••• /)) we should modify Part 2 of the above definition in the case p = a ov p = b. We shall say that lim f(x} exists if we obtain a definite finite limit for f(x) when x ap- proaclies a from above, since no other values of x would be admissible in su(;h a case. Similarly for lim/(.r). If a function f (^x') is continuous in the vicinity of every par- ticular value p which belongs to the interval (<*••• 6), it is said to be continuous in the whole interval. 458 LIMITS [Akt. 277 We have seen in Art. 273 that, if fix) is an integral rational function of a:, then lira f0 X — 1 x->0 X^ + X + 1 i->0 X 3. hin 5. lim " ' — j:_>1 X- + X + 1 x-^-l x^ — 1 x->- 1 X -1- 1 _ , . X^ + 1 1-^+7 8. hill 9. hm i_>.l x^ — 1 x-^\ X — 1 278. Indeterminate forms. We may summarize the prin- cipal results of Art. 277 as follows. If R{x) is a rational function of x, written in its lowest terms, and if x ajjproaches the limit p, then ^ ^ lim B(x) = Jl(p). Unless JO is a pole of RQx'), this limit will be a definite finite number. If R(x') is a rational function which is not in its lowest terms, and if we write /.^ x then /(a;) and gQx'), the numerator and denominator of IK^x^, will have a common factor dependinq- upon x. If we divide botli numerator and denominator of R(^x^ by tlieir highest common factor, we sliall obtain a new rational func- tion R^ix) which in in its lowest terms, and we shall have (1) IKx) = R,(x-) for all of tliose values of x for which this reduction is legiti- mate, that is, for all values of x except those whi.cli cause Art. 278] INDETERMINATE FORMS 461 the liigliest common factor of f(x) and gi^x) to assume the value zero. a;2 _ 4 Thus, the function R(x^ — ■ is equal to -'^iC^) = x + 2 for all values of x except for x = 2. For x = 2 these two functions are not equal, since the function /2(x) assumes the form ()/() for x = 2, and therefore is not defined for x = 2, while the value of Ri(x) for x ~ 2 is 2 + 2 = 4. But for all values of x, with this one exception, we have = X + 2. X — 4 If therefore we allow x to approach the limit 2, with the specification that X shall approach 2 tvithout reaching this limit, we shall have x'^ 4 lim — = liin (x + 2) = 4 (see theorem at end of Art. 2G8). x-^-2 X — 2 x^l This example is typical of a large and important class of cases. We have given a rational function R(x) which is not in its lowest terms. Consequently there are certain values of x, which cause both the numerator and denomina- tor of Rix) to vanish. Let x=p\ie one of these values of X. The function R(x) then assumes the form 0/0 for x=p and is therefore not defined for x=p. Consequently the statement \m\f{x) \imRix)= ^r> =.^ x^p lim g{x) would be meaningless. The question is : lias R(t) a limit when X approaches p, and if so what is the value of this limit '? We proceed as in the illustrative example. Reduce R{^x) to its lowest terms and let R-^{x) be the resulting fraction. We shall have (1) i2(^) = B,^(x) for all values of x in tlie ueigliborhood of x = p^ excepting only the value x = p itself. If x approaches j3 in such a way 462 LIMITS [Art. 278 as not to assume the value p during the approach, we shall have (Final Theorem Art. 268) (2) \{mR{x) = \imR^(x). But we have further either (3) \\m R^{x^ = R^{p) or (4) lim R-^(x^ = 00 x-^p according as R-^(x)^ which is in its lowest terms, is con- tinuous in the neighborhood oi x=p or else has 2;=^ as a pole. By combining (2) with (3) or (4) we obtain the desired limit. In Art. 87 and Art. 89 we introduced the notion of the derivative of a function. The variable whicli takes the j^lace of the a: of this article is the h which occurs there. Observe that we are there dis- cussing a problem of just the kind treated in the present article. This remai"k will serve to convince the student of the importance of such limits. The following examples will illustrate how to treat other cases in which the direct application of the theorems on limits of Arts. 273 and 274 give indeterminate or meaning- less results. 3.2 _ 2 Ex. 1. Find the limit which - — ^ —-^ — - approaches when x grows beyond bound. Solution. It is useless to write lim(x2-2) lim = ■ '-^^ °° • :_^ 5 x2 + 3 X - 7 lim (5 x^ + 3 x - 7) ^ because 5^ is just as meaningless or indeterminate as 0/0. But we may write q 1 -- x^ — 2 . ^~ lim - — = lim 5 n ,_^ 5 x2 + 3 a; - 7 .,^^ 5 4. § _ 1 Art. 278] INDETERMINATE FORMS 463 since the given fraction is equal to 1 --. for all finite values of x which are different from zero. As x grows be- 8 '> 7 yond bound -, ^, — all approach the limit zero, and therefore a:2 - 2 _ 1 Ex. 2. Evaluate the limit of (x^ + x - 2) • — -^ for x = 1. Solution. For x = 1 the given function assumes the indeterminate form • 00. But we may write 1 x2 + X - 2 (x2 + X - 2) 1 X - 1 which assumes the indeterminate form 0/0 for x = 1, and may therefore be treated by our first method. Ex. 3. Find lim [L — L- "/ f 1 x-^2Lx2-4 2x(x-2)J Solution. For x = 2 the function assumes the indeterminate form 00 — 00. But we may write X - 1 2 X - 3 - 3 X + 6 x2 - 4 2 x(x - 2) 2 x(x2 - 4) which assumes the indeterminate form 0/0 for x = 2 and may therefore be treated by our first method. EXERCISE CXXVIII Evaluate the following Ihuits. r2 1 x" — r;3 „ X'' — 1 1. lim-^^- 4- lim^^ ^. 7. lim ;;;r—r- ,_^i X — 1 x_>Kj X — a x_>.i X- — 1 o .. .r--n- R ,. x< - 16 « ,. X* - 1 2. hm 5- lim ^- 8- lim 77-— T" i^.a X- - a _>2 -r - J i_>.x •'- + 1 3. hm ^- 6. hm 9- bm t-^ ■ x_>.2 X - 2 x->.a X - a rr-^W. 2 X'' - 1 464 LIMITS [Art. 278 X + 1 T ^ liin 1 ■ ■'■''• I— V2 c^ — 3 X — 3x3-7x2+1 T- lim rl 11 r 3 x"* - 7 x-' + 1 3^4 iim fl 2 H ■ ^!^.ia:3 +5x2-7 x+l" ' '^^ Ix x(x + 2)J 12. lim (x2 - 1) . -^ . 15. lim ('^-±1- nV x->l -^ X — 1 ,(->.» V n / 16. lim i'^^+^-n). , „ lim ax" + i')x"~^+ •■• Ix + m ^"^^ a'x^ + 6'x"~^ + ••• + /'x + m' TO , . rtx" + ix"-i + ■ • ■ + Zx + m J-o. urn ^ — — — • x-^^ a'x" + &'x*-i + ••• +l'x + m' Distinguish the three cases n > k, n — k, n <« A=l 281. Convergence and divergence of infinite series. We now proceed to generalize the notions encountered in Arts. 279 and 280. Let Mj, u^, %, ••• be the terms of a non-termi- nating or infinite series, and let us denote by *S'„ the sum of the first n terms, so that S^ = Mj, S^ = U^ + U^i >S3 = U^ -{- U^ + W3, •»•, S^ = Wj + y.2 + W3 + ••• + "„-i + w„. Clearly the value of >S'„ will depend, in the first place, upon the nature of the series under consideration, that is, upon the law of formation of its terms, and, in the second place, upon the number of terms included in S^. We express this by saying that /S'„ is a function of n. If the sum of the first n terms of an infinite series approaches a definite finite limit as n grows beyond hound, that is, if lim S, = S ivhere S is a definite finite number, the infinite series Mj -I- ■?^3 + ••• to infinity is said to be convergent, and the limit S is said to be its sum. If the series is not convergent, /S'„ will not approach a defi- nite finite limit as n grows beyond bound, and we shall say Art. 281] CONVERGENCE AND DIVERGENCE 467 that the series is divergent. There are two chisses of diver- gent series ; those of the first class for wliich *S'„ becomes in- finite when n grows beyond bound ; and those of the second class for which *S'„ does not approach a definite finite number as a limit although 8^ does not become infinite. Series of the latter class are often called oscillating series. To avoid confusion, the student should note that some authors use the word divergent only for the case when lim >S'„ = oo. The word sum is here used in a new sense. (Compare also Art. 62.) Our original definition of a sum (Art. 2) only applies to the case where the number of terms is finite. The sum of an infinite series, as here defined, is not a sum at all iu the original sense of the word; it is the limit which such a sum approaches when the number of terms grows beyond bound. Example 1. The geometric progression 1 + * + i + i + - is a convergent infinite series. The sum of the first n terms is ^ - ^ ^/^" (See (1), Art. 279), so that proving that the series is convergent and that its sum is equal to 2. Figure 82 illustrates graphically how it happens that .S",, approaches the finite limit 2. The distance OB is equal to two units. The distances OSo, OSz, OSi, and so on represent the sum of two, three, four, ••• terms of the series respectively, and ^ '^ — >^-^' it is evident from the figure that 05„ has OB as p^^ g2 its limit. We have seen more generally, in Art. 279, that any geometric progres- sion, whose constant ratio is numerically less than unity, forms a convergent series, whose sum is equal to 1-r Example 2. Each of the series of Art. 280 is convergent. Their sums are 1, i, and ^ respectively. 1 - i 1-i s,. _ 2 _ On- i' lim 5„ = 2, n-^x 468 mrmiTE series [Art. 282 Example 3. The series 1 + 1 + 1 4- ••• + 1 + ••• is divergent. For, in this case 5„ = n and therefore S„ grows beyond bound as n becomes infinite. Example 4. The series 1 — 1 + 1 — 1 + 1 — l+--is also divergent. But in this case S„ does not become infinite. In fact S„ is equal to zero when n is even, and S,^ is equal to 1 when n is odd. As n grows beyond bound, S^ oscillates between the two values and 1. It does not be- come infinite, but neither does it approach a limit. The series is an oscillating one. The method of investigation illustrated so far is applicable whenever we actually know how to find an exact and simple expression for S„^ the sum of the first n terms of the given series. This is one of the reasons why it is a matter of great importance to be able to find the sum of a finite series. (See Arts. 262 to 266.) Moreover in all such cases we can do more than merely decide the question as to whether a series is convergent or divergent. If it is convergent, we can ac- tually find its sum. EXERCISE CXXIX Discuss the convergence or divergence of the following series, and find the sum if the series is convergent. 2. 1+2 + 4 + 8+.... ■ 1-2-3 2.3-4 3.4-5 3. •_) _ 2 + 2 - 2 + 2 ... . 7. 1 . 2 + 2 • 3 + 3 • 4 + ... . 4- 7 + 1 + J + I + ^'e- + ... . 8. 1 + 3 + 5 + 7 + ... . 5. -L + J_+ J_+ .... 9.2 + 4 + 6 + 8 + .... 1.22-33.4 282. Fundamental criteria for convergence. Whenever we have no explicit formula for *S'„ we cannot decide whether a series is convergent or divergent by the method of Art. 281. We must therefore seek for more general methods. Let the series (1) u^ + ^2 + ^3-1- ••• be convergent. Let S be the sum and let S^ be the sum of its first n terms. Then >S' — *S'„ must approach the limit zero Art. 282] FUNDAMENTAL CRITERIA 469 when )i grows beyond bound, and the same thing must be true of S—Sn-y Consequently the difference must also approach zero as a liuiit. But lS'„ = Wj + Wg -+- • • ■ + Wn-1+ "n^ SO that S^ — Sn-i is equal to m„. Consequently we obtain the following result. Theore^ni I. If a series is convergent its nth term must approach the limit zero ^ when n grows beyond hound; that is ^ the condition (2) lim M„ = must he fulfilled if the series (1) is to he convergent. The condition (2) is necessary for convergence^ hut hy no means sufficient. In other words, it may happen that con- dition (2) is satisfied and that the series is nevertheless divergent. Such is the case, for instance, for the following series, the so-called harmonic series, (3) i + J + i+l+-. In this case we have w,^ = — and lim Un = liin - = 0, n «->-» ?»->« n but the series is nevertheless divergent, as we shall now show. We may write (3) as follows : (4) 1 + ^ + (1- + \) + {\ + i + ^ + 1) + (i + A + ••• + h) + ■■; where the terms are collected in groups of one, two, four, eight, and so on in accordance with a law which is easily recognizable. Not counting the first term at all, the first group consists of a single term 1/2. The sum of the terms of the second group is I + \, which is greater than ^ + ^ or ^. The sum of the terms of the third group ^ + I + J + ^ is greater than ' + J + 1 + i, «• e. greater than \. 470 INFINITE SERIES [Art, 282 The mth group consists of 2'""^ terms, namely Each of these terms is greater than the last, whose value is equal to 1 ^ 1 ^ 1 Om-l _|_ >2m"l ~~ 2 • 2"'~1 ~ 2™ * The sum of the terms of the mth group is, of course, greater than the num- ber of these terms multiplied by the value of the smallest one among them, that is, greater than 2m 2 Consequently, if we denote by Sm the sum of those terms of (4) which are included in the first m groups, we have (5) .S„ > 1 + o' since the sum of the terms in each group is greater than 1/2. But according to (5), Sm becomes infinite when m grows beyond bound, that is, when n becomes infinite. Therefore the series (3) is divergent. We have seen that the condition (2) is necessary, but not sufficient for the convergence of an infinite series. The fol- lowing theorem gives a condition which is both necessary and sufficient ; but we shall merely state this theorem with- out proof since the proof is a little difficult for a beginner. Theoreisi II. In order that the series Wj -f ^2 + ••• may he convergent, it is necessary and sufficietit that, not only the nth term, but the sum of any number of terms following the 7ith term shall approach the limit zero when n grows beyond hound. EXERCISE CXXX Prove that the following series are divergent. 1. 1 + 2 + 4 + 8+ ••.. 2;(-^)- 2. 2 + 2 + 2 + 2+ -. ^ _j_ 3. (l + |) + (l + :) + (l + i) + lOUO * For the significance of this notation see Art. 264 and end of Art. 280. Arts. 283, 284] COMPARISON TESTS 471 283. Series all of whose terms are positive. We shall assume the following theorem without proof. The student will easily convince himself of its great plausibility. Theore:si I. If all of the terms of a series are positive, it cannot he an oscillating series. It is either convergent, or else the sum of the first n terms will become infinite as n groivs be- yond bound. We may also formulate this statement as follows. Theore^f II. An infinite series of positive terms is con- vergent if Sn remains finite for all values of n, that is, if there exists a finite positive number M, such that >S'„ < M for all values of n, no matter how great. For according to Theorem I, such a series is either con- vergent, or else lim >S'„= 30. But the latter possibility is excluded if >S'„ < J/ for all values of n. EXERCISE CXXXI 1. Formulate the theorems which correspond to Tlieorem T and 11 in tlie case of series all of whose terms are negative. 284. Comparison tests. The method outlined in the fol- lowing theorem often enables us to prove the convergence of a given series, by comparing it with another series whose convergence has been established previously. Theorem I. Let it be known that the series of positive terms (1) ^1+ ^2 + ^'3 -•-•••+ ^'n+ ••• is convergent, and let (2) ?/l + W2 + W3+ ••• +Mn+ ••• be a second series of positive terms, ivhose convergence is to be tested. If (3) w„ ^ v„, 472 INFINITE SERIES [Art. 285 for all of those values of n which follow a certain first value of n for which the inequality/ (3) is fulfilled, the series (2) is con- vergent. Proof. Since the series ?'i + re + ••• is convergent, the sura of any number of its terms following its nth. term will approach zero as n grows beyond bound. (See Theorem II, Art. 282.) Therefore we can make the sum of the p terms which follow r„, that is, »"n+l + ''n+2 + ••■ + Vn-^p, arbitrarily small by choosing n large enough. (See Art. 268, definition of a limit.) We may also assume that »i has been chosen so large as to insure at the same time the validity of all of the inequalities We shall then have Wn+l + Un+2 + ••■ + M„+p < l'„+l + V„+2 + •" + Vn+p. But this means that m + U2 + •■■ + u„+p may be made arbitrarily small by choosing n large enough. In other words, the sum u„+i + u„+2 + ••• + «n+p will approach the limit zero as ?i grows beyond bound, no matter how large or small j3 may be. But this means that the series wi + W2 + ••• is convergent. (See Theorem II, Art. 282.) Theorem II. Let the series of positive terms v^ + v.^+ ■•• he divergent, and let II ^ v "n ^ '^n for all of those values of n which follow a certain first value of n for which this inequality is fulfilled. Then the series u-^ + u^-\- •■• is also divergent. For, if this were not so, that is, if u^ + u^ + ••• were con- vergent, according to Theorem J, v-^ + v^+ ••• woukl also be convergent contrary to our assumption. 285. Some convenient comparison series. In order to be able to apply Theorems I and II of Art. 284, it is necessar}' to have some series at our disposal whose convergence or divergence has already been established. We have found some such series already. (See Arts. 279 and 280.) More- over, as soon as we have proved some new series to be Art. 285] SOME CONVENIENT COMPARISON SERIES 473 either convergent or divergent by this method we may make use of it for the purpose of examining still other series. The following theorem is particularly useful in connection with the comparison tests. The series (1) T- + ^ + ^ + -r+- +-+••• iP 2p 8p 4p 7t" iH convergent wlien p > 1. It is diveryent when p = 1, or ichcn p < 1. Proof. We have already shown that this series is divergent for /> = 1 since in tliat case it reduces to the harmonic series (2) 1+1 + 14-1+ ... +1+.... L: o 4 n (Compare Art. 282.) if yj < 1, we iiave n'><.n for all values of n except for n — 1. (See Theorem V, Art. 1(>3.) Therefore ->1, for n =2,3, 1. •••, nP n so that Theorem II of Art. 281 assures us that the series (1) is divergent for p < 1 . It remains only to show that (1) is convergent when p > 1. To do this we arrange the terms of (1) in groups of two, four, eight, and so on, as we did in Art. 282 for the harmonic series. We have 2p ;ip 2p 2p-i' 4p op Op 7p 4p 4p~^ 1 + i- -L ,\ -!- + _ + _+ ... <'_J__ + ^ + ■*■ + ... ^* ^ ^p^p^p 2P-1 <^p-\ %p-^ But the right member of (3) is a geometric progression whose first term is a = 1/2''-^ and whose common ratio is r = 1/2''-^ This progression is a convergent series if the common ratio is less than unity, that is, if p 474 INFINITE SERIES [Art. 286 is greater than one. Therefore, the series in the left member of (3) is convergent if jo > 1, as was to be proved. We may even draw a further conclusion. The sum of the non-termi- nating geometric progression in the right member of (3) is 1 n OP-1 1 1-r i__L 2*^1 -1 OP-l Therefore we see that (4) if;j>l. EXERCISE CXXXII Examine the following series for convergence or divergence. 1. 1 +i + l + i+ ... +i-+ ..., 2^ 33 44^ ^„«^ 2 3 • 2 4 . 22 5 . 23 (n + l)2"-i 3. 1 +_L. + ^_ + ^_^ ... +_J_+ .... 2 . 22 3 . 32 4-42 n. 71- 4. . 1 + _L + -L + J- + ... + J- + ... . v'2 V3 V4 Vn i^ 8. 2;ifor. 1, tlie series is divergent. The question of convergence or divergence remains undecided by this test, if either lim m„+i/?/„ = 1, or if Un+i/Un does not approach any definite limit as n groivs beyond hound. Proof. Let us consider first the case when lim ^^2+1 < 1. »!—><» Uy^ Let us denote by r the limit of w„-f.i/M„, so that (1) lim^*2+l = r< 1. According to the definition of a limit, tlie meaning of (1) may also be stated as follows. Let us consider the sequence of ratios -^■, -^, -^, '•'■, W'-f Wrt It'O and let us choose a positive number S, which may be taken as small as we please. Then there will present itself sooner or later a first one of these ratios, say which differs from r by less then 8, and such that w„+i/m„ for all values of n which are greater than m will also differ from r by less than 8. Thus, after B has been chosen, m can be determined in such a way that all of the ratios *, ■ «, ..., ', ... will be included between r — 8 and r + 8. Since r was, by hypothesis, less than unity, and since 8 was a positive number which could be chosen arbitrarily small, we may in particular choose 8 in such a way that r -\- 8 will also be less than unity. 476 INFINITE SERIES [Art. 286 In Fig. 83 the line-segment OU is one unit long, and 07? = r rep- resents the limit, less than unity, which u,^^Ju^ approaches as n grows beyond bound. If 8 is chosen as a positive number f\ J Tf "K XJ ^ ^i» r r+l \ such that ?• -f S is still less than unity, the line-seg- „ „„ ments OL and OK represent the numbers ?• — 8 and Fi<3. 83 „ . r 4- o respectively. The statement that «„ n/",, ap- proaches r as a limit is equivalent to saying that the various line-segments which represent the quotients M„r]/w„ for growing values of n will ap- proach OR as a limit, so that if m is taken large enough, all of the quotients w„+i/w„, for which n exceeds m, will lie between OL and OK. Now let us put r + S = A". Then A; < 1, and we have (2) ?^^tt2<^A-, m+l ^^A;, 'm+2 and so on. But from (2) we find ■^m+i ^ "^w„ \6) "MTO+ajS-^^m+i^^ ^TO? and so on. If we compare the sum of the left members of (3) to the sum of the right members, we conclude (4) w,„+j -f w,„+2 + ^™+3 + • • • :4 w^(^ + A'- + F -f • • •). Since k is less than unity, the geometric progression in the right member of (4) is convergent. Consequently the series Wm+i + w„+2 + '■■ ^^ convergent. If we add to this series the first 7W terms u^ + u^-\- ■■• -}- w„, we alter the value of the sum, but the series will remain convergent. Thus we have proved that our series u^-\-u<^-\- •••is con- vergent if lim ^^^ < 1. If instead lini !^2+l = r > 1, n— >.oc W„ Art. 286] RATIO TEST 477 a slight modification of our argument shows that vi may be chosen so large that u,n+\/'^m ^^^ 3-^1 of the ratios m„+i/w„ for which n > m will be greater than unity. In Fig. 83 the point R would be to the right of U and both points L and K will be to the right of [/ if S is chosen sufficiently small. Thus we shall have in this case and so on, so that liroving that w^ + Wj + ••• must be divergent if lim ^fii±i>l. n— >« W„ This last argument enables us to complete our theorem by the follow- ing statement : If lim ^^5+1 = ^ = 1, hut if all of the ratios of the sequence *'TO+l "'m+2 "'m+3 Urn Um^ I Ufn+2 which folloiv a certain first one u^j^jum^ are greater than or equal to unity, then we may still assert that the series is divergent. But the ratio test gives us no information whatever in case lim ?^^ = r = 1, n->oo Un while all of the ratios of the sequence are less than unity. Such a series may be convergent or diverofent. 478 INFINITE SERIES [Art. 286 Thus the harmonic series 'J o n is known to be divergent. In this case we have M„ n + \ n n + \ 1j-1 n This ratio is less than unity for all values of n, but it approaches unity as a limit. The reason that the proof of convergence for the case lim ^/i+i — ^^1 is not applicable to this series is easily seen. Although we have !f»±i j : If *S'„ has a finite limit, the same thing must be true of SJ. For, as we have just noted, S^ either has a finite limit or becomes infinite, and the latter possibility would contradict the inequality (10). A similar contradiction would arise if lim *S'„ = X unless we have also lim S„' = ^. Thus our theorem is proved. EXERCISE CXXXIV Examine the convergence or divergence of tlie following series: 1- ] +| + i + i + -- 2. i + i + i + l + .... 482 INFINITE SERIES [Art. 288 3. 1 [- — + — + •••. Distinguish the cases ^' > 1 and A: < 1. Ifc 3k 5fc 7A; • ^ 7 + 5 n ^ /i^ + 8 n2 + 4 n + 1 i. y •'^^-^ " 7. y ^ 5 n2 I 7 „ _ 1 A' 5?i2 + 7„_l frinVn + S 288. Series with positive and negative terms. Clearly the criteria which we have developed for series all of whose terms are positive are applicable, with very minor changes, to series all of whose terms are negative, or to series all of whose terms, excepting only a finite number, have the same sign. We may make use of these criteria also for many series which contain an infinite number of terms of either sign, as a consequence of the following theorem. Theorem I. An infinite series u^ + u<^+ %+ ••• whose terms are all real, but not necessarily/ all positive, will be convergent, if the series of positive terms, composed of the absolute values of the terms of the original series, is convergent. Proof. Let /^■v 00 Un lim Wn+l n-^o: Wn <1. > 1. The test fails to give any information if lim = 1. Proof. The first assertion is an immediate consequence of Theorem I. The second follows from the fact that the limit of «„ cannot be equal to zero if lim 9i— ^<0 Wn+1 >1, and we have seen (see Art. 282) that in any convergent series the limit of u,i must be zero. The third assertion is merely a reiteration of what we found before. (Theorem II, Art. 286.) EXERCISE CXXXV The following series are to be tested for convergence or divergence -+ .... 1! 2! ;]! 4! 5! 1! ■L + 1_JL + 3 ! 5 ! 7 ! 484 INFINITE SERIES [Arts. 289, 290 3. 1 11 1 2! 4! 6! 8! 4. 1 -. 1 + i - i + • 5. 1-1+^-1+ 22 ^32 42 ^ 289. Conditionally convergent series. A series with posi- tive and negiitivo terms may be convergent although it is not absolutely convergent. (See the definition of absolute convergence in Art. 288.) Such series are said to be con- ditionally convergent. A simple illustration of a condition- ally convergent series will be given in the next article. 290. Alternating series. An alternating series is one whose terras are alternately positive and negative. If Wj, Wgi Wg, W4, ••• denote positive numbers, (1) Wj — W2 + Wg — M4 -H W5 — Wg -I 1 • • • is an alternating series. Any alternating series may be expressed either in the form (1) or else in the form (2) — (Wj — M2 + Wg — 7/4-I — ••■). Since (1) and (2) are convergent or divergent at the same time, it sujBfices to consider a series of form (1). We have the following theorem due to Leibniz : An alternating series is convergent if each term is numericalli/ less than the preceding one, and if the nth term of the series approaches the limit zero when n becomes infinite. PuooK. Let j/j. u^, «.. ••• be positive numbers, such that (8) Uj>W2> "3> "4>--- >u„> w„+i> •••, and let (4) lim w„ = 0. n->co If n is an even number, we may write (o) S,^ -{ui - M,) + (M3 - M4) + ••• + (w„_i - u„) where each parenthesis is positive on account of (.3), so that S„ is surely ]iositive. We may also write (6) S^ = "1 - (",, - W3) - ("4 - W5) ("«-2 - "«-i) - ""• Art. 291] SERIES OF FUNCTIONS 485 Again each of the differences inclosed in a parenthesis is positive, so that (6) tells us that (7) S„<» But if n is even, n — 1 is odd, and we have Consequently lim 5„_i = lim S,^ — lim ?/„ = S n— >.<» n — >-» n > oo on account of (4) and (8). Thus 5,, approaches the same definite finite limit S whether n be even or odd, and therefore the alternating series is convergent. EXERCISE CXXXVI Investigate the following series for convergence or divergence : -^ C— l)"n n=l \ L+ 1 _ ^ , . v^(-i)"iO". ^ (-1)"10' *• jLf 10" + n l.I 1.02 1.00:3 1.0001 ^ 10" + ,1=1 5. I-L4--I---L + -L-.... \/2 v';3 Vl V5 291. Series whose terms are functions of x. Let ns con- sider a series v^iz) + u.^(y) + u^{x) 4- ••• wliose terms are fuiictioiis of x. If we i)ut for x some par- ticular value, such as x = a or x = b^ we may examine tlie convergence of the series in each of these cases. It may happen that such a series is convergent for some values of x 486 INFINITE SERIES [Art. 292 and divergent for others. If it is convergent for two differ- ent values of x^ x = a and x = b, we may expect it to have different sums in the two cases. All of those values of x for which a series of the form (1) is convergent, are said to constitute its domain of conver- gence. For all values of x in its domain of convergence, the series defines a function of x. 292. Power series. The simplest and most important case of this kind is that of a power series Uq -{- a^x -\- a^x^ + • • • -H «„a;" -|- •••, every term of which is a product of a constant a„ (whose value depends upon n but not upon x} multiplied by the power x", n being a positive integer. The nth and (^n -\- l)th terms of such a power series are M„ = a„_ja:"~i and m„^j = a„a:;". These terms may be positive or negative even if a„_j and a„ are both positive, since x may be positive or negative. If we wish to apply the ratio test, we must therefore use it in the extended form of Art. 288. We have Wn+1 _ a„a;» _ «n a„_ia;"-i «„-l Therefore, the series will be convergent or divergent for a given value of X, according as (1) ■'■■ "• \x\ lim n — ^(x> is less than or greater than unity. If we write (2) lii ■■n— 1 1 = - or r *n— 1 lim ^n-\ = r. we conclude that the poiver series «Q-|- a^x + a^x^ -f ••• is convergent for those values of x for which \x\ < r and diver- gent for those values of x for ivhich | a: | > r. Art. 292] POWER SERIES 487 It may happen that Un- does not approach any definite limit, finite or infinite. In that case our theorem conveys no inforjnation. But if such a limit exists, we have the following results: If the value of r obtained from (2) is zero, the power series is divergent for all values of x except for a:=0. If r is infinite, the series is convergent for all finite values of x. If r is finite and different from zero, the series is convergent for |a;| < r and divergent for | a; | > r. Our test tells us nothing, however, as to whether the series is convergent for \x\ = r^ that is, when x = ±r. Corollary. If a 'power series is convergent for x = k, it is also (ionvergent for every value of x for ivliicli \x\ < | A: |. EXERCISE CXXXVII Example 1. Investigate the convergence of the power series 4- + J + T + + i-" + Solution. In this case we have 1 1 a„ a- = a„_, = n — 1 «„_i so that lim J = l-i, = 1. Consequently equation (2), Art. 292, gives r = 1. The given series is con- vergent for all values of x for which | x| < 1 and divergent for | .i' | > 1. In this case we can also decide what happens when | a: [ = 1 . For, if X = + 1, the series reduces to UU1 + • 1 2 .} + • + This is the harmonic series (see Art. 282) and is divergent. If j: = — 1, the series becomes -i + i-i + i-+ •••• 488 INFINITE SERIES [Arts. 293, 204 This is an alternating series which is convergent on account of the theorem of Leibniz, proved in Art. 290. Thus the given series is con- vergent when I x I < 1 and also when x — — 1. For all other real values of X the series is divergent. Investigate the convergence of the following power series. _ . X , X- , x^ _ ,1 j:^ , 1 ■ .3 x^ , 1 • .3 • 5 x'^ , 1! 2 1 3! • 2 3 2.4 r> 2.4-67 7. X - - + - - 3 5 3! 5! 7! 3 3-^ 38^ 5. •:?_£!+ ^^_£!+ .... 9. 1 + x + 2!x2 + 3!xH4!x4+ .... 12 3 4 293. Equality of two-power series. The following theorem, which is really an extension of Theorem F of Art. 126 has many important applications. Let each of the two-power iSeries a(^+ a-^x + a^x^ + ■■■ + a„2:" + ••• and be convergent for some values of x which are different from zero. If the sums of the two series are equal to each other for all of those values of x which make both series convergenU the coefficients of like powers of x in the two series must be equal, that is Aq = Jq, aj = 5^, • • •, a„ = b^. A rigorous proof of this theorem would require a rather long chain of preliminary theorems. We therefore content ourselves with a statement of the theorem witliout giving a proof. 294. Expansion of functions as power series. We know from the theory of geometric progressions that the equation (1) l + :, + :^:2+^+ ... = 1_ 1 — X Art. 205] EXPANSION OF RATIONAL Fl'NCTIONS 489 holds for all values of x for which 1 2; | < 1. We may ex- press this by saying that the function (2) .-^ \ — X has been expanded into a power series, namely, (3) l4.:^+^:2_^^.3+ .... It is important to note that our proof of equation (1) assures us that its two members are equal to each other for all of those values of x for which \x\ < 1, but not for any other values of x. In fact the two members of equation (1) are not equal to each other for 2; = 2. F'or tlie left member becomes infinite when 2;= 2, while the right member is equal to — 1 when a; = 2. We may generalize these notions as follows. When a function f{x) is given, it is frequently possible to find a certain power series which is equal to fix') for those values of x for which the power series is convergent. In such cases we say that tlie function has been expanded as a poiver series. Tlie equivalence between the given function and the power series can never hold for values of x for which the power series does not converge. 295. Expansion of rational functions. We may use the theorem of Art. 293 to obtain the expansion of a rational function, whenever such an expansion exists, by the method of undetermined coefficients. (See Art. 14:2.) The follow- ing examples will help to explain this method. 1 — X • Example 1. Expand into a power series. 1 + x'^ Sdlulinu. It' such a power series exists, let us denote its coefficients (as yet unknown) l»y a^, ui, a.^, and so on, so that 1 — X (1) ^ = «o + ^1-^ + "2-^" + "3-^^ + ^^4^ + •■•• If this equation is true for all values of x for which the series is con- vergent, the following equation, obtained from (1) by clearing of frac- tions, must also hold for all such values of x; 490 INFINITE SERIES [Art. 295 \ — X = Gq + a^x + a^x'^ + a^x'^ + a^x* + ••• + aQX- + a-^x^ + rioX-* + •••• In other words, we must have (2) 1 - X = Qq + rtjx + («(, + rtj)^- + («! + f/3)x3+ (oj + a^)x* + •■• for all values of x which make the series in the right member convergent. According to the theorem of Art. 293, this can be so only if the coeffi- cients of like powers in the two members of (2) are equal to each other. Consequently we conclude that (2) can be true only if 1 = «o» — 1 = "d = «o + «2' = rtj + «3, = 02 + 04, •••, whence Oq = 1, «j = - 1, flj = - «0 = - 1' "3 = - "i = + 1, O4 = - ^2 = + 1» — • Consequently we have (3) i^^ = 1- x-x2 + x3+ .r* , ^ 1 + x^ if there exists a power series at all for (1 — .r)/l + x". In this particular example it is not very difficult to find the law of the coefficients, to prove that the series converges for ! .r| < 1 and to prove that equation (3) is actually true for all such values of x. However, that is a matter with which we are not primarily concerned just now. Example 2. Expand '- into a power series. x(x — 1) Solution. Let us try to use the same method as in Ex. 1 by putting 3 (4) — — = Qo + «i^ + "2^^ + •■■' x(x — 1) whence, clearing of fractions, 3 = — UqX — «j.rj — aox"' ••• + (loX- + Oj.r^ + •••. Equating coefficients of like powers we at once strike a contradiction, namely, 3 = 0. Therefore such an expansion is impossible in this case. The reason for this impossibility is very clear from (4). The function S/x(x — 1) has X = as a pole (see Art. 139) and therefore becomes infinite when x approaches zero as a limit. But the right member of (4) remains finite for x = 0, since it reduces to Qq. Therefore it is clear by inspection that an expansion of the form (4) is impossible. We may, however, write 3 ^3 1 x(x — 1) X X — 1 Art. 295] EXPANSION OF RATIONAL FUNCTIONS 491 and expand the second fraction. We find 1 X- 1 so that — \ — X — X- — x^ ^ = _ ? _ .3 - .3 X - 3 x2 - 3 x8 - •• x(x-l)- This is not an ordinary power series expansion on account of the term — 3/x which occurs in it. This term may be written — 3 x"^. It in- volves a negative power of x as factor. These examples will suffice to justify the following gen- eral statement. But we shall not attempt to give a formal proof of its correctness. If a fractional rational function of x does not have x = as a pole, it may he expanded into a power series of the form a^ -f a^x + a^x^ + ••■ which will converge for some non-vanishing values of x, but not for all finite values of x. The coefficients of this series may be obtained by the method of undetermined coefficients. If a; = is a pole of the rational function, no such poioer series exists, but the function may be expressed as the sum of such a power series and certain additional terms, each of these terms having a negative power of x as factor. The following remark is of importance, if it be desired to obtain the general law according to which the coef'licieuts of the expansion are formed, so as to be able to judge of the convergence of the resulting series. In order to be able to recognize the general laiv of the co- efficients in the expansion of a rational function, it is advisable to express the function as a sum of simple partial fractions (^see Arts. 142-144) ayid then to expand the several partial fractions separately . EXERCISE CXXXVIII Expand the following rational functions in powers of x. Compute at least four terms of the expansion, find the general term whenever you can, and then determine the values of x for which the resulting series are convergent : 492 INFINITE SERIES [Art. 296 1. 1 - X 2. 1 I + X 3. o 1 ~x 4. a \ - X 13. 1 1 + X 6. J_ 2 — x 7. J_ 6" — X 8. a h - X 1 9. h + X 10. 2 + 3a: 1 -a;2 11. 1 + a; + a;2 1 - X + a;2 12 2 a: + 3 x2 1 + 2 a: + 3 x2 2 + Sa- il + xr 16. i- -"^A (1 + xY 14. ^ 15. --! (1 + a-)^ (1 + x)3 296. Expansion of some irrational functions. The method of Art. 295 may be used with very small modifications in order to obtain the expansion of certain irrational functions. Illustrative Example. Expand Vl + x as a power series. Solution. Assume (1) Vl + X = 0(| + «ix + a.^x- + (i^x^ + •••. Then we have, squaring both members, (2) 1 + X = rig- + 2 itoU^x + 2an«2^^ -^• 2 rt^agX^ + ••• + «i-X'^ + 2 fljOiX^ + •••• Equating coefficients of like powers of x in the two members of (2), we find (3) 1 = a^'^, 1 = 2a^a^, = 2a^a.-^ + o,2, = 2 a^n^ 4 2a^a.^, and so on. The first equation (3) gives Aq =± 1- If by Vl + x we mean tlie positive square root of 1 + x, we must choose rt^ = + 1, since the positive square root of 1 + x reduces to + 1 when x = 0. Thus we find from (3), "0=1- «1 = 2' ^^2 = - i. ".■! = + A. so til at vnr;- = 1 + I ,- - 1 .r- + i, .,-3 + .... EXERCISE CXXXIX Expand the following functions to 4 terms : 1. Vl + x2. 4. vTT^-. 7. Vl + X 4- X-'. 2. V;5 + 2x. 5. v/1 - X. 8. xVl + 3. V4^r^-. 6. ^rT7^. 9. ^^ +^. 1-x Art. 297] (IKNKRAL BINOMIAL EXPANSION 493 297. The expansion of (1 + x)". The binomial theorem (see Art. 88 J shows us that (1) (]+.-)" whenever n is a positive integer. In that case the right member of (1) contains w + 1 terms. If n is a negative number or a fraction, we may still form a series like the one which occurs in the right member of (1), but in all such cases this series will contain an infinite number of terms. It is easy to show that this series will be convergent for |a;j< 1 by applying the ratio test. (Art. 286.) We have in fact ""' l-2.^...(k-l)k H-i _n(n-l)(n-2) ... (n-k+2') whence so that 1 . 2-3 ■•• (yfc-l) n —k+1 «i-i k k lim = 1. 'A-l I Thus, the quantity denoted by r in Art. 292 is equal to 1, and we conclude from the principal theorem of Art. 292 that the series is convergent for all values of x for which | a-j < 1. Thus, whenever |a:| < 1 the right member of (1) is a con- vergent series and therefore has a definite meaning. The left member of (1) also has a definite meaning. (See Arts. 156 and 157.) In the particular case when w is a i)Ositive integer the two members are equal, as we have actually proved. We now state without proof, that the two members of (1) are equal (for \x\ < 1) for all values of n. This gives us the general binomial theorem. 494 INFINITE SERIES [Art. 297 If n is any number^ not necessarily a positive integer^ we may expand (1 + a;)" according to the formula (1) (i + xy=\ + ^^x+ ''"^'\-'^K '^+... n(n-l)(n-^^ ■■■{n-k+l) ^ + - X + .... If n is a positive integer^ this expansion consists of a finite num- ber of terms and is valid for all values of x. If n is 7iot a posi- tive integer^ the expansion will be an infinite series, and equation (1) will be valid for all values of x which are numerically less than unity. In Exercise CXXXVIII, Example 2, Exercise CXXXIX, Example 4, and the illustrative example of Art. 296, we found a few terms of the expan- sions for -^ = (1 + x)-\ Vl + x = (1 + x)i v^rrr = (l + x)i The 1 + X student should verify that the expansions obtained by him for these functions are in agreement with the results which would be obtained by using formula (1). This may be regarded as a partial proof of (1). A complete proof may be found in Dickson's College Algebra, Chapter XV, according to a method due to Euler. The most convenient proof, however, depends upon methods developed in the calculus. Formula (1) may also be used to compute (a + by. For we may write a-\-b = a(l-{-- \ ay (2) Qa + by=a«(l+^y. The second factor may be expanded as a power series in X = b/a by means of (1) if 1 5 1 < | a |. If instead 1 6 1 > | a |, we write (a + by = b"(l + ^Y and put x= a/b. This method is very convenient for computing square roots, cube roots, nth roots of numbers. Art. 298] EXPONENTIAL SERIES 495 Thus, to find the cube root of 30 we write 30 = 27 + 3 = 27(1 + ^)= 27(1 + i), so that v^ = 9 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 51:32 5145 5159 5172 33 5185 5198 5211 5224 52:57 5250 5263 5276 5289 5302 34 5315 5328 5340 5:353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 55()3 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 57(53 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 (50(54 (i075 6085 (509(5 6107 6117 41 6128 (>13« 6149 61(50 (5170 (5180 6191 (5201 (5212 6222 42 ()232 6243 (J253 (52(53 6274 (5284 6294 (i:504 (5314 6325 43 6335 6345 6355 6:565 (5375 (i:585 6395 (5405 6415 6425 44 6435 6444 6454 6464 6474 (5484 6493 (5503 6513 6522 45 6532 ()542 6551 6561 6571 6580 6590 6599 6609 6618 46 (5628 6637 6(i46 6656 (!()()5 (5(575 (5(584 6(593 6702 6712 47 (1721 6730 6739 6749 6758 (5767 6776 (5785 (5794 6803 48 ()812 6821 6830 6839 6848 6857 6866 6875 (5884 6893 49 6902 ()911 (5920 6928 69:57 (;94() 6955 6;5 8000 8007 8014 8021 8028 80:55 8041 8048 8055 64 80ti2 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 81 2<) 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 .Sl4 8500 8506 71 85i;{ 8519 8525 8531 85:57 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8()03 8609 8(515 8(521 8627 73 8(>:5;i 8639 8645 8(551 8(>57 8(5(i3 86(59 8675 8(i81 8686 74 8(i'J2 8698 8704 8710 8716 8722 8727 87:53 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 88«5 8871 8876 8882 8887 889;$ 8899 849 8954 85K50 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 903<) 9042 9047 905:5 9058 9063 i)0(i9 9074 9079 81 1KXS5 9090 909(5 9101 9106 9112 9117 9122 9128 91 ;« 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 t)238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 92<)9 9:504 9:509 9:515 9320 9325 9:i:50 9335 9340 86 9;M5 9350 9:555 9:5()0 9:5(55 9370 9375 9:580 9:585 9390 87 9395 9400 i)405 425 94:50 m:55 9440 88 !H45 9450 9455 484 9489 89 9491 9499 9504 9.509 9513 9518 9523 9528 953;5 9538 90 9542 9547 9552 9557 9562 95(!6 9571 9576 9581 9586 91 95! K) 9595 9(500 9605 Domain of convergence . . . 48() Dopplers principle .... 80 Duplication of the cube . . . 18(5 e 208 numerical value of e . . . 299 Eliminant 380 Ellipse 404 Equation, conditional .... 211 of a straight line 75 linear, quadratic, etc. See adjectives. Equivalence, of fractional equa- tions 243 of in-ational equations . . . 262 of quadratic equations . . . 00 of systems of simultaneous equations 411 Errors of observation .... 375 Euclid 6 Euclid's algorithm .... 6 Euler .... 207, 209, 435, 494 Events 366 compound 368 dependent 368 exclusive • 3(i8 independent 3()8 Exclusive events 368 Expansion, binomial .... 493 Taylor's 148 Expectation 374 Exponential, equations . . . 301 functions 274 series 405 Exponents, fractional, negative. and vanishing 265 irrational 270 Factor, highest common . 229 of an integer 3 of proportionality .... 67 prime 6 Factored form of rational func- tion 2;S3 Factorial 353 Factor theorem 132 Falling bodies 66 False position, mt'thod of . . 158 Fermat 435 Ferrari 207 Ferro 201 Field 114 Floating spheres 182 Follows 11 Force 121 dimension of 124 Foster ' . . . . 200 Fourth order, equation of . 204 Fraction 8 rational 226 reduction to lowest terms . . 10 Fractional equations .... 241 Fractional exponents .... 267 Fractional rational functions . 226 improper 226 in lowest terms 228 proper 226 reduction to lowest terms . . 229 Function 55 linear, quadratic, etc. See adjectives. Functional notation .... 131 Fundamental laws . .1.2. .], 12, 13 for coiiiplex imiiibers ... 51 Fundamental theorem of algebra 208 Galois 208 Gauss 5:], 104, 209 Gauss plane 63 Gay-Lussac 246 General term of a sequence . . 429 Geometric means . . . . 87, 88 Geometric progression ... 86 common ratio 2 Transcendental function . . . 2(58 Transmission of light .... 314 Trial 3Gt> Trisection of an angle . . . 186 Uniform motion on a straight line 120 Uniformly accelerated motion . 05 Uniqueness of division ... 15 Van der Waals 240 Vanishing roots 222 Variable 55 bounded . • 448 independent or dependent . 55 which remains finite . . . 448 Variation 50 constant of 57 Variations of an equation . . 109 Vector 39 Vector addition 30 Velocity 03 Vernier 5!» Vibrations, dampened . . . 313 Vieta 2<)3 Volume 01 Weierstrass 19, 53, 204 Weld 375 Wessel 53 x-axis jr-intercept of a line y-axis 1/- intercept of a line Young 70 70 70 75 180 Zero 20 as exponent 267 division by 29 of a function 78 UC SOUTHERN "^'^'°'l'5JnE||l|jK AA 001318 354 6