:t,. \ i ELEMENTARY ALGEBRA J. A. GILLET Professor in the New York Normal College NEW YORK HENRY HOLT AND COMPANY 1896 •^ * .*. • • * i' «•> « s Copyright, 1896, BY HENRY HOLT & CO. ROBERT DRUMMOND, EI.ECTROTYPER ANP PRINTER, NEW YORK. .^c PREFACE. This book is designed to be at once simple enough for the beginner and complete enough for the most advanced classes of academies and preparatory schools. The first three quarters of it constitute an elementary algebra in the strictest sense of the term; the remainder may be regarded as an intermediate step between elementary and higher algebra, and includes the topics of the most advanced re- quirements in this subject for admission to American col- leges and technical schools. One of the main differences between this book and its American predecessors lies in the prominence given to problems and the consequent early introduction of the equa- tion. The statement of problems in the form of equations calls forth the pupil's intellectual resources and develops in him the power of concentrated thought. It is an invalua- ble mental exercise, and one, moreover, in which as a rule pupils take pleasure. Drill in algebraic operations, on the other hand, tends rather to strengthen the memory, to quicken the apprehension, and to cultivate habits of ac- curacy. Though absolutely necessary to secure facility in manipulating algebraic expressions, this drill is apt not to be interesting. For the sake, therefore, both of giving varied employment to the mental activities and of main- taining an equilibrium of interest, it seems desirable that ill 800537 iv PREFACE. problems and exercises should proceed together from the very outset. Problems are accordingly introduced at a much earlier stage than usual, and occur with uncommon frequency in every chapter. At first they are so simple that the resulting equations can be solved by elementary arithmetical processes, and they gradually increase in com- plication with the pupil's increasing knowledge of algebraic methods. The majority of them are either new or else the old ones with new data; the remainder have been selected from a great variety of sources. The book further differs from its predecessors (1) in the attention given to negative quantities and to the formal laws of algebra, known as the Commutative, the Associa- tive, the Distributive, and the Index laws. In presenting these laws the author has endeavored to be rigorous without sacrificing simplicity. (2) In the fuller development of factoring and in its more extensive application to the solu- tion of equations. The method of solving quadratic equa- tions has been based entirely on the principles of factoring. Certainly this method is more in harmony with the pro- cesses of advanced algebra, and it is the author's experience that, even for the beginner, it is quite as simple as the method of completing the square. The first steps in the book have been simplified for the pupil by building upon his knowledge of arithmetic and adding, one by one, the distinguishing features of algebra; — the use of letters as well as figures to express numbers, the use of equations in the solution of problems, the more extended and systematic use of signs, the meaning and use of negative numbers, and the general proof of theorems. In further recognition of practical requirements, the exer- cises in Part I have been divided usually into two sets, the first set being as a rule easier than the second. Careful provision is made in both sets for frequent review of topics already studied. PREFACE. V As the author and publisher cannot hope to have been entirely successful in their efforts to keep the text free from typographical and other errors, they will esteem it a favor to have their attention called to any that may have escaped their vigilance. J. A. G. Normal College, New York, December 10, 1895. f Digitized by tine Internet Arciiive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/elementaryalgebrOOgillricli TABLE OP CONTENTS. (The numbers refer to the pages of the text.) PART I. FUNDAMENTAL PRINCIPLES AND OPERATIONS- CHAPTER I. NOTATION AND SYMBOLS. Symbols of Operation — Algebraic Expressions, 1. — Exponents — Co- efficients, 2. — Numeric Values, 3. — Quantitative Symbols — Terms — Monomials and Polynomials, 4.— Similar Terms, 5. CHAPTER II. EQUATIONS AND PARENTHESES. Members of an Equation — Verbal Symbols — Axioms, 7. — Transposi- tion of Terms — Collection of Terms— Division by Coefficient of x — Solution of an Equation, 8. — Literal Coefficients — Solution of Problems, 9. — Clearing of Fractions, 11. — Symbols of Aggrega- tion, 14. — Signs of Parentheses, 15. — Parenthetic Factors, 16. — Note on Transposition, 18. CHAPTER III. NEGATIVE QUANTITIES. Counting — Signs of Quality, 21. — Scale of Numbers, 22. — Absolute and Actual Values — Addition and Subtraction of Integers, 23. — (Corresponding Positive and Negative Numbers — Special Signs of Quality, 25. — C'ommutative Law of Addition — Addition and Sub- traction of Corresponding Numbers, 26. — Associative Law of Addition — Oppositeness of Positive and Negative Numbers, 27. CHAPTER IV. ADDITION OF INTEGRAL EXPRESSIONS. Arithmetical and Algebraic Sums, 32. — Signs of Coefficients — Inte- gral Expressions — Extension of the Formal Laws of Addition — Definition of Addition of Algebraic Expressions, 33. — Addition of Monomials and Polynomials, 34. — Simplification of Polyno- mials, 36. — Aggregation of Coefficients, 38. vii Viii TABLE OF CONTENTS. CHAPTER V. SUBTRACTION OF INTEGRAL EXPRESSIONS. Definition of Subtraction — Rule for Subtraction of Integral Expres- sions, 41. — Operations on Aggregates, 43. — Compound Paren- theses, 45. CHAPTER VI. MULTIPLICATION OF INTEGRAL EXPRESSIONS. Multiplication of Integers — Two Cases of Multiplication, 49, — Law of (Signs in Multiplication — Commutation Law of Multiplication, 50. — Associative Law of Multiplication, 51. — Multiplication of Monomials, 52. — Changing the Signs of an Equation, 54. — Dis- tributive Law of Multiplication of Integers, 55. — Extension of the Distributive Law, 59. — Arrangement of Terms according to the Powers of a Letter — Multiplicati(m of Polynomials, 61. — Multiplication by Detached Coefficients, 64. — Degree of an In- tegral Expression, 66. — Product of Homogeneous Expressions — Highest and Lowest Terms of a Product, 67 — Complete and In- complete Integral Expressions, 68. CHAPTER VII. DIVISION OF INTEGRAL EXPRESSIONS. Definition of Division — Division of Monomials, 69. — Division of Polynomials, 71. — Freeing an Equation from Expressions of Di- vision, 77. — Synthetic Division, 79. CHAPTER Vni. INVOLUTION OF INTEGRAL EXPRESSIONS. Definition of Involution— Involution of Monomials, 87. — Squaring of Binomials, 88. — Squaring of Polynomials, 89. — Cubing of Bino- mials, 90. CHAPTER IX. EVOLUTION OF INTEGRAL EXPRESSIONS. Definition of Evolution — Inverse of Involution, 93. — Corresponding Direct and Inverse Operations do not always cancel, 94. — Extrac- tion of Root of Monomials, 95. — Square Root of Polynomials, 96. — Squaring Numbers as Polynomials, 97. — Square Root of Numbers, 99. — Cubing of Polynomials, 103. — Cube Root of Polynomials, 103.— Cubing Numbers as Polynomials, 104. — Cube Root of Numbers, 105. CHAPTER X. MULTIPLICATION AT SIGHT. Complete Expression of the First and Second Degree — Product of Two Linear Binomials, 108. — Product of x-\- a and x -\- b — Product of TABLE OF CONTENTS. IX x-\-a and x -{- a — Product ot x-\- a and x — a, 110.— Product of «a;4-&and cx-\-d, 111.— Product of Binomial Aggregates, 112. — Product of X 4" y and x^ — xy -f- y'^ — Product of x — y and x^ ■-{- xy -\- y'\ 114. — To con vert ;r- -|- ^^ ii^to a Perfect Square, 115. — To convert x^'^ + ^-c" into a Perfect Square, 116. — To con- vert x^ -\-bx-\-c into a Perfect Square, 117. — To convert ax^-\-bx into a Perfect Square, 118. CHAPTER XI. FACTORING. Resolution of an Expression into Factors — Resolution of an Expression in Monomial and Polynomial Factors, 120. — To factor the Differ- ence of Two Squares, 121. — Special Cases of factoring Quadratic Trinomials, 122, — Functions, 124. — Remainder Theorem, 126. — To factor the Sum and Difference of the Same Powers of Two Quantities, 129. CHAPTER XII. HIGHEST COMMON FACTORS. Definition of Highest Common Factor — H.C.F. of Monomials, 132. — H. C. F. of Polynomials by Inspection, 133. — General Method of finding the Highest Common Factor of Polynomials, 135. — Gen- eral Method for Three or More Polynomials— H.C.F. not neces- sarily the G. C. M., 139. CHAPTER XIII. LOWEST COMMON MULTIPLE. Definition of Lowest Common Multiple — L. C. M. by Inspection, 144. — L. C. M. by Division, 145. CHAPTER XIV. FRACTIONS. The Symbol — , 150. — The Denominator of a Fraction is Distributive, 151. — Theorem : — = — r-, loo. — Iheorem : -- z= , Sim- h mh b h -i- m plification of Fractions, 154. — Reduction of Fractions to a Common Denominator, 156. — Theorem : i- X -7 = rr. 158. — Corollary : b d bd '' - X c — c X -r = -fy 159. — Reciprocal of a Fraction — Theorem : a c ad a a a^e ... X Corollary : - -i- c = — = — - — , 161. — b d b c b Corollary : c -i- — = c X - , 162 b a of Two or More Fractions, 163. Corollary: c -i- — = c X -, 162.— To cancel the Denominators b a X TABLE OF CONTENTS. CHAPTER XV. CLEARING EQUATIONS OF FRACTIONS. Three Classes of Equations involving Fractions, 166. CHAPTER XVI. RADICALS AND SURDS. Rational and Irrational Numbers — Radicals— Surds, 176. — Imaginary Quantities — Rational Quantities expressed as Radicals — Orders of Radicals, 177.— Arithmetical Roots, 178. — Theorem : 4/^^ x y'b = ^ab, 179. — Reduction of Radicals — Pure and Mixed Surds. 180. — Theorem : '\/a -r- \^h = \/a-^h, 181. — Similar Quadratic Surds — Theorem : m \/a X n \/a = mna — Theorem : The Product of Two Dissimilar Quadratic Surds cannot he Ra- tional — Rationalizing Factor, 182. — Reduction of Fractional Radicals to Integral Radicals — Addition and Subtraction of Radicals of the Same Order, 183. — Rule for Addition of Radicals — Rule for Subtraction of Radicals — Addition and Subtraction of Radicals of Different Orders, 184. — Multiplication of Radicals of the Same Order, 185.— Simple, Compound, and Conjugate Radi. cals, 187. — Rationalization of Polynomial Radicals, 188.— Ra. tionalization of the Denominator of a Fraction— Division of Radicals of the Same Order, 189.— Theorem : ( 1/aY = l/'a" , 191. — Theorem : V \^a = ^j/a — To change Radicals from One Order to Another, 193. — Multiplication and Division of Radicals of Different Orders, 193. — Radical Equations, 194. — Reduction of Radical Equations by Rationalization, 196. CHAPTER XVII. THE INDEX LAW. Meaning of Fractional Exponents, 198. — Meaning of Zero Exponent — Meaning of Negative Exponents, 200. — The Index Law holds for all Rational Values of m and n, 201. CHAPTER XVIII. ELIMINATION Simultaneous and Independent Equation, 207. — Two Unknown Quantities require Two Independent Equations for their Solution, 208. — Elimination — Three Methods of Elimination, 209. — n In- dependent Equations are required to solve for n Unknown Quantities, 216. CHAPTER XIX. QUADRATIC EQUATIONS OF ONE UNKNOWN QUANTITY. Trinomial and Binomial Quadratics — Factors of a;^ _|_ ^^ 222. — Factors of a Trinomial Quadratic, 224. — Quadratic Equation of One tJn- TABLE OF CONTENTS. xi known Quantity — Roots of an Equation, 237. — Solution of a Quadratic Equation, 228.— Formation of Quadratic Equations, 280. — Interpretation of Solutions, 234.— Solution u:t:^ -\- bx -\- c = 0, 238. — Solution of Equations which are Quadratic in Form, 243. CHAPTER XX. QUADRATIC EQUATIONS OF TWO UNKNOWN QUANTITIES. Special Cases of Elimination, 246. CHAPTER XXI. INDETERMINATE EQUATIONS OF THE FIRST DEGREE. Indeterminate Equations— Solution of Indeterminate Equations of the First Degree in x and y, 259. — Solution of Indeterminate Equations of the First Degree in x, y, and 2, 263. CHAPTER XXII. INEQUALITIES. Definition of Greater and Less Quantities — Inequalities, 267. — Ele- mentary Theorems, 268. — Type Forms, 273. CHAPTER XXIII. RATIO AND PROPORTION. Definition of Ratio — Expression of Ratio, 276. — Terms of a Ratio — Kinds of Ratios — Ratio of Equimultiples and Submultiples, 277. ,„. a-\-x a . a — X ^ a . ^ , . — Iheorem : ; — ; < ,-, and 7* > -, when a > b and b-\-x b b — X b 7 rni a 4- X a . a — X a . X < b — Theorem : -; — -— > — , and < — , when a < b b.-j-x b b — X b and X < b, 278. — Compound Ratios — ^Definition of Proportion — Test of the Equality of Two Ratios, 279. — Permutations of Pro- portions, 280. — Transformation of Proportions, 281. — Solution of Fractional Equations, 28-5. — Direct Variation, 288. — Inverse Variation, 289. — Constant of Variation, 290. CHAPTER XXIV. LOGARITHMS. Definition of a Logarithm — Working Rules of Logarithms, 293. — Systems of Logarithms — Common Logarithms, 295. — Character- istic and Mantissa, 296. — Logarithmic Tables, 297. — Method of using Logarithmic Tables, 299. — Cologarithms, 303. — Multiplica- tion by Logarithms — Division by Logarithms, 304 — Involution by Logarithms — Evolution by Logarithms, 305: — Theorem : logi,m = \oga?n . logfort, 307. Xii TABLE OF CONTENTS. PAET II. ELEMENTARY SERIES. CHAPTER XXV. VARIABLES AND LIMITS. Constants and Variables — Functions, 311.— Limit of a Variable — Axioms — Theorem : If c denote any finite quantity, then, by taking x great enough, — < c — Theorem : If c denote any finite quantity, then by taking x small enough, - > c, 312. — Infinites — Infinitesimals, 313.— Approach to a Limit, 314. — Theorem : If k be any fixed quantity and s denote a quantity as small as you please, then, by taking x small enough, kx < s — Theorem : Two equal functions must have the same limit, 315. — Theorem : The limit of the sum of several variables is the sum of their limits, 316. — Theorem : The limit of the product of two functions is the product of their limits, 317. — Theorem : The limit of the quotient of two functions is the quotient of their limits — Defini- tion of f[-n=a and — Theorem: Lim. x-a _\^^ x-aj = w«"~i for all values of n, 318. — Definition of Series — Theorem: The limit of Ao -\- AiX-\- A.x'' + Asx"^ . . . = Ao , 320.— Theorem : In the series Ao + A^x' -\- A-iX^ -f . . . , by taking x small enough we may make any term as large as we please com- pared with the sum of all the terms which follow it, and by taking x large enough ^e can make any term as large as we please compared with the sum of all the terms that precede it, 321. — Vanishing Fraction, 322. — Discussion of Problems, 324. CHAPTER XXVI. THE PROGRESSIONS. A ARITHMETICAL PROGRESSION. Arithmetical Series — The nth. Term, 333. — Problem : To find the common difference, and any other term when two terms are given — Arithmetical Means, 334. — Problem : To find the sum of w terms of an A. P., 337.— The Average Term, 340. — Two Allied Series, 342. B. GEOMETRICAL PROGRESSION. Geometrical Series, 844. — Type Form of the Series — Geometrical Means, 345.— Problem : To find the sum of n terms of a geo- metrical series — Divergent and Convergent Series, 347. — Value of Repeating Decimals — Values of Recurring Decimals, 350. TABLE OF CONTENTS. Xlll C. COMPOUND INTEKEST AND ANNUITIES. Compound Interest-— Problem : To find the amount at compound in- terest, 851. — Present Worth, at Compound Interest— Problem : To find present worth, at compound interest, 353. — Problem : To find the amount at compound interest of a fixed sum invested at stated intervals, 353. — Annuities — Problem : To find the pres- ent value of an annuity, 354. — Problem : To find the amount of an annuity purchasable at a given sum — Problem : To find the amount of an annuity to begin after m years purchasable for a given sum, 355, D. HARMONIC PROGRESSION. Harmonic Series— Theorem : If three quantities are in harmonic progression, their reciprocals are in A. P., 357. — Harmonic Mean — Theorem : The geometric mean of two quantities is the geometric mean of the arithmetic and harmonic means of the quantities — Problem : To insert n harmonic means between a and 6, 358. CHAPTER XXVII. BINOMIAL THEOREM. Binomial Formula, 360. — Binomial Coefficients, 362. — Recurrence of the Coefficients — Exponent — Signs, 364. — Practical Rules, 365. — General Term, 366. — Binomial Theorem for any Rational Index, 368. CHAPTER XXVIII. PERMUTATIONS AND COMBINATIONS. Permutation — Combination, 369. — Symbols of Combination and Permutation, 370. — Number of Permutations, 371. — Problem : To find the number of permutations of 7i dissimilar things r at a time, 372. — Problem : To find how many of the permutations "Pr contain a particular object, 373 —Problem : To find the number of permutations of n things all together when u of the things are alike, 375. — Problem : To find the value of "Cr— Problem : To find the number of times a particular object will be present in the combinations "^Cr , 376, — Meaning of the Bi- nomial Coefficients, 378. CHAPTER XXIX. DEPRESSION OF EQUATIONS. General Equation of the nth Degree in x — Theorem : If a is a root of the equation x^ -j-aiX^-^-\- a^x^^-^-\- . . . an'-ix4- «« = 0, the first member is divisible by x — a, 381. — Converse of the Theorem — An Equation of the nth. Degree has n Roots, 382. XIV TABLE OF CONTENTS. CHAPTER XXX. UNDETERMINED COEFFICIENTS. Theorem : An integral expression of tlie nih. degree in x cannot vanish for more than n vahies of x, except the coefficient of all the powers of x are zero — Theorem : If Aic" -\- Bx'^~ i -f- . . . = A'x^ -\- B'X*^-^ + • • • for all values of a*, both functions being of finite dimensions, then A = A', B = B' , etc., 386 — Definition of par- tial fractions, 387. — Separation of a fraction into its partials, 388.— Theorem : If Ax^' -\- Bx»-^ -\- . . . = A'x^ + J5'a;«-i -}-..., for all values of x which make the series convergent, both func- tions being of infinite dimensions, then A = A', B = B', etc., 393. — Expansion of Functions, 393. CHAPTER XXXI. CONTINUED FRACTIONS. Definition of a Continued Fraction— The Convergent of a Continued Fraction, 399.— Theorem: ^-^ = ^'''Pr-i + Vr-2 ^ ^^^ _ ^^^^:^^^ qr arqr-i-i-gr-2 and Complete Quotients— Theorem: ^ - ^^^ = tlU!! 402. Qn Qu-l gnQn-l Theorem: Each convergent is nearer in value to the continued fraction than any previous convergent, 404. — Theorem: The V 1 value of a continued fraction difEers from — ^ by less than — r qn qn' and by more than r-^ Theorem: The last convergent pre- ceding a large partial quotient is a close approximation to the value of a continued fraction, 405. — Theorem: Every fraction whose numerator and denominator are positive integers can be converted into a terminating continued fraction, 406. — Periodic Continued Fractions — Theorem : A quadratic surd can be ex- pressed as an infinite periodic continued fraction, 410. — Theorem: An infinite periodic fraction may be expressed as a quadratic surd, 412. PART I FUNDAMENTAL PRINCIPLES AND OPERATIONS ELEMENTARY ALGEBRA. CHAPTER I. ALGEBRAIC NOTATION AND SYMBOLS. 1. Symbols of Operation. — Algebra treats of the prop- erties and relations of numbers. In this respect algebra agrees with arithmetic. The fundamental operations of algebra are the same as those of arithmetic. These are addition, subtraction, multiplication, division, involution, and evolution. These operations are also indicated by the same signs in algebra as in arithmetic. These are -|- (plus) for addition, — (minus) for subtraction, X for multiplication, -^ for division, a figure placed above at the right (called an ex- ponent) for involution, and |/ (radical) for evolution. These are called operative symbols, or symbols of operation. Multiplication is also indicated by a dot between the factors. Thus, 4 . 5 means that 4 is to be multiplied by 5. 2. Algebraic Expressions. — Numbers are denoted in algebra by letters as well as by figures. This is one respect in which algebra differs from arithmetic. When figures are written one after another in arith- metic, the expression denotes the sum of the different orders of units denoted by the figures separately. Thus, 334 = 300 + 20 + 4. 2 ALOEBBAIG NOTATION AND SYMBOLS. When letters are written one after another in algebra, the expression formed denotes the product of the numbers denoted by the individual letters. Thus, abc — axh X c. When figures are used in algebra, they are combined to form numbers in the same way as in arithmetic. When figures and letters are written one after another, the expression denotes a product of which the numeral and literal parts are factors. Thus, 12^c = 12 x ^ X c. Literal expressions are more comprehensive than nu- meral expressions. Thus, 324 means one number only, while al)c represents every product that is composed of three factors, and these factors may be integral, fractional, or surd. Owing to this comprehensiveness of its expressions, algebra is sometimes called generalized arithmetic. To find the value of an algebraic expression is to find the number which it represents on the supposition that its letters stand for particular numbers. 3. Exponents. — When the same letter enters more than once as a factor in a product, the number of times that it enters as a factor is indicated by writing a figure after it at the top. Thus, aU^c^ — a X h X i X c X c X c. The expression is read '' a, h square, c cube," or ^' a, l second, c third." The number used to denote how many times the same factor occurs in a product is called an exponent. 4. Coefficients. — The number used to denote how many times a single letter or a product of two or more letters is taken is written before the letter or product and on a line with it. The number thus used is called a coefficient. Thus, 6x denotes that the number x is taken 5 times. That is, 6x = x-]-x-]-x-]-x-\-x; while x^ {x fifth) = x X x X x X x X x. 7abc = abc + abc -{- ahc + ahc + ahc + aic + aho. ALGEBRAIC NOTATION AND SYMBOLS. 3 When no coefficient or exponent is expressed, the number one is to be assumed. EXERCISE I. Find the value of the following expressions when a = d, h — b, and c = 7 : 1. abc. 2. 5abc. 3. ab^c. 4. 4.a'bc\ 5. Gd'b^c. 6. 12a3^,V. 7. 2oab''^c^. 8. 4:0a^Pc. 9. 75aHc^ 10. 250a^h. 11. Find the cost of a oranges at 5 cents a piece. 12. Find the surface of a rectangular board 10 ft. long and a inches wide. 13. There are twenty pages in a book, and on each page there are m lines, and in each line n words. How many- words in the book ? 14. There are a drawers in a case, a compartments in each drawer and c specimens in each compartment, and there are 25 cases in a room How many specimens in all the cases ? 5. Numeric Values. — A magnitude is any thing which has size or extent, and which is doubled when added to itself. Thus, lengths and distances are magnitudes. Magnitudes are measured by comparing them with some other magnitude of the same kind, to see how many times they contain it. The magnitude with which other magnitudes are com- pared in measurement is called the unit of measure^nent^ or the iniit magnitude. When the magnitude contains the unit an exact number of times the number which expresses how many times a 4 ALGEBRAIC NOTATION AND SYMBOLS. magnitude contains the unit is called the numeric value of the magnitude. This term is also extended to the cases in which the value can he expressed only by a fraction or a surd. Numerical expressions, whether composed of figures or letters or of both, are called quantities. Every algebraic expression is numerical; that is, it represents some num- ber. Hence every algebraic expression is a quantity. 6. Quantitative Symbols. — The symbols which express number are called quantitative symbols. In algebra they are both numeral and literal. 7. Terms. — When an algebraic expression is made up of parts separated by signs of operation, the parts separated by the consecutive signs are called terms. Thus, in the expression bci^b + c — 12 -f ati^c, 6aH, c, 12, and alP'c are terms. It will be noticed that a term may be a single letter, a number expressed by one or more figures, or a product com- posed of literal or of literal and numeral factors. The nu- meral factor of a term is commonly called its coefficient, and when no numeral factor is expressed the coefficient is to be regarded as one. Thus, in the expression 7a^?/ — 5« + ^•^^.' the coefficient of the first term is 7, of the second 5, of the third 1. 8. Monomials and Polynomials. — An algebraic expres- sion which contains no signs of operation is called a mo- nomial, or a one-term expression; one composed of two terms separated by a sign of addition or subtraction, a hi- nomial, or a two- term expression ; one composed of three terms separated by signs of addition or subtraction, a tri- nomial, or a three-term expression. Expressions which contain more than three terms are sometimes called multi- ALGEBRAIC NOTATION AND SYMBOLS. 5 nomials, and all expressions which contain more than one term are usually classed together as polynomials. To find the value of a polynomial, we must find the value of each of its terms and then add or subtract these values according to the signs before the terms. Every minus term of a polynomial must be subtracted from the sum of the plus terms or from some individual plus term. AVhen no sign is placed before the first term of a poly- nomial it is understood to be a plus term. EXERCISE II. Find the value of each of the following polynomials, when « = 3, ^ = 1/2, and c = 2/3 : 1. 5 4- a^c - 2abc - Wc + lOa^c^. 2. 9ac^ - 24:bh - (Jab^ - ISabc^ + 7a^c. Find the values of the following polynomials when a = 2, b = d, c = 4:, and d — b: 3. Qabcd — 5«^6' — 7«^^^+ 'da^cd^* 5. ^a^cd^ — lab'^d -\- iSabcd — 6d^c. 6. - 5a^c + Sa^cd' + (Jabcd - lab^d. Note that the value of a polynomial remi^ins the same in whatever order its terms are written. Note also that the value of a polynomial may be found by first adding together the values of its plus terms, and also of its minus terms, and then subtracting the latter sum from the former. 9. Similar Terms. — Similar terms are those which agree both in their letters and in their exponents. They need not, however, agree either in their signs or in their coefficients. Thus a^xy^, ba^xy^, — Zc^xy^, are all similar terms. 6 ALGEBRAIC NOTATION AND SYMBOLS. The similar terms of a polynomial may be combined into one term by performing upon their coefficients the operations indicated by the signs of the term, and using the resulting number as the coefficient of the common literal factors of the terms. Dissimilar terms cannot thus be combined into one. Similar plus terms are combined into one plus term by adding their coefficients, similar minus terms are combined into one minus term by adding their coefficients, and a plus and a minus term, when similar, are combined into one by subtracting their coefficients. EXERCISE III. Eeduce the following polynomials to simpler forms by combining their similar terms : 1. 9a^^ + lOa^^ - ^a%^ - ^aW + 12. 2. 12« - 5^2 _ 6^ _ 7^2 _ 2« _ 3^ _^ 6 - 3. 3. - Qxhj + 8 - ^x^y + Ibx'y - 10 + 7 - 5Z»3. 4. 7«2^ - l^ahj + ^ay^ + 'da'y - a^y - 7. 6. - 7 A + 12A - 5A3 _ Qa^x + %a^x + 15-9. CHAPTER II. EQUATIONS AND PARENTHESES. A. EQUATIONS. "10. Members of an Equation. — An algebraic expression of equality is called an equation. It is composed of two members separated by the sign of equality. The part be- fore the sign of equality is called the/rs^ member, and the part after the sign, the second member. Thus, 7x — 2^; + 6 =: 26 + ^ is an equation. Ix — 2x -[- 6 is its first member, and 26 + 2; is its second member. 11. Verbal Symbols. — The signs =, >, <, .'. stand for the phrases ''equal to," "greater than," "less than," " therefore" or "then," and are hence called verbal signs. 12. Axioms. — A mathematical truth so evident as to be generally accepted without proof is called an axmn. The following are important axioms about equations. 1°. If the same quantity or equal quantities be added to equals, the sums will be equal. 2°. If the same quantity or equal quantities be sub- tracted from equals, the remainders will be equal. 3°. If equals be multiplied by the same quantity or by equal quantities, the products will be equal. 4°. If equals be divided by the same quantity or by equal quantities, the quotients will be equal. 5°. The same powers of equals are equal. 6°. The same roots of equal quantities are equal. 7 8 EQUATIONS AND PARENTHESES. The two following axioms are applicable to all algebraic expressions. 7°. The subtraction of any quantity from an algebraic expression neutralizes the effect of its addition to the ex- pression. 8°. The division of an algebraic expression by any quantity neutralizes the effect of multiplying the expression by the same quantity. 13. Transposition of Terms. — It follows from axioms 1° and 2° that a term may be omitted from one member of an equation and written with the opposite sign in the other without destroying the equality of the members. Thus, if 7a; - 2a; + 6 = 26 + a;, then, by axiom 2°, 7x — 2x — X -\- 6 — 26 -{- X — X, and, by ax. 7°, 7x — 2x — X -{- Q = 2Q. Again, by axiom 2°, 7a; - 2a; - a; + 6 - 6 = 26 - 6, and, by ax.7°, 7a; - 2a; - a; = 26 - 6. When a term is omitted in one member and placed with the opposite sign in the other it is said to be transposed. A plus term is transposed by subtracting it from each member, and a minus term by adding it to each member. Combining the similar terms in the last equation, we get 4a; = 20. 14. Collection of Terms. — The combining of the similar terms in an equation is called collecting the terms. 15. Division by the Coefficient of x. — Dividing each member of the equation 4a; = 20 by 4 we get, by axiom 4°, X = 5. 16. Solution of an Equation. — To solve an equation is to find the value in terms of known quantities of the letter in it which represents an unknown quantity. EQUATIONS. 9 It is customary to represent known quantities by the first letters of the alphabet and unknown quantities by the last letters, x, y, z, etc. Among the steps necessary to the solution of an equa- tion are transposition, collection, and division by the coef- ficient of the unknown quantity. EXERCISE IV. Solve each of the following equations, and name and explain each step taken: 1. ^x + ^x - 12 = bx + 72. 2. 14^ + 8 — "ly =99 — 2/, 3. 8;^ - 5 + 6 + 2^ = 3^ + 53. 4. l/2x + 3/2:?; - .T + 7 = 27 - l/3a;. 5. 7/52; - l/3x - 18 = 72 + d/4:X, 6. Sx -\- a = b -\- 5a. 7. ax -{- d -\- 3ax — c — hax. 17. Literal Coefficients. — In the seventh example, a may be considered as the coefficient of x in the first term, 3« as the coefficient of x in the third term, and 5« as the coefficient of x in the last term. Coefficient means felloiu factor, and in any literal product all the factors but one may be taken as the coefficient of that factor. 18. Algebraic Solution of Problems. — To solve a prob- lem algebraically, wo must first obtain an equation in terms of the known and unknown quantities of the problem, and then solve the equation to find the value of the unknown quantities in terms of the known. e.g. 1. Divide the number 105 into two parts, one of which shall be six times the other. Let X = the number in the smaller part; . •, 6x = the number in the larger part. 10 EQUATIONS AND PARENTHESES. and Qx-{- X — the whole number. Also 105 = the whole number; . • . Qx^x^ 105. Collfccting, rx = 105. Dividing by 7, a; = 15. .• . Qx = 90. The numbers are 15 and 90. e.g. 2. Eight times the smaller of two numbers is equal to 143 minus the larger, and the larger is three times the smaller. Find the numbers. Let X = the smaller number ; . •. 3x = the larger number. .-. Sx= 143 -dx. Transposing, 8x -\- dx = 143. Collecting, llo; = 143. Dividing by 11, a; = 13. .-. 3x= 39. The numbers are 13 and 39. EXERCISE V. I. 1. rind two numbers whose difference is 9 and whose sum is 63. 2. Divide 103 into two parts whose difference shall be 13. 3. Find two numbers such that the larger shall be 4 times the smaller, and that 6 times the smaller shall equal 60 plus the larger. 4. Find two numbers such that the larger shall be 5 times the smaller, and that 7 times the smaller shall equal 374 minus 3 times the larger. 5. Divide 450 into three .parts such that the second shall EQUATIONS. 11 contain twice as many as the third, and tlie first tliree times as many as the third. II. 6. 120 marbles are arranged in 3 piles so that there are twice as many marbles in the first pile as in the second and tliree times as many in the second as in the third. How many marbles in each pile ? 7. In a scliool there are three grades, and there are three times as many scholars in the lowest grade as in the middle grade and five times as many in the middle grade as in the highest. The whole school numbers 735. How many scholars are there in each grade ? 8. A man bought a horse, a carriage, and a harness for 450 dollars. He paid three times as much for the horse as for the harness, and twice as much for the carriage as for the horse. What was the cost of each ? 9. A boy bought a speller, an arithmetic, and a history for $2.30. He gave twice as much for the history as for the arithmetic, and three times as much for the arithmetic as for the speller. How much did he pay for each ? 10. A boy is three years older than his sister, and has a brother who is five years older than himself. Their united ages are 41 years. How old is he ? 19. Clearing Equations of Fractions. — Since a fraction is reduced to its numerator when it is multiplied by its de- nominator, and since botli members of an equation may be multiplied by the same number without destioying their equality, an equation may be freed of a fraction by multi- plying both its members by the denominator of the frac- tion. Zx e.g. Free the equation — - = 6 of its fraction. 12 EQUATIONS AND PARENTHESES. ^X5 = 6X5. (Why?) .'. ^x = 30; a; = 10. Note that 8 + 4 multiplied by 2 = either 12 X 2 = 24 or 8 X 2 + 4 X 2 = 16 + 8 = 24. Also that 8-4 mul- tiplied by 2 = either 4x2 = 8 or 8x2-4x2 = 10 -8 = 8. So in general a -\- h multiplied oy 2 = 2a + 2b, and o, — b multiplied by 2 = 2a — 2b. That is, to multiply any algebraic expression by a number, we must multiply each term of the expression by the number. If an equation contains two or more fractions it may be freed of all of them by multiplying both its members by the product of all the denominators at once. 2x ^'x e.g. Free the equation — - -|- — — = 8 of fractions. Multiplying both members by 12, we get ^+^-96 or Sx + 9a: = 96. Instead of multiplying both members by the product of all tlie denominators, we may multiply by the least common multiple of the denominators. 2x 3^/ A:X e.g. Free the equation -;j- + t — H To" ~ ^ ^^ fractions, o o 12 The L. C. M. of 3, 5, and 12 is 60. Multiplying both members by this, we obtain 120a; , 180a; , 240aj _ or 40a; + 36a; + 20a; = 120. EQUATIONS. 13 Ex. 1. Divide 150 into two parts such that the first shall be 2/3 of the second. Let Hence or X = the number in the second part; 2x 3 = a a S' to 2" is three times the distance from M to S. Find the distance from M to JV, from JV to S, and from S to T, CHAPTER III. NEGATIVE QUANTITIES. 24. Counting. — The fundamental relations of numbers are determined by. counting, and the fundamental opera- tions of arithmetic and algebra, when they are performed on integers and result in integers, are simply abbreviated methods of counting. Numbers may be counted forward or backward. In the former case the numbers obtained are always increasing and in the latter case decreasing. In arithmetic we may count forward indefinitely, but backward only to zero. Counting forward is counting on, or addition ; counting backward is counting off, or subtraction. In arithmetic subtraction is impossible when the number to be subtracted, or counted off, contains more units than the number from which it is to be subtracted, or counted off. 8 — 12 rep- resents an operation which is arithmetically impossible. In algebra the operation is generalized, and counting off is considered to be as unlimited as counting on. Num- bers, instead of running only forward from zero as in arith- metic, are considered as running backward from zero as well. 25. Signs of duality. — In arithmetic the scale of num- bers begins at zero and runs forward only, while in algebra it runs both ways from zero at the centre. To indicate in which part of the algebraic scale a number belongs, the forward part of the scale is called tlie positive part, and the 31 22 NEGATIVE QUANTITIES. numbers in this part of the scale are either written without a sign or are preceded by a plus sign. The numbers are called positive numbers, and the plus sign so used is called the positive sign. The backward part of the scale is called the negative part, and numbers in this part of the scale are written with a minus sign before them. These numbers are called negative numbers, and the minus sign so used is called the negative sign. The signs -|- and — perform a double office in algebra. They indicate the operations of addition and subtraction, and also whether a quantity is to be taken in the positive or the negative sense. In the former case they are properly called plus and minus, and are symbols of operation and in the latter, positive and negative, and are symbols of quality or sense. When a term stands alone the sign before it is to be regarded as positive or negative. A term standing alone without a sign is understood to be positive. 26. The Algebraic Scale of Numbers. — Counting along the algebraic scale towards the positive end is counting on, or in the positive direction, and counting along the scale towards the negative end is counting off, or in the negative direction. The algebraic scale may be represented by a horizontal line of numbers with zero at the centre and the consecutive numbers differing by a single unit, those to the right of zero being distinguished by the positive sign, and those to the left of zero by the negative sign. Thus, \ 13, 12, 11, fo, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, + + + + +•+ + + ++ + rf 4- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Counting along this line from any point towards the NEGATIVE QUANTITIES. 23 right is counting forward, or positively y and from any point towards the left is counting backward, or negatively. e.g. Beginning at minus five and counting positively, we have minus five, minus four, minus three, minus two, minus one, zero, one, two, three, four, five, etc. In this case each new number mentioned is one greater than the last, minus four being one greater than minus five. Beginning at five and counting negatively, we have five, four, three, two, one, zero, minus one, minus two, minus three, minus four, minus five, etc. In this case each new number mentioned is one less than the last. Whatever a positive unit may be, the corresponding negative unit is something just the opposite. 27. Absolute and Actual Values of Numbers. — The absolute value of a number is the number of units in it ir- respective of their sign, while its actual value is its value due to the number and sign of its unit. As the absolute value of a positive number increases, its actual value also increases, but as the absolute value of a negative number i7icreaseSj its actual value decreases. 28. Algebraic Addition and Subtraction of Integers.— -f- 4 or simply 4 means the number obtained by beginning at zero and counting four steps forward, and — 4 means the number obtained by beginning at zero and counting four steps backward. In general -\- a or a means the number obtained by be- ginning at zero and counting a steps forward, and — a means the number obtained by beginning at zero and counting a steps backward. 6 -|- (-[- 4) m.eans the operation of beginning at plus 6 on the scale and counting four steps forward, or in the direction indicated by the sign of the number to be added. 6 + ( — 4) means the operation of beginning at plus 6 on the scale and counting four steps backward. 24 NEGATIVE QUANTITIES. 6 — (+4) means the operation of beginning at plus 6 on the scale and counting four steps backward, or in the opposite direction to that indicated by the sign of the number to be subtracted. 6 — (— 4) means the operation of beginning at plus 6 on the scale and counting four steps forward, or in the op- posite direction to that indicated by the sign of the number to be subtracted. Note. 6 + (+ 4) and 6 + (— 4) having the meanings given, which are really definitions of addition of a positive and a negative quantity, 6 — (+ 4) and 6 — (— 4) must have the meanings given them because of subtraction being the inverse, or opposite, of addition. In general, the placing of one number after another with a plus sign between indicates the operation of begin- ning on the scale at the first of the two numbers and counting as many steps as there are units in the number to be added and in the direction indicated by the sign of that number. The placing of one number after another with a minus sign between indicates the operation of beginning on the scale at the first of the two numbers and counting as many steps as there are units in the number to be subtracted, and in the opposite direction to that indicated by the sign of that number. EXERCISE X. Find by actual counting on the scale the values of the following expressions : I. 1. 12 + (+6): 2. 12 + (^6). 3. 6 + (+12). 4. -6 + (+12> 5^ 6 + (--12). 6, -12 + (+6), NEGATIVE QUANTITIES. 25 7. - 6 + (- 12). 8. -12 + (-6). 9. 12 - (- 6). 10. - 12 - (- 6). 11. 4 - (+ 4). 12. 4 +(-4). 13. « - (+ a). 14. «+(-«). 15. - 6 - (+ 12). 16. - 6 - (- 12). 17. a ~ (- a). 18. - ft - (+ ft). 19, Designate the pairs of operations above which give precisely the same result. 29. Corresponding Positive and Negative Numbers. — Every positive number in algebra has a corresponding neg- ative number, that is, a number the same distance from zero on the opposite side. The sum of a positive number and its corresponding negative number is zero. Thus, 6 + (- 6) = 0, ft 4- (-ft) = 0. 30. Special Signs of duality. — To indicate whether the number to be added or subtracted is positive or nega- tive, instead of enclosing the number with an ordinary plus or minus sign before it within a parenthesis, we may simply put a' small plus or minus sign before the number at the top, and when the number is positive the small plus sign may be omitted. Thus, ft 4- (+ ^) may be written a -{- '^h or a-\- i. a-\- (— b) may be written a + ~b. ft — (+ ^) may be written ft — '•"J or ft — d. a — {— b) may be written a — ~h, — «—(—&) may be written ~a — "b, etc. 26 NEGATIVE QUANTITIES. To indicate that the a and h may represent either posi- tive or negative numbers we may write ^a -\- "^b. 31. Commutative Law of Addition. — From examples 1 and 3 in Exercise X we see that a -\- b = b -\- a; from examples 7 and 8, that ~a -\- ~b = 'b -\- ~a; from examples 5 and 6, that ~a -\- b = b -{-~a; and from examples 2 and 4, that a -\- ~b = ~b -\- a. Whence we have the following general law : ^a + ''b= ^Z* + ="«. In words, the algebraic sum of two numbers is the same no matter in what order the numbers are taken. This is known as the Commutative Law of Addition. 32. Addition and Subtraction of Corresponding Num- bers. — Show by actual counting on the algebraic scale that 8 + -4 =: 8 - +4, or 8-4 = 4 -8 + +4 = -8 - -4 = -4. Also that 8 + +4 = 8- -4 = 12 and -8 + +4 = -8 - -4 = - 4. In general, ^a + -b = ^a - n, or *« - 5 and "-a + +Z* = *r/ - "6. Whence ="« -f H^ *a - n. In words, tlie addition of any number has precisely the same effect as the subtraction of the corresponding number NEGATIVE QUANTITIES. 27 toith the reverse sign. And the subtraction of any number has precisely the same effect as the addition of the corre- sponding number with the reverse sign. This is one of the most important theorems of algebra. 33. Associative Law of Addition. — Show by actual counting on the algebraic scale that 8 + 5-3-4 = (8 + 5) -3- 4, • = 8 + (5 - 3) - 4, = (8 -f 5 - 3) - 4, = 8 + (5 -3 -4), === 8 + 5 - (3 + 4) = 6. In general, -^^ + ^b- ^c- *^ = (-=« + n) - ^c- ^d, = *« 4- (^b - ^c) - "-d, = {^a-\- ^b- ^c) - ^d, = =^« + (*^ - ^c - ^-d), = ^a-]- H- (^6?+ ^d). In words, the sum of three or more numbers is the same in whatever way the numbers may be aggregated. This is known as the Associative Laiv of Addition. N.B. — When terms are associated with a negative sign before the sign of aggregation, the signs of all the terms within the sign of aggregation must be reversed. (21.) 34. Oppositeness of Positive and Negative Numbers. — Positive and negative signs always imply oppositeness. In case of abstract numbers, a negative number is simply the opposite of a positive number; that is, a number which L 28 NEGATIVE QUANTITIES. would produce zero when added to its corresponding posi- tive number. Positive and negative numbers always tend to cancel each other. In the case of concrete numbers, a negative number is the result of a measurement in the opposite direction to that which gives a positive number. Thus, distances measured to the right or upward are usually regarded as positive, and those measured to the left or downward as negative. Dates after a certain era are regarded as positive, and those before the era as negative. Degrees of temperature above zero are positive, while those below zero are negative. Assets are usually regarded as positive, and debts as negative. A surplus is positive, and a deficiency negative. The following quotation is from Dupuis' Principles of Elementary A Igehra : "It an idea which can be denoted by a quantitative symbol has an opposite so related to it that one of these ideas tends to destroy the other or to ronder its effects nu- gatory, these two ideas can be algebraically and properly . represented only by the opposite signs of algebra. '^ If a man buys an article for b dollars and sells it for s dollars, his gain is expressed by s — b dollars. So long as s > b, this expression is -f, and gives the man's gain. ^'But if s < b, the expression is — . It denotes that whatever his gain is now, it is something exactly opposite in character to what it was before. And as he now sells for less than he buys for, he loses. In other words, a neg- ative gain means loss. " Thus, gain and loss are ideas which have that kind of oppositeness which is expressed by oppositeness in sign. If a man gains + a dollars, he is so much the wealthier: if he gains — a dollars, he is so much the poorer. ** Whether gain or loss is to be considered positive must NEGATIVE qUANTITIES. 29 be a matter of convenience, but only opposite signs can denote the opposite ideas. ' ' Among the ideas which possess this oppositeness of character are the following : '^ (1) To receive and to give out; and hence, to buy and to sell, to gain and to lose, to save and to spend, etc. '• (2) To move in any direction and in the opposite direc- tion; and hence, measures or distances in any direction and in the opposite direction, as east and west, north and south, up and down, above and below, before and behind, etc. ^^(3) Ideas involving time past and time to come; as, the past and the future, to be older and to be younger than, since and before, etc. "(4) To exceed and to fall short off; as, to be greater than and to be less than, etc." EXERCISE XI. Give the meaning of the following expressions: 1. — 6 A.D. 2. ~n A.D. 3. "40 B.C. 4. ~« B.C. 5. — (— 30) B.C. 6. — ~h B.C 7. — "50 A.D. 8. — (— c) A.D, 9. The temperature is — 20°. 10. The temperature has risen — 12°. 11. The temperature has fallen — 16°, 12. The temperature has fallen — ( — 7°). 13. The temperature has fallen — ~8°„ 14. The temperature has risen ~ ~«°o 30 NEGATIVE QUANTITIES. 15. It is — 17° colder to-day than yesterday. 16. It is — 8° warmer to-day than yesterday. 17. It is — ~12° warmer to-day than yesterday. 18. Howard lives — 3 miles east of Albert. II. 19. Louis lives — 5 miles north of Horace. 20. Ethel is — 4 years older than Edith. 21. Mabel is — 6 years younger than Florence. 22. Hilda is — (— 2) years younger than Margaret. 23. Hermon owes the grocer — 3 dollars. 24. Hilda weighs — 7 pounds-more than Louis. 26. Mr. Crane is — 20,000 dollars richer than Mr^ Weston. EXERCISE Xil. 1. A man having c dollars paid out a dollars to one person and h dollars to another. Express in two ways what he had left. 2. A man bought at a market tomatoes at a cents a peck and potatoes at h cents a peck, and paid 7n cents for an equal number of pecks of each. How many pecks did he buy ? 3. Two cities are 42 miles apart. Two men start at the same time from the two cities and walk towards each other. The first travels four miles an hour and the second three miles an hour. In how many hours will they meet and how far will each have travelled ? 4. Two cities are a miles apart. Two men start at the same time from the two cities and travel towards each NEGATIVE QUANTITIES. 31 other, the first at the rate of m miles an hour, and the second at the rate of n miles an hour. In how many hours will they meet, and how far will each have travelled ? 6. Find two numbers whose sum is 108 and such that 10 times the greater minus 5 times the less shall be less than 762 by 4 times the sum of the numbers. CHAPTER IV. ADDITION OF INTEGRAL ALGEBRAIC EXPRESSIONS. 35. Arithmetical and Algebraic Sums. — The sum, or amount, of two or more integral numbers is the number obtained by counting all the numbers together. The oper- ation of finding the sum of two or more numbers is called aclditio7i. Since the numbers of arithmetic are all positive, the addition of a number in arithmetic will always increase the number of units in the number to which the addition is made, and the sum of two or more numbers will contain as many units as all the numbers together. The arith7netic sum of two or more numbers is the sum of the numbers without regard to their signs. That is, it is the sum of the absolute values of the numbers. In algebra, the addition of a positive and a negative number will tend to diminish the number of units in the number which has the greater absolute value. The alge- braic sum of two such numbers is the arithmetical difference of the numbers with the sign of the one which has the larger absolute value. The algebraic sum of two numbers both positive or both negative is the arithmetic sum of the numbers with then common sign. Thus, 8 + 10 = 18, -8 + -10 = - 18, 8 + -10 rr: ~ 2, "8 + 10 = + 2. 32 ADDITION OF INTEGERS. 33 The algebraic sum of two or more numbers is the sum of the numbers regard being had to their signs. That is, it is the sum of the actual values of the numbers. 36. Signs of Coefficients. — The sign of a term may be regarded as belonging to its coefficient only. That is, plus terms may be regarded as those whose coefficients are posi- tive. The reason for this will appear farther on, under Multiplication. 37. Integral Algebraic Expressions. — It has been learned in arithmetic that numbers are not only integral, but also fractional and surd. In any algebraic expression the letters may stand for any kind of number. An algebraic expression such as x^ + 6x^ - 4:x^ - 3x^ + 2:c + 1, or . , l-{-2x-3x^ - 4:X^ + 5x^ + x^, in which the exponents of the letters are all positive inte- gers, and in which none of the letters occur in the denom- inators of fractions, or in the divisors of an indicated division, are called ifitegral algebraic expressions. The co- efficients of the various terms may be fractional. 38. Extension of the Application of the Formal Laws of Addition. — In the addition of integral algebraic expres- sions it is assumed that the commutative and associative laws already established for integral numbers apply equally to fractional and surd numbers. This is in accordance with the generalizing spirit of algebra. 39. Definition of Addition of Algebraic Expressions. — To add integral algebraic expressions is to combine their various terms into a- single algebraic expression, each term to be preceded by its own proper sign. The resulting ex- pression should be given in its simplest form. 34 ADDITION OF INTEGERS. 40. Addition of Monomials and Polynomials. — Similar terms are analogous to concrete numbers of like denomina- tions, and dissimilar terms are analogous to concrete num- bers of unlike denominations. Similar terms may be added by finding the algebraic sum of their coefficients and writing after this the common literal factors of the terms. Thus, the sum of ba^b, 1la%, and — ^a^h is 4a^^. Dissimilar terms can be added only by placing them one after another in a polynomial expression each with its own sign. Thus, the sum of ^a%, — 4:al), and 5c is 3a^ — 4:ab + 5c. The sum of these dissimilar terms is really 3a^ + ~^(ib -\- 5c, but, as we have seen, to add ~4:ab is the same as to subtract +4abx ^xY — \ahx ~ QxY - %abx Wy^ IQabx ^ 9^y ■— 7abx — 5(?^y^ l%abx ADDITION OF INTEGERS. 35 To add similar terms tcith unlike signs, find the arith- metical sum of the coefficients of the plus terms, and of the coefficients of the minus terms, and tlie arithmetical differ- ence of these tivo sums, anyiex to this difference the common literal factors, and prefix the common sign of the terms whose coefficients produce the larger arithmetical sum. Ex. 3. a Sax b — 4tby — c —hd a -\- b — c Sax — 4:bg — 5d To add dissimilar terms, write them one after another, each ivith its oivn sign. Ex. 4. —a Sx'y -b -7x^g da -Qxy^ -2b - Sb -5 -'Sxy^ 2a-db- 5 - 4:Xh/ - QXlf - ■ Sb To add terms some of which cere similar a7id some dis- similar, combine the different sets of similar terms into single terrns, and write the resulting terms together icith the remaining terms one after another in a 2^oly7io7nial ex- pression each icith its own sign. Ex. 5. 2cd- 3cx^ + 2c^x - Scd - ex' - 5c^x -f- cx^ 12cd + lOcx' - Qc'x - 11 Qcd + Qcx' - 9c'x + cx^ - 11 To add polynomials, combine the different sets of similar 36 ADDITION OF INTEGERS. terms in the polynomials into si^igle terms, and write these and the remaining terms as a polynomial. In the addition of polynomials, it is convenient to ar- range the terms so that the similar terms will fall in verti- cal columns. 41. Simplification of Polynomials. — When any polyno- mial contains one or more sets of similar terms, it may be simplified by combining these sets into single terms. EXERCISE XIII. Find the sum of the following terms: I. 1. 3«, 7«, 2«, a, 12a. 2. 7a^x, da^x, c^x, 20a^x. 8. - 6ab^ - ab\ - lal)\ - llaJ)% - 4.ay^, - 8«5l 4. — Ix, — 2x, — 8x, — X, — 12Xf — llic, — 15a;. 6. Zx^, - bx\ 82^2, - 12x^. 6. — bac^x, ac^x, — %a(?x, \^a(?x. 7. 5?/2, 4«c, — ac, — 7y^, — 6ac, iy"^, — 5. 8. 7a^x. — Aad, — ax^, — da^x, — 8, — 5abo Simplify the following polynomials : I. 9. 4:X — 5al) -]- 7x -{- c -\- llab — 20a;. 10. daH^ -7x^- 5-{- 12x^ - 4:a^^ + 12 - c. 11. l/'dx - l/2x + d/4cx + X. 12. 2/'6y - S/4.y - 2y - l/3y + 6/6y + y. 13. 9(a + 5) + 10{a + ^) - (« + ^) ~ 2{a + 5). ADDITION OF mTEGBm. Zl II. 14. 7« - 3(^ + 2^) + 8« - (a; + 2/) + 3(a: + y) - 16«. 15. 2(m + n) + 3(a + Z») + (« + ^») - {m -f /^) + (r^ + /;) - 6(m + 7^. 16. 3r/,(/; + a:) + 5r/(^' + a;) + r^C^* ^ x) - lla{b + a;). 17. 2C(«2 _ J2) _ 3^(^2 _ J2) _^ 6^.(^2 _ ^2) _ ic(^2 _ J2). Add the following polynomials: I. 18. Mz — ^hy - 8, - 2az + bhy + 6, 6az + 6 J?/ - 7, and — Mz — 7% + ^^ 19. Soa:; — dcz^, — 5ax + 5c;2;^ ax + 2c2;^, and — iax — 4:CZ^. 20. 8^, + h,2a-i + c,- 3« + SZ* + 2^, - 65 - 3c + dd, and — 5« + 7c — 2d II. 21. 7a: — G?/ + 5;^ + 3 — ^, — a; — 3?/ — 8 — ^, — a: + 2/ ~ '"^^ - 1 + '^ff^ - 2x ^ dy -{- 3z - 1 - g, and a; + Sy-5z + ^+g. 22. 2«'^ + 5ffZ> - xy, - 7a^ + dab - dxy, - 3a^ - ^ab -\- 6xy, and 9a^ — ab — %xy, 23. ^a^b^ - MV^ + x}y + xa/. ia^'^ - la^ - dxif + 6x2?/, 3^^3^2 _^ 3^^2^3 _ 3^2^ _|_ 5,^^2^ ^nd '^a^b^ - a%^ - dx'^y — '6xy'\ I. 24. A lady bought three yards of ribbon at a cents a yard, 10 yards of tape at c cents a yard, and five spools of thread at d cents a spool. She paid x cents on the bill. How much remains due ? 38 ADDITION. 25. One morning tlie mercury in the thermometer stood at X degrees. During the next 24 hours it rose h de- grees and fell c degrees. The following day it rose d de- grees. What was its height then ? 26. A father divided his property of 27,000 dollars among his four children, giving 500 dollars less to each in succession from the eldest to the youngest. How much did he give to each ^ ir. 27. A father gave his eldest son x dollars, his second son 7 dollars less, his third son 9 dollars less than the sec- ond, and his fourth son 1 1 dollars less than the third. How much did he give to all ? 28. A father divided his property among his four chil- dren. To each of the first three he gave 1/4 of his prop- erty plus 200 dollars, and to the fourth he gave 1400 'dol- lars. What was the value of his property ? 29. A man left his five children x bonds worth a dol- lars each, and x acres of land worth i dollars each ; but he owed m dollars to each of q creditors. W^hat was each child's share of the estate ? 42. Aggregation of Coefficients. — When two or more terms of a polynomial contain one or more common factors, whether numeral or literal, the terms may be collected into one by enclosing the terms within a parenthesis and placing the common factors outside. When the common factors are numeral and literal, it is customary to place the numeral factor and the letters which belong to the first part of the alphabet before the parenthe- sis, and the letters which belong to tlie last part of the alphabet after the parenthesis. e.g. bacx + bbcx — bcdx = bc{a -{- h — d)x. ADDITION OF INTEOEBS 39 EXERCISE XIV. Collect the coefficients of x and y in tho following ex- pressions : I. 1. ax -\-hy -\- mx -|- ny. 2. 7nnx -\- 2by -{- pqx — 4:by. 3. 3x — 2y -{- (Jbx — 4// + 7ax -\- m -\- n. 4. ^ax + ^hx + hy + 1x - by ^ x - 5y. 5. Howard is twice as old as Albert. If x represents Albert's age now, what would represent their respective ages eight years hence ? 6. Howard is now twice as old as Albert, but 12 years from now he will be only 3/2 as old. How old is each ? 7. Two cities, A and B, are on a straight road and 18 miles apart. Two couriers, P and Q, start at the same time from the respective cities and travel in the same direc- tion, P from A towards P at the rate of eight miles an hour, and Q from B at the rate of six miles an hour. In how many hours will P overtake Q, and how far will each have travelled ? 8. Divide the number a into two parts, one of which shall exceed the other by b. II. 9. ax -j- by -\- rz — mx — ny — 2^^- 10. "idx + ^ey + 4A - 2/:r - ?>dy + Uz. 11. ^/oay - 2x + ^/\by + Qax. 12. ^ax — by — 3bx — 4:ay. 40 ADDITION. 13. Horace is now twice as old as Herbert, but a years from now he will be only 4/3 as old. How old is each ? 14. Two towns, A and B, are a miles apart. Two cour- iers, P and Q, set out at the same time from the respective towns, and travel in the same direction. P travels from A towards B at the rate of h miles an hour, and Q from B at the rate of c miles an hour. In how many hours will P overtake Q, and how far will each have travelled ? CHAPTER Y. SUBTRACTION OP INTEGRAL ALGEBRAIC EXPRESSIONS. 43. Definition of Subtraction. — Subtraction is the in- verse of addition, or the process of undoing the operation of addition. In addition, two numbers are given and their sum or amount required. In subtraction, the sum of two numbers and one of the numbers are given, and the other is required. The given sum is called the mimie^id, the given num- ber the subtrahend, and the required number the difference or remainder. Since the minuend is the sum of the subtrahend and difference, we may prove our subtraction by adding the subtrahend and difference to see if their sum agrees with the minuend. 44. Rule for Subtraction of Integral Algebraic Ex- pressions. — We have already seen in section 15 that the addition of any number produces the same effect as the subtraction of the corresponding number with the reverse sign, or, conversely, the subtraction of any number is equivalent to the addition of the corresponding number with the reverse sign. Hence we have the following rule for algebraic subtraction : Add the subtraliend with its signs reversed to the minu- end, 41 42 SUBTRACTION. In the operation of subtraction it is better not actually to change the old signs, but merely to think of them as changed in the addition. If the new signs are written, it is better not to change the old into the new, but to write the new as small signs before the terms at the top. EXERCISE XV. I. 1. From "Zx -\- y -\- 1 z take 6x -]- 2y — 7z. 2. From 9a — 4:b -\- Sc take 5a — db -{- c. 3. Subtract 3a^ - a^^la-U from lla^ - 2a^ + da^ — Sa. 4. From 10«V + Uax^ + 8A take - lOa^x^ + 15ax^ - 8 A. 5. Subtract 1 — a -\- a^ — da^ from a^ — 1 -\- a^ — a. 6. From 2/3^:2 _ ^/^x - 1 take - 2/3^2 -{- x - 1/2. 7. From a take b — c. 8. What must be taken from 6a + 5 — 3J to produce 8a + 6Z» + 13 ? 9. What must be taken from 2x^ — 3a^x^ + 9 to pro- duce x^ -\- ba^x^ — 3 ? 10. What must be added to a + 5Z» + 9 to produce 3a - 2^* + 6 ? 11. Ethel is twice as old as Edith, and six years ago she was four times as old. What is the age of each ? 12. A and B have together 150 dollars. If A were to give ^35 dollars, B would have three times as much as ^. How much has each ? II. 13. What must be added to x to produce y ? PARENTHESES. 43 14. By how much does 6x — 7 exceed 3a; + 4 ? 15. From what must 5a; -f- 4?/ + 7ft — 12 be subtracted to produce unity ? 16. From what must x^ — x^ -\- x — IhQ subtracted to produce %x'^ -|- 2 ? 17. From l{a + I) take 3(ft + h), 18. From 3«(6' — x) take a{c — x). 19. From la^(l) —-x) — ab(a — b) take 5a^{h — x) — bab(a — h). 20. Howard is x years old. How old was he eight years ago? 21. Divide the number m into two parts such that, when a is taken from the first and given to the second, the second will be five times the first. PAREKTHESES. 45. Operation upon Aggregates. — Every algebraic ex- pression, however complex, represents a quantity, and may be operated upon as if it were a single symbol of that quantity. When an expression is to be operated upon as a single quantity it is enclosed within parentheses or brackets, but the parenthesis may be omitted when no ambiguity or error will result from the omission. Thus, one polynomial may be added to another or to a monomial by writing it, enclosed within a parenthesis and preceded by a plus sign, after the expression to which it is to be added; and a polynomial may be subtracted from a polynomial or monomial expression by writing it, enclosed within a parenthesis and preceded by a minus sign, after the expression from which it is to be subtracted. Since terms written after one another each with its own 44 SUBTH ACTION. sign in a polynomial expression are to be considered as added, and since in addition there is no change of signs, a parenthesis preceded by a plus sign may be omitted without any change of signs; and since the subtraction of any quantity produces the same effect as the addition of the corresponding quantity with the reverse sign, a parenthesis preceded by a minus sign may be omitted if the sign of every term be changed. N.B". — It must be carefully borne in mind that the sign before the parenthesis is not the sign of the first term within it, but of the parenthesis as a whole. This sign really goes with the parenthesis when the latter is removed. When no sign is expressed with the first term within the parenthesis, the term is understood to be plus, and its sign must be written on the removal of the parenthesis, as plus when the parenthesis is plus, and as minus when the parenthesis is minus. EXERCISE XVI. Clear the following expressions of parentheses and re- duce the results to the simplest form : 1. I. ah — (m, — 'Sal) + 2ax) — 7ab. 2. X — {a — x) -\- (x — a). 3. 2b+{h- 2c) - {b-\- 2c). 4. 4:X - 3y -\- 2z - {- 7x + 5y - Sz 5. II. 7ax — 2% — {8ax + Sbi/) — {Sax ■ 6. {a — x) — (a -{- x) -\- 2x. 7. -(a-b)-(b-r)-(,- a). 8. - (Sm + 2») - {3m - 2n) + 9m. Zby) PARENTHESES. 45 22. Of course in forming aggregates preceded by a minus sign, the sign of every term enclosed within the parenthesis must be changed. EXERCISE XVII. Keduce the following expressions to the form x — (an aggregate) : I. 1. X — a — h. 2. X — 7)1 — n. 3. a-{- X — 'dx-\-^y. 4. - 3^ + a; 4- 2c + 56?. 6. 2x-2a-i- 2b. 6. x + S - (a-\-b). 7. X -{- a — (b ~ o) -{- (m — n). II. 8. 2x -\- a — b. 9. 3x - 2m + 2n. 10. ox -{- ab — m — Sab -\- 2m. 11. X — 2m — {Sa — 2b). 12. X — {am -\- b) — {p — q) — (am — n). 13 X — (a^b) — (p — q) — (m — n). 46. Compound Parentheses. — An algebraic expression having parentheses as a part of it may be itself enclosed in parentheses with other expressions, and this may be repeated to any extent. Each order of parentheses must then be made larger or thicker, or different in shape, to distinguish it. e.g. Suppose we have to subtract a from b, the remain- 46 SUBTRACTION. der from c, that remainder from d, and so on. We shall have : First remainder, h — a. Second remainder, c — {h — a). Third remainder, . o . . d — [^c — {h — a)]. Fourth remainder, . e — {d — \g — {i — a)]]. Fifth remainder, ^ — [e — \d — [^ — (^ — «)] }]. Such parentheses are called compoimd pare^itJieses. Compound parentheses of addition and subtraction may be removed by removing separately the individual paren- theses of which they are composed. AVe may begin either with fhe outer ones and go inward, or with the inner ones and go outward. It is customary to begin with the inmost. e.g. Clear of parentheses: ^_[«_ {j_ [c_ (^_e)]}]. Beginning with the inmost, the expression takes, in succession, the following forms : ic- [«- |&- [c- ^ + e]j] = X — \a — [h — c -\- d — e}'] = X— [a — I)-{-G — d-{-e] = X — a -{■ b — G -\- d ~ e. Beginning with the outmost, we have x-[a- \b- [G-(d- e)]}] = x-a-\-{l)-{c- {d-e)']} = X — a~\-h — \_c — {d ~ e)'] = X — a-\-l) — G-{-{d— e)=. X — a -\- b — G -\- d — e. Again, x — {— (a -]- b) -\- {c -\- d) — {e — z)"] PARENTHESES. 47 gives, when we begin with the inner parentheses, X — \^— a — h -\- c -\- d — e -\- z\ = x-{-a-\-b — c — d -\- G — z\ and when we begin with the outer parentheses, x^- {a-^h) - {c^ d) ^ {e - z) =^ x-\-a-\-h — c — d-^-e — z, EXERCISE XVIII. Remove the parentheses in the following expressions, and combine the terms containing x, y, and zi I. 1. rn-{-i-{p-q)-^{a-l)-\-{-c-\-d)]. 2. m.-l- {a-h)- {p-\-q)-]- (n - h)], 3. 'Tax - [(2ax + by) - {Sax -by) + (- "^ax + 2%)]. 4. a—\a— [a —\a— {a — «)]}]• 5. p—\a — h — {s-\-t-\-a)-\-{—m — n)]. 6. A father left 80,000 dollars to his four children. The eldest was to receive four times as much as the youngest less 1800 dollars, the second was to receive three times as much as the youngest less 1200 dollars, and the third was to receive twice as much as the youngest less 600 dollars. How much did each receive ? 7. Divide a into three parts such that the second shall equal the first minus h and the third shall be c less than twice the first. II. 8. 2aa; — \^ax — by — (7 ax + 2by) — {5ax — Sby)]. 9. ax -\- by -\- cz -\- [2ax — 3cz — {2cz -\- 5ax) — {7by - dcz)]. 48 SUBTRACTION. 10. X — \^x — y — [3:r — %y — (4a; — 3«/)] }. 11. ax — bz — {ax -\- bz — [ax — bz — {ax -f- bz)]]. 12. my — \x-}- Sy -\- [2my — 3{x — y) — 4:ab] -\- 5]. 13. Divide 186 into five parts such that the second shall exceed the first by 12, the third shall exceed twice the first by 24, the fourth shall exceed three times the first by 36, and the fifth shall exceed four times the first by 48. CHAPTER VI. MULTIPLICATION OF INTEGRAL ALGEBRAIC EXPRESSIONS. A. LAW OF SIGNS, OF COMMUTATION, AND OF ASSOCIATION. 47. Multiplication of Integers. — Multiplication is the operation of finding what number is obtained by counting a number over a given number of times. The number to be counted over is called the multipli- cand, the number which indicates how many times the multiplicand is to be counted over is called the multiplier, and the number obtained as the result of the operation is called the product. The multiplier and the multiplicand are called / or a — b units, and the c rows to- II ~ab I +ab o o o o o o o o o o o o o o o +b +b o oooo o o o o o o o o o o o o o o o -a o o o o +a ~a +a O O o o o O O o o o o o o ~b ~b o o o o o o o o o o o o o o o o o o o o +ah III O O O O -ab IV {a -\- b) X c = ac -{- be ac -\- be oooo . o o c o o o o . o o oooo . o o a + & -be + «^ oo o o o o c oo o o o o o o oooo 'b a 58 MULTIPLICATION gether represent the })roduct of {a 4- ~h) and r, or (a—b)c. This product is evidently the sum of the two products ac and — be, and is equal to ac -\- ( — be), or ac — be. The two expressions ac -\- {— be) and ac — be are not identical in mean- ing. The former represents two sets of units, one positive and one negative, and indicates that they are to be com- FiG. 7. bined into one ; the latter represents one set of units and indicates that it has been obtained by uniting two sets of units, one positive and one negative. Of course the products ac and — be tend to cancel each other wholly or in part, but the actual cancellation can be expressed only when the products are numerals or similar terms with numeral coefficients. In the actual illustration ac represents 12 positive units and — be 6 negative units, and ac — be represents 6 positive units obtained by cancel- ling 6 of 12 positive units by 6 negative units. Were be>ac, the result of the cancellation would have been a number of negative units equal to the arithmetical difference of the two products. So long as ac > be, the expression ac — be, as a whole, is positive, and denotes that the operation produces a sur- plusage of the kind of units employed ; and when ac < be, the expression ac — be, as a whole, is negative and indicates that the operation produces a deficiency of the kind of units employed. EXERCISE XXI. 1. Arrange the units to represent the case (a-^b) X ~c and show that it equals — ac — be. 2. Arrange the units to represent {~a -\- ~b) X c, or {— a — b) X c, and show that it equals ■ - ac — be. LAW OF ASSOCIATION. 59 3. Arrange the units to represent {~a + ~h) X ~c, or (— ft — ^) X — c, and show that it equals ac -\- be. Ex. 2. (6 + 4)(3 + 2) = 10 X 5 =1 50, and 3 . 3 + 4 . 3 + 6 . 2 4- 4 . 2 = 18 + 12 + 12 + 8 = 50. In general. ad o o o o hd o o o + To represent the case {a -\- h){c + d), arrange c -{- d rows containing a -\- b units each in the first quadrant (Fig. 8). The c + d rows will represent the product of a -\- b and c -\- d. This prod- uct is evidently equal to ac -\-bc -\- ad -\- bd. The product of a poly- nomial and a polynomial is the sum of the products of the first polynomial and each term of the second. ac, o o o o o o o o o o o o o o o o o o o o a + Fig. 8. o o o he o o 55. Extension of the Application of the Distributive Law. — The distributive law of multiplication which we have demonstrated for integers is assumed to hold for all kinds of numbers which can be expressed by letters. Hence the last two definitions hold for all integral algebraic ex- pression in which the multiplicand is an integral polyno- mial. 60 MUL TIP Lie A TION. EXERCISE XXII. I. 1. Arrange the units to represent the case (a + h){c + -d), or {a + h){c - d), and show that it equals ac -\- he — ad ~ hd. Show by a similar arrangement that 2. (a-\-h)('~c-\-d), or {a-\-h){ — ('-\-d) = —ac—hc-\-ad-\-hd. 3. {a-\-h){~c-\-~d), or {a-\-h){—('—d) — —ac—bc—ad—hd. 4. (a-\-~b)(c-{-d), or (a—b){c-\-d)=ar—bc-\-ad—bd. 6. (a-\-~b){c-\--'d), or (a—b)(c—d)=ac—bc—ad-\-bd. 6. («H-~^)(~c+f?), or («— Z>)( — ^-|-^) = — «6'-|-^c-|-«!<:/— M. II. 7. {a-{-~b){~c-{-~d),ov{a—b){—c—d) = —ac-\-bc—ad-\-bd. 8. (~a-h&)(c+fZ), or {—a-^b){c-]-d):=^—ac-\-bc—ad-\-bd. 9. (~«+Z>)(c+"rZ), or (—«H-^)(C — <-/)=:— ft6'+^C+«^—M 10. (~a-\-b){~c-\-d), or (— ^-j-Z')( — 6*-f-6Z)=«c— Z>6'— «fZ+^>r?. 11. {-a-{-b){-c-\--d),or{-a-\-b){-c-d)-^ae-bc-\-ad-bd. 12 ("rt+~^)(6'+^?), or {—a — b){c-\-d) = —ac—bc—ad—bd. 13. (~«4-~^)(^+~^)jOr ( — <-«— /^)(6'—fZ) = —«6'—Z»c+r^r/+J^. 14. ("«4-~^)(~^+fO'Oi'(— <^' — ^)(— ^+^0=^^^-r^^— «^^— *^- 15. (~a-{-~b)(-c-\-^d),OY{—a — b){—c—d)=ac-{-bc-{-ad-^bd. "N^ote that the numbers in the adjacent quadrants tend to cancel each other, while those in the opposite quadrants tend to augment each other. The expression finally ob- tained will be positive or negative according as the sum of LAW OF ASSOCIATION. 61 the units in the first and third quadrants is greater or less than the sum of those in the second and fourth quad- rants. 56. Arrangement of Terms according to the Powers of a Letter. — A polynomial is said to be arranged accord- to the powers of some letter when the exponents of that letter either ascend or descend in magnitude in regular order. Thus, ba — iSbx -\- 'dcx^ — 4:a^x^ is arranged accord- ing to the ascending powers of x; and 3^^ — ^ax"^ -{- ex — 7 is arranged according to the descending powers of x. 57. Multiplication of Polynomials. — (a) To multiply a polynomial hy a monomial, multiply each term of the poly- nomial by the monomial, and tv7'ite the result as a poly- nomial reduced to its simplest form. EXERCISE XXIII. Multiply together: I. 1. 3xy -\- 4:yz and — VZxyz. 2. ab — be and a^be^. 3. -• X — y — z and — ^x. 4. a^ — b'^ -\- c^ and abc. 5. — ab -\- be — ca and — abc. 6. -2a^ ~ 4:ab^ and - 7d^b\ 7. 5x^y — Qxy'^ + Sx^y^ and dxy. 8. ~ 7x^y — bxy^ and — %x^y^. 9. — bxyh -{- dxyz^ — Sx^yz and xyz, 10. ix^yh^ — Sxyz and — 12x^yz^. , 11. — 13a;y^ — Ibx^y and — 7c(^y'^. 62 MULTIPLICATION. II. 12. ^xyz — lOx'^yz^ and — xyz, 13. ahc — a%c — ab'^c and — «^c. 14. — (v^bc + ^^6'r« — c^ab and — 0^6. Find the product of 15. 2rt. — 'db-\- 4:0 and — 3/2«. 16. 3x — 2// — 4 and — 6/Qx. 17. 2/3r« — J/6^ — c and d/Sax. 18. 6/7«V _ 3/2f/.r3 ^nd - 7/Sa^x. 19. - 5/3«V and - 3/2«2 + «a: - 3/5^1 20. - 7/-Zxy and - 3.^2 + 2/7x1/ . 21. - 3/22-y and - l/3i;2 + 2if. 22. - 4/7.x^-\-x- ^ and - ?>x^. 4. :c2 + a; + 1 and cc2 - 1. 5. x^ + 2a; + 3 and x^ -x^\. 6. X? -hx^^ and x? + 5:r + 6. 7. X? -\- xy -{- y^ and x — y. 8. x^ — xy -\- y^ and x -\- y. 9. x^ + xy -\- y^ and x^ — xy -{■ y^. 10. .'2;^ + 3x^ H- Sa; + 1 and x^ + 2a; + 1. 11. 3(a; - 4) = 361 + 8(2a; - 12) - 5(4a; + 40). Clear of parentheses and find the value of x. 12. A man bought three houses. He paid for the sec- ond 8000 dollars less than three times as much as he paid for the first, and for the third five times what he paid for the first less the cost of the second. Five times the cost of 64 MULTIPLICATION. the first minus the cost of the second is equal to 192,000 dollars minus three times the cost of the third. What was the cost of each house ? 13. A man started to give 50 cents apiece to some beg- gars and found he had not money enough within 7 cents. He then gave them 45 cents apiece and had 18 cents left. How many beggars were there ? II. Multiply together: 14. x^ — 2ax^ + 2 A — da^ and x^ — Sax + 2a^. 15. ^ — ax^ — 2 A -j- a^ and x'^ -\- ax — a^. 16. x^ -\- iix^y + Qx^y'^ + 4.^'?/^ + y^ and x'^ — 2xy -\- y^, 17. X — a, X + ci, and x^ -\- a^. 18. X — a, X -{- b, and x — c. 19. \-\-x-\-Q^,\ — x-\- o;^, and 1 — x -\- x^. 20. a — i, a -\- b, a^ — ah -\- W, and c? ^ ab -\- W'. 21. ^x? 4- Vlxy + \^y^ and 3cc — 4?/. 22. 25«V - IhaWxy'^ + UHf and 5 A + Wf. 23. 16«V +.20rt//li-^''^ + 25^ V and ^az^ - Wx. 24. A man bought three horses. He paid 50 dollars less than twice as much for the second as for the first, and for the third three times the cost of the first less the cost of the second. Seven times the cost of the first minus twice the cost of the second is equal to 1700 dollars minus twice the cost of the third. What was the cost of each ? 25. A man gave some beggars 30 cents apiece and had 12 cents left. He found that he needed four cents more to enable him to give them 32 cents apiece. How many beggars were there ? 68. Multiplication by Detached Coefficients. — When two expressions contain one and the same letter and both LAW OF ASSOCIATION. Q)^ are arranged according to the ascending or descending powers of that letter, much labor of multiplication can be saved by writing down the coefficients only. Thus, to multiply q-? — 5a* + 6 by x^ -\- bx-\- Q, we write 1-5+6 1 + 5+ 6 1-5+6 5-25 + 30 6-30 + 36 1 + 0-13+ + 36 The highest power of x in the result is x'^, and the rest follow in order. Hence the required product is ^4 _^ ox^ - l^x^ + Oa; + 36, or a;^ - 13^2 _^ 35^ When some of the powers of the letter are wanting, the coefficients must be written down as zeros in their jiroper places. Thus, to multiply x^ + '6x^ + 3a; + 1 by a;^ + 'Zx^ + 1, we write 1+0+3+ 1 1 + 2+ + 1 1+0+3+ 3+1 2 + 0+ 6 + 6 + 2 0+ 0+0+0+0 1+0+3+3+1 1 + 2 + 3 + 10 + 7 + 5 + 3 + 1 Hence the product is xi + 2a;« + 3a;5 + 10.^:* + 7.^3 + bx^ + 3a; + 1. The method illustrated above is known as the method of detached coefficients. QQ MULTIPLICA TIOK EXERCISE XXV. Do the following multiplications by the method of de- tached coefficients. Multiply : I. 1. 3a:2 - a: 4- 2 by ^x^ + 'Zx - 2. 2. ic* - 2a:2 + a: - 3 by 2;* 4- 2-3 - :^^ - 3. 3. ^ -5x^ + 1 by %x^ + 5a; 4- 1. 4. ^x^ -3x^-{-x-2 by x^ -2x^-x-^% 5. 1 - 2x + a;2 by 1 + 2x + 3x^ + 4:X^ + 6x\ 6. 1 + 'Zx + 3a;=^ + 4^-3 + 5a;4 + Gx^ by 1 - 2.C 4- or^. 7. 1 - 2x 4- 'da^ by 1 4- 3:z; - 52;^. 8. Z -\- Sx — 2x^ by 2 — 3a; 4" ^^^« 9. x^ -2x^-{-x-\-l by .^2 4- 1. 10. x'' -2x^ + 3 by 2x' - xK II. Examples 1-10 of Exercise XXIV. 69. Degree of an Integral Expression. — 'the degree of an integral term in any letter is the number of times that letter is contained as a factor in the term, and is equal to the exponent of the letter. The degree of an integral term in two or more letters is the number of times all together that these letters occur as factors in the term, and is equal to the sum of the expo- nents of the letters in the term. The degree of a term in any letter or letters is often called the dime?isio7i of the term in that letter or those letters. The degree of any integral algebraic expression in any LAW OF ASSOCIATION. 67 letter or letters is the degree of the term in it which is of the highest dimensions in that letter or those letters. e.g. The term 5a^b^x^ is of the fifth degree in x, of the nintli degree in bx, and of the twelfth degree in abx. The expression ba^x'^ + Qa^a^ — llax^ is of the sixth degree in x and of the seventh degree in ax. It will be noticed that in the last example every term is of the same degree in ax. When all the terms of an ex- pression are of the same degree in any letters, the expression is said to be homoge^ieous in these letters. 60. Product of Homogeneous Expressions. — The prod- uct of tivo homogeneous expressions must be homogene- ous. — For each the terms of the product is obtained by multiplying some one term of the multiplicand by some one term of the multiplier, and the number of dimensions of the product of two terms is clearly the sum of the number of dimensions of the separate terms. Hence, if all the terms of the multiplicand are of the same degree, and all the terms of the multiplier are also of the same degree, it follows that all the terms of the product must be of the same degree. It also follows from the above consideration that the degree of the product is the sum of the degrees of the fac- tors. When the two factors to be multiplied are homogene- ous, there must be some error if the products obtained are not homogeneous. 61. Highest and Lowest Terms of a Product. — It is im- portant to notice that, in the product of two algebraic ex- pressions, the term which is of the highest degree in any particular letter is the product of' the terms in the factors which are of the highest degree in that letter, and the term which is of the lowest degree in that letter is the product of the terms which are of the lowest degree in that letter in 68 MULTIPLICATION. the factors. Thus there can be obtaiued only one highest- degree term and one lowest-degree term. 62. Complete and Incomplete Integral Expressions. — It is also important to notice that if each factor in mnltiplica- tion is complete in any letter, that is, contains every degree of that letter from the highest one given down to zero, the product will be complete in that letter. Thus the product oi x^ -\- x^ -\- x^ -\- 1 and x'^ -\- x -\- 1 is x^ -f 2r^ + ^x^ + ^x^ + 3a;2 + 2.t + 1. If an expression is incomplete in any letter it may be completed by filling in the blank spaces with terms of the proper degree having zero as their coefficients. Thus x^ -f a:'-^ -f 1 may be written x^ + Ox^ + Ox^ -\- x^ -\- Ox -^ 1. CHAPTER VII. DIVISION OP INTEGRAL ALGEBRAIC EXPRESSIONS. 63. Definition of Division. — Division is the inverse of multiplication, or the process of undoing multiplication. In multiplication two factors are given and their product is required. In division the product and one of the factors are given and the other factor is required. The product of the two factors is called the dividend, the given factor the divisor, and the required factor the quotient. Since the dividend is the product of the divisor and quotient, we may prove our division by multiplying together the divisor and quotient to see if their product agrees with the dividend. 64. Division of Monomials. — The rules for division are obtained by studying the corresponding cases of multiplica- tion. Take the following cases of the multiplication of monomials : Note: 1°. That the sign of one factor is + when the signs of the product and of the other factor are alike, and 70 DIVISION. — when the signs of the product and of the other factor are unlike. 2°. That the coefficient of one factor is the quotient obtained by dividing the coefficient of the product by the coefficient of the other factor. 3°. That the exponent of any letter in one factor is the difference between its exponent in the product and in the other factor, and that when this difference is zero the letter does not appear in the other factor. When any letter which appears in the product does not appear in one factor, its exponent in that factor is to be regarded as zero. From these observations we obtain the following rule for the division of a monomial by a monomial : Divide the coefficient of the dividend hy that of the divi- sor for the coefficient of the quotient, subtract the expoiient of each letter in the divisor from its exponent in the dividend for its exponent in the quotient, and place before the term in the quotient the plus sign when the sights of the divisor and dividend are alihe, and the minus sig7i when the signs of the divisor and dividend are tinlike. EXERCISE XXVI. Divide : 1. 20a^y by 4a;l 3. 5^a*b^c by 6aH^c., 5. blaxh by — 3azx^. 2. 21a^ by 7b. 4, 4:9a^yh by 7xyh, 6. - 132a^yh by 12^2^. II. 7. — Sbx^yh"^ by - 9. l/5.cy by l/lOa^y. 10. 11. - 2/3ay by - 5/6a^y. 12. -'27a^c^\)j-Sabc^. l/^a^b" by - \/Vlab^, - Oary^s by %/3xt'. Division. tl Multiply : I. 13. b{x-[- ijfz by 3(a; + i/)V. 14. 13(rt - Ifx by - 3(^/ - VfT?. 16. - 5c(a + Z>)4a:y by U{a + ^)3r2. II. 16. - na%{c - d)y^ by - ^a})\c - dfx. Divide: 17. 45(rt + ifx^ by 9(« + ^).r2. 18. Q'dac\b - d^xf by - 7c(^> - ^)2a:^. 19. - ^'lc^d{h + c)2:^2 by _ 3c2(j _|_ c)^^ Simplify: I. 20. (V'h'^C, X (- 8r«3J4^,5) _^ _ 4^6j6p4^ 21. - ^xhf X (- 12.c. {a-b^ cY={a-b + c){a - b + c) = «2 ^^2_^^_ 2ab + 2ac - 2bc. {a — b — c)'^ = (a — b — c)(a — b — c) . =: «2 _|_ j2 _|_ ^2 _ 2(ih - 2ac + 2bc. {— a — b — cy= (—a — b — c)(—a — b — c) = ^2 _|_ j2 _|_ ^2 _|_ 2ab-{- 2ac + 2bc. 90 INVOLUTION. Note that in each of these cases the square consists of the square of each term of the polynomial and, in addition, twice the product of the terms of the polynomial taken two by two in every possible way, regard being had to the signs of the terms. The surest way to get every possible combination of the terms two by two is to combine each term of the poly- nomial with each term which follows it. The law stated above holds whatever be the number of the terms in the polynomial to be squared. Hence we have the following rule for squaring a polynomial : Square each term of the polynomial, and take twice the swn of the products of each term and the terms which follow it, and ivrite the terms thus obtained as a polynomial, each with its own sign. EXERCISE XXXIV. Form the squares of: 1. 1 -f 2^ + 3x\ 2. 1 + 2a; -f 3a;2 + 4a;3. 3. 1 + 2^; + dx^ -h 4a:3 _j_ 5^5^ 4i. a — b -{- c — d. 6. da-i-2b - c-i-d. 75. Cubing of Binomials.— ^x.(a-^bY=(a-]-b){a-\-b)(a + b) = a^-}- da^ + 3« J2 _|. j3^ (a-bY=(a-b)(a-b)(a- b) = a^ - da^ -h dab^ - b\ i^^ a + by^ (- a-\-b)(- a-\- b)(- a-i-b) = -a^-i-'Sa^-'Sab^-\-b\ (-a-bY=(-a-b)(-a-b){~a-b) = ^a^- 'da% - 3ab^ - b'\ INVOLUTION. 91 Note that in each case the cube of a binomial is a quad- rinomial, and that two of its four terms aie cubes of the two terms of the binomial, and each of the other two terms is three times the product of one of the terms of the bino- mial and the square of the other. Hence we have the fol- lowing rule for cubing a binomial : Cuhe the first term, take three times the product of the square of the first term a7id the seco7id term, also three times the product of the first term and the square of the second, and the cube of the second term, and write the terms obtained as a polynomial, each with its oivn sign. e.g. (dx - 2«2)3 ,^ {3xY-3(3xY . 2a^-^ 3(3x)(2a^y-(2a^y = 27a;3 - 54«V2 -|- 'SQa'x - Sa\ EXERCISE XXXV. Write down the cube of each of the following expres- sions : I. 1. X + a. 2. X — a. Z. X — 'Zy. 4. 2x-\-2j. 5. 3x - 5y. II. 6. ab' -\- c. 7. 2al) — 3c. 8. 5a — be. 9. x^ + 4?/l 10. 4:X^ — ^y^- EXERCISE XXXVI. 1. Divide 9x^ - I. .Qx^-6x' + x'-x + 2 by x^~3x-\~2. 2. Divide 1/43^^ + l/72xy^ + 1/Uy^ by l/2x + 1/3?/. 3. Find two numbers whose difference is 5, and such that the square of the smaller plus 9 will equal the square of the laro^er minus 56. 92 INVOLUTION. 4. Find two numbers which shall differ by 3, and such that the square of the smaller plus 15 shall equal the square of the larger minus 24. 5. Find two numbers that shall differ by 2, and such that the cube of the smaller increased by six times its square shall be 44 less than the cube of the larger. 6. A farmer bought some cattle at 30 dollars a head. Had he bought three more for the same money, they would have cost him 2 dollars less a head. How many did he buy ? CHAPTER IX. EVOLUTION OP INTEGRAL ALGEBRAIC EXPRESSIONS. 76. Definition of Evolution. — Evolution is the inverse of involution. In involution we have given the factor and the number of times it is employed, and are required to find the product, or the power, of the factor. In evolution we have given the power, or product, and the number of times a factor must be employed to produce it, and are required to find the factor. The factor whose involution will produce a power or number is called the 7'oot of the number, and the number of times the factor is to be employed is called the mdex of the root. The operation of finding the required factor is called extracting the root of the number. The operation of evolution is indicated by the radical sign, V , with a bar extending over the expression whose root is to be extracted, unless that expression be a numeral or single literal factor. The index of the root is written in front of the radical at the top. Thus : iV, V^^. When the index is 2 it is ordinarily omitted. A parenthesis may be used in any case instead of the bar. 77. Inverse of Involution. — Involution is not com- mutative, that is, 2^ does not equal 5^. In subtraction, the inverse of addition, there are two questions that may be asked. For example, we may ask what number must be added to 5 to make 9, or to "what number must 5 be added 94: EVOLUTION. to make 9 ; but as addition is commutative, there is only one inverse operation. Each of the above questions is an- swered by subtraction. Also in division, the inverse of multiplication, two questions may be asked. For example, we may ask how many times is 4 contained in 20, or what number is con- tained 4 times in 20. This is equivalent to asking *'20 is how many times 4, or 20 is 4 times what number. ^^ But since multiplication is commutative, there is only one in- verse operation. Each of the above questions is answered by division. In evolution, the inverse of involution, two questions may likewise be asked. For example, we may ask what is the fifth root of 32, or what root of 32 is 2. As involution is not commutative, these questions cannot be answered by one and the same operation. The former is answered by evolution, and the latter by logarithms. The former is the only inverse operation that we shall consider here. 78. Corresponding Direct and Inverse Operations do not always Cancel each Other. — Corresponding inverse and direct operations usually cancel each other. Thus the addition and subtraction of the same number cancel each other, the multiplication and division by the same number cancel each other, also the extraction of a root and raising to the corresponding power cancel each other. Thus : It must, however, be borne in mind that roots are more than one- valued, and hence the statement with reference to the inverse operations of extracting roots and raising to powers need restriction. It is true, "necessarily and uni- versally, that [l^aY = a, but not that \/a'' = a. For instance, Va^ = "^a. Wliile the statement that the extrac- tion of a root is cancelled by raising the result to the cor- EVOLUTION. 95 responding power is true necessarily and universally, the inverse statement that the raising an expression to a power is cancelled by the extraction of the corresponding root of the result is not necessarily true. 79. Extraction of Roots of Monomials. — Since evolution is the inverse of involution, we extract the root of an ex- pression by doing just the opposite to what we do in finding a power. Thus, we find the power of a monomial by raising its numeral factor to the power indicated by the exponent, and multiply the exponent of each literal factor by the exponent of the power. e.g. (4a;V)3 := UxhK Hence we extract the root of a monomial by extracting the indicated root of the numeral factor and dividing the exponent of each letter by the index of the root. e.g. V^^x^z^ — 4A^ N.B. — Since (=^«)^ = 6^2, .'. Vd' = "^a. That is, the square root of a positive quantity is either -f- or — , and the square root of a negative quantity is im- possible, or imaginary. The same is true of any even root. The odd root of a positive quantity is -{-, and of a negative quantity — . EXERCISE XXXVII. I. Find the indicated roots of the following monomials : 1. Va^h^c'\ 3. Vlla^1)'c\ 4. f- Uda'^'^ 5, Vx''yH\ 6. V'-x'Y^ 96 EVOLUTION. 80. Extraction of the Square Root of Polynomials. — To obtain a rule for extracting the root of a polynomial, let us examine the square of a polynomial. e.g. (rt + J + c + ^0' = a^ + ^,2 ^ ^2 + 6?2 + ^ab + %ac-^'^ad-\^Uc^^hd-]-^cd = a' + 2ah + *^ + ^ac + 2bc + c^ + 2ad -\- 2M -\- 2cd -}- d'' = a^-\-(2a-\-b)b-]-{2a-{-2b-\-c)c+(2a-{-2b-\-2c-\-d)d = a^ -{- {2a -]- b)b + [2(a + b) -\- c]c-{-ma-\-b -}- c) -{- d]d. From the last of the above equations we may derive the following rule for writing at sight the square of any poly- nomial : Write the square of the first term, then the product of twice the first term pl^is the second multiplied by the second, then the product of twice the first tivo terms plus the third multiplied by the third, then the product of twice the first three terms plus the fourth multiplied by the fourth, etc. If now we take the second member of the second equa- tion and compare it with the second member of the last, we may readily obtain a rule for extracting the root of a polynomial. a^-{-2ab-^b''-{-2ac-Jf-2bc-\-c''+2ad-\-2bd-{-2cd-{-d'^ I a+b-]-c-^d 2a-{-b 2ab-{-b^ 2ab 4- 6' 2a 4- 2& + c 2ac + 3&C 4- c« 2ac 4- 26c 4- c^ 2a^2b-{-2c-\-d 2ad + 2bd + 2cd + d^ 2cd 4- 2bd 4- 2cd + d^ First arrange the terms of the p)olynomial according to the powers of some letter ; theii tahe the square root of the first term, place it in the root or quotient, square, subtract, and bring dotvn one or more terms; then double the root EVOLUTION. 97 already found and place the resiilt in the divisor, find how many times this is contained in the first term of the re- 7nainder, place the result in both the root atid in the divi- sor, multiply, subtract^ and briny dotvn; then double the root already found and proceed as before ; and so on to the end. EXERCISE XXXVIII. Extract the square roots of: I. 1. a^ + 4:a^ + 2a^ - 4a + 1. 2. x^ - 2x^y + 3xY - '^xy^ -\- t/, . 3. 4ft« - rZa^X -\- 5ff4^2 _|_ 6^3^3 _^ ^2.^4_ 4. 9x^ - 12.cy + IQxY - 24:xy + 4?/6 + lQxy\ 5. 4^8 + 166'8 + IQa'c^ - 32«V. 6. 4:X^-}-9 - SOx - 20ic3 + 37:^2. 7. 162;^ - Uabx^ + 16^»2a;2 + la^b'^ - 8ab^ + 4:b\ II. 8. x^ + 25x^ + 10^-4 - 4a;5 - 20a^ + 16 - 24:r. 9. a;^ + SxY — ^^y — ^xy^ + %xh/ — lOxY + y^- 10. 4 - 12a - lla^ + 5^2 _ 4«5 4- 4a« + 14a^. 11. 25a;« - 'dWy'^ + 34a;y _ 'dOx^y^y^-^xy^^lQxY- 12. ^c"^ — a;^?/ — 7/4a:y + a:^^ + ?/*. 13. x^ - 4:a^y + 6xY - Qxy^ + 5/ - ?^ + ^. 81. Squaring Numbers as Polynomials. — Every number composed of two or more digit's may be written as a poly- nomial. Thus: 25 = 20 + 5, 234 = 200 + 30 + 4, etc. 98 EVOLUTION. Hence (234)2 = (200 + 30 + 4)'^ = (200)2+(2.2004-30)x30+(2.230+4)4. 40000 + 12900 + 185G = 5475G. EXERCISE XXXIX. In a similar way find the squares of the following numbers : I. 1. 327. 2. 3789. 3. 845. II. 4. 5006. 5. 19683. 6. 5083. Observe that the square of a number contains either twice as many or one less than twice as many places as the number itself. Ex. .234 = .2 + .03 + .004. (.234)2 = (.2 + .03 + -004)2 = (.2)2+(2x.2+.03).03+(2x.23+.004).004 = .054756. In a similar way find the square of : I. 7. .0304. 8. .0028. Observe that when a number is a decimal, its square is a decimal and contains twice as many places as the num- ber. Ex. 23.4 = 20 + 3 + .4. (23.4)2=3(20+3+. 4)2=(20)2+(2x20+3)3+(2x23+.4).4 ^ 547.56. EVOLUTION. In a similar way find the squares of : I. 9. 69.4. 10. II. 43.21. 11. 37.89. 12. 8.008. Observe that when the number is composed of an integer and a decimal, its square is composed of an integer and a decimal, and that the number of places in the integral part of the square is either twice as great or one less than twice as great as that in the integral part of the number, and in the decimal part of the square twice as great as in the deci- mal part of the number. 82. Extracting the Square Root of Numbers. — Observe, in all the cases of the last section, that if we begin at the decimal, point and divide the square into periods of two places each, the square root of the largest square in the left- hand period will be the left-hand figure of the number squared, and the number of this left-hand period, counting from the decimal point, will be the order, or place, of the figure in the root, or in the number squared. Hence the first step in finding the root of a number is to divide the number into periods of two figures each, be- ginning at the decimal point. The periods thus obtained correspond to the terms of a polynomial whose square root is to be found, and the pro- cess of finding the square root of a number is precisely analogous to that of finding the square root of a poly- nomial. e.g. |/387420489. 100 EVOLUTION. 3 - 87 - 42 - 04 - 89il0000 + 9000 -f 600 + 80 + 3 1 00 00 00 oo' 20000 + 9000 29000 2 2 87 61 42 00 04 00 89 00 38000 + 600 38600 26 23 42 16 04 00 89 00 39200 + 80 39280 3 3 26 14 04 24 89 00 39360 + 3 39363 11 11 80 80 89 89 19683. It appears from the above example that, after the first step, the extraction of the square is a case of division, in which the divisor varies with each remainder, and in which the exact or complete divisor is unknown. It also appears that the incomplete or trial divisor in each case is double the part of the root already found. Evidently the work in the above example might be made more compact by omitting the ciphers, and writing the root at once in the usual form, instead of in the form of a polynomial. Thus : 3- 1 -87 - -42- -04- -89 19683 29 2 2 87 61 386 1 26 23 42 16 3928 ~ 3 26 14 04 24 39263 1 11 11 80 80 89 89 From the above considerations we may deduce the fol- lowing rule for extracting the square root of a number: EVOLUTION. Ml Divide the number into periods of two places each, be- ginning at the deci7nal point; find the largest perfect square in the left-hand period, subtract it from this period and place its root in the quotient, and bring down the next period; double the root already found for a trial divisor, and seek how many times this is contained in the remainder ■exclusive of the last figure, and place the result in both the divisor and the quotient; multiply, subtract, bring down, and proceed as before. As the trial divisor is smaller than the real divisor, we must guard against taking too large a figure for the quo- tient. Of course this figure can never exceed 9. Should the trial divisor not be contained in the remain- der after the last figure has been excluded, place a cipher in the divisor and quotient, and bring down the next period and try again, and so on till a significant figure is obtained. In the actual work, after the number has been separated into periods, the decimal points may be disregarded. It should be placed in the quotient, or root, when its position has been reached, but farther than this it may be entirely neglected. When the number is not an exact square, its root may be obtained to any required degree of approximation by bringing down two ciphers for each new period. Of course care must be taken to place the decimal point in the right position in the quotient. EXERCISE XL. Find the square roots of : I. 1. 14356521. • 2. 25060036. 3. 25836889. 4. 16803.9369, 1 W^ii ■':' EVOLUTION. II. 6. 4.54499761. 6. .9. 7. 6.21. 8. .00852. 83. Cubing of Polynomials. — {a + bf = «3 + 3«2^ + 'dah^ + h^ (« + Z» + 6f = «3 + ^,3 _^ c^ 4_ 3«2^ 4. 3^2^ _|_ 3^,2^ _^ 3^^2 + 3«c2 + 3^c2 + 6«Jc = «« + (3«2 + 3«^ + V")}} + [3(« + Z*)2 + 3(« + ^)c + o'^c. By means of the above formulas the cube of any poly- nomial may be written at sight. First, write the cube of the first term ; then the product of three times the square of the first term plus three times the product of the first and second terms plus the square of the second term multi- plied by the second; then the product of three times the square of the first two terms plus three times the product of the first two terms and the third plus the square of the third multiplied by the third; etc. EXERCISE XLI. Cube the following polynomials by the above method : I. . 1. a-\-\. 2. X -\- 2. 3. ax — y^. 4. 2m - 1. 5. 4a - U. e. 1 + a: + x\ n. 7. 1 - 2a; + ^x\ 8. a-\-U-c. 9. U^ - 3a + 1. IQ, l-x-\-x'^ - x\ EVOLUTION. 103 84. Extracting the Cube Root of Polynomials. — If we arrange the terms of {a-\- b -{- cY according to the descend- ing powers of a and the ascending powers of c, we have Comparing this with a' + (3«2 -f- Mb + b^)b + \^{a -\- bf -^ Z(a -\- b)c + c^c, we may readily extract the cube root of the first expression. Thus: a8_|_3^25^3^;,2_|.53_|_3«2c_^6«jc+3&2c4-3ac2-j-3&c2+c3|a+6+c 3a»+3a6+62 3«*6+3a*2+63 3«--»+6a6+362+3ac+36c+c- 3V and l/4aV + \/Za%^o? + 4/9&V. 12. 1/5 «V + 1/6^V and l/'lha^'x^ - l/ZWl^x^ + l/36^»Vo. Write at sight the missing factor of the following examples : 13. (3«V - l/3aa;)( ) = 27A» - X/Vta^^, 14. (1/46^32:5 + 1/6^V)( ) = 1/64A15 + l/216J«a^2i_ 95. To Convert x^ + Zia; into a Perfect Square. — The square of a binomial of the first degree of the form x -\- a, that is, of one having a constant term and unity as the co- efficient of its first-degree term, is a complete quadratic trinomial. The first-degree term of this trinomial is twice the product of the two terms of the binomial, and the con- stant term of the trinomial is the square of the constant term of the binomial, or the square of half the coefficient of the first-degree term of the trinomial. e.g. {x + 4)2 = a;2 + 8a; + 16. Here 16 = (|)l {x - 1/2)2 ^x^ -x-^ 1/4. Here 1/4 = (1/2)2. Hence a quadratic binomial of the form x^ + ^^? that is, one having a first- and a second-degree term in a letter and unity as the coefficient of its second-degree term, may be converted into a perfect square by adding as a constant term the square of half the coefficient of its first-degree term. e.g. The quadratic binomial :i^ — ^x becomes a perfect square on the addition of (3)2 to it as a constant term. When thus completed it becomes the trinomial x?'—^x-\-^. 116 MULTIPLICATION AT SIGHT. EXERCISE XLIX. Convert the following quadratic binomials into perfect squares : I. 1. a? + 8:^;. 2. m2 - 10m. 3. a^-^x. 4. n^ — hn. 6. x^ + lx. 6. II. f - %. 7. x^ - Z/^x, 8. z^ + h/^z. 9. x^ + Ix. 10. x^ - 6bx. 11. a?-{-x. 12. y'-y- 96. To Convert x^ + bx"" into a Perfect Square.— Bi- nomials of a similar form but of a higher degree may be converted into perfect squares in the same way. The form of the expression will be similar when the degree of one term in any letter is twice that of the other term in the same letter, and the coefficient of the term of the higher degree is unity. e.g. x^ — 8x^ becomes a perfect square on the addition of (4)2. It will then be x^ - Sx^ + 16. This is the square of x^ — 4. Of course in any of these cases an aggregate may take the place of a single literal factor.' EXERCISE L. Convert the following plete squares: binomial expressions into com- I. 1. x^ + 6x^ 2. m* - 12m\ 3. x' - 5x^. 4. a^ + 7a\ 6. x'-\-bx^ MULTIPLICATION AT SIGHT. 117 II. 6. z^ - z^. 7. ^^« - 2/3:?:^ 8. 71^ - 3/4^^3. 9. {x + 2)2 + Q{x + 2). 10. {x - 5)2 - '6{x - 5). 97. To Convert x? -\- bx -\- c into a Perfect Square. — Quadratic trinomials of the form x^ -{- hx -\- c may be con- verted, without change of value, into perfect squares plus or minus a term which may be either simple or complex, by the addition and subtraction of the square of half the coefficient of x. It is best to make the addition and sub- traction immediately after the second term, and then to combine the last two terms into one. e.g. cc2 + 4a; — 8 = a;2 -f- 4a: -|- 4 — 4 — 8 = a;2 _j_ 4^ _|_ 4 _ 12. The first three terms of the last polynomial are a perfect square. x^-^^x-\-10 = x^-\r 6x + 9-9 + 10 = a;^ + 6a: + 9 + 1. x^J^bx- 7=za:2+5a; + ?^-^-7 4 4 = ^' + 5a; + — - — . EXERCISE LI. Convert each of the following trinomials into a perfect square plus or minus a constant term, without change of value : I. 1. a;2 - 8a; - 2. 2. x^ - 12a; + 30. 3. x^ + 7a; ^ 3/4. 4. a;^ - 7a; + 3/5. 5. Divide l/32a;5 - 1024 by l/2a; - 4. 118 MULTIPLICATION AT SIGHT. 6. A workman was employed for 60 days, on condi- tion that he should receive 3 dollars for every day he worked, and forfeit 1 dollar for every day he was absent. At the end of the time he received 48 dollars. How many days did he work ? 7. A can do a piece of work in 10 days, and B can do it in eight days. After A has been at work on it for three days, B comes to help him. In how many days will they finish ? II. 8. / - % + 3. 9. z^ + Hz - 7. 10. x^ -\-'bx -\- c. 11. y^ — hy — c. 12. Divide 32/243a;5 + 3125 by 2/3:c + 5. 13. A privateer, running at the rate of 10 miles an hour, discovers a ship 18 miles off running at the rate of 8 miles an hour. How many miles can the ship run before she is overtaken ? 14. A cistern has two supply-pipes respectively capable of filling it in 4| and 6 hours. It also has a leak capable of emptying it in 5 hours. In how many hours would it be filled when both pipes are on ? 98. To Convert ax^ -j- hx into a Perfect Square. — Quadratic binomials of the form ax^ -\- hx may be converted into perfect squares by first dividing them by the coefficient of x^ and then adding the square of half of the resulting coefficient of x. e.g. Zx^ + 12a; becomes, on division by 3, x^ + 4a:, and then, on addition of the square of half of 4, x^ -\- ^x -{- 4, which is a perfect square. Similarly, ^x^ — 5x becomes x^ — 6/Sx, and then x"^ — 5/3x -f 25/36, which last is a perfect square. MULTIPLICATION AT SIGHT. 119 EXERCISE Lll. Convert the following quadratic binomials into perfect squares, and solve the given equations : I. 1. Qx^ + l%x. 2 Sa^ - 15a;. 3. 6x^ - 15a;. 4. 7a:2 _|_ 63^^ 6. d{x-^aY-5(x^a). 6. ^^ZT^^^^^' II. 7. ax^ + ix. 8. my^ — ?i«/. 9. 2x* + 3a;2. 10 ^sz^ _ 9;23 a. 7(.-5)^+3(.-5)^. X.. :-±i=:-^. EXERCISE Llil. Convert the following quadratic trinomials, without change of value, into expressions which shall be a perfect square plus or minus a constant term : I. 1. 2a;2 -f 32; + 6. 2. Sx^ - 18a; - 12. 3. 4a;2 -Qx + 7. 4. 5a;2 + 25a; - 20. 5. 6a;2 + 42a; + 50. 6. Find the square root of 2 to four places of deci- mals. 7. Find the cube root of 3 to three places of decimals. II. 8. 7a;2 - 63a; + 49. 9. Sa^ - 40a; - 12. 10. 9ar^ - 81a; + 63. n. lOa;^ _^ 70a; - 80. 12. lla;^ — 2a; + 3. 13. ax^ -\- bx -\- c. 14. mz^ — 7iz + p. CHAPTER XL PACTORING. 99. Resolution into Factors. — To factor an expression is to resolve it into its component factors. To be able to factor algebraic expressions readily and accurately is a mat- ter of very great importance. Other things being equal, the one most skilful at factoring is the best algebraist. 1°. To Resolve an Expression into a Monomial and a Polynomial Factor. — When every term of a polynomial contains a common factor, it may be resolved into a mono- mial and a polynomial factor. The factor common to all the terms will be the mono- mial factor, and the quotient obtained by dividing the ex- pression by this factor will be the polynomial factor. e.g. 6:^2 _^ i2x - 18 = Q{x^ + 2^; - 3). a^x — a^ = a^(x — 1). EXERCISE LIV. Eesolve each of the following expressions into a mono- mial and a polynomial factor : I. 1. Gab + 2ac, 2. 2a^x^ - %d^hx + Wh\ 3. ^l)^(?x + bbh^y - 6b^c\ 4. 7a - 7a^ + UaK 5. Qx^ + 2x^ + 4:X^. 120 FAGTOniNQ. II. T. bu^- - lOA^ - 9. ^^- -x^^x. 121 6. 15a2 - 225^^ T. ^x^ - lOA^ - 15aV. 8. 38«V + 57aV. 10. '^x^y^ — 'Sx^y^ -\- 2xy^. 2°. To Facto?' the Difference of Two Squares. — The difference of two squares is equal to the product of the sum and difference of their roots. EXERCISE LV. Factor each of the following expressions : I. 1. x^ — a^. 2. x^ — 9. 3. 4^2 - 64. 4. 9 A2 _ 25^2, 5. 81 - 16A*. 6. 49fi^V - lQa^z\ 7. {x^ + Ux + 36) - 49. 8. y^-8y + 16 - 81. 9^ («2 _ 4rt + 4) _ 16. 10. (^2 ^ 24.b + 144) - 121. 11. The head of a fish is 9 inches long, the tail is as long as the head and half the body, and the body is as long as the head and tail together. What is the length of the fish? Note. — In solving problems concerning numbers composed of digits, the student must bear in mind that a number composed of two digits is equal to 10 times the left-hand digit plus the right-hand digit; that a number composed of three digits is equal to 100 times the left-hand digit plus 10 times the middle digit plus the right-hand digit. Thus, 46 = 10 X 4 + 6, and 387 = 100 X 3 + 10 X 8 + 7. 12. A number is composed of two digits, and the left digit is 4/3 of the right. If 18 be subtracted from the number, its digits will be reversed. What is the number ? II. Factor : 13. 12 - 3a2. 14. 48«3 _ lOSabK 122 FAGTORINO. 15. 27«5 - 75«.'r^ 16. 125«V _ 45:ry. Convert the following trinomials into the difference of two squares and then factor: 17. 0? + 14a: + 40. 18. x^ - l^x - 17. 19. x^ - lOx - 11. 20. x^ + 30x + 29. 21. A and B together can do a piece of work in 12 hours, A and together can do it in 16 hours, and A alone can do it in 20 hours. In what time can they all do it together, and in what time could B and C together do it? 22. A number is composed of two digits whose sum is 13, and if 9 be added to the number its digits will be reversed. What is the number ? 3°. Special Cases of Factoring Quadratic Trinomials. — We have seen that the product of two binomials of the first degree in any letter is, in general, a quadratic trino- mial in the same letter, and that the coefficient of the sec- ond-degree term of the letter is the product of the coeffi- cient of the first-degree terms of the letter in the binomials, the coefficient of the first-degree term of the letter in the product is the sum of the products of the coefficient of the first-degree term of the letter in each binomial multiplied by the constant term of the other binomial, and the con- stant term of the product is the product of the constant terms of the binomials. Hence a quadratic trinomial in any letter may be re- solved into two binomial factors of the first degree in that letter whenever we can discover four numbers such that the product of the first two will be the coefficient of the second- degree term of the trinomial, the product of the last two will be the constant term of the trinomial, and the alge- braic sum of the cross-products of the numbers will be the FACTORING. 123 coefficient of the first-degree term of the trinomials. The first two numbers will then be the coefficients of the first- degree terms of the factors, and the last two numbers will be the constant terms of the factors. It is best to arrange diagrammatically the four numbers selected for trial, as in the corresponding case of sight mul- tiplication. e.g. Resolve Qx^ + '^•^ ~ ^0 into binomial factors. 3x2 = 6, the coefficient of x^; 2 X -4^-8; ^' 3 X 5 = 15; 15 + (— 8) = 7, the coefficient of x; 5 X the constant (_ 4) = - 20, term. Hence ^x'^ + 7:?; - 20 = (2^ + 5)(3a; - 4). Notice that the complete test involves two trials, if first be unsuccessful: e.g. 3 above '^ ' and 2 below as well as 2 above and 3 below. Again, resolve 3a;^ — l^x — 63 into binomial factors. The required factors are {x — 7) and {3x + 9). Resolve x'^ — 2x — 63 into bino- mial factors. The factors are {x -\- 7) and (X - 9). The case in which the coefficient of the second-degree term of the tri- nomial is unity is of frequent occur- rence and of great importance. 124 FAGTOBING. EXERCISE LVI, Eesolve the following quadratic trinomials into binomial tors.- I. 1. X' -^ l^X -\- ^b. 2. x^ - 12a; + 27. 3. x^ -^x- 32. 4. x^ + lx- 30. 5. X^ — X — 42. 6. x^ + x- 20. 7. %x^ - lOx - 48. 8. 3.^2 _|_ 26a; + 55. 9. Qx" - 17^ + 7. 10. 202;^ + 37^ + 8. 11. 'dbx^ + 39^ - 36. 12. 56a;2 - lOOx - 100. 13. A, B, and together can do a piece of work in 5 days, A and B together can do it in 8 days, and B and together in 7 days. In what time can each do it alone ? II. Factor the following expressions : 14. 12 + 10^ - %xK 15. 48 - 128^ + Mx\ 16. 35 4- 41a; + 122;2. 17. 6a;2 + (21 - ^a)x - 7«. 18. ahx^ -\- {"Ha — 5b)x — 35. 19. acx^ + (^c — ad)x — M. 20. x^^^bx-a^-^y"' 21. {d^ - y^)x^ - 2{a + db)x - 8. 22. 3a;2 + 9a; - 54. 23. 7x^ - 7x - 210. 24. 102;2 + 50a; - 140. 26. 75flV - 5 A - 30^2. 26. Find a number composed of two digits whose sum is twelve and which will have its digits reversed by adding 63 to the number and dividing the sum by 4. 100. Functions. — In mathematics, one quantity is said to be a function of another when its value depends upon the value of the other and changes with it. FACTORING. 125 e.g. The value of the expression x^ -{- ^x — 6 depends upon the value of x and changes with the value of x. Hence the expression a:^ -f- 6^ — 6 is a function of x. The symbol f{x) means' any algebraic expression con- taining X. This is a very convenient notation when we wish to indicate any expression containing x without des- ignating any particular expression. f{a) indicates the algebraic expression obtained by substituting a for x in f{x). Thus \if{x) = x'^-\-'dx^Q, then f{a) = 6?2 + 3a + 6. EXERCISE LVII. I. 1. lif(x) =x^-{- dx^ - 10, find/(3). 2. ltf{x) ^x^ + 3x^ - 10, find/(- 3). 3. If /(^) = x^ — 5x -\- Q and y = 3 — x, ^ndf{y) in terms of x. 4. ltf(x) =x^-irx-Jrl, find /(a; - 1). 6. Itf{x) = x^-]-2x- 7, find/(5). 6. If /(ft) = (ft + J + c)3 - «3 _ J3 _ ^3^ find/(-^»). 7. If /(^) = x'- y\ find f\y). 8. \if{x) = x'-y\^^^f{y). II. 9. \if{x)=x--y\^v.^f{y), 10. If/(a:) = a;5 + ^^find/(-^). 11. \lf{x) = x^^y\^T,^f{y). 12. ^/(2;) = a:^ + y^find/(-2/). 13. If/(a:)=:a;^-fy*, find /(«/). 126 FACTORING. 14. lif(x) = x"" + y'' and n is odd, find/(- y). 15. If /(^) = a;" + 2/" ^^^ ^ is even, find/(— y). 16. If/(a;) = a:~ + ?/%find/(2/). 101. Remainder Theorem. — When f{x) is divided by X — a, the process of division being continued till the re- mainder, if there be one, does not contain x, the remainder will =/(«). Proof. — Denote the remainder, which is supposed not to contain x, by R and the quotient by Q. Then we have X — a X — a or f{x) ^ Q(x - a)-\- R. If now we substitute a for x in each member, R must remain unaltered since it does not contain x, and x — a will become a — a = 0. Hence f{a) = R. e.g. Letf(x) = x^ -\- 2x^ — 5x — 6, and let a = 4. Then /(«) = 43 + 2 X 42 - 5 X 4 - 6 = 64 + 32 - 20 - 6=70. By division, x^-\-2x^- 6x- 6\x -4: a^-\-6x-\- 19 6x^ - 5x ex^ - Ux 19.T - 6 l^x - 76 70 Again, let f{x) = x^ -{- 32, and let a Then f{a) = 32 + 32 = 64. FACTORING. 127 By division, x> + 32 - ^x" \x -2 X' x^ + 2x f 32 - 4:c3 ^ + 4a; 32 8:rM Sx'- 2 + 8a; + 16 2x^- 2x'- 4^3 + -32 - 16a: 16a; + 32 16a; - 32 Again, let f{^) = a;5- -32, and let a - = 2. Then Ao) = 32 -32 = 0. By division, x^- 32 \x- 2 x" - 2x^ nA 1 Oo,3 1 1 /1/V.2* 1 Q/>. 1 1 2a;* - 32 2a;*- 4a;'^ 4a;3 - 32 4a;3- Sa;^ 8a;2 - 32 8a;2 - 16a; 16a; - 32 16a; - 32 The theorem proved and illustrated above is a fun- damental theorem in factoring. By it we can readily de- termine whether x — a is a factor of /(a;). We have merely to substitute a for x in the given expression, and see whether it reduces to zero or not. In the former case the 128 FACTORING. expression is divisible hy x — a without remainder, and therefore :r — « is a factor of it. In the latter case the expression is not divisible hy x — a without remainder, and therefore x — ai^ not a factor of it. EXERCISE LVIII. Find in each of the following examples whether or not the given binomial is a factor of the given expression : 1. x-b of x^ - Ix^ + 7a; + 15. 2. X -\- 1 oi "^x^ -\- X — 1. 3. X — 1 of x^ -\- x^ — 2. 4. a; - 3 of 'Zx^ + 10a;2 - ^x - 40. 6. X — h oi x^ — h^. 6. X -{-!) of x'' -\- V, 7. X -{-h of x^ — W. 8. X — h of x^ + y^, 9. X — h olx^ — b^. 10. X -\- b of x^ — b^. 11. x — b of a;^ + b^. 12. is + 2» of x^ + i^ .13. OJ — Z> of a;" + Z*" when w is odd. 14. X — b of a;" + ^" when w is even. 16. X -{- b of x'' -{- ^" when ^ is odd. 16. X -\- b of x^ + ^" -when ?^ is even. 17. X — b of X" — ^" when ^ is odd. 18. X — b of x^ — J" when ^i is even 19. X -\- b of ic" — &" when 7i is odd. 20. X -\- b of a:" — ^" when ^ is even. 21. Divide x* — b~ hy x — b. 22. Divide x^ — b^ hy x -\- b. 23. Divide x^ -^ b^ hy x — b. 24. Divide a;^ + b^ by x -\- b. FAC TORINO. 129 26. Divide x^ -\- h^ ^-^ x — h. 26. Divide x^ -\- IP hj x -\- h. 27. Divide x'^ — h^hj x — h. 28. Divide x^ — Whj x-\- h. 102. Factors of the Sum and Difference of the Same Powers of Two Quantities. — From examples 13-20 it ap- pears : 1°. That the sum of the same odd powers of two quan- tities is divisible by the sum of their roots, but not by the difference of their roots. 2°. That the sum of the same even powers of two quan- tities is divisible by neither the sum nor the difference of their roots. 3°. That the difference of the same odd powers of two quantities is divisible by the difference of their roots, but not by the sum of their roots. 4°. That the difference of the same even powers of two quantities is divisible by both the sum and difference of their roots. From examples 21, 26, 27, 28, it appears: 1°. That when the difference of the same powers is di- vided by the difference of the roots, the terms of the quo- tient are all positive ; and that when the sum or difference of the same powers is divided by the sum of the roots, the terms of the quotient are alternately positive and negative. 2°. That in any case the first term of the quotient is the letter of the first term of the dividend with its exponent diminished by one, and that the exponent of this letter de- creases by one in each of the succeeding terms of the quo- tient; and that the letter of the second term of the dividend occurs in the second term of the quotient with unity for its exponent, and that the exponent of this letter increases 130 FACTORING. by one in each subsequent term till it becomes one less than its exponent in the dividend. These two laws enable us in these cases of division to write the quotient at sight. EXERCISE LIX. Write at sight the quotient in each of the following cases : 1. {x^ - if) ^ (x- y). 2. (x^ - f) -^{x- y). 3 (^6 _ ^6) ^(^x^ y). 4. {x' + f) ^{x + y). 5. {x^ - 27) ^{x- 3). 6. {x^ - 81) -^{x- 3). 7. (.T^ - 16) ^ (;?: + 2). 8. {x? + 32) ^ (x + 2). Find the remainder when — I. 9. {x — %af + (%x — (if is divided by re — «. 10. {x -\- a -\- Hf ^ x^ is divided by 2; + a. 11. {x + 2«)2" + (2.^ + of" - 2«'^" is divided by a: + a. 12. (« + ^ + ^Y — ^'^ — Ifi ~ & is divided by a + h. II. 13. {a ^h -\- cf — ci} — h~ — c' is divided hy a -\-d. 14. {a-^b^cY - {b-[- cy- (c + aY - (« + hy + ^^ + ** + c* is divided by a + ^. 15. «"(^ — c)+ J"(^ — «)+ ^"(^ — ^) is divided by h—c. Show that the given binomial is a factor of each of the following expressions, and find the other two factors : I. 16. 3a:3 + a;2 - 22:2:- 24; a:- 3. 17. x^ + 2.^2 - 13a: + 10; x - 2. 18. x^-]-%x'^ - 11a; - 12; ;r + 1. FACTORING. 131 II. 19. 3a;' - %0x^ + 36:c - 16; x- L 20. ^x^ + 13a;2 - 32.i- + 15 ; x + 5. EXERCISE LX. I. 1. A cistern can be filled by one pipe in five hours and by another in eight hours, and it can be emptied by a third pipe in four hours. Were the cistern empty and all three pipes opened together, in what time would it be filled ? 2. Suppose the cistern in the last example could be emptied by the third pipe in three hours. Were the cistern full and all three pipes opened together, in what time would it be emptied ? 3. A man does 3/5 of a piece of work in 30 days and then calls in another man and they together finish it in 6 days. In what time can they do it separately ? II. 4. A marketwoman bought a number of eggs at the rate of two for a penny, and as many more at the rate of three for a penny, and sold the whole at the rate of four for 3 cents, and found she had made 24 cents. How many of each kind did she buy ? 5. A person hired a laborer on condition that he was to receive 2 dollars for every day he worked and forfeit 75 cents for every day he was absent. He worked three times as many days as he was absent, and received $47.25. How many days did he work ? 6. A sum of money was divided between A and B, so that the share of A was to that of B as 5 to 4. The share of A exceeded 5/11 of the whole by 300 dollars. What was each man's share ? CHAPTER XII. HIGHEST COMMON FACTORS. 103. Highest Common Factor. — A common factor of two or more expressions is a factor which is contained in each of them, and the highest common factor of the expres- sions is the product of all their common factors. Thus, Stt^Z'V and Qa^Vc have 2, a^, W, and c as common factors, and 2«^J^c as their highest common factor. The abbreviation H. C. F. stands for highest common factor. The highest common factor is sometimes called the greatest common measure, and denoted by G. C. M. 104. The H. C. F. of monomials may be found by in- spection. It is necessary merely to factor the expression, select the common factors and find their product, using each of these factors the least number of times that it occurs in any of the expressions. e.g. Find the H.C.F. of \WIHH, 9«^^>V, and Vla^WdK Factoring, we have Z.'^.^.a.a.h.h.'b.c.c.c.c.d^ S.d.a.a.a.b.b.c.c.c.c.c, and Z . % . % . a . a . a . a . h . h . h . h . d . d . d . d. The factors common to all the expressions are, 3, a, and J). The least number of times that 3 occurs in any of the expressions is oncej that a occurs in any of the expres- 189 niQHEST COMMON FACTORS. 133 sions is twice; and that b occurs in any of the expressions is twice. Now 3 . a . a . b . b — 3a^^, and this is the highest common factor of the expressions. Of course we might have seen at once that the highest common factor of the coefficients is 3, that the common letters are a and b, and that the lowest dimension of these letters in any of the expressions is 2. Hence the H. C. F. would be da^b"^. EXERCISE LXI. Find the H. 0. F. of the following expressions: I. 1. 5x^1/, Ibx^ii^z. 2. 7x^yh, 2Sx^yh^. 3. lSab^c% 36a^cd\ 4. 2xY, 'dxY, 4xY:f. 6. ITa^^V, 51a''b^c\ QSa^bh\ II. 7. Multiply ^x"^ + 6a:" - 5xPy^ by 3cc" - 4:X^ + Qxy^. 8. Divide 6a;"* + ^ -f 9a;'" + ^ + 12a;" + ^ -\- 18a;" + ^ — 8x^ - 12x^ by 2a;2 + 3a;. 105. To Find Highest Common Polynomial Factor by Inspection. — In a similar way we may find the H. C. F. of two or more polynomial expressions by inspection when we are able to resolve them into polynomial factors. We have simply to resolve the expressions into their polynomial fac- tors, select the factors common to all the expressions, and combine them into a product, using each factor the least number of times that it occurs in any of the expressions. e.g. Find the H. C. F. of x^ -{- x - 6, x^ -\- Qx + 9, and a;^ — a; — 12. Factoring, we obtain {x 4- 3)(a; - 2), {x + 3)(a; + 3), and {x + 3)(a; - 4). 134 HIOHEST COMMON FACTOHS. The only common factor is x -\- 3, and the least number of times that this occurs in any of these expressions is once. Hence the H. C. F. of these three expressions is a; -f- 3. When any of the polynomial expressions contains a monomial factor, this factor should be removed before searching for polynomial factors ; and if this factor is com- mon to all the expressions, or contains a factor common to them, the common factor should be set aside to be made a factor of the H. C. F. e.g. Find the H. C. F. of 3rtV + 3A - 60^2, Qa^x^ - d6a\ and Ua^x^ - lOSa^x -f 240«^J. Removing the monomial factors, we have 3a%x^ + x- 20), 6a%x^ - 16), and na^{x^ - 9x -\- 20). Sa^ is the H. C. F. of the monomial factors thus re- moved. Factoring now the three polynomial expressions, we have (x - 4:)(x + 5), (x - 4:){x + 4), and {x - 4.){x - 5), the highest common factor of which is 2: — 4. Therefore the H. C. F. of the three given expressions is 3«2(a: - 4) = 3alT - na\ EXERCISE LXil. Find the H. 0. F. of the following expressions: I. 1. x^-l,x^-\-3x-{-2. 2. x^-]-5x-\- 6, x^-{-7x-{- 12. 3. x^ -9x- 10, x^-\-2x- 120. 4. x^-^Hx- 18, x^ - 8. 6. x^ -\- {a -\- I?)x + CL^, x^ -\- {a — b)x — ab. 6. x^ — Ixy -\- 6?/^, x^ — xy^. 7. x^ — X, 2a:''^ — 4:X -\- ^, x? -\- x^ — 2x. HIGHEST COMMON FACTORS. 135 II. 8, or^ + y^, {x + yY-, x^ + 2:r^y + '^xy'^ + 2/"- 9. 120^ + ly, 6(.7-^ - 1)3, 18(:r + 1)4. 10. 2^ - f, 3(.T-^ - /), 7(./-« - /). 11. x^ — 3A — 2rt^ x^ — 'dax^ + 4«^ x^ — ax — 2a^. 12. 2:^ + a:?/ — 2;^^, x^ — 3.^1?/^ + 2?/^, x^ -\- 3x^y — iy'^. 106. The method of finding the highest common factor of two or more expressions which cannot readily be resolved into factors is based on the three following theorems: 1°. If tic exjn'essions have a co?nmon factor, any mul- tiples of these expressions will contain this factor. Let A and B represent any two expressions which have a common factor, and let this factor be represented by /; let p denote the quotient resulting from dividing A by /, and q the quotient obtained by dividing B by/. Then A — pf and B — qf. Let m and n be any integral expres- sions whatever. Then mA will represent any multiple whatever of A, and nB any multiple of B. But 7nA — mpf and uB — nqf. Hence / is a factor of both mA and nB. 2°. If two expressions have a co7nmon factor, the sum and difference of the expressio7is or of any multiples of the expressions will contain this factor. Use the letters as in 1°. Then A — B = pf — qf = (p — q)f which contains the factor/. Also A -\- B = pf -\- qf = {p -{- q)f, which contains the factor/ Again, niA — ?iB — mpf — nqf — (mp — nq)f, which contains the factor / Also mA -{- nB = mpf + nqf = (mp -\- nq)f, which contains the factor/. 136 HIGHEST COMMON FACT0B8. 3°. If two expressions have a cornmon factor, and one of them be divided by the other and there be a remainder, this remaitider will contain the common factor. Let A and B represent the two expressions which have a common factor, Q the quotient obtained by dividing B by A, and E the remainder. Then B= QA-^ E. By hypothesis B and A have a common factor /, and by 1°, QA contains /as a factor. But since B is divisible by /, and one term of its equivalent expression (QA -j- E) is divisible by f the other must be also. Hence the remainder E must contain / as a factor. OoK. — If now we divide A hy E and denote the remain- der by S, then the common factor of E and S will be the same as that of A and E and, therefore, of A and B. If this process be continued to any extent, the common factor of any divisor and the corresponding dividend will be a common factor of the original expressions. In other words, the remainder ivill always contain the common fac- tors of the original expressions. If at any stage there is no remainder, the divisor must be a factor of the corresponding dividend, and therefore, since it is evidently the highest-factor of itself, it must be the H. C. F. of the original expressions. By the nature of division the remainders are necessarily of lower and lower dimensions, and hence, unless at some stage the division leaves no remainder, we must ultimately reach a remainder which does not contain the common let- ter. In this case the given expressions have no H. C. F. As the process we are considering is to be used only to find the highest common polynomial factor, it is evident that any dividend or divisor which may occur in the pro- cess may be multiplied or divided by any monomial factor without destroying the validity of the operation; for such HIGHEST COMMON FAG TOM. 13? multiplication or division will not affect the polynomial factors. Ex. 1. Find the H. C. F. of ^3 _|_ ^2 _ 2 and x^ + 2a:2 - 3. Q^ + 2a;2 - 3 \x^ + x^-% a;3 + a;2 - 2 ' a;3 _|_ ^2 _ 2 x^ -1 1 ^^ - ^ ^ + 1 x^ + x- 2 ^ - 1 x^-1 x-1 ^'-^ ^4-1 x-1 x-1 The H. C. F. is a; - The work might be shortened by noticing that the fac- tors of the first remainder, x^ — 1, are x — 1 and x -\- 1, and that of these^ only a; — 1 is a factor of x^ -\- x^ — 2. Ex. 2. Find the H. C. F. of x^ + 4.x^y - Sxy^ + 'Mif and 4:X^ - 4cxhj + d^xY-^'^xY- The second expression is divisible by 4:X^, which is evi- dently not a common factor. We have therefore to find the H. C. F. of x^ — x^y -\- Sxy^ — 8?/* and the first expression. X* - x^y + '^xy^ - %y^ | a;^ -[- 4:X^y - ^xy'^ -\- a;4 j^ 4^3y _ 8^2^2 ^ 24a;^^ V^ITy - hx^y + ^xY - ^^xy^ - Sy^ - bx^y - %OxY + 40:r«/3 _ noif ^SxY - 562;«/3-f 112^4 138 HIGHEST COMMON FACTORS. Rejecting the factor 28^/^, we have rj. j^ 4^^y _ 8:?:/ + 24?/3 | a.^ - 2xy + 4:2/ ^x^y - 12.t/ _^ 24?/3 6:?;^^?/ — 1 ')lxy'^ + 24?/ Hence the H. C. F. \% x^ — "^xy -\- 4y^. Ex. 3. Find the H. C. F. of To avoid fractional coefficients, the second expression may be multiplied by 2 and then divided by the first. Za^+lbx^-^r bx^^lOx-^'^ 12^:4 + 9.^3+14:^ + 3 2 V, ■ Qx^ + 30a:3 _^ iQ^.2 _^ 20a: + 4 62;4 + 27a:3 + 42a; + 9 %x!^ + 9a:3 + 14a: + 3 | 3^:^ + lOa;^ _ 22^; - 5 3 2a: +7 6.T*+27a:3+42a: + 9 6a:4+20a:3-44a;2-10a; 7a:3+44a:2+ 52a:- +9 3 21a:=^ + 132a:2 + 156a: + 27 • 21a:3 + 70a;2 _ 154^ _ 35 62a:2 + 310a: + 62 | 62 3a:3 + 10a:2 - 22a; - 5 | x^ -\- 5a: + 1 3a:3 + 15a:^+ 3a: 3^ _^ — 5a:^ — 25a: — 5 - 5a:2 _ 25a: - 5 The H. C. F. is a:^ + 5a: + 1. From the above theorems and examples we may derive HIGHEST COMMON FACTORS. 139 the following rule for finding the H. C. F. of two expres- sions : Arrange the two expressions according to the descending potvers of some common letter and, if the expressions are oj the same degree i7i that letter, divide either hy the other, hit if they are of different degrees in that letter, divide the one ivhich is of the higher degree by the other. Take the remainder after division, if any, for a new divisor, and the former divisor as dividend; and continue the process till there is no re7nai7ider. The last divisor ivill be the H. C. F. required. If the two expressions contain common monomial fac- tors, their H. C. F. must be obtained by inspection, and this must be multiplied by the last divisor found by the above rule. Any divisor, dividend, or remainder which occurs may be multiplied or divided by any monomial factor. 107. To find the H. C. F. of three or more polynomial expressions, we first find the H. C. F. of any two of them, and then of this and a third, and so on. Let the expressions be J, B, C, D, etc. First find the H. C. F. of A and B, and denote it by^. Then since the required H. C. F. is a common factor of A and B, it must be a factor of E, which contains every com- mon factor of A and B, and so on. 108. Note. — The highest common factor of algebraic ex- pressions is not necessarily their greatest common measure. For if one expression is of higher dimensions than another in a particular letter, it does not follow that it is numeric- ally greater. In fact, if a be a positive fraction, c? is less than a. 140 HWHEBT COMMON FACT0M8. EXERCISE LXIII. Find the H. C. F. of— I. 1. x^-{-2x-{-l and x^ + 2x^ + 2^; + 1. 2. x^ - 8x^ + 7:r + 24 and x^ - 6x^ + 8a; - 6, 3. x^ - 5x^ + dx + 6 and x^ - dx^ -\- ix - 4=, 4. 2x^ — 7x-2 and 6x^ - 3x^ - ISa:^ 5. 4:X^ + Sx^ - 6Qx^ - 12a;3 and 6^:3 - 6x^ - S6x. 6. 12«V + 120aV - 132a^x and SaV - 27aV + 7. 7:^;* - lOa.'^s + 3aV - 4^^^ + 4«4 and 82;'* - 13aa^ 8. 25a;* -i-5x^-x-l and 20a;4 + x^ - 1. 9. 1 - 4a;3 + 3a;4 and 1 + a; - a;^ - 6x^ + 4a;*. II. Work the last nine and also the following examples by synthetic division: 10. 11a;* + 24a;3 + 125 and x^ + 24a; + 55. 11. 2a;5 - lla;2 - 9 and 4:X^ + 11a;* + 81. 12. a;5 + lla;3 - 54 and ^ + 11a; + 12. 13. x^ — 2x^ — x-\- 2, x^ — x^ — 4:X-\- 4, and a^ — 7x-\- 6. 14. a;* - Qx^ + 8a; - 3, a;* - 2a:3 _ 7^2 _j_ 20a; - 12, and a;* — 4a;2 + 12a; — 9. 15. Multiply 3a;"* - 4a;'" " ^ + 5a;" + ^ by 6x^ + 7a;"* + K 16. Multiply a;" - Sx^ + 5a;3 by 4a;* - 6:^-^ EXERCISE LXIV. I. Ex. At what time after 5 o'clock will the minute-hand of the clock be ten minutes ahead of the hour-hand ? HIGHEST COMMON FACTORS. 141 In examples about the position of the hands of»a clock, it is best to draw a circle to represent the clock-dial, and to mark on it the positions of the hands at the beginning of the hour specified. Then note the number of minute-spaces between the hands at this time, and let x denote the num- ber of minute-spaces that the minute-hand must pass over before it comes into the required position. Then, since the minute-hand goes 12 times around the dial while the hour- hand is going once around it, x/1^ will denote the number of minute-spaces passed over by the hour-hand in the same time. Then x will equal the number of minute-spaces between the hands at the beginning of the hour plus x/1% minus the number of spaces the hands are required to be apart when the minute-hand is required to be behind the hour- hand ; and x will equal the number of minute-spaces be- tween the hands at the beginning of the hour plus x/1% plus the number of spaces the hands are required to be apart when the minute hand is required to be ahead of the hour-hand. Thus, in the example, the minute-hand will be at XII at the beginning of the hour specified, and the hour-hand at V, and there would be 25 minute-spaces between them. While the former is moving over i\\Q x spaces to its required posi- tion of 10 minute-spaces ahead of the hour-hand, the hour-hand will move over x/1% spaces. Therefore a; = 25 + ^/12 4- 10; 11/12^; = 35, X = 38^. That is, the minute-hand would be in the required po- sition at dSf^ minutes past five. XII 142 HIGHEST COMMON FACTORS. Had tiie question been, at what time after 5 o'clock will the minute-hand of the clock be ten minutes behind the hour-hand, we would have had x = 25-{- x/12 - 10; .-. ll/12a: = 15, X = Uj\. 1. At what time after 3 o'clock is the minute-hand of the clock 18 minutes ahead of the hour-hand ? 2. At what time after 7 o'clock is the hour-hand 20 minutes behind the minute-hand ? 3. At what time after 9 o'clock is the hour-hand 15 minutes behind the minute-hand ? 4. At what time nearest to 2 o'clock is the minute- hand 15 minutes behind the hour-hand ? 5. At what time between 4 and 5 o'clock are the hour and minute hands at right angles ? 6. The sum of the two digits of a number is 8, and if 36 be added to the number the digits will be interchanged. What is the number ? 7. If the first of the two digits of a number be doubled it will be 3 more than the second, and the number itself is 6 less than five times the sum of its digits. What is the number ? 8. A courier who goes at the rate of 40 miles in eight hours is followed after 10 hours by a second courier who goes at the rate of 72 miles in 9 hours. In how many hours will the second overtake the first ? II. 9. A courier who goes at the rate of 31^ miles in five hours is followed, after eight hours, by a second courier HIGHEST COMMON FACTORS. 143 who goes at the rate of 22^ miles in three hours. In how- many hours will the second overtake the first ? 10. Ten years hence a boy will be four times as old as he was ten years ago. How old is the boy ? 11. One man is 60 years old, and another man is 2/3 as old. How long since the first man was five times as old as the second ? 12. A father is four times as old as his son, and four years ago the father was six times as old as his son. What is the age of each ? CHAPTER XIII. LOWEST COMMON MULTIPLE. 109. Lowest Common Multiple. — A common multiple of two or more expressions is an expression which is exactly divisible by each of them. The loivest common multiple of two or more expressions is the expression of the lowest dimensions which is exactly divisible by each of them. The lowest common multiple is usually denoted by the letters L. C. M. 110. To Find L. CM. by Inspection. — The lowest common multiple of two or more expressions must evidently contain every factor of each, and each of these factors the greatest number of times that it occurs in any one of them, otherwise it would not be divisible by each expression. e.g. Let ^a^V^c, Qa%^G^d, and ^h^c^e be the numbers whose L. C. M. is required. To be divisible by each of these expressions the required expression must contain the factors 2, 3, a, h, c, d, and e, and it must also contain the first of these once, the second twice, the third four times, the fourth four times, the fifth three times, the sixth once, and the seventh once. The L. C. M. is iMh^c^de, Hence we have the following rule for finding the lowest common multiple of two or more expressions which may be factored by inspection : Find all the different factors of each expression, a7id take ■ each of these factors the greatest number of times which it occurs in any of the expressions, or to the highest degree that it has in any of the expressions, and find the product of these factors, X44 LOWEST COMMON MULTIPLE. 145 EXERCISE LXV. Find the L. 0. M. of the following expressions: I. 1. lSa%% U%H\ and taHK • 2. 3a;^?/2^ bxyh^y 16x^yh, and 20a!^yh\ 3. x^ — y^, xy — y"^, and xy -f- y"^. 4. :z;^ — 2a; — 15, x^ — 9, and x^ — ^x -{■ 15. 6. 5a; H- 35, x^ - 49, and x^ + 14^; + 49. II. 6. x^ -X- 20, a;2 + 3a; - 40, and j? -\- V2x 4- 32. 7. 2x^ -X- 1, 2a;2 -|- 3a; + 1, a;^ - 1, 4a;^ - 5a;2+ 1. 8. 12a; - 36, x^ - 9, x^ - 6x -\- 6. 9. x^ — dx + 2, a;'-^ — 5a; -|- 6, and a;^ — 4a; + 3. 10. x^ — Qax -f 9a^, 7? — ax — 6a^, and 3a;^ — 12^^^. 111. To Find L. C. M. by Division. — Since the highest common factor of two expressions contains every factor common to the expressions, if two expressions be each di- vided by their highest common factor, the quotients ob- tained will contain no common factors. Hence the L.C.M. of the two expressions will be the product of these quotients and their H. C. F. e.g. Find the L. C. M. of a;3 4- a;2 - 2 and x^ + 2a;2 - 3. The H. C. F. of these two expressions is a; — 1. (a;3 + a;2 - 2) -^ (a; - 1) = a;^ + 2a; + 2, and (a;3 + 1x^ - 3) -=- i^x - l) r= x^ + 3a; + 3. a;3 + a;2 - 2 = (a; - \){p? + 2a; + 2), and a;3 + 2a;2 - 3 = (a; - l)(a;2 + 3a; + 3). 146 LOWEST COMMON MULTIPLE. Since x^ -\- 2x -\- 2 and x^ -\- dx -\- 3 have no common factor, {x - l){x^ + 2a; + 2)(a^ + 3a; + 3) must be the L. 0. M. otx^ + x^-2 and a^ + 2x^ - 3. In general, let A and B stand for any two expressions, and let h stand for their H. 0. F. and I stand for their L. 0. M., and let P and Q be the quotients when A and B respectively are divided by ^; so that A = P.h and B= Q.h. Since h is the H. C. F. of ^ and B, P and Q can have no common factors. Hence the L. 0. M. of ^ and B must hQ P X Qxli, or I = PQh; or ^, on , B h h Hence the L. C. M. of two expressions may he found hy dividmg either one of the expressions by their H. C. F., and multiplying the quotient by the other expression. Also, since T_ AX B I X h = A X B. That is, the product of any two expressions is equal to the product of their H. C. F, and L. C. M. EXERCISE LXVI. Find the L. 0. M. of the. following expressions: I. 1. Qx^ — 5ax — (ja^ and 4:X^ — 2ax'^ — Qa^. 2. 4«2 - 5ab + b^ and 3a^ - da^ + ab^ - b^ 3. Sx^ - 13x2 + 23x - 21 and Qx^ -^ x^ - Ux + 21. 4. x' - lla;2 + 49 and 7x^ - iOx^ + 7ox^ - ^Ox -f 7. LOWEST COMMON MULTIPLE. 147 5. x^ 4- Qx^ -I- ll.^. _|_ G aud x"- + x^ - 4x^ - 4x, 6. x^ - x^ + 8ic - 8 aud x^ + 4:6-3 - Sx^ + 24a;. II. 7. 8^3 - l8ab^ Sa^ + Sd^b - Qah^, and 4ft2-8«& + 3^>l 8. x^ -Ix^ 12, 3a:2 - 6:?; - 9, and ^x^ - 62;2 - Sx. 9. 8a;3 + 27, 16a;* + 36^2 + 81, and Qx^ - hx - 6. 10. x^ — Qxy + 9^'^ x? — xy — 6?/^, and 3^;^ — 121/2. 11. Multiply x"^ + a;" by x"^ — x^. 12. Multiply 'Sa^'x " — ^oTx'' by 3a"a;"* + 4(x'"iC". 13. Divide a; - »" + ^ by a:"* + ^ 14. Divide 4'-\-b ^^ _{a I b\ a -\- b _a b c ~ c c' It is thus seen that this theorem is a consequence of the assumption that the distributive law of multiplication holds for fractional symbols. Hence the denominator of a fraction is distributive throughout the terms of the numerator. And, conversely, the algebraic sum of any number of fractions with the same denominator is the fraction whose numerator is the algebraic sum of the numerators of several fractions, and whose denominator is their common denominator. The sign before a fraction may always be regarded as belonging to the numerator as a whole, and it must be so regarded in finding the algebraic sum of the numerators of fractions which have the same denominator. Thus, 152 FRACTIONS. + r = —f—y and — ^ = —r-- The value of a fraction is to be regarded as a quotient, and when the divisor is positive the sign of the quotient is the same as that of the dividend. Hence, if a and h both represent positive quantities, — - = — — = — -, but is not = — -. That is, 0—0 — the minus sign before a fraction may be regarded as be- longing to either the numerator or denominator as a whole, but not to both. The same is evidently true when both a and h repre- sent negative quantities, or when one of them represents a negative quantity and the other a positive quantity. For — — , — zT-j and — — each evidently represent the same negative quantity; and — :p7, , and — jpr each evidently represent the same positive quantity, as do also 7, — 7- , and - -b. To illustrate by numerals : - ^- - 2, =^= - 2, and -1^ = - 2; _:^8_2, -:^:=2, and ^=2; It must be borne in mind carefully that, in finding the algebraic sum of the numerators of fractions which have the same denominator, all the signs of the numerator of every fraction which has a minus sign must be changed. FRACTIONS. 153 EXERCISE LXVIII. I. 1. Write ^ as the sum of three separate fractions. 2. VVrite ^— r— 7 ' — as the sum of four a -\- b fractions. 3. Write - — — ^ — - — —7 + -; — — Y as one fraction. '^a -{-b 2a-{- b 2a -\- b ,-,^ ., 3.T + 5 4:^; + 6 6x — a , 7x — c 4. Write — -r r a :: as one 4c 4c 46* 4c fraction. II. 6. Write 2a -\- 3b -c 5a -7b ^11 da -{- 5b - 7 lla - d x^-3 ^ x^-3 x^-S x^-3 as one fraction. 6. Write 3x - 4t(a -\-b) 5x^ 7{a ■\-b) 7x - 5(a- c) a2 -J)i a^- ])i a" - b^ as one fraction. 114. Theorem II. The value of a fraction is not altered by multiplying its numerator and denominator by the same quantity. It IS required to prove T~~i- By the commutative law t • mb — -r- . bxm = am = ma. 154 FRACTIONS. By definition ma mb . mb ■■ = ma a 1' mb — ma mb' mb. a ma I " mb' 115. Theorem III. The value of a fraction is not altered by dividing its numerator and denominator by the sa7ne quantity. T. ' . /, X a ^ m a It IS required to prove -^ = 7. ^ ^ b -r- m b -o XT, 1 X XI, a^m (a-^ 771)771 By the last theorem = -77 { — . •^ b -^ m {b -^ 7n)m ^ , - , ^ ., . (a -=- m)m a But by definition 77 ^-. — ^. -^ {b -i- m)7n b 116. It follows from Theorem III that a fraction may be simplified without altering its value by the rejection of any common factor from its numerator and denominator. Thus the fraction -73- takes the simpler form —7-3, when the factor Xy which is common to its numerator and denom- inator, is rejected. A fraction is said to be in its loivest terms when its numerator and denominator have no common factors. A fraction may be reduced to its lowest terms by re- moving, or cancelling, the common factors one after an- other from the numerator and denominator by inspection, or by dividing the numerator and denominator by their H. C. F. When the numerator and denominator of a fraction are polynomials which can be factored by inspection, it is FRACTIONS. 155 best to write them as factored, and then to cancel their common factors. 3a;^ + a; - 2 _ (3a; - 2)(a; + 1) _ 82; - 2 ®*^' '■Zx^ -x-d~ {2x - d)(x + 1) ~ 2:r - 3 • It is not worth while to divide the numerator and de- nominator by their H. C. F. except in cases where their common factors cannot be discovered by inspection. EXERCISE LXIX. Reduce the following fractions to their lowest terms. 12A UaWc X — a 4. x^ — a' ax — a^ ' x^ -\- ax 3x^ - 9x^y Saf^x^ - 16a^x^ 7. 7x^ - 21xY ^ali^x^ - 16Z»V ' x^ -\-x-'2Q x^ - 36 x'' -\\x^ 28* ®- ^:i"3^~irT8- II. 4a7^ — 16 ^x^ — 7.T 10. %x^ - 2x - 12' x^ + 3x - 28 11. 2 ■ o ^- 12. 2x? + X -6 ' 2:3 + 27 X^~^' 4.x? - ^x + 3 . Qx^ -\- xy — y^ ^^' 8^2 _^ 2x1/ -'^" ^*- 4:7-2 _^ 4^ _ 3 • 117. Reduction of Fractions to a Common Denominator. — Two or more fractions may be reduced to equivalent fractions with a common denominator by finding the L. C. 156 FRACTIONS. M. of the denominators for the common denominator, and dividing this by each of the old denominators in turn, and multiplying each numerator by the corresponding quotient for the numerator. N.B. — This is equivalent to multiplying the numerator and denominator of each fraction by the quotient obtained by dividing the L. C. M. of all the denominators by its own denominator; and hence the value of the fractions will not be altered. (Why ?) An integer may be regarded as a fraction whose denom- inator is one. Hence an integral term may be reduced to a fraction with any denominator by multiplying it by the re- quired denominator and placing the product obtained over the denominator. Of course any fraction may be reduced to an equivalent fraction with any required denominator (which is a mul- tiple of its own) by multiplying the denominator of the fraction by the factor which will produce the required de- nominator, and the numerator by the same factor. Such a factor may be obtained by dividing the required denomina- tor by the old one, or, often, by simple inspection. EXERCISE LXX. I. 1. Reduce -^ to an equivalent fraction whose denom- inator is 9«c^. 2. Reduce — to an equivalent fraction whose denominator is 'iWu^. X — S 3. Reduce to an equivalent fraction whose de- X -J- i nominator is 2;^ -|- :^ — 42. FRACTIONS. 157 4. Reduce ~ ~ to an equivalent fraction whose cle- nominator is I'^x^ -\- x — Q. 5^ ij" 6. Reduce ^r to an equivalent fraction whose ZX — D denominator is %x^ — 34a; + 30. 6, Reduce ^a^x to an equivalent fraction whose de- nominator is ba^x^. II. 7. Reduce 2^V to an equivalent fraction whose de- nominator is 3 — laW^. 8. Reduce 3:?; — 5 to an equivalent fraction whose de- nominator is 7r?; + 8. 9. Reduce 5a; — 7 to an equivalent fraction whose de- nominator is 6 — Zx. 10. Reduce 3a; + 8 to an equivalent fraction whose de- nominator is 9 — hx. 5 Ix 11. Reduce ^r— 7^ and —3- to equivalent fractions with a common denominator. 12. Reduce and to equivalent fractions X ~j~ TC X 4: with a common denominator, and find their sum. Reduce the following terms to equivalent fractions with a common denominator, and then the whole to a single fraction : , , a;+ 7 a;- 8 13. 1 + — ~- r-T- 158 FRACTIONS. 6x + 6 14. ^r- 15. dx 2a 4:a^ 2x-3 4a: - 6 3x-\-4: 5x-2 16 Reduce — ^j-^ + - to a single negative fraction. 4fl2 ' Of II. 257,2 >yQ 17. Reduce - ^tts + ^ ^0 a single negative fraction. Sba'* oa 18. Reduce 1 ■ — j-r to a single positive frac- tion. 19. Divide x^"" — x^"- by x"^ + x"". 20. Divide x^"^ + x^"" by x"' + a;". 118. Theoeem III. The product of two fractions is the product of their 7iumerators divided hy the product of their denominators. ^ . . -, , a c ac It IS required to prove t'^-3 — -ti- By the commutative law -X-j.hd = -.h X -^. d = ac. •^ h d b d ac By definition -:j—,.bd=ac. ^ bd b d bd a c _ ac b d~'bd' Hence the product of two fractions is another fraction whose numerator is the product of their numerators, and whose denominator is the product of their denominators. FBAGTI0N8. 159 The product of any number of fractions may be found by first finding the product of any two of them, and then of the resulting fraction and a third, and so on to the end. The resulting product evidently will be the fraction whose numerator is the product of the numerators of all the given fractions and whose denominator is the product of their denominators. Thus, b d f h bd f h bdf h bdfU Hence - \b)=b''b=¥' ^^^ \b) ^¥' Cor. 1. A fraction may be multiplied by a quantity by multiplying its numerator by that quantity. For let 7- be a fraction and c be the quantity by which b it is to be multiplied, c may be written as the fraction -. a a c ac b bib Also, by the Commutative Law, 7- X c = c X 7-. a ac Cor. 2. A fraction may be multiplied by a quantity by dividing its denominator by that quantity. For let - be a fraction, and c be the quantity by which it is to be multiplied. 160 FRACTIONS. Then - x c = j-. Multiplying both the numerator and denominator by -, we have 1 ac . — c a a or , 1 b ' ^^ i^c .— — c c EXERCISE LXXI. N.B. — In multiplying fractions by integers or fractions it is best to cancel common terms as in arithmetic. Find the following products ; dz 4:X^y^ z ' c? — x^ a^x -\- ax^ 2{a — x) 2ax (T — 2ax + x;^ c? -\- ax a^ 4- ax a^ — a^ 3. -^ — 5-X a!'^ — x^ ax{c? -\- ax-\- x^)' c^ — x^ c? — if- I ax \ 4. 1 X r^X^H . a-\- y ax -\- x^ \ ' a — xJ II. ax — x^ «2 _|_ ff^x a^ — 2ax -{-x^ a^ + 2ax + x^' FBACTI0N8. 161 •■ e+^^)(:-+«--)' 10. 42)2 - 16a; + 15 ^ Q? -6a; -7 ^ 4a;2 2a;2 + 3a; + l 2a;2 - 17a; + 21 4a;2 - 20a; + 25* 119. Reciprocals. — The reciprocal of a fraction is the c . d fraction inverted. Thus, the reciprocal of -^ is -. (t c 120. Theorem IV. To divide one fraction hy another is equivalent to multiplying the first fraction hy the recip- rocal of the second. It is required to prove that - -^ - = - x — . ^ ^ d b c By definition of division, - ^ - x - = 7-. ^ 1) d d h By Theorem III and the associative law of multiplica- tion, a f? c _ « cd _a h G d b' cd b' (a ^c\c _fa d\c [b ' d)d~[b^c)d' a _^G ^ a d b ' d b c ' Hence to divide one fraction by another, we invert the divisor, and then proceed as in multiplication. Cor. 1. A fractio7i may be divided by a quantity by multiplying its denominator by the quantity. For, let T- be a fraction and c be the given quantity. Then will ? ^ c = h b xc 162 FRACTIONS. c,. c , a a 1 a Since c = -, we nave — -hc = -X-=7-. 1 be CoE. 2. A fraction may he divided hij a quantity hy dividhig its numerator hy the quantity. For, let T- be a fraction and c be the given quantity. Then - -^ c — -r-, and multiplying the numerator and de- nominator by -, we have 1 a a . - - a c c a -7- c -j_ c = =: — - or -— . h ., 1 h h he . - c Cor. 3. To divide a quantity hy a fraction we multi- ply the quantity hy the reciprocal of the fraction. Let fl^ be a quantity, and ^ be a fraction. Then will «^^ = « xf a c a c a d ad d - X - = — = aX -0 Ice c EXERCISE LXXII. Perform the operations indicated in the following ex> amples : * I, Ux'^ - 7x 2x-\ *• 12^:3 + 24:^2 ' x^ + %x a^^ + dab ^ ah^Z r- r : —m 4a^ - X • 2a -1- 1* 3. FRACTI0N8. «2- -121 « + 11 a' -4 • « + 2 * 2a^ + 13x + 15 . 2^2 _^ 11^. _^ 5 4:^2 _ 9 4a;^ - 1 • x'- - 14:« - - 15 x^ - 12a; - 45 x^ -^x- -45 • x^ - ex - 27 * (10 + lla; - Qx^) 9a;2-4 ' 4 -3a;* (15. ,;2_ 19a; + (3) 18 - 18a; - 20a;2 2a; + 7 {x'^ -%x- -63) -^ a;2 + 2a; - 35 163 8. 121. To Multiply Several Fractions by a Factor which will Cancel all their Denominators. — If each of several fractions be multiplied by the L. C. M. of their denomina- tors, there will be introduced into the numerator of each fraction a factor which will cancel its denominator, and the resulting products will be the product of the numerator of each fraction and all the factors of the L. C. M. of the de- nominators except the denominator of the fraction. We may therefore obtain these products by dividing the L. C. M. of the denominators by each denominator and multiply- ing the numerator of each fraction by the resulting quo- tient. e.g. Find the product which would result from multi- plying each of the following fractions by the L.C.M.D. : a; + 7 a;-8 ^ a; + 9 and Q?-\-'^x- 10' a;2 - 8a; + 12 x^ - x - 30' Factoring the denominators, we get x-\-l a;— 8 a;+9 (x - 2)(a; + 5)' {x - 2){x - 6)' (x + 5)(a; - 6)' 164 FRACTIONS. Hence the L. 0. M. of the denominator is {x - %){x + b){x - 6). Multiplying each fraction by this L. C. M., and cancel- ling the common factors, we obtain {x^l){x-^){x-^6){x- 6) \x-^)\x^6) (g;- 8)(a;- 2)(a;+ b){x - 6) {x-%){x-Q) and (a;+9)(:.-2)(a;+5)(:.-6) {x+b){x-Q) or x^-^x- 42, x^-'^x- 40, and x'^-\-'ix- 18. EXERCISE LXXIII. Find the products obtained by multiplying each frac- tion of the following sets by the L. C. M. of the denomina- tors: a;-4 a?-\-x-6Q' x^.-\- 11x^^4:' x^-4:x-21' dx-7 5a; - 4 a; + 11 10a;2 - 43a; + 28 ' 16x^ + 8a; - 16' 6x^ - 13a; - 28* 5a; — 8 6 — 7a; 3 — a; 66a; - ISa;^ _ q^> 24:X^ - 90a; + 54' 40a;2 _ 86a; + 42* 11. x-S x-^ 7 *• 0^-64:' (^'i-4:X-32' a^ -\- 4:x -\- 16' x-\-7 a;4-6 15 ^' ie8 + 216' a;2-36' 3a;2 - 108' 6. Divide x^"" ~ x> by x^ — re*. FRACTIONS. 165 EXERCISE LXXIV. I. 1. A is four times as old as B and 6 years ago he was seven times as old. What is the age of each ? 2. At what time after 3 o'clock are the hands of a watch opposite each other for the first time ? 3. Divide 45 into two parts such that one of them shall be four times as much above 20 as the other is below 19. 4. A man had $13.55 in dollars, dimes, and cents. He had 1/7 as many cents as dimes, and twice as many dollars as cents. How many of each kind had he ? 6. Divide 313 into two such parts that one divided by the other may give 2 as a quotient and 19 as a remainder. II. 6. A is m times as old as B, and in c years he will be n times as old. What is the age of each ? 7. At what rate of simple interest will a dollars amount to h dollars in c years ? 8. The denominator of a fraction is equal to four times the numerator, diminished by 41, and if the numerator be diminished by 6 and the denominator be increased by 9, the value of the fraction will be 5/12. What is the frac- tion ? 9. At what time after 5 o'clock are the hands of a watch together for the first time ? 10. Divide n into two parts such that one divided by the other will give g' as a quotient and r as a remainder. CHAPTER XV. CLEARING EQUATIONS OF FRACTIONS. 122. Three Classes of Equations Involving Fractions. — As we have seen, an equation may be cleared of fractions by multiplying both members by the least common multi- ple of the denominators of all the fractions in the equa- tion. Equations involving fractions may be divided into three classes : 1°. Those m loliicli we should clear of fractions at once or after maMng some slight reductions. 2°. Those ivhich might he cleared of fractions partially and then simplified. 3°. Those in which some or all of the fractions had hetter he reduced to a mixed form. Case 1°. In clearing equations of fractions, it must be borne in mind that every term of both members, integral as well as fractional, must be multiplied by the L. C. M. D. In clearing equations of fractions, it is best to express the L. C. M. of the denominators as factors, and also to indicate the work of multiplication before actually perform- ing it. In this way like factors in tlie numerators and denominators may be cancelled, and the work much shortened. e.g. Solve — -— -— -\ — - = 0. 166 CLEARING EQUATIONS OF FMACTlONS. 167 L. 0. M. D. (x + l){x + 2){x + 4). {x -j- l){x -^ 2){x + A) _ 2(x + l)(x+2)(x + ^) x+l x-^2 {x + l){x-^2)(x + ^) _^ ' X -\- 4: ... (x-i-2)(x + 4)-2{x-\-l){x-\-4)-\-{x-^l){x-\-2) = 0, or a;2 _^ 62; + 8 - 2a;2 _ lOa; - 8 + ^c^ + 3a: + 2 = 0. -x-\-2 = 0. x = 2. When all of the fractions are written as decimals, it is best first of all to reduce these to the form of vulgar frac- tions. e.g. Solve •^^"•^^'^ - (.03 - .02x) = .03. Reducing the decimals to vulgar fractions, we have _ /J 2^\ _ _3^ \100 100/ ~ 100' ^' IO-ro-100 + -100=^100- L.C.M.D. = 100. 5 X 100 a; X 100 3 X 100 2x X 100 _ 3 X 100 10 10 100 "*" 100 ~ 100 ' or 50 - 10a; - 3 + 2a: = 3. -8a:r=-44; X = 5i, 5 X 100 100 1 10 168 GLEAMtNQ EQUATIONS OF FRACTIONS. EXERCISE LXXV. I. 1. \x-m^^[^x-i)-l{x-\-i)-ii. 2. |(2^-7)-|(^-8)=?^^ + 4. 3. ,^(4^ + 1) - ^(217 - ^) = 45 - IZi^. 4. .03a; + .02 = .0%x - .06. 5. .Ql{x - 10) + .542: = .2(.l - .Ix) - 3(.05 - .02). ^ — X 6 — X ^ x^ — 2 6. :; i^ = 1 — 1-x 7-x 7-8a; + a;2 3 1 ^ + 10 _ 2a; - 4 x-{-2 ' 2x^ — S 2 ^ - 1 4a;2 - 1 **• 1 -2x 2x-7 4:x^- IQx + 7' II. 9. (1 - 22;)(.01 - .03a;) - .23 = (.6a; + .l)(.la; - .1) - .03a;. .Ola; X .Ola; , ^ „, ''• -^-30 = ^^^'^^' .03a; - .01 .02(a; - 1) .Ola; - .03 , .21 ^^' .02 .03 .4 ^ .2 12. (.la;+.2)2 + .7(.3a;-.l) = .06(2a; + 4) + (.la; - .2)2-. 65. 4-a; 6-a; ^ 2a;2 + 8 13. H o = 2 - 2 - a; 8 - a; 16 - 10a; + a;^' 5 2 11a; ^ ^^' 3a; - 9 a; + 3 ~^ 3a;2 - 27 CLEARING EQUATIONS OF FRACTIONS. 169 _ __3 3^ _ ^x{^x - 17) ^*- l-'dx dx-7 ~ ^x^ - 24a; + 7 Case 2°. 123. When the L . C . M. of the denominators of all the fractions which occur in the equations is inconveniently large, it is easier to multiply both members by the L.O.M. of two or more of the denominators, and then reduce as much as possible before proceeding farther. e.g. Solve ^-^- ^,_^^^^ ^^^-^+l. Multiplying by 2 (a; — 1), we get 2^ __ 3(2_+|!) ^ 3 _ 22; + 2a; - 2, • a; — 1 or 2. ^(^+f) = i. a; — 1 .-. 2a;(a; - 1) - 2(2 + a;^) = a; - 1. •. 2a;2 _ 2a; - 4 - 2x'^ = x - 1. .-. 3ai=-3. .-. a;=-l. EXERCISE LXXVI. Solve the following equations: I. I + - ; - 2a; a; - 3 1 10 22 ~ 5* 2. 4a; 4- 9 3 8a; + 19 7a; - 29 __ 18 ' 5a; - 12 ■" * Z h^'^ 3a; + 10 X ^^"^^ 10a; -50 -5' 170 GLEAMING EQUATIONS OF FEACTI0N8. II. 8a: + 5 3 - 7a; IQx -\- 15 _ 2i *• "~Ii 6a; + 2 28 ~ 7 ' 6a;- 7i 1 + 16a; _ 53 - 24a; _ 12g - 8a ; ^' 13 - 2a; "^ 24 ""12 3 ' Case 3°. 124. When the degree of the numerator of any of the fractions equals or exceeds that of the denominator, it is best in most cases to write the fraction in the mixed form obtained by dividing the numerator by the denominator and writing the remainder in the form of a fraction after the integral quotient; thus: x-l ^ 2 a;+l a; + l' 5a; + 4 ^ 2 X — 'S X — S' After writing the fractions as mixed numbers, the equa- tion may generally be considerably reduced before finally clearing of fractions. , ^, , , .^ + 9, 3a;2 + 6 e.g. 1. Solyea: + 3~^-_,^^=3^-^. Writing the second fraction in the mixed form, we have a; + 9 , a;+6 3(a; - 1) ' 3a; - 1 0; + 9 a; + 6 • • 3(a; ■ -1)" ■ 3a; - 1* (a; + 9)(3a;- -1) = : 3(a; + 6)(a; -1). 3a;2 + 26a; — 9 = 3(a;2 -f 5a; - -6). ,*, 11a; = : -9. • . a; = 9 11* or CLEA1UN6 EQUATIONS OF FRACTIONS. 171 „^, X — 1 X — 2 X ~ 4: X — 5 2. Solve ^ — = -. X — Z X — 3 X — 5 X — b Writing each fraction as a mixed quantity, we have l + -^-(l+^ = l + ^--fl + -^). ■x-2 \^x-SJ ^x-6 \^x-Qj 1 1 1 _ 1 X — 2 X — 3 ~ X — 6 X — 6' We may now write each member as one fraction and get X -3 -x-{-2 _x -Q-x-^5 {x - 2)(x - 3) ~ Jx'- 6){x - 6)' - 1 _ -1 {x - 2)(x - 3)~ (x- 5){x - 6)* .-. (x - 2){x - 3) = (x - 6){x - 6). .-. x^ - 6x + = a;2 - II2; + 30. .-. 6a; = 24. . % X = 4:. EXERCISE LXXVil. Solve the following equations : I. x — 1 x — 2 a: + ^ ,^ — '^..o ^\ ^ZT^-x-b' ^' x-3'^x-Q x-\-3x-4._ X 3x _ ^ 3a; +5 , 2a; + 4 _ ^- 3a; - 5 + a; - 2 '^' 2x 5 2a; - 5 __ ®- 2it; + 1 "^ 2a; - 1 + 2a; -f 1 172 CLEABINQ EQUATIONS OF FRACTIONS. 7. 10. 11. 12. II. X — 1 2 . x-^ x-3 X-4: x-4: X - 6' X — x-^~ x^ x-Y 1^ + 6 x-\-2 x-\-3 x + 5 ^ x + Q 8- hx 4.x + 3 = lh 2a;- - 1 ' x + 3 ^ + « x-\-b X — « ' X — V X X -\- a — b a(a — h) X — a x-h ~ {x- c){x - d)- X — la X — a 5a _ X — a X — 9a X — 3a x — 2a x -{- 'Za EXERCISE LXXVIII, 1. A vessel can be emptied by three taps : by the first alone in 3 hours and 40 minutes, by the second alone in 2 hours and 45 minutes, and by the third alone in 2 hours and 12 minutes. In what time would it be emptied were it full and all three taps were opened together ? 2. A cistern can be filled in 15 minutes by two pipes, A and B, together. After A has been opened for 5 minutes B is also turned on, and the cistern is filled in 13 minutes more. In what time would it be filled by each pipe sep- arately ? 3. A man invests one third of his money in 3-per-cent bonds, two fifths of it in 4-per-cent bonds, and the remain- der of it in 5-per-cent bonds. His income from his invest- ment is 1180 dpllars. How much had he invested ? CLEARING EQUATIONS OF FRACTIONS. 173 4. A man invested one quarter of his money in 3 - per-cent bonds, two sevenths of it in 4-per-cent bonds, and the remainder of it in 4^-per-cent bonds. His income from his investment was 3450 dollars. How much had he in- vested ? 5. Two men, A and B, 66 miles apart, set out, B 45 minutes after A, and travel towards each other, A at the rate of 4 miles an hour and B at the rate of 3 miles an hour. How far will each have travelled when they meet ? 6. The second figure of a number composed of three figures exceeds the third by 5, and the first digit is one fourth of the second. If the number increased by 3 be divided by the sum of its digits, the quotient will be 22. What is the number ? 7. A number is composed of three digits. The second digit is one half of the third and 2 smaller than the first. If the number be diminished by 18 and then divided by the sum of its digits, the quotient will be 37. What is the number ? 8. A banker has two kinds of money. It takes a pieces of the first to make a dollar and b pieces of the second to make a dollar. He was offered d dollars for c pieces. How many of each kind would he give ? 9. A and B start in business at the same time, A putting in 3/2 as much capital as B. The first year A gains 150 dollars and B loses 1/4 of his money. The next year A loses 1/4 of his money and B gains 300 dollars;, and they now have equal amounts. How much had each at first ? II. 10. Two couriers, A and B, set out from the same place and travel along the same road in the same direc- 174 CLEARING EQUATIONS OF FRACTIONS. tion, A starting 8 hours before B. B rides at the rate of 8 miles an hour, and A at the rate of 6 miles. How far will each have travelled when B has overtaken A ? 11. A and B find a sum of money. A takes $2.40 and 1/6 of what is left; then B takes $3.52 and 1/7 of what is left; and they find they have taken equal amounts. What was the sum found and what did each take ? 12. A fox is pursued by a greyhound, and has 60 of her own leaps the start. The fox leaps three times while the greyhound leaps twice, but the hound goes as far in 3 leaps as the fox does in 7. How many leaps does each make before the hound catches the fox ? 13. A hare takes 4 leaps to a greyhound's 3, but two of the hound's leaps are equivalent to three of the hare's. The hare has a start of 50 of her leaps. How many leaps must the hound make to catch the hare ? 14. A man and a boy agreed to do a piece of work for $5.25, the boy to receive 1/2 as much per day as the man. When 2/5 of the work was done the boy left, and, in con- sequence, it took the man 1^ days longer to complete the work than it would otherwise have done. How much did each receive per day ? 15. In a mixture of spirits and water, half of the whole plus 25 gallons is spirits, and a third of the whole minus 5 gallons is water. How many gallons are there of each ? 16. A garrison of 1000 men was provisioned for 60 days. After 10 days it was reinforced, and from that time the provisions lasted only 20 days. What was the number of the reinforcement ? 17. A laborer was engaged for 36 days on condition that he should receive 2s. 6d. for every day he worked and should forfeit Is. 6d. for every day he was idle. At the CLEARING EQUATIONS OF FRACTIONS. 175 end of the time he received 58 shillings. How many days did he work ? 18. At a cricket match the contractor provided dinner for 24 persons, and fixed the price per plate so as to gain 12i per cent upon his outlay. Three of the cricketers were absent. The remaining 21 paid the fixed price for their dinner, and the contractor lost 1 shilling. What was the price per plate ? CHAPTER XYI. BADICALS AND SURDS. 125. Rational and Irrational Numbers. — A numerical quantity which can be exactly expressed as an integer or a fraction whose numerator and denominator are integers is called a commensurable or a rational number, and one which cannot be so expressed, an incommeyisurdble or an irrational number. 126. Radicals. — Any algebraic expression which con- tains a factor under a radical or other root sign is called a radical expression, or simply a radical^ and the factor under the root sign is called the radical factor. Any algebraic expression which contains no radical factor is called a rational quantity. To rationalize an expression is to free it of radical or other root symbols. 127. Surds. — A surd is an incommensuratle root of a commensurable number. In other words, it is the root of an arithmetical number which can be found only approx- imately. While every surd is an incommensurable number, there are many incommensurable numbers which are not surds, or due to any finite combinations of surds. As examples of these we have 3.1415926 . . . , the ratio of the circum- ference to the diameter of a circle, and 2.7182818 . . . , the base of the natural or Napierean system of logarithms. 176 RADICALS AND SURDS. 177 A radical expression which cannot be freed from root symbols is called an irrational or surd expression, or simply a surd. The symbol of a surd is \^n, in which n denotes any positive integer, and a any integral algebraic expression. A surd may be expressed as a radical quantity, but every radical quantity is not a surd. Thus, V'6, r5 are surds, but VT, Vs are not surds. The expression V 2 + V2 is not a surd according to definition. 128. Imaginary Quantities. — Since no even combina- tion of negative factors can produce a negative product, an even root of a negative quantity is called an imagiiiary quantity. Thus, V— 2, V-^, ^/-a are imaginary quan- tities. The value of the expression \^a will be real or imag- inary according to the values assigned to n and a. It will be imaginary when n is even and a is negative. In all other cases the value will be real. When ^ is a perfect wth power, \^a is rational and in all other cases irrational or surd. 129. To Express a Rational Quantity as a Radical. — Any rational quantity may be expressed as a radical by first raising it to the power indicated by the index of the radical and then placing it under the radical sign. e.g. 4 = Vie", 3 = f27^ 130. Orders of Radicals. — A radical is said to be of the first, second, or nth. orders according as its index is 1, 2, or n. 178 RADICALS AND SURDS. EXERCISE LXXIX. Express the following quantities as radicals of the second order: I. 1. m. 2. n. 3. 3a. 4. hob. 5. 7^^ 6. 6a:^y^ 7. 1/4A. II. 8. i/3ay. 5aV . 10. a-\-b. 11. x-y. 12. 3«2 + 7. Write the following as radicals of the third order: 13. X. I. 14. Sa^x. II. 16. 1/3«V. 16. x^h. 17. a- 3. 18- -rr- 131. Arithmetical Boots. — We have already seen that Va/^ has two values, + a and — a; also, that Va has two values which differ only in sign, one being positive and the other negative. In higher algebra it is shown that Va has three values, one of which is real and the other two imaginary; also, that Va has 9i values, one, or at most two, of which may be real, and the others imaginary, and that when there are two real roots they will differ only in sign. When a root symbol is placed before a number it de- notes the arithmetical root only, but when placed before RADICALS AND SURDS. 179 an algebraic expression it denotes one of the roots. Thus Va has two values either of which is denoted by the sym- bol, but V^ is supposed to denote only the arithmetical root, unless it is written ± V^. In the demonstrations in the present chapter the sym- bol Va in all cases must be taken in a restricted sense, — to mean the real root of a whose sign is the same as the essen- tial sign of a. Thus Va" must be taken to mean a, and Va to mean the one real root of a which has the same sign as a. The theorems established in this chapter do not necessarily apply to other real roots than the one specified above, or to imaginary roots. In this chapter it is assumed that the associative, dis- tributive, commutative, and index laws, which have been established for integers, and applied to rational algebraic expressions, also apply to surds. 132. Theorem I. The product of the same roots of ttuo factors is equal to that root of the product of the factors. By definition \a used n times as a factor will give a as a product. .-. C^ay = a. Similarly, ( VhY = h, and ( "x/abY = ah. But ( y~a X "i^lY = ( VaT X ( ^Y = ah, and ( VahY = ah. /. iVaxny=(VabT. (Why?) /. -\^axVI= 'i^^. (Why?) Cor. The product of the same roots of any number of factors is equal to that root of the product of those factors. 180 RADICALS AND SURDS. Note. — It should be borne in mind that \^~a, taken arbitrarily, x V~b, taken arbitrarily, does not = Vab, taken arbitrarily. Thus the negative root of 2 multiplied by the positive root of 3 does not equal the positive root of 6. The equation Va X yb=^ Vah is true when the mean- ing of the symbols is restricted as in 131. It is also true that any one of the n roots of a multiplied by any one of the n roots of i will be equal to some one of the n roots of ab. 133. It follows from Theorem I that, when the quantity under the radical sign can be separated into factors one or more of which is an exact power of the order of the root indicated, the product of the indicated roots of these factors may be placed as a factor outside the radical. e.g. l/l92 = 1^16 X 4 X 3 = VU X Vlx V^= 8 V^. f 864 = \/Wx 8X4 = f 2f X ^^8 X ^4 = 6 1^4: 134. Pure and Mixed Surds. — The factor without the radical sign may be regarded as the coefficient of the radi- cal. A ptire surd is one that has no rational coefficient except unity. A mixed surd is one that has a rational factor other than unity. A surd is said to be in its simplest form when it has no rational factor under the radical sign. EXERCISE LXXX. Write the following as mixed surds in their simplest forms : I. 4. ^7357 5. VWi^. 6. ^5677 7. fl35. MABICALS AND SURDS. 181 II. 8. V448. 9. i/5632. lo. 4/48a2J. 11. 4/l25«V. 12. VUlaH\ 13. l/4«& + 8«^^ + 4a3^l 14. 1/I22;y - 24a;y + 12a^y\ A mixed surd may be reduced to the form of a pure surd by raising its coefficient to the power indicated by the order of the surd and placing it as a factor under the radi- cal sign. e.g. 7 1/5 = 1/72^5 = '^245. EXERCISE LXXXI. Express the following as pure surds: I. 1. '^Vn. 2. 41/13; 3. 6 1/7. 4. 2 V^. 5. 4 \^6. 6. 6 t^4. 7. 3^ Va — b. 8. {x + ij) V'Sx. 9. 'Sa(a — b) Vbab. 136. Theorem II. The quotient of the same roots of two qtiantities is equal to that root of the quotie?it of the two quantities. Expressed algebraically, Va -^ Vb = \^a -^ b, ( V^^ ny = ( ;/ay - ( f^)" = a~.b. But ( \^a -- by = a^ b, •/. ( 4^« -- \^by = ( f ^TT^)'. (AVhy ?) /. i^a -4- yT = 1^«T~^» (Why ?) Cor. a / ~ = ,, _= That is, any root of a fraction « /a Va ,p 182 RADICALS AND SURDS. may be indicated by placing the corresponding radical over the fraction as a whole, or over its numerator and denomi- nator separately. 136. Similar and Quadratic Surds. — Similar surds are those whose radical factors are identical, e.g. Vb, 3 Vb, are similar surds. So also are a ^x and c ^x. Surds of the second order are called quadratic surds. 137. Theorem III. The product of two similar quad- ratic surds is a rational quantity. mVaXnVa — 7mi Va^ = mna. The product of the coefficients is necessarily a rational quantity, and the product of the similar radical factors is necessarily the square root of a perfect square, and, there- fore, rational. 138. Theorem IV. The product of two dissimilar quadratic surds canyiot le- rational. Let ^a and ^h be the surd factors. Since the surds are dissimilar, a and h cannot be composed of the same prime factors, and hence their product ah cannot be com- posed of square factors only. Therefore 'fab cannot be rational. 139. Rationalizing Factor. — Any factor which will convert a radical expression into a rational one is called a rationalizing factor. It follows from Theorem II that the surd factor of a pure or mixed surd is a rationalizing factor. e.g. l/5'xl^==5. 3 V3"x |/3 = 3 X 3 = 9. h^a-hy.^a-h = h{a-V). RADICALS AND SURDS. 183 140. To Reduce a Fractional Radical to an Integral Radical. — A fractional radical may be reduced to an inte- gral radical with a fractional coefficient, by writing its nu- merator and denominator each as a separate radical, and then multiplying each by the rationalizing factor of the denominator. e.g. ^5-= -^ = -^£^=1/5^. /^ _ _V« _ V^x Vb _ Vah _-\^ y h~ Vb~ VbxVb~ b -i^^^' EXERCISE LXXXII. Reduce the following to integral radicals : I. 1. VT/2. 2. VT/b. 3. Vy^. . vm, . /|- a. /f±|. II. 7. ./x + 4 ^x — 5 a/6x — 2 /4:X - 6 Jb - 2x 10. f . 11. V . 12. —Zx-^1 L-^-4^ 4 141. Addition and Subtraction of Radicals. — Similar radicals may be added and subtracted by combining their coefficients in the same way as similar rational terms. The common surd factor must be written after the coefficient resulting from the combination. e.g. The sum of 3 V5, V5, and - 7 VE is 6 VE. The difference of 3 V2 and 9 1^ is - 6 1^2. 184 RADICALS AND SURDS. Dissimilar radicals can be added and subtracted only by writing them one after another, each with its proper sign, as in the case of dissimilar rational terms. Thus, Vy added to Vx = Vx + V^, and never Vx -{- y, unless either x or y i^ zero. 143. Rule for Addition of Radicals. — To add surds of the same order, reduce them to their simplest forms and add the coefficients of the resulting surds which are similar, and write those which are dissimilar after one another. e.g. ^V^ -\-VT^-{-%^/n - V^-\-^Vb 143. Rule for Subtraction of Radicals. — To subtract two radicals of the same order, reduce them to their simplest form, and then, if they are similar, subtract their coefficients, and if they are dissimilar, write them one after the other with the proper sign between. e.g. From 3 Vb take 2 Vv^. = 3 V5 - 10 1/5"= - 7 y^. From 3 \^ take 2 VSO. 144. Addition and Subtraction of Radicals of Differ- ent Orders. — Radicals of different orders can be added and subtracted only by writing them one after another with the proper signs between. EXERCISE LXXXIII. Find the sum of the following sets of radicals : I. 1. 4/18', -^32, 1^50, and Vn. 2. 2 VS, 3 /50, and 6 4^18. RADICALS} AND 8UBD8. 185 3. i/3/5, ^1/15, and ^15/49. 4. 2/3^^279; l/ev'ITse, and d/b\/yZ2. 5. xVVZa% 2a^V27x\ 3« V48«V, and |/75aV. II. 6. 2V3, 1/2 Vl2, 4^27, and 4/T27I6. 7. I'54^^ 7«y'2^^ and 8b \^2a^^ 7nn and y (n — s 71 — S \ {^ ~ ^) ^^ ~ EXERCISE LXXXIV. I. 1. From 2 1/320" subtract 3 I^SO. 2. From « |/646f^Z>4 subtract i VWda%. 3. From Va% + 2a52 _^ J3 subtract Va% - 2aW + ¥. 4. From |/2a34-46j2^+2«J2 subtract ^Iw^-^d'l^laV^. II. 5. From 2/3 1^2/9 + 3/5 1^3/32 subtract 1/6 Vl/36. 6. From l/289a3j subtract 3 VlUa^, 7. From 2 l/Sc^ + 5 4/72c3 subtract 7c VlSc + I^SOc^^, 8. From (c — :c) Vc'^ — x^ subtract a / -^ — . \J c — X 145. Multiplication of Radicals of the Same Order. — To multiply together two radicals of the same order, multiply together their coefficients for the new coefficient, and the quantities under the radical sign for the new radical. 186 RADICALS AND 8UBD8. EXERCISE LXXXV. Perforin the following multiplications and reduce the results to the simplest form : 1. 3 f^ X 2 Vm, 2. 7^2/81 X 3/2 \/y^^. 3. 4 i/l2 X 3 V% 4. ^^1727 X 3/4 ^12. 5. 5 Vc^ X 1/2 V^bix. 6. ^ t^2a^ X a ^Sab. 7. (2 V2 - 3 V3 + 4 4/5) X (3 4/5 + 4 1^3). 8. (3V5-4: V2) (24/5 + 3 4/2). 9. ( 4/7"+ 5 4^) (2 1^ - 4 VS). 10. ( 4/2 + 4/3 - 4^) ( 4^+ 4/3 + 4^). 11. (3 4/^ - 2 ^) (2 4^+ 3 4/^). 12. Multiply 4/f + 9 by4^ - 6. 13. Square 4^5 + 3. 14. Multiply 4^ - 6 by 4^ ~ 8. 16. Square VY — 5. 16. Multiply t^ + 4 by V^ + 3. 17. Square Vx -{- 9. 18. Multiply Vx -{- Q hj Vx — 5, 19. Square Vx — Vs. 20. Multiply 4^ + 4/7 by VE - VY. 21. Square tY + 4^8. 22. Multiply 4^2; + 5 by 4/:^ — 8. RADICALS AND SURD8. 187 23. Square Vx — 4= -\- Vx -\- 6. 24. Multiply 4/a; + 7 by Vx - 7. 25. Square l^a; — 3 + Vo; -|- 3. II. 26. Multiply 3xVa — ij by 5x Va — 7. 27. Square 2a VQx-\-2 Vb. 28. Multiply 5aVx-{-7 by 7Z> 4/3; + 7. 29. Square d Vx -{- 6 — 4:Vx — 7. 30. Multiply 7^/^ Va; — 4 by 9^ Vx -f 4. 31. Square 3a Va -\- 3 -\- 5a Va — 5. 32. Multiply Vx — 4: — 5 by 4/a; — 4 + 5. 33. Multiply 1^2' + 8 + VQ by i^o; + 8 - Vq. 34. Multiply Va; - 5 + i^a; 4- 8 by 4/3; - 5 - 1^2; + 8. 35. Multiply 3 Vx -\- 6 -^ 4: Vx -{- 5 hy 3 Vx -\- 6 - 4: Vx -i- 6. 36. Multiply 3a^x Vx-S ~ 5xWx-\-7 by 3 A Vx - S + 6x^ Vx + 7. 146. Simple, Compound, and Conjugate Radicals.— A smjy/e radical expression is one which contains only one term, and a compou?id radical expression is one which con- tains more than one term. Thus, Vx, V a-\- X, a Vab, (a -\- b) Vx -\- 4, are simple radicals, a -\- Vx, Va + Vx -\- b, are compound radicals. Two binomial quadratic radicals which have the same 190 RADICALS AND SURDS. ^ 4. ^ ^ . 5 + 2^ ^2 9 + 2V1T .. a^Va'- b^ II. V3-\-a^- V3 -«2 Vd-i-a^-\- V'd -a^- '• V5 + a;2 4-2 3 + 4/6 6. ^a^ ^y^-y 2Vx-i-3-^SVx- 3 2 V^"+3 - 3 Vx'^^' 6V3 -2 Vi2 - VS2 + VW Divide the following radicals at sight : I. 11. VT8"by Vq. 12. I^2rby \^. 13. 12V35by3i^. 14. a^ Vb^hj a"^ Vb. 15. Vx^ - 49 by Vx + 7. le. Ya^^ - 8 by i^a; - 2. 3/ 17. i^x^ 4- 27 by ya;2 _ 3a; + 9. II. 18. Vx^-\-2x- 15 by Vx-}-6: 19. Vx"- 13a; + 42 by Vx - Q. 20. Vx^ -x-72 by Vx + 8. 21. V6x^ + 17a; - 14 by V2x + 7. 22, V6x - 2a; - 7 by fa; + 1. RADICALS AND SURDS. 191 Divide the following radicals by first expressing the division in the form of a fraction and then rationalizing the denominator. I. 23. 29 by 11 + 3 Vl. 24. 17 by 3 i^ + 2 Vd. 25. 3|/2 - 1 by 3|/2 + 1. 26. 2 i^+ 7 1/2 by 5V3-4.V2. II. 27. ^x — Vxy by 2 Vxy — y. 28. (3 + Vl)( 1/5 - 2) by 5 - Vb, Va , Va-\- Vx by 2 1^15"+ 8 8 1/3 - 6 V5 5 H- VTS" ^^ 5 V3 - 3 VS * 150. Theorem IV. The 71th power of the root of any quantity is the same root of the nth power of the quantity, 71 and the index of the root both being positive integers. 1°. When the index of the root is the same as the exponent of the power. By definition, (^ya)" = af and r «" = a. ,'. (i/aY = te 2°. When the index of the root is not the same as the exponent of the power. (ni — v )n n Va") = a , 19S RADICALS AND SURDS. ■^^^^ yi^Vaj) means that Vfl^ is to be used mn times as a factor, and (( "Va )") means that "Va is to be used mn times as a factor. ... ((?«)•")" =((i^)"r. But ((?«)")" = «». .-. ((r«)r=«"- ... ((v«)T=("v;?r. 151. Theorem V. The mth root of the nth root of a quantity is equal to the mnth root of the quantity. Imf n _\m By definition, \V Va) = V^. ... {(VWrY=(^'a)''=a. Also, ("'f^)»»=«, and {(Vl^TY=(Vl^r. 152. To Change Radicals from One Index to Another. — It follows from Theorems IV and V that a radical may be changed from one index to another by multiplying both the index of the radical and the exponent of the quantity under the radical by the number which will produce the re- RADICALS AND SURDS. 193 quired index. For the former of these operations would extract a root of the radical quantity, and the latter would raise it to the corresponding power, and these two opera- tions would neutralize each other. e.g. \/a^^"''\/^^='\/^. To change radicals of different orders to those of the same order with the smallest possible indices, multiply each index by the quotient obtained by dividing the least com- mon multiple of all the indices by that index and raise the. quantity under the radical sign to the corresponding power. This will, of course, make the index of each radical the least common multiple of all the indices. e.g. Reduce 4^5, Vd, and r 2 to radicals of the same order with the smallest possible index. The L. 0. M. of 2, 3, and 4 is 12. 2X6 1/5= T5«= ?15635. 163. Multiplication and Division of Radicals of Different Orders. — Radicals of different orders may be multiplied to- gether by first reducing them to the same order and then multiplying together their rational and their irrational factors. Similarly, radicals of different orders may be divided by each other, by first reducing them to radicals of the same order and then dividing their integral and radical factors. EXERCISE LXXXIX. 1. Reduce r 10, 1^5, and r 11/12 to a common index. 2. Reduce Va + h, Va — b, and Vd^ + a;^ to a com- mon index. 194 RADICALS AND SURDS. 3. Multiply V^ by Vb. 4. Multiply \/yYhy VyI. 6. Divide Va^ by \d^. 6. Divide 2 V'Zac by t^4^c^. 7. Divide 1/2 ^273 by 1/3 I'lTsT 164. Radical Equations. — An equation which contains radicals is called a radical equation. Such equations are solved by first clearing them of radicals, or rationalizing them. If the equation contains fractions it should be cleared of them first of all. In the case of a quadratic radical equation, after it has been cleared of fractions, it is best to transpose all the terms into the left-liand member and place this equal to zero. Each member should then be multiplied by the con- jugate of the first. If the first member contains more than two terms, they should first be collected into a term and an aggregate, or into two aggregates, and the terms arranged, if possible, so that the aggregate shall contain no radical. Multiplying then by the conjugate expression will square each of the terms or aggregates, and place the minus sign between the squares obtained, and the result will be rational. If either aggregate contains a radical, the result of the first squaring will be irrational. In this case a new pair of aggregates must be formed and the operation must be repeated. e.g. 1. Vx — Q -4=9. Transposing, we get V^^^Q - 13 = 0. Multiplying by the conjugate expression Vx — ^ -j- 13, we get RADICALS AND SURDS. 195 X- 6 - 169 = 0, 2. V^^^ + 2 1^-5 = 3. Transposing, we get V^^^d + 2 1^ - 8 = 0. Writing this as the sum of two aggregates, thus, Vl^^d + (2 ^^ - 8) = 0, and multiplying this by the conjugate expression V¥x'^3 - {2Vx — 8), we get 4:X-S -4:X-{-32Vx-64: = 0. Collecting, we get 32 1^-67 = 0. Multiplying again by the conjugate 32 V^ + 67, we get 1024a; - 4489 = 0. EXERCISE XC. Solve the following radical equations : I. l^Vx- 5 = 3. 2. V4:X-7 = 5. 3. 7 - Vx-4: = 3. 4. 2 |/5x + 4 = 8. 6. VSa; - 1 = 2 Va; + 3. 6. 2 |/3 - 7a; - 3 VSx - 12 = 0. 7. Vx-\-25 = l -\- Vx. 8. V8a; + 33 - 3 = 2 V2x. 196 RADICALS AND SURDS, 9. Vx- 10 - Vx- V^-^Vx = 5. 10. - V25 4- 9 -4 + 3 = x^'dVx. 11. Vx + 11, 12. V9a; - 8 = 3 |/a; + 4 - 2. II. 13. Vx -\- Aab = 2« + Vx, 14. Vx + V4:a -\-x = 2Vb-\-x. 15. "^x^ + ^4:r2 + x + Vfx^'-\-'V2x = 1 ^ x. 16. '^« + Vax — Va — Va — Vax. 17. Vx -\- Vax — a — 1. 18. 19. 21. Vs-c + '' .= Vbx^ 6. ' Vsx + (j 4^- _ 237 - 4 + 10:?; 4/^ • Va- 0^ — T- — ^ Va- - a; Vx- a 7^+3 = :c-4 i^^+2 * Vx -\-Va-x. V,T- -!/«-:?; = 155. Reduction of Radical Equations by Rationaliza- tion. — When a radical equation contains but one radical fraction, it is often best to rationalize the denominator of that fraction before clearing of fractions. RADICALS AND SURDS. 197 Va-\-x-\-Va — X e.g. '_ — , = h. ra-\-x — Va — x Rationalizing the fraction, we get 2a 4- 2 Va^ - x^ , a 4- Vif - x^ = b, or = h. 2x X Clearing of fractions and transposing, we get a — bx -{- Va^ — x^ — 0. Multiplying by the conjugate, we have «2 _ 2abx + b^x^ - «2 -\.x^ = 0, or (Z>2 + l)a;2 = 2abx. .-. {b^^l)x =z2ab. _ 2ab ''' "^ ""F+~l* EXERCISE XCI. Solve the first four of the following equations by ration- alizing the denominator: I. V3 + X A- V'S - X , V6 + ^ + |/6 — = 4-. 2 ^ — _= \/S -^ X — V3 — X V6 i- ^- — V6 Vx-^^-{-Vx ^ Vx^^-\-Vx _ • ;= = 5. 4. , -j^— lU. |/a; _^ 4 _ ^x Vx-\-Q — Vx II. l^a; + « + i^ Vx + fl'- -|/^ -|/^ 1^2 + 2: 7 V2-\-x + Vx 12' 6. 1^2 + a; - l^a; _ 5_ V2~-^ -i- Vx 9 ' V2-\-x — Vx _b_ V2-\-x-]- Vx ~ ^ CHAPTER XVII. THE INDEX LAW. 156. Meaning of Fractional Exponents. — It has been shown that, when 7n and n are positive integers, or xa'' = or + ''. (1) Also as a corollary to this, when m> n, or ^ a"" = ar-"". And as a consequence of (1) it has been shown that {ary = «»»« = (a^^)^, (2) and («^)" = a'^b^ (3) These three laws follow from the definition that an ex= ponent denotes the number of times a quantity is employed as a factor. The law expressed by equation (1) is known as the In- dex Law. The definition of an exponent becomes meaningless if the exponent, or index, be other than a positive integer. The spirit of algebra is to generalize, and the use of indices cannot be restricted to the particular case of inte- gers, but it must be extended to the case of fractional, zero, and negative indices. All of these indices must be governed by the index law, and they must be interpreted in accordance with this law. We will proceed first to find the meaning of a fractional index in which both numerator and denominator are positive integers. 198 THE INDEX LAW. l90 Let this index be denoted by - . q Since the equation a"^ . a" = (fi + n jg ^^ -^^ ^^^^ ^^^ ^y[ values of m and n, we may replace each by — . We then have P P 2p and multiplying each member by a**, we get and so on up to q factors, when we should have P_ P_ P_ OP a*^ , a*^ . a*^ . . . . q factors = a'^ = a^» . •. (aT) = a». Therefore, by taking the g'th root of each member, we have p_ p_ or, in words, ft ** is equal to ^'the 5'th root of a to the ^th power." If ^ = 1, we should have 1 or ft" is equal to the nth root of a. For the present the meaning of the symbol «»» must be restricted to the real nth. root of a whose sign is the same as the essential sign of a, or to what may be called the arithmetical root of a. If this strict limitation is departed from, we are led to various paradoxes. e.g. By the interpretation of fractional indices 200 THE INDEX LAW. But icV2 z= x^, which is right if we take x^f^ to stand for the positive value of V^', but leads to the paradox x^ = — x^ if we admit the negative value. Again, according to the index law, and (9V2)2 ^ (92)1/2^ or (±3)2 =±9, or 9 = ± 9, if both values are admitted. 157. Meaning of Zero Exponent. — Since al^ , a^ = oT -^ '^ is to hold for all values of 7n and n, we may replace 7n by zero. We then have aO.«" = «« + " = a". Therefore, by dividing each member by a", we get ^" 1 a" Therefore a quantity with zero index is equal to 1. 158. Meaning of Negative Exponents. — Since a^ . «"= ^m + n ^g ^Q ^^o\^ for all values of m and ?^, we may replace m by — n. We then have Therefore by dividing each member by a" we get fi~^ — "" -a-- or' Also, dividing each member by a~" we get a" = a~" a THE INDEX LAW. 201 Hence a quantity with a negative exponent is equal to the reciprocal of the same quantity with the corresponding positive exponent. Cor. Any factor may he transposed from the denom- inator to the numerator of an expression, and the reverse, hy simply changing the sign of its exponent. 159. The Index Law holds for all Rational Values of w and n. — Now that we have found what, in accordance with the index law, indices must mean for all rational values of m and n, we must show that, with these meanings, the three laws ft'" . a« = «»"+«, (1) {cd^Y = a^^^ (2) and («^J)" = a"^>" (3) must hold for all rational values of m and n. I. To show that a"^ . a"" = «""+* for all rational values of w and n. t) r 1°. Let m and n be any fractions — and -, in which j3, q, r, and s are positive integers. Then a^'^ . a''" = ^aP • f a^, by definition; = YaP' .7a^, by 152; y^ps+rq^ by 132; pa+rq = a ^^ , by definition — ^plq+rfs __ ffti+n If either m or 7i is a positive integer while the other is a fraction with positive integers for its numerator and de- nominator, the integer may be expressed in a fractional form, and the demonstration just given will hold. 202 THE INDEX LAW. We know already that the law holds when m and 7i are positive integers. Therefore for 'all positive rational values of m and n, 2°. Let m and n be essentially positive, either fractions or integers. 11 1 Then m fi-n a ".a or or a' by definition. And if w — w be positive. and or . a'"" .«" = «"*. — . «" = a"». a" . *. a . a-"", a"" = a"*"" . a". Hence if m — w be negative, that is ?^ — m be positive. Therefore for all rational values of m and n «,«*. «"■= «*" + ". Cor. Since «*""''. «" = r?"" for all rational values of m and n, it follows, by dividing both sides by a^, that ^m _^ ^n -. f^m-n f qj, ^jj ^atlonal valucs of m and 7i. II. To prove that («"*)" = a"*" for all rational values of m and n. 1°. Let m have any value whatever, and let 7i be a positive integer. THE INDEX LAW. 203 Then, by definition, {cry = or .or ' a"^ . - 'to n factors = oT''. 2°. Let m have any value whatever, and let ti be a fraction -, in which p and q are positive integers. Then {aTy^ = ^{oTY, by definition; = ^a-^byII, 1°; mp = tt 3 , by definition; = a"*' 7 = «"»". 3°. Let n be any rational negative quantity and = —i?. ThenK)-^= (^ = J^p = «-""• We know already that the law holds when m and n are positive integers. Hence for all rational values of ?n and n III. To prove (aby = «"Z>" for all rational values of 7i. 1°. Let n be any positive rational quantity which may be denoted by a fraction -, in which p and 5' are positive integers. Then (aby = (ab)^ = ;{/{abY, by definition; = ^a^b^, by (2). 202 THE INDEX LAW. We know already that the law holds when m and n are positive integers. Therefore for "all positive rational values of m and 7i. 2°. Let m and n be essentially positive, either fractions or integers. Then a""* . a"" = J- . 1- = -^^ = a-"*-*^. by definition. And if m — ^ be positive. and «*".«-".«" = «"*. — . «" = a™. a"* .-. a . a-"", a"" = a"*-", a". . •. a"^ . " for all rational values of 7i. 1°. Let n be any positive rational quantity which may be denoted by a fraction -, in which p and 5' are positive integers. Then (aby = (ah)^ = ^{aby, by definition; = ^a^h^, by (2). ^04 THE INDEX LAW. Also, {a'^lf'Y for all values of 7i = fl^"^" . ft"Z»^ . a'^h'' ioq factors — aJ^ . oT' . a"" . . ,io q factorsX ^'' .&".&"... to §* factors . •. a^'lf = ^«"«^»««. But, since n = -- or nq = p. Therefore for all positive rational values of n (aby = a''b\ 2°. Let n be any rational negative quantity and = — p, p being a positive integer. Then w = {"i}-' = p)-. = i= »-" • "-" = «"*"• We know already that the law holds when m and n are positive integers. Hence for all rational values of n (ahy = a/'b\ EXERCISE XCII. I. Find the values of: 1. 642/s. 2 16-8/2^ 3 25-1/2. 4. Q-f. 5. (100000)-/= e. (4)"^. THE INDEX LAW. 205 Simplify : 9. {a-y%-^)-\ 10. {d'by'')-yK Express with fractional or negative indices : 11. Va-\- Vh -\- \^x^. 12. V^^ + ^^ay\ 13. 4/«V + VaY- 14. VxYz^ + VaV. Express without fractional or negative indices : 16. xV^ - z-^ 16. a-^-y^ 17, ^3/4J-2 _ ^-3/4J2, 18 a-SJ-2/3 + 3^1/3^,-3/4. Multiply : I. 19. a;2/5 _^ y2/5 ^ ^2/5 _ y2/5^ 20. 1 + a;V5 -f a;2/5 by 1 - a;V5. 21. «V2 _ ^1/4^,1/4 _|_ Jl/2 by «l/4 4. ^1/4. II. 22. icVe _ a;V6 _|_ :^;i/2 _ ^1/6 _|_ ^-1/6 _ ^-3/6 i^y ^^/e _|_ 23. x^ + :z;3/2 + 1 by ^"^ + x-^/^ + 1. '^*- ^^ - 3^^^'^^^'^ + i«^'^^^'^ - i' ^^ r^'^ 4 Divide : I. 25. x^ — y^ by ^/^ — y^ '^. 206 THE INDEX LAW. 6w 6n 2n 2n 26. x^ —y^ by x'"* —y^. 27. x^ + y^ by 2;V3 .f ^4/3_ 28. x^ + 32?/5/^ by x^/^ + 2«/V4. II. 29. x''/^ - 2 + ^-V3 by .^2/3 - x-y\ 30. «'/2 - X by «Vio _ ^4/5, 81. a;^/^ — 2;^^/^ + xy^y — 3/^/^ by a;V2 _ yV^, CHAPTER XVIII. ELIMINATION. 160. Simultaneous and Independent Eq[uations. — Two or more equations are said to be simultaneous when they are satisfied by the same values of their unknown quanti- ties. The equations are independent when one cannot be de- rived from the other. When an equation contains two or more unknown quantities, an indefinite number of values of their quanti- ties may be found which will satisfy the equation. e.g. Let 3x + 4?/ = 18. Transpose the term containing y and solve for x, and we have ^- . 3 • If in this result we put ?/ = 3, we get 18-12 ^ and if we put ^ = 4, we get 18 - 16 2 From this it appears that when an equation contains two unknown quantities, it can be satisfied by an unlimited number of pairs of values of these quantities, for by assign- 807 208 ELIMINATION. ing any value whatsoever to one of these quantities we ob- tain an equation from which the other may be found. In general terms, if ax -{- hy -\- c = 0, we may give y any value m. Then ^^ ax -\- hm -f- c = 0, hn + c x= — . a The values y — m, and x— ^^, evidently satisfy the given equation. That is, in an equation of the first degree in x and ?/, to every value of y there is a correspond- ing value of X which will satisfy the equation. 161. Two Unknown duantities require two Inde- pendent Equations for their Solution. — If, however, we have two independent equations in x and y, of the indefi- nite number of pairs of values of x and y which will satisfy either equation alone, there is only one pair which will satisfy both. To obtain this pair of values, we may solve each equa- tion for the same letter, and put the resulting values equal. e.g. Let 3a: + 4?/ = 18, (1) and 2:c+5«/=:19. (2) From (1), we have x — — ^, o and from (2), x — — ^. Now as we are seeking the value of x, which is the same in both equations, we may put 18-4y _ 19-5y ELIMINATION. 209 As this is a simple equation of the first degree in y, we may solve it for y, and then find the value of y which will give the same value of x in the two equations. Solving (3) for y, we obtain y = 3. Substituting this value of y in (1), we get 3:?; + 12 = 18. .-. 3:^^ = 6, and X = 2. The same value of x would have been obtained had we substituted the value of y in (2). 162. Elimination. — The general method of solving si- multaneous equations of two or more unknown quantities is to get rid one after another of all the unknown quantities but one, so as to obtain an equation containing that un- known quantity alone; then to find the value of this quantity from the resulting equation, and afterwards of the remaining unknown quantities by substitution. The process of getting rid of the unknown quantities is called elimination. 163. Three Methods of Elimination. — There are three general methods of elimination, known respectively as the methods by comparison, by substitution, and by addition or suMradion. The first has been illustrated already. It consists in finding the value of the same unknown quantity from each of the two equations, and putting their values equal to each other. e.g. 2a; + 3^ = 19; (1) • - 3x-\-'ly = 16. (2) From (1), X = — --^-, 210 ELIMINATION. and from (2), x = -— ^ . o 19 - 3y _ 16 - %y 2 ~ 3 ' or 57 - 9?/ = 32 - 4.y, or 5«/ = 25. .-. y= b. Substituting this value of ?/ in (1), we get 22; + 15 = 19; x = 2. The second method consists in finding the value of one of the unknown quantities from one of the equations, and substituting that value in the other. e.g. 2x-{-'dy = 19; (1) Sx + 2y = 16. (2) From (1), we obtain x = ^r — -. Substituting this value for ^ in (2), we get "V^*^ 2 ^^ y = 16, or 57 -9.V + 4^: = 32, or by = 25. *• y = ^' Substitute this value in (1) or {% and we find x= 2. The third consists in multiplying each of the equations by some number which will make the coefiicients of one of ELIMINATION. 2 1 1 the unknown quantities the same in both, and adding the equations when these coefficients have opposite signs in the two equations, and subtracting the equations when the coefficients have the same signs in both. e.g. 2a: + 3?/ = 19, (1) dx-\-^ = 16. (2) Multiplying the first equation by 3 and the second by 2, we have Qx -{- 2y = 57, (3) and 6:r + 4?/ = 32. (4) Subtracting (4) from (3), we get by = 25. .-. ^ = 5, and X = 2. The third method is the one usually employed, and the first is least used. The student should, however, be familiar with the use of all three. EXERCISE XCIII. Solve the following equations by each of the three methods : I. 3:c + 4:X- y= 9, %y= 2. 2. bx- %y= 5, 2a; + y = 11. %x- %x- 6?/ = 10, 72/ = -> 3. 4. Ix + lly = 17, 2x - by = 13. 9x- 3^ + 5^ = - 1, 6?/ = 15. 6. 11a; + ly=- 5, 4a; - by^- 32. %x-\- 7x-\- 2/= 4, 8i/ = - 13. 8. 8a; - 1/ = - 6, a;+ By =-17. 212 ELIMINATION. 9. 14a; — dy — 45, lo. 6x — 7y — 0, 6x + 17^ =1. 7x-\- 5ij = 74. ' Solve the following by any method of elimination : 11. -3- + 2/ = 10, 12. 2x - ^—j- = 4, , y ^ o r^ ^ — ^ ^+1= 5. 3^ = 9--^. 13. Find the first four terms of the square root of 1 — a:. 14. Find the cube root of cc^ — ^x^ + 15^;* — 20:^3 + 15x^ -6x-\-l. II. 3x^1/ , „ 2x —Sy Sx —by 1 ^_X=s :r + 4y 5.T - 4y _ 3 8 11 "^ 7 - 17. -yC^ + 2/) = 5(^ - «/)^ —(a; + 2/) = 35-(^ - «/) - y 18. :?:(2/ + 7) == y{x + 1), 2:^; = 3^/ - 19. 4 5 2 ^3~ - ^^ 2 • 20. «.'^ = %, 21. :^ + y = h X -\- y — c. ax -\- hy — c. 22. ic + ?/ = a + J, 23. a; +?/ = « + ^, X -\- a _ h ax -\- hy = a;^ -\- h^. y -\-h ~"^' ELIMINATION. 213 a h X A- a , b ax+ by = c. y -\- b a EXERCISE XCIV. Solve the following problems by two unknown quan- tities : Ex. 1. Find two numbers whose sum is 17 and whose difference is 3. Let a; = the larger number, and y = the smaller number. Then x + y = 15, (1) and X - y = 3. (2) Add equation (2) to equation (1), and we get 2x = 18. .-. x= 9. Subtract equation (2) from equation (1), and we get 2y = 12. .-. y= Q. Hence the numbers are 9 and 6. 2. Find a fraction such that when 5 is added to its numerator and 2 is added to its denominator, its value is 3/4; and if 1 be subtracted from its numerator and 5 be subtracted from its denominator, its value is 3/5. Let X = the numerator, and y = the denominator. Then — {— = -, y^2 4 214 ELIMINATION. X - 1 3 and k~'k- y - b 5 Clearing of fractions, we have 4a: + 20 = 3«/ + 6, or 4:X - dy = - 14, (1) and 6x - 6 = 3y - 15, or 5x - Sy = - 10. (2) Subtracting (1) from (2), we get X = 4:. .'. 16 - 3^ = - 14, or dy = 30. .-. y = 10. Hence the fraction is 4/10. 3. There is a number composed of two digits. The sum of the digits is 7, and if 9 be added to the number the digits will be reversed. Let X = digit in the tens^ place, and y = digit in the units' place. Then the number is 10a: + y. When the digits are reversed the number is lOy -\- x. Then x -\- y = 7, ^ (1) 10x + y + 9 = 10y + x, or 9a; — 9i/ = — 9, or X — y = — 1. (2) Adding (1) and (2), we get %x = 6. .\ X = 3, ELIMINATION. 215 Subtracting (2) from (1), we get 2^ = 8. .-. y = ^> Hence the number is 34. I. 1. The sum of two numbers is 8 and their difference is G. What are the numbers ? 2. There is a certain fraction, such that if its numer- ator be increased by 4, its value is 4/5 ; and if its denom- inator be increased by one, its value is 1/2. What is the fraction ? 3. A certain number of two digits is equal to five times the sum of its digits, and if 9 be added to the number, its digits will be reversed. 4. A number consists of two digits whose difference is 1 ; if it be diminished by the sum of its digits, the digits will be reversed. What is the number ? 5. Eight years ago A was five times as old as B, and in two years he will be three times as old. What are their present ages ? 6. A alone does 3/5 of a piece of work in 30 days, and then with B's help finishes it in 10 days. In what time could each do it alone ? II. 7. A man buys 8 lbs. of tea and 5 lbs. of sugar for $2.39 ; and at another time 5 lbs. of tea and 8 lbs. of sugar for $1.64, the price being the same as before. What were the prices ? 8. Two vessels contain mixtures of wine and water. In the first there are three times as much wine as water, and in the second five times as much water as wine. How many gallons must be drawn from each vessel to fill a third, which 216 ELIMINATION. holds 7 gallons, with a mixture which shall be half wine and half water ? 9. Two vessels contain mixtures of wine and water. In the first there are 4 gallons of wine to 3 gallons of water, and in the second there are 5 gallons of water to 2 gallons of wine. How many gallons must be drawn from each ves- sel to fill a third, which holds 12 gallons, with a mixture which shall be 1/3 wine ? 10. A man buys 2 lbs. of tea and 6 lbs. of sugar for 81 cents, and at another time 4 lbs. of tea and 9 lbs. of sugar for $1.51|, the price being the same as before. What were the prices ? 164. To Solve for n Unknown Quantities requires 7i In- dependent Equations. — We have seen that we need two si- multaneous equations in order to find the value of two un- known quantities. Similarly, we need three independent simultaneous equations in order to find the value of three unknown quantities, and n independent simultaneous equations in order to find the value of 7i unknown quanti- ties. With three unknown quantities, we first combine any pair of the three equations so as to eliminate one of the un- known quantities, and then another pair so as to eliminate the same unknown quantity. We shall then have two equations with two unknown quantities. Then we combine these two equations so as to eliminate one of the remaining unknown quantities, and thus obtain one equation with a single unknown quantity. From this we obtain the value of this quantity, and then, by successive substitution, the values of the other two. e.g. Qx-{-%y -bz = 13, (1) ^x-\-^ -2z = 13, (2) '7x-\-by -^z = 26. (3) ELIMINATION. 217 Eliminate y from (1) and (2) by subtraction, multiply- ing (1) by 3 and (2) by 2. 18a; -\-Qy - 15z = 39, 6x-i-6t/ - 4.z = 26. .-. 12a; -11^ = 13. (4) Next eliminate y from (1) and (3) by subtraction, mul- tiplying (1) by 5 and (3) by 2. 30a; + 10?/ - 26z = 65, 14a; + Wy - 6z = 52, .-. 16a; -19^ = 13. (5) Next eliminate x from (4) and (5) by subtraction, mul- tiplying (4) by 4 and (5) by 3. 48a; - 44^; = 52, 48a; - 57z = 39. .-. 132=13, and z = 1. Remember that the equations may be combined in any order, and that those combinations are best which will pro- duce the required result in the simplest and most direct way. EXERCISE XCV. x-Jr2y-^2z = 16, 2a; -f 1/+ z = n, 3a; + 42/+ ^ = 22. 2. a; + 3^/ + 4^ = 7, a; + 2^+ z = 0, 2a; + 2/ + 2^ = 6. X + 4.y + 3^ = 14, 3a; + 3^+ 2 = 21, 2a; + 22/+ z = Id, 4. 3x-2y-\- z = 10, 2a; + 3«/ + z = 18, 218 t:LIMINATI0n. 5. Sx-^4:y = 0, 6. 5:?; + 2^ = 8f , 2y — 4:z= — 14, Sz-y = 1|, x-i-dy-\-2z=- 1. 8a; - lOz = 3f. II. 7. .-1 = 12, y + z z + x x-\-y ^ 5 - 4 - 3 ' ^-1=14, X + y-\-Z = 18. .-1=15. 9. % =% =5.- -3x, 10. a; + 16 = ^ + 14 2^ + 2 = 32; - 3. = 3^ + 9 n n ''[^ n n 11. Multiply 3rr2 + ^^^ — 5^* ^J ^^ — 2a;*. 6n 6to, n m 12. Divide a; 2 — a;^ by x^ — x^, 13. Square 2a;V3 _ 3^2/3 _j_ 4^^ Note. — When tliere are more than three unknown quantities, the process of elimination is similar. EXERCISE XCVI. Work the following examples by three unknown quan- tities : I. 1. The sums of three numbers, taken two by two, are 20, 29, and 27. What are tlie numbers ? 2. The sum of three numbers is 78, 1/3 the difference of the first and second is 4, and 1/3 the difference of the first and third is 7. What are the numbers ? 3. A person bought three silver watches. The price of ELIMINATION. 219 the first, with 1/3 the price of the other two, was 40 dol- lars, the price of the second, with 1/4 the price of the other two, was 42 dollars, and the price of the third, with 1/2 the price of the other two, was 44 dollars. What was the price of each watch ? 4. A, B, and C together have $2100. Were B to give A 300 dollars, A would have 380 dollars more than B, and if B received 200 dollars from C, they would both have the same sum. How many dollars has each ? 5. A, B, and C can perform a piece of work in 20 days, A and B in 30 days, and B and in 40 days. How long would it take each to do it alone ? 6. A and B together can do a piece of work in 6 days, B and in 6f days, and A and in b^^ days. How long would it take each to do it alone ? 7. A number is composed of three digits whose sum is 9. The digit in the units' place is twice the digit in the hundreds' place, and if 198 be added to the number, the digits will be reversed. What is the number ? 8. A number is composed of three digits whose sum is 10. The middle digit is equal to the sum of the other two, and if 99 be added to the number its digits will be reversed. What is the number ? 9. A number is composed of three digits whose sum is 14. Seven times the second digit exceeds the sum of the other two by 2, and if the first and second digit be inter- changed the resulting number will be less than the given number by 180. What is the number? II. 10. A and B can do a piece of work in r days; B and C in s days; and A and C in / days. In how many days can each do it alone ? / 220 ELIMINATION. Do the following by two unknown quantities: 24. A crew can row 10 miles in 50 minutes down stream and 12 miles in an hour and a half up stream. What is the rate in miles per hour of the stream, and of the crew in still water ? Let X — the rate in miles per hour of the crew in still water, and y = the rate in miles per hour of the current. .*. X -\- y = the rate in miles per hour of the crew down stream, and X — y = the rate in miles per hour of the crew up stream. Since the number of miles rowed, divided by the rate in miles per hour, is equal to the time in hours, we have 10 5 x-^y 6' and 12 3 x-y- 2' .*. X = 10, and ?/ = 2. 25 A crew can row 20 miles down stream in an hour and 20 minutes, and 18 miles up stream in 2 hours. What is the rate of the current in miles per hour, and what is the rate of the crew in still water ? 26. Two trains start from two stations at the same time, and each proceeds at a uniform rate towards the other station. They meet in twelve hours, and one has gone 108 miles farther than the other, and then if they continue to travel at the same rate they will finish their journey in 9 hours and 16 hours respectively. What is the rate of the trains, and tlie distance between the towns ? ELIMINATION. 221 27. Two trains start from two stations at the same time, and each proceeds at a uniform rate towards the other station. They meet in six hours, and one has gone 30 miles farther than the other, and then if they con- tinue to travel at the same rate, they will finish the journey in 7 hours and 12 minutes, and in 5 hours, respec- tively. What is the rate of the trains, and what is the distance between the towns ? 28. A certain number of persons paid a bill. Had there been 10 more, each would have paid $2 less, and had there been 5 less, each would have paid $2.50 more. How many were there, and how much did each pay ? 29. A sum of money is divided equally between a cer- tain number of persons. Had there been m more, each would have received a dollars less ; if n less, each would have received h dollars more. How many persons were there, and how much did each receive ? CHAPTEK XIX. QUADRATIC EQUATIONS. A. SUED AND IMAGINAEY FACTORS. 165. Trinomial and Binomial Quadratics. — A complete quadratic exj^ression in one unknown quantity contains three terms, one containing the square of the unknown quantity, one containing the first power of the unknown quantity, and the third without the unknown quantity. The most general form of such an expression is ax^ -\- bx -\- c. The term which does not contain the unknown quan- tity is called the constant term of the expression, and the complete expression is called a trinomial quadratic. When the term containing the first power of the unknown quantity is wanting, the expression becomes a binomial, and is called an incomplete or a binomial quadratic expression. 166. Factors of x^ + c. — Every binomial quadratic of the form x^ -\- c may be factored as the difference of two squares, since it may be written in the form x^ - {- c). SURD AND IMAGINARY FACTORS. 223 The factors will be X -{- V — c and x — V — c. 1°. When c represents a positive number, these factors are imaginary. 2°. When c represents a negative number which is not a perfect square, the factors are surd. 3°. When c represents a negative number which is a perfect square, the factors are rational. e.g. 1. x^^b=x^-(-b) = (x - V'^){x^ V^), a;2+4=^-2_(_4) = (a; - V^{x + V-l) = {;x - 2 i/^)(:^:4-2 V^. 2. a;2 + (- 3) = 3-2 - 3 = {x - V^){:x + V^). 3. x^ + (- 9) = a:2 - 9 = (a; - Z){x + 3). When the expression is in the form ax^ -\- c, a may be taken out as a factor first, and then the remain- ing factor may be factored as the difference of two squares. Thus, ax^ + o = a[x^ + '-)=a(^x^-[-'-]^ e.g. 1°. 3r?;2 + 6 = 3(a;2 + 2) = 'd{x^ - (- 2)) =3 3(:r - V^^2)(a; + V^^^). r. ix^^{-^0) = 4.(x'-b) = 4:(x -f V5)(x - Vb). 224 QUADRATIC EQUATIONS. 3°. 6a;2+(-20) = 5(2;2-4) = b{x - 2)(^ + 2). = ^x - VbJ^){x + 1/573). 4a;2 4. (_ 3) := 4(a; _ |/374)(2; + VsTT) =4-irt){.+i^). EXERCISE XCVII. Factor the following quadratic expressions : 1. a;2 + 5. 2. x^ - 7. 3. a;^ + 16. 4. Zx^ - 9. 5. 5a;2 - 25. 6. '^x^ + 14. 7. 2a;2 _ 3, 8 3^2 _|_ 5^ 9 5^2 _ 2. 10. 4a;2 + 3. 11. dx^ - 4. 12. Ix^ + 5. 167. Factors of a Trinomial Cluadratic. — Every tri- nomial quadratic expression may be factored as the differ- ence of two squares. We first take out the coefficient of the square of the unknown quantity, and after the second term of the ex- pression we add and subtract the square of half the coef- ficient of the first power of the unknown quantity. This will give a polynomial of five terms, the first three of which will be a perfect square. The last two terms must be com- bined into one with a minus sign before it. The factors will both be real when this term is essentially positive, rational when it is an exact square, and surd when it is not SURD AND IMAGINARY FACTORS. 225 an exact square. The factors will both be imaginary when this last term is essentially negative. e.g. 1°. Factor ^x^ + 15a; + 18. First, we have Zx^ + 15:^; + 18 = ^x^ + 6x -\- 6). Then, after the second term of the second factor, add and subtract (5/2)^, and we get ('+l+l)('+l-8 . •. 3a;2 + 15^; + 18 = ^x + 3)(a; + 2). 2°. Factor ax^-\-bx^c. First, ax^ -{- hx -{- c = aix^ A — x A — ). \ a al Then ^ + ^ + ^=-' + ^ + £-£ + „- = x^ -^-x^ — - ^' ~ ^^^ a 4«^ 4a^ h , Vl>^ - 4:ac\f , b Vb^- 4.ac' = ("+2^+ 2^— jl^ + 2^- / , b + Vb^- 4:ac\/ , 2« b- Vb^- 4ac 2a ax^ -\-bx -\- c b^-S/W - 4.ac\f , b- Vb^ - 4:ac\ = T + 2a k + 2a Whether these factors be rational, surd, or imaginary depends upon the radical Vb'^ — 4ac, 226 QUADRATIC EQUATIONS. If the quantity under the radical be positive, the factors will be real. If also the quantity under the radical be a perfect square, the factors will be rational; and if this quantity be not a perfect square, the factors will be surd. If the quantity under the radical be 0, the factors will be equal. If the quantity under the radical sign be negative, the factors will be imaginary. Since ax^ -\-'bx-\- c is the general form of a trinomial .quadratic expression, I , ^ + y^^ ^(^c \( , h - Vh^- 4:ac \ may serve as a formula by which all such expressions may be factored. e.g. Factor ^x^ -f- 4a: + 5. Comparing this with ax^ -\-bx-\- c, we see that a = 3, b = 4:, and c = 5. Substituting these values in the formula, we get 4 + VU - 60\/ , 4 - Vl6 - 60^ ^^+— — ^ — U^+ or or In this case the binomial factors are imaginary. EXERCISE XCVIII. Factor the following trinomial quadratic expressions by the formula: I. 1. 4:X^-\-7x-6, 8, 2x^-]-6x + 2, ROOTS OF AN EQUATION. 227 3. 6x^ — ^x — 7. 4. 'ox^ — ^x — '6. 5. ^x'^'dx-\-Q. 6. 2:^2 + 10a; + 8. 7. A man bought 175 acres of land for 6000 dollars. For a part of it he paid 40 dollars an acre, and for the remainder 25 dollars an acre. How many acres in each part ? II. 8. 7;?;2 + 9a; + 2. 9. lx^-\-'^%x- 7. 10. ^x^ + 7a; - 6. 11. 4^2 - Mx + 12. 12. Ux^ -\- x-Q>. 13. 3a;2 - 10.C + 6. 14. A man bought m acres of land for s dollars. For a part of it he paid a dollars an acre, and for the remainder 1) dollars an acre. How many acres were there in each part ? V4a; + 1 + 2 Vx ^ 15 Solve — =z=r — = 9. V4:x + 1 - 2 Va; Vx -\- a 4- Vx 16. Solve — ^=^= — : = c. Vx 4- a — Vx B. ROOTS OF AInT EQUATIOI^. 168. Quadratic Equations. — A quadratic equation of one unknown quantity is an equation whose first member is a complete or an incomplete quadratic expression in that letter after the equation has been reduced to its simplest form and all its terms have been transposed into its first member. After reduction and transposition the equation takes either the form ax^ -{-hx -^ c = (1) or ax^ 4- c == 0. . (2) 169. Boots of an Equation. — A root of an equation is a value of its unknown quantity which reduces its first 228 qUADBATIG EQUATIONS. member to zero, after it has been reduced to the form of (1) or (2). 170. Solution of a Quadratic Equation. — To solve a quadratic equation is to find its roots, or the values of its unknown quantity which will reduce to zero the first member of the equation after it has been brought into its type form. Since a product is zero when any one of its factors is zero, the values of its unknown quantity which will reduce to zero the factors of the first member after it has been brought into its type form are the roots of the equation. Hence, to solve a quadratic equation, reduce it to the type form, factor its first member, equate each factor to zero, and solve for its unknown quantity. e.g. Solve a:^ — 6x = — 8. Reduced to the type form this becomes x^ - Qx -\- S = 0, or (x - ^)(x - 4) = 0. Put X -% = 0, and we have a; = 2. Put X- 4 = 0, and we have a; = 4. Hence 2 and 4 are the roots of the equation, for either of these values of x will reduce the first member of the type form to zero. We have seen that every quadratic expression in one letter may be resolved into two factors of the first degree in that letter. Hence every quadratic equation has two roots. Moreover a product cannot vanish unless one of its factors vanishes. Therefore a quadratic equation has only two roots. These roots will be rational when the factors of the first member of the reduced form are rational, and ROOTS OF AN EQUATION. 229 equal when the factors are identical; surd when the factors are surds; and imaginary when the factors are imaginary. e.g. 1. Solve x^ — Qx = — ^. When reduced to the type form this becomes a:2 _ 6^ + 9 = 0. .*. {x-d){x-'d)=Q. Therefore the roots are 3 and 3, and are rational and equal. The roots of a quadratic equation are equal when the first member of the reduced form is a perfect square. 2. Solve x' - lla: = - 28. Transposing, we have x^ - 11a; + 28 = .-. {x-^){x-l) = 0. .'. a: = 4 or 7. Therefore the roots of the equation are 4 and 7, and are rational and unequal. 3. Solve x^-4:x + l = 0. Bring the first member of this equation under the case of the diiference of two squares by adding and subtracting the square of half the coefficient of x, and we have iK2 - 4x + 4 - 3 - 0. .-. (x-2-\- V3)(x -2 - V3) = 0. ,', a: = 2 - V3 and 2 -f V3. Therefore the . roots of the equation are 2 — V3 and 2 + V'd, and are surd and unequal. 4. Solve a;2 - 6:r + 11 = 0. 230 QUADRATIC EQUATIONS. Bring the first member under the case of the difference of two squares by adding and subtracting the square of half the coefficient of x, and we have a;2- 6:?; + 9 - (- 2) = 0. (a; - 3 + y - 2)(« -'d- V ~ 2)=0. . •. a; = 3 — |/ — 2 and 3 + V — 2. Therefore tlie roots of the equation are 3 — i^ — 2 and 3 4- 'Z — 2, and are imaginary. EXERCISE XCIX. Solve the following quadratic ( 3quations by factoring: 1. a;2 - 32; - 18 = 0. X. 2. x^ -\- 4:X = 45. 3. x^ + 13a; + 25 = - 15. 4. a;2 - 12a: - 5 == - 40. 6. a;2 4- 4x + 20 = 4 - - 4a;. 6. x^ — 5x = 5x — 25. 7. a;2 - 3 = 6. 8. II. a;2 - 2«2 ::.: _ «2. 9. x^-{-{a-^'b)x-\-al)^0. 10. a;2+(«-J)a;-«J=0. 11. 2a;2 4- a; - 3 = 0. 12. 3a;2 + 5a: = 12. 13. 15a;2 + 14a; = 8. 14. 7a;2 + 15.T = - 8. 15. 12 + 2a;2 == 11a;. 16. - 3x^ + 17a; = 20. 171. Formation of Quadratic Equations. — Since we ob- tain the roots of a quadratic equation by equating to zero each factor of the first member of its type form, it follows that these factors are the unknown quantity of the equation minus each of its roots in turn. Hence we may obtain a quadratic equation in x whose roots shall have given values by using as factors x minus HOOTS OF AN EQUATION. ^31 each of the given roots in turn, finding the product of these factors, and equating this product to zero. e.g. 1. Form the quadratic equation in x whose roots are 4 and — 7. The factors of the first member of its type form will be {x — 4) and {x + 7). ... (^_4)(.'^ + 7) = 0, or a;2 + 3a;-.28 = 0, which is the required equation. 2. Form the quadratic equation in x whose roots are 3 + V5 and (3 - Vb). Here the factors are x — (3 + i^5) and a: — (3 — V6). ... (2; - (3 + V^)){x - (3 - 1/5)) = 0, or a;2 - 6a; + 4 = 0. EXERCISE C. Form the quadratic equations in x whose roots have the following values : I. 1. 3 and 7. 2. 4 and — 6. 3.-7 and — 1. 4. and 2. 5.-9 and 0. 6. 7 and — 7. 7. —8 and— 8. 8. 11 and 11. 9. 3 and 3/4. •3 + 4/7 ,3-1/7 7+4/^5' ,7-4/5 ,0. -^_ and — ^. 11. -^- and —-. 12. 4 + 4/- 6 and 4 - 4/- 6. II. 13. - 2/3 and - 5/6. u. 3/2 and - 1. 15. 7 and - 2/5. 16. 3 + 4/5 and 3 - 1^, 232 QUADRATIC EQUATIONS. 17. 2 + l/8~and 2- l/8. is. 5 + ^3 and 5 - l/3. 19. 9 + y- 4 and 9 - y- 4. 7 + 1/3-3 7 _ |/3^ 20. _^^— and— ^^ . „, 11 + 4/"=^ , 11 - V^l 21. 1^— and j^_. I. 22. Reduce — — ^ -| — to a single negative fraction. 23. Reduce , — ~ — a; to a single fraction. ic + 2 ^ 22;2 ^x 24. Reduce 2x — - to a sinde fraction. EXERCISE CI. 1. (a: - 2)2 - 1 = |(:r + 2). 2. 2.^2 _^ 2(a; + 1)2 =: :^2r(a; + 1). 13 3 3. (2 - xf - (2 - x)(x - 3) + (a^ - 3)2 = 1. 4. ^ + i = 4i. 5. ^^+-^ = 2i. a; + 2 2; + l _ 26 4 3 _ 17 ®' a: + l + « H- 2 ~ y' "'^ ic- 3 ~ a; + 5 ~ 10* II. HOOTS OF AN EQUATION. 233 4.x -d ., , 2a; - 3 x-1 x-\-l 6x 10. w- ^ = 3 + -. 11. — — : + 3a: - 7 'x-1' " ' x+1^ x-1 x^ -1' ^x-1 o^ + 1 _ « 2x-l 13 _ 3a; + 5 EXERCISE Cll. 1. Solve ^{3 - 4:x) + 4/(2 + 5x) = ^(5 + x). Transposing, we have |/(3 - 4.x) + |/(2 + 5a;) - |/(5 + x) = 0. Multiplying by the conjugate, we have 3 _ 4^ _}_ 2 |/(3 - 4a;) V(2 + 5a;) + 2 + 5a; - 5 - a; = 0, or 2 |/(3 - 4) 4/(2 + 5a;) = 0. .-. |/(3 - 4a;) |/(2 + 5a;) = 0. .-. (3 - 4a;)(2 + 5a;) = 0. .-. a; = 3/4 and - 2/5. 2. i/(5 - 7a;) + |/(4a; - 3) = |/(2 - 3a;). 3. Vi^ + «) + V(^ -^) = V(^^ -{-a-b). 4. |/(3 + 4a;) - i/(4 + 2a;) = |/(7 + 6a;). a. 4/(2 - 3a;) - |/(7 + a;) :=. 4/(5 + 4.x), 6. V(a;2H-3a;-54)- 4/(a;2-3a;-54)= |/(2a;2 -108). II. 7. 4/(a;2+4a;-60)- |/(a;2-4a;-60)= ^{2x^-120). 8 |/(12a;2-a;-6)- |/(12a;2 + a;-6)= i/(24a;2-12). 9. 4/(36a;2+24a;+l) + V(36:i;2-24a;+l) = t/(72a;2+2). 234 QUADRATIC EQUATIONS. 172. Interpretation of Solutions. — Ex. 1. A man sold a watch for 24 dollars and lost as many per cent as there were dollars in the cost of the watch. What was the cost of the watch ? Let X = the cost in dollars. Then X = the lost per cent, and ^•ioo=ioo=i°'''°'i°"«'^- Also, X — 24t = loss in dollars. ••• 100=^ '''■ Solving this , we get x=QO or 40. That is, the cost was either 60 dollars or 40 dollars ; for either of these values satisfies the conditions of the problem. 2. A farmer bought a number of sheep for 80 dollars. Had he bought 4 less for the same money, they would have cost him 1 dollar apiece more. How many did he buy ? Let X = the number bought. 80 Then — = the price per head in dollars, 80 and J = the price per head, if there had been 4 more. 80 80 X X — 4: - 1. Solving this equation, we get a; = — 16 or + 20. Only the positive value will satisfy the condition of the problem. Therefore the number of sheep was 20. In solving problems which involve quadratics, tliere ROOTS OF AN EQUATION. 235 will be, in general, two values of the unknown quantity, both of which may not answer to the conditions of the problem. This is due to the fact that the symbolic lan- guage of algebra is more general than ordinary language. So that the equations which correctly represent the con- ditions of the oral problems may represent other allied conditions also. The equation is entirely general, while the verbal statement is more or less restricted. Verbal statements are supposed generally to be restricted to an arithmetical sense which admits only of positive numbers ; while there is no restriction on the numerical symbols of an algebraic equation. A little consideration will enable the pupil to determine whether or not both values of the unknown quantity will fit the conditions of the verbal problem, and which one to select in case both will not answer. It will be found also a valuable exercise to interpret negative results when possible. Thus in the last example, to buy — 16 sheep has no meaning in the arithmetical sense, but algebraically it means to sell 16 sheep. To buy 4 less than — 16 would mean to sell 20. In the first case he would have paid — $5 a head for the sheep; that is, he would have sold them for $5 a head. In the second case he would have bought them for 1 dollar more a head, or for — 4 dollars; that is, he would have sold them for 4 dollars a head. When one of the solutions is negative the wording of the problem may be changed, in general, so as to make that solution positive and arithmetically true. Thus, a farmer sold a number of sheep for 80 dollars. Had he sold 4 more for the same money he would have received 1 dollar a head less for the sheep. How many did he sell ? e.g. 1. The length of a field is 12 rods and its breadth is 236 QUADRATIC EQUATIONS. 10 rods. How many rods must be added to the length of the field that the area may be 100 square rods ? Let X = number of rods to be added. Then (1^^ + a;)10 = 100. 10:^ = 100 - 120. x= -2. Hence the number of rods to be added to the length is — 2. This is possible algebraically, but impossible arith- metically. In the arithmetical sense, to add means to increase; and as the area of the field at first was 120 square rods, no increase in its length could make its area 100 square rods. But algebraically, to add — 2 means to subtract 2 arithmetically; and were the statement, *' How many rods must be subtracted from the length of the field to make its area 100 square rods ?" we should find the 2 to be positive and, therefore, true in the arithmetical sense. e.g. 2. A's age is 40, and B's 35. How many years hence will A's age be twice B's ? Let X = number of years hence. Then ^0 -\- x = 2(35 + x), x= —30. This is impossible arithmetically, but perfectly true algebraically, since — 30 years hence means 30 years ago. Had the question been worded, " How many years ago would A's age have been twice B's ?'' the solution would have been positive and the problem would have been possible arithmetically. When imaginary results are obtained in the solution of a problem, there is either an impossibility in the con- ditions of the problem or an error in the formation of the equation. BOOTS OF AN EQUATION. 237 e.g. Divide 12 into two parts whose product shall be 37. Let X denote one part. Then x{l% - x) = 37. nx -x^ = 37. x^ - 12x + 37 = 0. a;2 - 12a; + 36 - 1 = 0. x-Q ± V~^^ = 0. x = Q- V -1, or 6 + y - 1. \<^ - X -^ Q -\- V - I, or 6-4/-1. That is, 12 cannot be divided into two parts whose product is 37. EXERCISE cm. I. 1. Find two numbers whose difference is 7 and whose sum multiplied by the greater is 345. 2. Find three consecutive numbers whose sum is equal to 3/5 the product of the last two. 3. Find two numbers whose difference is 12 and whose sum multiplied by the greater is 560. 4. Find three consecutive numbers whose sum is equal to 3/7 the product of the last two. 6. Find two numbers whose sum is 6 and the sum of whose cubes is 72. 6. Find four consecutive numbers such that the prod- uct of the last two shall be equal to the number composed of the first two used as digits. 7, Find four consecutive numbers such that the prod- 238 QUADRATIC EQUATIONS. uct of the last two shall be 2^ times the product of the first two. II. 8. A merchant bought a quantity of flour for 120 dollars. Had he bought 10 barrels more for the same money, the cost would have been 2 dollars a barrel less. How many barrels did he buy, and at what price ? 9. A merchant sold a quantity of wheat for 16 dollars, and the loss per cent was equal to the cost in dollars. What was the cost of the wheat ? 10. A merchant sold a quantity of cloth for 96 dollars, and the gain per cent was equal to the cost in dollars. What was the cost of the cloth ? 11. A crew can row 10 miles down stream and back again in 2 hours and 40 minutes; and the rate of the stream is 2 miles an hour. What is the rate of the crew in still water ? 12. A crew can row 20 miles down stream and back again in 7 hours, and the rate of the stream is 3 miles an hour. What is the rate of the crew in still water ? 173. Solution of the General Quadratic Equation. — The most general type of a quadratic equation of one un- known quantity is ax^-\-lx-\-c = 0. (A) If we divide through by a, then x^ 4- -X A- ~ = 0', a a and if we substitute j) for — , and q for — , the equation C(/ (t becomes x'-^px^q = 0, (B) ROOTS OF AN EQUATION. 239 which is the quadratic equation reduced to its simplest form. P If in equation (B) we add and subtract the square of ^, we get or x^ which factors into +^, + Z_Z^ = o, Therefore x = l/2(- j9 + Vp^ - ^), and l/2(-j9- Vp^ - ^). On account of the double sign of the root symbol, y", both values are included in the one expression x^l/%{-p±Vf-4.q), (1) which is the solution of (B). h c If in this equation we write - for p and — for q^ we have a ^ a " 2\ a^^ a" ar 1/ ^ , Jh^ 4«c\ or ^=o ± r -T 2"' 2 \ « a'^ of I ( h Vh^ - 4.ac \ or ^ = o . 2\ a or x= ^i-{— b ± S/IP- — ^ac), Zci which is the solution of (A). (2) 240 QUADRATIC EQUATIONS. Formulae (1) and (2), for the solution of quadratic equa- tions, should be so thoroughly memorized that the roots of any quadratic equation may be written down at sight. Formula (1) is most convenient for use when the coefficient of x^ is unity, and formula (2) when the coefficient of x^ is not unity. e.g. 1°. Find the roots of x^ -{- %x - 35. l/2(-2± 1/4+ 140), or l/2(- 2 ± 12). Hence x^ = 5, and X2= — 7. 2°. Find the roots of 2x^ -\- 6x - 12. l/4(- 5 ± 4/25 + 96), or l/4(- 5 ± |/121), or l/4(- 5 ± 11). Hence x^ = 3/2, and a^g = — 4. 3°. Find the roots of 3x^ -{- 7x - 25. l/6(- 7 ± 1/49 + 300), or l/6(- 7 ± |/349). - 7 + ^^'349 ^ - 7 - 1/349 Hence x = , and Xo = 6 ' ' 6 Whether the roots be rational, surd, or imaginary de- pends upon the radicals Vp^ — 4g and Vb'^ — 4ac. When p'^ = 4g or P = 4:ac, the roots are equal, since the radical then becomes zero. EXERCISE CIV. I. 1. x^-\-Qx-\-S = 0. 2. x^ - Ux - 120 = 0. 3. 2x^ - bx^ 25. 4. 3x^ - 17a: + 14 = 0. ROOTS OF AN EQUATION. 241 6. 7^:2 = 22a; - 15. 6. {^x - 'df = 2a; + 3. 7. a;2_|_|^i8. 8. x^-^ = l. Sx^ 4a; 1 ^ 10. (a; - 2)2 = 1 + |(a; + 2). 11 ^ a;+l a; + 2 _ • a; + l"^a; + 2'^a; + 3 12. a;2 -|- 2«a; — V^ — a^. 13. a;2 + «(1 -I- 3^')a; + 3^^^ = 0. 14. ax^ + ^(1 - a^)x = ab\ II. 16. {a - xY -(a- x){h -x)^{x- If = (a - b)\ 16. a\x — hy = li^{x — of. 17. (2a- b- xf + 9(« - ^)2 = ((« + b)- 2x)\ ,1 ,1 x , a a b 18. a; 4- - = i2 + n^)^ 3a:2 _ 7^2 ^ _ ge, o;2-9«/2= _20?i(3m -f 4'/^;. MS QUADRATIC EQUATIONS II. 6. 4:X-16=z 17 Vx. 6. x^/' + a;3/5 = 702. 7. Multiply at sight ^-f- — \- c by j- -{- c, and ex- press the result without fractions. 8. Factor 5x^ — lOax -{- Sbx — 16ab. Case 2°. JVhen one equatio7i is of the second degree and the other of the first. All equations of this kind may be solved by finding the value of one of the unknown quantities from the first-degree equation, and then substituting that value in the second- degree equation. The resulting equation will be a quadratic of one un- known quantity which may be solved. When the value of one unknown quantity has been found thus, the values of the second must be found by substituting the values of the one already found in the first-degree equation. e.g. 1. Solve the equations '6x^ — xy = ^y. (1) 2a; + y = l. (2) From (2), we have i/ = 7 — 2:r. (3) Substituting this value in (1), we get 3^2 _ ^(7 _ 2x) = 2(7 - 2a;), or dx^ -7x-{- 2^2 z= 14 - Ax, .'. 5a;2- 3a;- 14 = 0. .». {x - 2){6x + 7) =0. Whence x = 2, or x = — 7/5. Substituting these values in (3), we get y = 3, OY y = +49/5. OF TWO UNKNOWN QUANTITIES. 249 Therefore: 1. ^ = 2, 2/ = ^• 2. x= - 7/5, y = 49/5. Certain examples in which one equation is of the third degree and the other of the second degree may be solved in a similar way. e.g. 2. Solve the equations ^3 _|_ ^3 = 152^ • (1) X -\-y =^. (2) From (2), we obtain y = 8 — x. (3) By substituting this value of ^ in (1), we get ^3 _^ (g _ ^y ^ 152, or x^ + 512 - 192^; + 24:X^ - x^ = 152, or 24:X^ - 192x + 360 = 0, or a;2 _ 8a; 4- 15 = 0. .-. {x-5){x-S) = 0. . •. a; = 5, or a: = 3. Substituting these values of x m (2), we get 6 + y=8, (4) and' 3 + 2/ =8. (5) From (4), we have «/ = 3, and from (5), y = ^^ Therefore a; = 5 or 3, and ?/ = 3 or 5. 1. X = 5, y = 3. 2. X = 3, y = 6. 250 QUADRATIC EQUATtOm EXERCISE CIX. Solve the following equations : I. 1. 3a^ — xy = 2y, 2. x -\- y = — 2, 2x -{- y = 7, xy = — 24:. 5. X — y = 2, ^. x^ -{- xy — y^ = — 11, x^ -\- y^ = 34:, ^ — y = — 4:. 6. x^ -y^= - 296, 6. x^ -{- y^ = 152, X — y = — 2. X -\- y = 8. II. 7. x-y = l, xy = a^ -\- a. 2A + 3/y = 1. 9. 8^ -y^= -7, 2x-y=-l. 10. x/y-^y/x = l0/3, 3x-2y=- 12. 11. 2a;2/« + 3:cV» - 56 = 0. 12. Factor Ibax — lOx + 6«J — 4 J. Case 3°. An expression is said to be symmetrical with respect to any of its letters when any two of them can be interchanged without altering the value of the expression. e.g. The expression ab -\- be -{- ca is symmetrical with respect to the letters a, b, and c; for if any two of them, as a and b, be interchanged, the expression becomes ba -\- ac -\- cb, which is the same as the original expression in meaning. The equations x -\- y = 2, xy = 3, are symmetrical in x and y. OF TWO UNKNOWN QUANTITIES. 251 The equations x — y = a, xy= I, are symmetrical except in their signs. When tlie given equations are sym7netrical in x and y, and one of them is of the second degree and the other of the first.) they may he solved hy combining them in such a luay as to oMain the values of x -\- y and x ~ y. e.g. Solve the equations x-\- y = 1, (1) xy=-Q. (2) Squaring (1), we have x^ + '^^y + ^^ = !• (3) Subtracting 4 times (2) from (3), we get x^ — 2xy -\-y^ = 25, which is the square oix — y. Extracting the square root of each member, X — y = ± 5. (4) Adding (4) to (1), we have 2x = Q or — 4. a; - 3 or - 2. Subtracting (4) from (1), we have 2y= -4: or 6. y= -2 or 3. 1. x = 3, y=-2. .', 2. x=~2, y = d. This method may be used in many cases when the equa- tions are symmetrical except with respect to the signs of the terms. e.g. Solve the equations x^ -\- y^ = G5, (1) X - y = -3. (2) 252 QUADRATIC BQUATIOm Multiply (1) by 2, and subtract the square of (2) from the result: 2x^ + 2?/^ =130 x^ — %xy -f 1/2 = 9 a;2 •+ ^xy -^ y'^ = 121 .-. x^y=±ll. (3) Add (3) to (2), and we get 2^ = 8 or - 14. . •. X = 4: or — 7. Subtract (3) from (2), and we get — 2?/ = — 14 or 8. «/ == 7 or — 4. 1. X = 4:, y = 7. .-. 2. x^-1l, «/=-4. Certain examples in which one equation is of the third degree and the other is of the first or second may be solved by the methods of this case. e.g. Solve the equations x^ -\- y^ = 189, (1) x^-xy-\-y^^ 21. (2) Divide (1) by (2), and we get ^ + 2/ = 9. (3) Square (3) and subtract (2) from the result: x^ H- ^xy + y^ = 81 x^ — xy -\- y^ = 21 3xy = 60 .-. - xy= - 20. (4) OF TWO UNKNOWN QUANTITIES. 253 Add (4) to (2), and we get x^ — %xy -[- ?/^ = 1. ,-, x-y=±\. (5) Add (5) to (3), and we get 1x = 10 or 8. .'. x= 5 or 4. Subtract (5) from (3), and we get 2y = S or 10. .'. y = 4: or 5. 1. X = 5, 2/ — ^• .-. 2. X = 4:, y = 5. In solving examples under this case, it must be borne in mind that, in every instance, we must combine the given equations in such a way as to obtain the values of x -\- y and X — y. e.g. Solve the equations x^ -{- y^ = 13, (1) xy= 6. (2) Multiply (2) by 2, and add the result to (1), and also subtract it from (1), and we get x^ + 2xy + y^ = 25, and x^ — 2xy + / = 1. .'. x + y = ± 5, and X — y = ± 1. .-. 2a; = 5 ± 1 or - 5 ± 1. .*. X = 3 or 2, or — 2, or — 3. And 2^ = 5 =F 1 or -5^1. . *. ?/ = 2 or 3, or — 3, or — 2. 254 QUADRATIC EQUATIONS Therefore; 1. x = d, ?/ = 3. 2. 2- = 2, y = '6. 3. x= -% y = -3. 4. X = - 3, y = - 2. A few examples in which both equations arc of the third degree may be solved by the methods of this case. e.g. 1. Solve the equations x^ — y^ — 26, (1) ':i?y — xy^ — 6. (2) Multiply (2) by 3 and subtract the result from (1), and we get x^ - 3x^y + 3a;?/2 -^3^8. (3) Extract the cube root of (3), and we get x-y = ^. (4) Divide (2) by (4), and we get xy = 3. (6) From (4) and (5), we get a; -f- 1/ = ±4. (6) .-. 2a; =6 or -2, and X ■- = 3 or - 1. Also, "^y- = 2 or - 6. .'. y: = 1 or - 3. Therefore : 1. x = 3, y = l. 2. x=-l, EXERCISE y = -d. ; ex. Solve the following equations; 1. xy = 42, x-{-y = 13. 1. 2 xy ~ 24, x + y "• 11. OF TWO UNKNOWN QUANTITIES. 255 8. x^ + if = 29, 4. x^ + ?/^ = 58, X -\- y = 1. X -{- y = 10. 6. x^-\-y'^= 26, 6. x^-\-y'^ = 68, X — y = — A:. X — y = — Q. 7. xy = — 18, 8. xy = — 72, a; — ?/ = 11. a; — ?/ = — 18. 9. x^-y^ = 279, 10. a;3 _|_ ^3 ^ 152^ x^ -{- xy -{- y^ = 93. x^ — xy -\- y^ = 19. 11. x^-y^= 152, 12. a;3 -f ?/3 ^ 637, X — y = 2. X -\- y = Id. II. 13. a;3 + ?/3 = 243, 14. rc3 -y^^ 386, a;^?/ + :ci/^ = 162. x^y — xy^ = 126. 15. x^-y^ = Wb + W, xy(x — y) = 2b(d'^ — b^). 16. x'-^xy-\-y^ = Id' - Idab + W, x?-xy^y^ = Za^- dab + Sb\ Case 4°. An expression is said to be homogeneous when each of its terms is of the same degree. Certain equations which are of the form : a homogeneoits expression in x and y of the second degree equals a constant, may be solved by the methods of cases 1° a7id 3°. When such equations can be solved by neither of these methods, they may be solved by putting y = mx, and solving, first for m, then for x, and finally for y. e.g. Solve the equations a;^ — %xy = — 8. (1) x'-\- y'=rd. (2) Putting y = mx, we have x^ - 2mx^ = -S, or x' = —^—-, (3) 27n — 1 ^ ' 256 QUADRATIC EQUATIONS 1 ^ and x^ + wV = 13, or x" = , , 1 + m^ 8 13 2m — 1 1 -{- m^ .-. 8 + 8m2 = 26m - 13, or 8m2 - 26m + 21 = 0, or (2m - 3) (4m - 7) = 0. .-. m = 3/2 or 7/4. Substituting the first of these values in (3), we get x^ - ^ - 4 .-. X = ± 2. Substitute these values of x in (2), and we get ^ =±3. Substituting the second value of m in (3), we get a_ 8 _16 ^ ~ 7/2 - 1 ~ 5 * .'. ' X = ± 4/5 1/5. Substitute these values in (2), and we get y = ±1/b V6. Then: 1. x = 4^/5 V5, y = l/bVb, 2. X =.^/bVb, y=- 7/5 V6. 3. ic = - 4/5 4/5", y = 7/5 \''5. 4. X = - 4/5 V5, y = - 7/5 Vb. In each case the value of y might have been obtained by substituting the values of m and xm y = mx. OF TWO UNKNOWN QUANTITIES. 257 EXERCISE CXI. Solve the following equations : I. 1. a:2 _^ 3^^ ^ 28, 2. x^-\-xy-\- 2tf = 74, xy + %2 = 8. 2a;2 + 2xy + y^ = 73. 3. x^ -\- xy — 6y^ = 24, 2^2 -j- 3xy - lOy^ = 32. II. 4. a;2 + a;^ — 6?/2 = 21, 5. ^^ — »;«/ + 2/^ = ^1* a:?/ — 2?/^ =4. 2/^ — 2a;?/ = — 15. 6. x^ + xy + 2?/''^ = 44, 2ay^-xy-\-y^= 16. EXERCISE CXII. I. Solve the following problems by using two unknown quantities : 1. The sum of two numbers is 8, and the sum of their squares is 34. What are the numbers ? 2. The difference of two numbers is 3, and the differ- ence of their squares is 33. What are the numbers ? 3. The sum of the squares of two numbers is 106, and the product of the numbers is 45. What are the numbers ? 4. The difference of two numbers is 6, and their prod- uct is 40. What are the numbers ? 6. The sum of two numbers is 7, and the sum of their cubes is 91. What are the numbers ? 6. The difference of two numbers is 4, and the differ- ence of their cubes is 316. What are the numbers ? 7. rind two numbers such that the square of the first 258 QUADRATIC EQUATIONS. and twice the square of the second shall together equal 32, and the square of the second and three times the product of the two shall equal 27. II. 8. Find two numbers such that three times the square of the smaller and the square of the larger shall together equal 7, and the square of the smaller shall be 7 less than four times the product of the two. 9. A man bought 8 cows and 5 sheep for 255 dollars. He bought 3 more sheep for 39 dollars than cows for 300 dollars. What was the price of each ? 10. A number is composed of two digits. If its digits be inverted, the sum of the new and original numbers will be 44, and their product 403. What are the numbers ? 11. Multiply a A -, by J r—i- ^•^ a — ^ a -\- b 12. Factor l^x^ — Sxy — 9x^y^ + 6y^. 13. Reduce — ^r^-^ -|- 75— to a single negative fraction. obd oCi 14. Simplify {l/VZb)-y\ 15. Multiply Vl by V^. 16. Express the following without fractional or nega- tive indices : ^2/3^-1 _ a-y^b. 8-51^ '•17, Rationalize the denominator of 3-21^2 CHAPTER XXL INDETERMINATE EQUATIONS OP THE FIRST DEGBEE. 176. Indeterminate Equations. — Equations are inde- terminate when the number of independent equations given is less than that of the unknown quantities which they contain. For when such equations are solved for any- one of their letters, the value obtained will contain con- stants and one or more of the letters which represent the other unknown quantities. Hence the value of the letter found will vary with the value assigned to the other letters. Thus, if 2a; -|- 5«^ = 8, a: = 4 — 5/2?/, and y may take as many values as we please, and to every value of y will correspond a single value of x; and, conversely, to every value of X will correspond a single value of y. Unless some restrictions be placed on the values of the unknown quantities, the equation may be satisfied in an indefinite number of ways. If, however, the values of the unknown quantities are subject to any restriction, n equations may suffice to determine the values of more than n unknown quantities. In the present chapter we shall consider only indeter- minate equations of the first degree in which the values of the unknown quantities are restricted to positive integers. 177. Solution of Indeterminate Equations of the First Degree in x and y. — Every equation of the first degree in X and y may be reduced to the form ax ± by = ± c, in 859 260 INDETERMINATE EQUATIONS which ay ^, and c are positive integers, and have no com- mon factor. The form ax -\- by = — c cannot be solved for positive integers ; for it a, b, x, and y are positive integers, ax -f- by must also be a positive integer. The remaining forms, ax ±by = c and ax — by— — c, cannot be solved for positive integers when a and b are commensurable. For if x and y are positive integers, the common factor of a and b must also be a factor of ax + by, and therefore of c, which contradicts the hypothesis that a, b, and c have no common factor. The form ax — by= — c becomes by changing its signs by — ax = c, which is essentially the same as ax — by = c, a and b and x and y being interchanged. Hence the two type forms ax-\-by = c and ax — by — c are the only ones that need be considered, and those only in the cases in which a and b are prime to each other. Ex. Solve hx -f- 12y = 263 in positive integers. Divide through by 5, the smaller coefficient, and we get ^ + ^y + ^=52 + |. .-. :. + 2^ + ?^^ = 52. (1) Since x and y are both integers, and the whole of the first member is an integer, therefore -=^-r — = an integer. Multiplying this fraction by the integer which will make the coefficient of y one more than the denominator (5), or than a multiple of the denominator, we get 6?/ - 9 -^^—^ — = an integer; OF THE FIRST DEGREE. 261 11 — 4 that is, y — 1 -\- '^—z — = an integer. V — 4 . , . •. —— — = an integer = p. o .'. y — 4: = 5p, or y = 5p -\- 4:. (2) Substituting this vahie of y in (1), we get or x+ lOp -\-S + 2p+l = 52, or x-\-12p = 43. .-. X = 4:3 -Up. (3) From (2) and (3) it is evident that x a^nd y will be integral when p is an integer and only when p is an integer; for they will both be integers when 5p and 12p are both integers and in no other case, and 6p and 12^0 will be integral when p is integral and in no other case. From (3) it is evident that x will be negative when p exceeds 3, and y will be negative when p is negative. Hence p must be a positive integer less than 4. Hence the only possible values of p are 0, 1, 2, 3. Thus the only positive integral values of x and y are obtained by putting in (2) and (3) p = 0, 1, 2, and 3. The corresponding values of x and y are shown in the following table: p = 0,l, 2, 3, X = 43, 31, 19, 7, . 2/ = 4, 9, 14, 19. Note that the coefficients ofp in the values of x and y in (2) and (3) are the coefficients of y and x respect- 262 INDETERMINATE EQUATIONS ively in the given equation, and that one of the signs is changed. Hence when the given equation has the type form ax -\- by = c, the term in p in the value of x ov y must be negative, and the integral values of p and therefore of X and y must be limited. Ex. 2. Solve Sx — Sy — 28 in positive units. Dividing by 3, the smaller coefficient, we get ,.+ !_, .9 + 1. .-. 2x-y + ^^^ = 9. (1) 2x-l — - — = an integer. Multiplying by 2 so as to make the coefficient of x greater by one than 3, 4:X-2 — an integer. 3 ■x-2 2 X ~\ — = an integer. o an integer = p. 3 .-. X — "2 = 3p, or x = ?>p -h 2. (2) Substituting this value of x in (1), we get or 4 + 6iJ-y+l + 2jo = 9, OF THE FIRST DEGREE. 263 or 8p — y = 4:. .: y = 8p-i. (3) From (2) and (3) we see that p may be any positive integer except zero. When p = 1, 2, S, etc., :zj = 5, 8, 11, etc., and y — ^> 1^? 20, etc. In this case the term in p is positive in both (2) and (3), and the number of solutions is unlimited. This will be the case always when the equation has the type form ax — hy = c. 178. Solution of Indeterminate Equations of the First Degree in x, y, and z. — To solve two equations in three unknown quantities for positive integers: first eliminate one of the unknown quantities so as to get one equation in two unknown quantities; then solve this for positive integers and obtain the value of each of the two unknown quantities in terms of p and constants; and finally sub- stitute these two values in one of the original equations to find the value of the third unknown quantity in terms of m and a constant, observe what values of p will make each of these three positive integers, and find the corresponding values of each of the unknown quantities. e.g. Solve 2a; + 3?/ - 5^ = - 8, bx- y-\-4:Z = 21, (1) for positive integers. Eliminating y by addition, we get 17:^:4-7^ = 55. (2) ... 2^ + . + ^=7 + |, 264 INDETERMINATE EQUATIONS or o i , 3:?^ - 6 ^ 3a: ~ 6 , ^ — — = integer. 15:?; - 30 2^-4 + = integer, — integer. x-2 integer = p. x-2 = 7p, or x = 7p-^2. (3) Substituting this value of x in (2), we get 119jo + 34 + 7;? = 55, or 119;? + 7;2 = 21. .% 17i?.+ ^ = 3. .-. ;2 = 3-17i?. (4) Substituting (3) and (4) in (1), we get 35J9+ 10 - 2/ + 12 - 68j9 = 21, or — 'SSp — y = — 1, y = l- 33p. (5) The only value of p that can make z a positive integer is 0. Substitute this value in (3), (4), and (5), and we get x = 2, and z = d. OF THE FIRST DEOBEE. 265 EXERCISE CXIII. Solve the following equations in positive integers: I. I. 7a; +15?/= 59. 2. 8a; + 13.y = 138. 3 7^; -f 9^ = 100. 4. 13a; + lly = 200. 5 Find the number of solutions in positive integers of 11a; + 15^ = 1031. Solve the following equations in positive integers : 6. Qx -\- 7y -\- 4:z = 122, 7. 12x - Uy -\- 4.z = 22, 11a; -{-Sy -6z = 145. - 4a; + 6y -\- z=17. II. 8. 20a; - 21y = 38, 9. 7a; + 4^ + 19z = 84. 3^ + 4^ = 34. 10. 23a; + 17y + 11^ = 130. Find the general integral solutions of the following equations : II. 7a; - Idy = 15. 12. 9a; - lly = 4=. Solve in least positive integers : 13. 119a; - 105y = 217. 14. 49a; - 69y = 100. 15. How can a length of 4 feet be measured by means of two measures, one 7 inches long and the other 13 inches long? 16. How can 45 pounds be exactly measured by means of 4-pound and 7-pound weights ? 17. In how many different ways can the sum of $3.90 be paid with fifty- and twenty-cent pieces ? 266 INDETERMINATE EQUATIONS. 18. In how many different ways can the sum of $5.10 be paid with half-dollars, quarter-dollars, and dimes, the whole payment to be made with twenty pieces ? 19. A farmer purchased a number of pigs, sheep, and calves for 160 dollars. The pigs cost 3 dollars each, the sheep 4 dollars each, and the calves 7 dollars each ; and the number of calves was equal to the number of pigs and sheep together. How many of each did he buy ? 20. Find the least multiples of 23 and 15 which differ by 2. 21. Find two fractions whose denominators shall be 113 respectively 9 and 5 and whose sum shall be . ^ . CHAPTER XXII. INEQUALITIES. 179. Definition of Greater and Less Quantities. — One quantity is said to be greater than another when the remain- der obtained by subtracting the second from the first is positive; and one quantity is said to be less than another when the remainder obtained by subtracting the second from the first is negative. N.B. — Throughout the present chapter every letter is supposed to denote a real positive quantity, unless the con- trary is stated. In accordance with the definition just given a is greater than J) when a — h \^ positive, and, conversely, when a is greater than h, a — h m positive. Also, a is less than h when « — J is negative, and, conversely, when a is less than h, a — J is negative. Thus 2 is greater than — 3 because 2 — (— 3), or 5, is positive; also — 2 is greater than — 3 because — 2 — (— 3), orl, is positive. Again, — 2 is less than 1 because — 2 — 1, or — 3, is negative; and — 4 is less than — 2 because — 4 — (— 2), or — 2, is nega- tive. According to this definition zero must also be regarded as greater than any negative quantity. 180. Inequalities. — An inequality is an algebraic state- ment of the fact that one of two expressions is greater than the other. The two expressions compared are connected together by the sign >, "greater than, ^' or <, "less than," 267 268 INEQUALITIES. the open end of the symbol always being directed towards the larger member of the inequality. Two or more inequalities are said to be in the same sense, or of the same species, when the first member of each is the greater or the less, and two inequalities are said to be in the opposite sense, or of the opposite species, when the first member of the one is the greater, and of the other is the less. Thus a > h and c > d are two inequalities in the same sense, or of the same species. So also are m h and c < d, or m < n and p> q are inequalities in the opposite sense, or of opposite species. The working rules for inequalities differ in some re- spects from those for equations. They are based upon cer- tain elementary theorems of inequality which are readily deduced from the axioms of equality. Theoeem I. If equals be added to unequals, the sum will be unequal in the same sense. Let a > b, and let their difference be denoted by r. Then a = b -\- r. Adding x to each member of this equation, we get a-\-x = b-\-x-\-r. .'. a -\- X > b -{- X. Theorem II. If equals be tahenfrom unequals, the re- 7nainders will be unequal i7i the same se^ise. Let a > b, and let their difference be denoted by r. Then a = b -{- r. Subtracting x from each member of this equation, we get a — X = (b — x) -\- r. ,\ a — x > b — x. INEQUALITIES. 269 Cor. From these two theorems it follows that we have the right to add equals to the members of an inequality, and to subtract equals from the members of an inequality, without altering the sign of inequality. Also, that we have the right to transfer a term from one member of an inequality to the other by changing its signs, without altering the sign of inequality. Theorem III. If imequals be suUracted from equals, the remainders loill he unequal in the reverse sense. Let a> by and let their difference be denoted by r. Then a = b -\- r. Subtracting each member of this equation from x, we get X — a = X — {b -{- r) = {x — b) — r, .'. X — a bx. Theorem V. If unequals be divided by equals, the quotients will be unequal in the same sense. Put a = b -\- r 2^% heretofore. 270 INEQUALITIES. Dividing each member of this equation by x, we get a _h r X ~ X x' a h Cor. From Theorems IV and V it follows that we have the right to multiply or divide both members of an inequality by the same positive quantity without altering the sign of inequality. If, however, both members of an inequality be multi- plied or divided by a negative quantity, the signs of both members will be reversed. This reversal of signs is equiv- alent to an interchange of the members, and therefore it reverses the character of the inequality. Hence, on such multiplication, the sign of inequality must be reversed. Theorem VI. If equals ie divided iy unequals, the quotients will he unequal in the opposite sense. Put as before a = h -\-r. Dividing x by each member of this equation, we get X X OX ox -j- rx — rx a ~ ~ b-i-r~b{b-]- 7')~ b{b + r) x{b -{- r) rx -b{b + r)- b{b^r) X rx ~b b{b-^ r)' X X Theorem VII. If two inequalities of the same species be added together, the results will be unequal in the same sense. Let a> b and c > d. INEQUALITIES. 271 Put a = ^ -|- r, and c = d -{• s. Then, by addition of equals, a-\-c = b-\-d-\-r-{-s, .'. a -\- c > b -\- d. Note. — By subtraction we would get a — c = b — d-\-r — s\ from which we cannot infer whether « — c > 6 — t?, or a — c s, a — c > b — d; but it r < s, a — c < b — d. Hence addition of corresponding members of inequali- ties of the same species without changing the sign of in- equality is always admissible, but not subtraction. CoE. It a > b, > dj e > f, etc., then a -{- c -\- e -{■ etc. > b -\- d -\- f -\- etc. Theorem VIII. If two mequalities of the same species be multiplied together, the results will be unequal iu the same sense. Let a> b, and c > d. Put a = b -\- r, and c = d -{■ s. Then, by the multiplication of equals, ac = (b -\- r)(d -j- s) = bd -\- bs -\- dr -{■ rs, .'. ac > bd. Cor. 1. It a > b, c > d, e > f etc., then a . c . e . etc. > b . d .f. etc. Cor. 2. It a > b, then a'" > b"^. Cor. 3. If « > b, then a-"' < b-"^, EXERCISE CXIV. 1. For what values of x is 5^--< — + 6? 272 INEQ UALITIE8. Multiplying both members by 5, we get ^bx - 16 < 10:?; + 30. By transposition, 15a; < 46. .-. x<^^. This inequality holds for all values of x less than S^^g 2. For what values of x and y are 4:c + 3?/ > 27, 3a; + 4^ = 29 ? Multiplying both members of the inequality by 4, and of the equation by 3, we get 16a; + 12?/ > 108; 9a; + 12?/= 87; .-. 7a; > 21; .-. X > 3. Multiplying both members of the inequality by 3, and of the equation by 4, we get 12a; + 9?/> 81; 12a; + 16?/ = 116; " 7?/ > - 35. .-. 7?/<35. .-. ?/<5. Hence the values are all of those of x greater than 3, and of y less than 5, which make 3a; + 4?/ == 29. N.B. — The values of x and y obtained as above are called the limits of x and y. That is, they are the values which bound the possible values which x and y can have under the given conditions. INEQUALITIES. 273 Find the limits of x in the following cases : 3. (42; + 2)2-29> (22; + 2)(8a:-4). 4. Cdx - 2)(4^ + 3) > {'^x - ^){ 35 - bx, and 4a; - 12 > 6a: - 31. Find the limits of x and y in the following case : 6. 3a: + 7«/ > 46, X - y zzz -1, 181. Type Forms. — Inequalities among algebraic quan- tities are usually established by reference to certain stand- ard forms. The following is a very important standard form : For all values of x and y except equality, x^ -\- if > 2xy. (A) Proof. — {x — yY is essentially positive and hence > 0. . •. x^ -\- y^ - 2xy > 0. .'. x^ -\- y^ > 2xy. e.g. The sum of a number and its reciprocal is > 2. Let X denote the number. Then will a:+i > 2. X Multiplying both members by x, we get a;2 + 1 > 2a:, or a:^ + 1^ > 2a: . 1. That is, the first inequality is true if the last is. But we know that the last is true by reference to standard (A). Hence we infer that the first is also true. Theorem I. The product of two positive quantities whose sum is constant is greatest when the qua^itities are equal 274 INEQUALITIES. Denote the two quantities by a + a: and a — x. Then, whatever value be assigned to x, the sum of the quantities will be 2«, and their product a^ — x^. Evidently the product will be greatest when a; = 0; that is, when the quantities are equal. If a and l be two unequal quantities, the two halves of their sum would be two equal quantities whose sum would be the same as that of a and b. Hence a 4- h a 4- h , la + JV , ^ > ^^. .-. a^h> 2Vab. (B) Theorem II. The product of any number of positive quantities ivhose sum is constant is gi'eatest when the quan- tities are all equal. For, if any two of the factors are unequal, their product would be increased by making them equal without chang- ing their sum. This would necessarily increase the whole product without altering the sum of the factors. If a, b, c, . . . . up to n quantities be unequal, by tak- ing the ^th parts of their sum we should obtain n equal quantities whose sum would be the same as that of the n unequal quantities. Hence (a-^b-\-c-\- . . .y or • > Vabc . . . a-\-b-\-c-\- . . ,> n Vabc . . , (0) INEQUALITIES. . 276 e.g. a^-\-h'^> 2ai, and . a^-^b^-i-c^> Ulc, in all cases when a, h, and c are positive and unequal. EXERCISE CXV. I. 1. For what value of x would 16:^2 -f 25 = 40a: ? Show that for all other values of ^, 16a;^ -f 25 > 40a;. 2. Show that for no positive integral value of x is x^-{- — <^x - —. 5 5 8. Show that for no positive value of a can (3ff + 2Z>)(3« - 2^) < U{ 4:abxy. 6. Show thsit {b -]- c)(c -\- a) (a -{- b) > 8abc, II. 6. If a^-{-P=l, and x^-^y^=l, show that ax-\-by 9x^y^z^. 9. Show that «* + Z*'' > rt^^ -|- ab^, except when a and b are equal. 10. Show that a^ -\- ¥ > a^b -\- ab^, except when a and b are equal. CHAPTER XXIII. RATIO AND PROPORTION. A. EATIO. 182. Definition of Ratio. — The term ratio denotes the relation which one quantity bears to another of the same kind in magnitude. The magnitude of one number compared with another is ascertained by dividing the number by the one with wliich it is compared. When the number is a multiple of the one with which it is compared its ratio to it may be expressed by an inte- ger, otherwise the ratio may be expressed by a mixed num- ber or a fraction. e.g. The ratio of 12 to 4 = 12 ^ 4 = 3; the ratio of 3 to 5 = 3 -4- 5 = 3/5; the ratio of 13 to 4 =: 13/4 or 3^. The ratio of one number to another might be defined as the number by which the second must be multiplied to produce the first. e.g. 5 must be multiplied by 4 to produce 20. There- fore the ratio of 20 to 5 is 4. Again, 5 must be multiplied by 3/5 to produce 3. Therefore the ratio of 3 to 5 is 3/5. 183. Expression of a Ratio. — The ratio of one number to a second may be expressed either by writing the numbers in the form of a fraction with the first number as the nu- merator, or by writing the second number after the first with a colon between. RATIO. 277 e.g. The ratio of 2 to 3 may be expressed thus: |, or 2:3. 184. The Terms of a Ratio. — The first term of a ratio is usually called the antecedent, and the second term the consequent. When either term of a ratio is a surd the ratio cannot be expressed exactly either by an integer or by a rational fraction, though it may be expressed to any required degree of approximation, by carrying out the extraction of the indicated root to a sufficient number of places. e.g. The ratio of the V'S to 4 cannot be expressed exactly by any rational integer or fraction. Thus, :^ = ^^^:^««^ =.559017... 4 4 By carrying the decimals further a closer approxima- tion may be obtained. 185. Kinds of Ratios. — When the antecedent of a ratio is equal to its consequent, the value of the ratio is one, and the ratio is said to be a ratio of equality ; when the ante- cedent is greater than the consequent, the value of the ratio is greater than one, and the ratio is said to be a ratio of greater inequality ; and when the antecedent is less than the consequent, the value of the ratio is less than one, and the ratio is said to be a ratio of less inequality. 186. Ratio of Equimultiples and Submultiples. — Since ma ' mV multiples Also, ratio as their equi-submultiples, equi-submultiples being the --——:, two numbers have the same ratio as their equi mo ^ Also, since t — ~n — '- — » two numbers have the same ^ m 278 RATIO AND PROPORTION. quotients obtained by dividing two or more numbers by the same number. 187. Theorem I. If the consequent of a ratio of greater inequality le positive, the ratio will he diminished hy adding the same positive quantity to both of its terms, and increased hy subtracting the same positive quantity (less than the conseque^U) from hoth of its terms. Let h be positive and a > h, then will , , < 7-. ^ h -\- X h For a-{- X a h(a-{- x) — a(h -|- x) x(h — a) h-^x h ~ h{h + x) ~h{h ^ x)' Now since a, h, and x are positive by hypothesis and < a, the fraction ,;-. , — { is neffative. . •. y— ' — < 7-. h(h ~\- X) ^ b -\- X h . . a — X a x(a — b) But, since a > b, a — b is positive, and since x <.b, b — X is positive. Hence the fraction ~ !- is positive. h(b- x) ^" ^ ^* ' ' h-x^ h' 188. Theorem II. If the consequent of a ratio of less inequality he positive, the ratio will be increased by adding the same positive quantity to both of its terms, and dimin- ished by subtracting the same positive quantity (less than the consequent) from hoth of its terms. a I X a Let h be positive and a < b, then will , , > 7-, and t) -]- X a — X a b — X Prove these cases in the same manner as those of the last section. PROPORTION. 279 189. Compound Ratios. — When the antecedents and also the consequents of two or more ratios are multiplied together the ratios are said to be compounded, and the ratio of the products is called the compound ratio of its compo- nents. Thus, ac -.hd is the compound ratio of a : b and c : d. When a ratio is compounded with itself its terms are squared, and the result is called the duplicate ratio of the original. Thus, a^ : b^ is the duplicate of a : b. Similarly a^ : b'^ is called the triplicate ratio ot a :b. B. PROPORTION". 190. Definition of Proportion. — Four abstract numbers are said to he 2^ropo)'tionaL or to form 2i proportion^ when the ratio of the first to the second is equal to that of the third to the fourth. Thus, \i a :b ^= c : d, the four quan- tities a, b, c, and d form a proportion, which may be written in any one of the following ways: a c a -.b — c -.d. 7- = -:, or a \b \\c \d. b d The first and last terms of a proportion are called the extremes, and the second and third terms, the means. Thus, in the above proportion a and d are the extremes, and b and c the means. If a, b, c, d, e, etc., are such that a :b — b : c = c : d = d : e, then a, b, c, d, e are said to be in continued propor- tion. If three quantities, a, b, c, are in continued proportion, so that a '.b = b : c, then b is said to be a mean proportional between a and c. If a : b = b \ c = c : d, then b and c are said to be two mean proportionals between a and d, and so on. 191. Test of the Equality of Two Ratios. — Since a 280 RATIO AND PROPORTION. ratio is virtually a fraction, we test the equality of two ratios in the same way that we test the equality of two fractions. Two fractions are equal if, on reduction to a common denominator, the resulting numerators are equal. Thus, ft c take the two fractions ^ and -: , reduce them to a common a donominator, and we have -j-z and j^ . These resulting fractions will be equal when ad = be. Hence the four quan- tities a, b, c, d are proportional when the product of the first and fourth is equal to the product of the second and third ; and, conversely, li a -. h = c : d, then ad = he. In any p^^oportion the product of the extremes is equal to the product of the means. This is the great numerical law of proportions. 192. Permutations of Proportions. — Any interchange of the terms of a proportion is permissible which does not destroy the equality of the product of the extremes and means. The various interchanges of the terms of a pro- portion are called permutations. If we write the four terms of a proportion in the four corners formed by two lines which cross at right angles, so that the first ratio shall be at the left and the second at the right, the two antecedents will be at the top and the two consequents at the bottom, and the extremes will be in one pair of opposite corners and the means in the other. Thus a c in the form , « : J is the first ratio and c : d the sec- h d ond ; a and c are the antecedents and h and d the conse- quents; a and d are the extremes and h and c the means. The letters a and d and h and c, which stand in opposite corners in the above form, may be called the opposites of a proportion, and we may make the general statement that pitopoBTioir. ^81 The terms of a proportion may he ivritten in any order, provided the opposites remain the same. An interchange of antecedent and consequent in each ratio is called an inversion, an interchange of an antecedent of one ratio with the consequent of another is called an al- ternation, and an interchange of one ratio with another a transpositioyi. There are seven permutations of an ordinary proportion, so that when four quantities are proportional they may be arranged in eight different ways. d Thus, by inversion — , and by mov- c C 1) — becomes — d a ing the terms of each of these successively around to the right eacli of the above may be changed three times by alternation. Thus a c I — becomes — a d I c — , and — d h d d c c a a b d . a — becomes — h c > a d , and c c c d d b b a And Write out in the ordinary form each of the proportions given above, and state by what change each proportion is obtained from the last. 193. Transformations of Proportions. — Besides these eight permutations there are other transformations which a proportion may undergo. li a :b — c '.d, then a -\- b : b = c -\- d \ d. Let — = X. Therefore a = bx. Then, also, ;t = ^- (Why ?) Therefore c = dx. 282 RATIO AND PROPORTION, Then — 7 — becomes, by substitution, hx±h_h{x^l)_ —j~ - ^— _ a: + 1. Also, — ^ — will become j — = (x 4- 1). a a ^ ^ Therefore — -, — = x 4-1 = — -^ — . a Hence a -\~ b : b = c -{- d : d. This change is called composition. EXERCISE CXVI. Prove the following cases by methods similar to the above : 1. a — b : b = c — d : d. This change is called division. 2. a -{- b : a — b = c .-\- d : c — d. This change is called composition mid division. 3. a -\- b : a ^=^ c -\- d : c. 4. a — b\a — c — d\c. 6. If a -.b — c '. d = e \f= etc., then a-{-c-{'e\b-\-d-\-f=a:b. This change is called addition. 6. li a '.b = c \ d, then ma : mb — nc : nd. 7. Write the last proportion in eight different ways. 8. \ia\b = c\d, then «" : ¥ - c" : d"". 9. \t a '.b ^=^ c \ d, and m '.n ^=^ r \ s, then am : Z'm = cr : ds. PROPORTION. 283 10. li a :h =^ c \ d, then la -\- mb : pa -{- qb = Ic -\- md : pc + qd. 11. If a :h — c : d^ and m : 71 = r : s, then a Vm — bVn:cVr — dVs = a Vm -\- b V71 : c Vr -\- d Vs. EXERCISE CXVII. Ex. Which is the greater ratio, a'^ -]- b^ : a -\- b or a^ — b^ : a — b, a and b each being positive ? Write each ratio in the form of a fraction, and subtract the second from the first, and show that the result is essen- tially negative. Hence the second ratio must be the greater. Thus, a''-{-¥ _ a'-b' _ {a' + b'){a - b) - {a' - b'){a + b) a-\- b a — b ~ {a -\- b)(a — b) _ 2ab' - ^a'b . ~ {a-^b){a- b) "la^b - 2ab' (a + b)(a - b) 2ab{a^ - b^) (a + b){a - b) 2ab{a^ + ab + b^) a^b Now since a and b are both positive, both the numera- tor and the denominator of this fraction must be positive. ^4 _ 54 Hence the result obtained by subtracting 7- from ^4 _|_ J4 ffi _ ^4 -—r- is negative. Therefore 7 must be larger than a-\-b *^ a — b ® a' + i' a-\-b' ^84 RATIO AND PROPORTION. 1. Which is the greater ratio, 5 : 7 or 151 : 208 ? 2. Which is the greater ratio, 6 : 11 or 575 : 1056 ? 3. Which is the greater ratio, 7 : 12 or 589 : 1008 ? 4. Which is the greater ratio, x^ -\- y"^ : x -\- y or x^— y'^ '. X — y, X and y both being positive ? 5. Which is the greater ratio, x^ -\- y^ : x -\- y or x^— y^ '. X — y, X and y both being positive? 6. Which is the greater ratio, x"^ -\- y'^ : x -{- y or x^ — y"" : X — y, X and y both being positive ? 7. In one city a man assessed for $10,000 pays $72 tax, and in another city a man assessed for $720 pays $4.50 tax. Compare the rate of taxation in the two cities. 8. Two men can do in 4 days what three boys can do in 5 days. Compare a man's working capacity with that of a boy. 9. For what vahie of x will the ratio b -\- x -. S -{- x become 5 : 8, 6 : 8, 7 : 8, 8 : 8, 9 : 8 ? 10. What number added to both antecedent and con- sequent will duplicate the ratio 3:4? n. If X -^ 1 \& io %(x -{- 14) m the duplicate ratio of 5 : 8, what is the value of a; ? II. 12. Find two numbers in the ratio of 7 : 12 such that the greater exceeds the less by 275. 13. What number must be added to each term of the ratio 5 : 37 to make it equal to 1 : 3 ? 14. If a; : y = 3 : 4, what is the ratio of Ix — 4y \ 3x + 73/? PROPORTION. 285 16. If 15(2a;2 — y^) = "^xy^ what is the ratio oix \ y'i 16. If 3(7a;2 _ 24«/2) = _ 29a;«/, what is the ratio of x\y'i 17. What is the least integer which added to both terms of the ratio 5 : 9 will make a ratio greater than 7 : 10? 194. Solution of Fractional Equations. — When an equa- tion consists of two fractions only, or can be expressed in the form of two fractions, its solution may be simplified by a judicious application of one or more of the following principles of composition and division. Lel l\^% Then h d 1°. a — c a c h-d^h~ d~ a -{- c -b-^d' 2°. a-\- b c -\- d h ~ d ' 3°. a — h c — d h ~ d ' 4° a-^h c-\-d a — b c — d' ft Q Prove the first of these cases by letting — = ;, = ^'- The remaining three have already been proved. e.g. 1. Solve the equation — — - = — — -. • X -\- ^ a-{- b {x-4.) + {x^ 4) _ (^ - 5) + (^ + 5) (a; - 4) - (a; + 4) ~ (« - 5) - (« + 5)' X — 4: -\- X ~\- 4: _a — 6 -\- a -\- 5 ay — 4: — X — 4: ~a — 6 — a — 6' 2x _ 2a Applying 4°, or or 286 RATIO AND PROPORTION. or X _a 4 ~5' . '. bx = 4:a, e.g. 2. Solve the equation Applying 1' or X -\- 4: — b X -\- 4: {x-4:-\-b)-{x — 4:) X- 4: {x^4-I))-{x-\- 4) ~ x-i-4:' b _x — 4: ■=l"'^~+4* a:-4 -1 X-\-4: 1 Applying 4°, .^ ^ _ = o. .-. x = 0. e.g. 3. Solve (x^2){x+5){x-\-3){x^S) = {x+l){x+Q){x+4:)(x+7). Dividing both sides by (x + 3)(x + 8)(^ + 4)(x + 7), we have (x-\-2)(x+5) _{ x-\-l)(x+e) {x + 4.)(x+7)-{x+'S)(x-j-8y x^-\-7x-\- 10 _ x^-\-7x-\-6 *** a;2 + lla; + 28 ~ x^ -{- 11a; + 24* Applying 1°, we have (a;2 + 7a; + 10) - {x^ -\- 7x -^ Q) _ a:^ + 7a; + 10 (a;2 + 11a; + 28) - {x^ + 11a; + 24) ~ a;^ + 11a; -f- 28* 4 arJ + 7a; + 10 1 or 4 ~ a;2 + 11a; + 28 ~ 1 • i«r« + 7a; + 10 = a;2 _|. 113, _|. 28, PROPORTION. 287 or — 4a; = 18. . •. X — — ^. e.g. 4. Solve [x - l)(2a; - 3)^ = {x - 3)(2a; - 1)1 Dividing both sides by (2a; — 3)2(2a; — 1)^, we have a; — 1 a; — 3 or or or 7. (2a; - If ~ (2a; -3)2' X — 1 a; -3 4a;2 - 4a; + 1 ~ 4a;2 - 12a; + 9' Applying 1°, we have {X -1)- {X - 3) a;-l (4a;2-4a; + l)- (4a;2 - 12a; + 9) 4a:2 _ 4a; + 1' 2 a;- 1 1 8(a ^-1)" - 4:^2 _ 4:^ _|_ ■l-4(a:-l)- -• . 4(a; - 1)2 = 4a:2 - 4a; + 1, 4a;2 - 8a; + 4 =: 4a;2 _ 4a; + 1. .' •. - 4a: = - - 3. .♦. a: = 3/4. EXERCISE CXVIII. Solve the following equations X — a a ~ h- c c -I. 2. x-b l-l 5 ~ 7 * x-1 . x-\-l- 1- a 4. a- - 3 3 - c a; + 3 ~ 3 + c* 2a; + 3 2.2: - 3 5 "2" 6. 3a; -7 7 3a: + 7 ~ 3 * mx + n & + c — a ^. A 3a; + 4 c^a — l mx — n 6' -|- « — ^' ■ 3a; — 4 a-\-l) — g 288 RATIO AND PROPORTION. 2x-\-l 1 3ic - 1 9. ^ o , J — r-T, = — r-T- 10. 15. 2a;2 + 2a; + 3 "~ ic + 1* Sa:^ - 3^; + 5 a; - 1 II. 11. (i»+l)(22; + 5)2 = 4(a; + 2)3. 12. (a: - l){x - %){x + 6) ^ (a; + 2)(,r -- 3)(a: + 4). 13. {x - l){x - ^y{x - 5) = x{x - '6f{x - 4). &x^ + 5x^ + 6a; + 2 _ 2a;^ 4- a; H- 1 ^^- 6a;2 4- 5a; + 3 ~ 2./: + 1 * W -f 4a;^ + 8-^ + 4 _ 3a;^ + '^^ + 1 9a;2 4- 4a; + 5 ~" 3a; + 2 * C. VAKIATION. 195. Direct Variation. — Suppose x and y to represent two variable quantities which depend upon each other in such a way that when one changes its value, the other must also change its value; and let x and y be so related that y = mx (m being a constant), whatever be the value of x; and let x^, x^, x^, etc., and ?/, , y^, y^, etc., be corre- sponding values of x and y, so that y^= mx^ , y^= nix^ , etc. Since y = mx and y^ = wa;,, y _ mx _ X y^ mx^ x^ ' Whence y : y^= x -. x^, or x : y = x^ : y^. When two quantities are thus related, one is said to vary as the other. Since the relation is mutual, we may say that y varies as x, or that x varies as y. The symbol oc denotes this relation, and is read " varies as " or '^ varies directly as." Thus y a a; is read '' y varies as a;"; and x oc y, " X varies as y." To say that y varies as x is to say that one is a constant multiple of the other, or that they so vary that their ratio VARIATION. 289 remains constant, or that any two values of x and the corre- sponding values of y are in proportion. It is a law of Optics that the intensity of the illumina- tion upon a surface varies directly as the sine of the angle which the rays from the light make with the surface. Tliat is, the larger the sine of this angle, or the more nearly per- pendicular the rays are to the surface, the more intense is the illumination. If two surfaces are held at the same dis- tance from the light, but one so as to make the angle-sine for the rays twice as great as for the other, the illumina- tion of the former will be twice as intense as that of the latter; if the surface were held so as to make the angle- sine three times as great, the illumination would be three times as intense; and so on. While the illumination in- creases with the size of the angle, it does not increase in the same ratio. Hence the illumination does not vary as the angle. 196. Inverse Variation. — When y varies as x, or di- rectly as X, as we have already seen, y = mx, m being a constant. When y — m— , y is said to vary inversely as z. That is, y increases as z decreases, and vice versa, and both change at the same rate. In the case of the light, the intensity of the illumina- tion on a surface varies with the distance of the surface from th^light, the intensity becoming less as the distance becomes greater, and the intensity changes at the same rate as the square of the distance changes. Hence we say that the intensity of the illumination varies inversely as the square of the distance from the light. If y denote the in- tensity of the illumination, z the distance from the light, and m the intensity of the illumination at a unit distance from the source, then y = m-^ , and y cc —. z z 290 EATIO AND PROrORTIOK 1 X . . When y = mx . — or m . —, y varies directly as x and inversely as z. In the case of the light already considered, if y denote the intensity of the illumination, x the sine of the angle which the rays make with the surface, and z the X distance from the light, then y—7n—^. That is, the intensity of the illumination varies directly as the angle-sine and inversely as the square of the distance. AVhen y = mwx, y varies jointly as w and x. If to denotes the intensity of the source of light, y the intensity of the illumination on the surface, x the angle- 21) X sine, and z the distance from the source, then y = —5-. ^ z^ Express this relation in words. 197. The Constant of Variation. — In all the cases of variation, the constant (m) may be determined when any set of corresponding values is given; and when the con- stant and all but one of a set of corresponding values are known, the remaining one can be calculated. e.g. 1. A ex B, and when A = 8, B = 6. What will A equal when B = 24:? A = mB. ,\ S = 6m. .'. m = 3/4. .-. ^ = 3/4x24 = 18. 2. ^ a -^, and when A = S, B = Q. What will A equal when ^ = 24 ? . 1 A=m.^. VARIATION. 291 ,\ 48 = m. 3. A (X BC, and when ^ = 2, 5 = 6 and (7=4. What will A equal when i? = 18 and C = 6 ? A = m. B. C. 2 = m X 6 X 4. .-. m = 1/12. .-. A = 1/12 X 18 X 6= 9. 4. A cc B . -^, and when ^ = 2, B = 6 and C = 4. What will ^ equal when B = 18 and C=Q? A = m . B . Yf. .-. 2 = m.6.^. 4 .-. m=4/.3. .-. A =4/3. 18. 1/6 = 4. EXERCISE CXIX. I. 1. A varies as B, and when A is 6, 5 is 4. What is ^ when 5 is 9 ? 2. Jf varies inversely as iV, and when if is 4, iV^is 13. What is M when iV^ is 20 ? 3. A varies as B and C jointly, and A = 3 when B = 6 and (7=4. What is ^ when 5 is 8 and (7 is 3 ? 4. A varies as ^ and inversely as C, and A = 4: when 5 = 6 and (7 = 8. What is the value of A when B = IS and (7 = 6 ? 292 RATIO AND PROPORTION. 5. The area of a circle varies as the square of its radius, and the area of a circle whose radius is 10 is 314.16. What is the area of a circle whose radius is 20 ? II. 6. The volume of a sphere varies as the cube of its radius, and the volume of a sphere whose radius is 1 foot is 4.188 cubic feet. What is the volume of a sphere wliose radius is 5 feet ? 7. The volume of a cone of revolution varies as its height and as the square of the radius of its base jointly, and the volume of a cone 7 feet liigh with a base whose radius is 3 feet is 66 cubic feet. B'ind the volume of a cone 14 feet high with a base whose radius is 18 feet. 8. The volume of a gas varies as the absolute tempera- ture and inversely as the pressure upon it, and when the temperature is 280 and the' pressure 15 the volume of a cer- tain mass of a gas is one cubic foot. What would be its volume were the pressure 12 and the temperature 600 ? 9. The distance of the offing at sea varies as the square root of the eye above sea-level, and the distance is 3 miles when the height of the eye is 6 feet. What is the distance when the height is 72 yards ? 10. The intensity of illumination varies as the sine of the angle which the rays make with the surface and in- versely as the square of the distance from the source, and when the sine and distance are each unity the illumination is 40. What will be the illumination when the sine is 3/4 and the distance 8 units ? CHAPTER XXIV. LOGARITHMS. 198. Definition of a Logarithm. — In the expression cv^ zzz y^ X is called the logarithm of y to the base a. This relation is indicated also by writing x — log„ y. The base a being some fixed positive number, to every value of y there is a corresponding value of x, and to every value of X there is a corresponding value of y, but these values are often incommensurable, so tliat they can be ex- pressed only approximately. The logarithm of a number may be defined in words as the index of the power to which a given base must he raised to jyroduce the number. A. GENERAL PROPERTIES OF LOGARITHMS. 199. The Working Rules of Logarithms. — Let a"^ — m., and a- — n. Then x — ^og^v:, and y = log^w. From these two equations we may deduce four impor- tant theorems : 1°. mn = r?^. a^ = cf^^', and \og^{mn) = .t + ^; or log„(w/0 = loga^ + logan- That is, the logarithm of the product of two numMvs is the sum of the logarithms of the numbers. Of course this theorem may be extended readily to the product of any number of factors, and in its general form it would be : 293 294 LOGARITHMS. The logarithm of any product is the sum of the loga- rithms of its factors. 2°. m-^ n = a'' -^ a^ = a'^-y, and loga(m -i- n) = x — y, or loga(m -^n) = log„m - log^n. That is, the logarithm of the quotient of two numbers is the logarithm of the dividend minus the logarithm of the divisor. 3°. m^ = {a^'Y = «^^, and loga{m^) = px, or \oga(m^) =p\ogam. That is, the logarithfn of a power of a number is the logarithm of the number multiplied by the index of the power. and log„(mV^) = 1/p . x, or log^(mV^) = l/p log^m. That is, the logarithm of a root of a number is the loga- rithm of the number divided by the index of the root. These four theorems are the working rules of logarithms as applied to numhers. From these four theorems we see that addition of loga- rithms corresponds to multiplication of numbers, subtrac- tion of logarithms to division of numbers, the multiplica- tion of logarithms by numbers to the raising of numbers to powers, and the division of logarithms by numbers to the extraction of roots of numbers. There are no operations on logarithms which correspond to the addition and sub- traction of numbers, and there is no operation on numbers in ordinary arithmetic which corresponds to the raising of logarithms to powers or to the extraction of their roots. GENERAL PROPERTIES OF LOOARITHMS. 295 200. Systems of Logarithms. — The general properties of logarithms are the same for all bases, and any positive number, rational or irrational, may be taken as a base. Certain numbers, however, otter special advantages as bases in working with logarithms and in calculating them. The base which is most advantageous for numerical computation is 10, and the one most advantageous for theoretical inves- tigation is the incommensurable 2.7182818 .... The for- mer is the base of the system of logarithms in common use, and the latter of the Napierean, or natural, system of loga- rithms. 201. Common Logarithms. — When the base of the sys- tem is 10, the 10 is omitted after the abbreviation "log." Thus, log 100 = 2, means that 10 must be raised to the second power to produce 100. Written in full the expres- sion wo uld be logiolOO = 2. 1 = 100, . •. log 1 = 0. 10 =r lOS .-. log 10 = 1. 100 == 10^ . •. log 100 = 2. 1000 = 103, .-. log 1000 = 3. etc. Whenever a number is an integral power of ten, its logarithm is a positive integer, and is equal to one less than the number of places in the number to the left of the deci- mal point. ••l-li^-^^-^ .-. log .1=-1. .01=1., =10- .-. log .01 = -2. 001 - ^, = lo-^ .-. log .001 = -3. = .4, = 10-% . *. loof. = — 00 10 296 LOGARITHMS. The logarithm of is negative infinity. Tlie logarithm of a negative number is imaginary. Whenever a number is a decimal and equal to 1 divided by an integral power of 10, its logarithm is a negative integer and is equal to one more than the zeros to the right of the decimal point. Inasmuch as the logarithm of any number to base 10 or any base greater than 1 increases with the number, it is evident from the above that the logarithm of any number greater than one is positive, and the logarithm of any num- less than one is negative; also that the logarithm of any number between 1 and 10 lies between and 1, and is a positive decimal; that the logarithm of any number be- tween 10 and 100 lies between 1 and 2, and is 1 plus a positive decimal ; and so on. It is further evident that the logarithm of any number between 1 and .1 lies between and — 1, and is — 1 plus a decimal; that the logarithm of any number between .1 and .01 lies between — 1 and — 2, and is — 2 plus a decimal ; and so on. 202. The Characteristic and Mantissa of a Logarithm. — In general, the logarithm of a number is composed of two parts, an integer and a decimal. The decimal part of a logarithm is incommensurable, and therefore cannot be expressed exactly. It is called the mantissa of the loga- rithm, and is always taken as positive. llie integral part of a logarithm is positive or negative according as the number is greater or less than one. It is called the characteristic of the logarithm. The method of computing logarithms cannot be con- sidered here. Its discussion is a matter of Higher Algebra. It has been found that 6742 = 103'8276+^ . .. log 6742 = 3.8276 +. Now 6-7420 = 6742 X 10 == lO^'S^^e x 10^ = W'^\ .'. log 67420= 4.8276; GENERAL PROPERTIES OF LOGARITHMS. 297 and 674200 = 6742 X 100 = lO^-^^^e x 10^ = io^-^r.\ .'. log 674200 = 5.8276. Also, 674.2 = 6742 ^ 10 = lO^-^^^e ^ iqi = io2.8276^ .-. log 674.2 = 2.8276; and 67.42 = 6742 -f- 100 = lO^-^s^e ^ iq2 ^ ioi'8276^ ..-. log 67.42 = 1.8276, etc. We see from the above that so long as the figures of a number and their arrangement are the same, the mantissa of the logarithm is the same no matter what position the group of figures may occupy in the scale of enumeration. The shifting of the group of figures one place to the left increases the logarithm by unity, because it multiplies the number by 10, and the shifting the group of figures one place to the right diminishes the logarithm by unity, be- cause it divides the number by 10. This property of logarithms is peculiar to the system whose base is 10, and is of very great practical importance. 203. Logarithmic Tables. — The mantissae of the loga- ritlims of all numbers from 1 to 99999 have been calculated, and published in the form of tables. In these tables the approximation in the mantissae is carried sometimes to four, sometimes to five, sometimes to six, and sometimes to seven decimal places, giving rise to tables of four-place, five-place, six-place, and seven-place logarithms. The characteristics of the logarithms are not given in these tables, because these can be found by inspection of the numbers. The following table contains the mantissae of the loga- rithms of all integers from 100 to 1000, calculated to four places of decimals, and from it can be found approximately the logarithms of all numbers. 298 LOGARITHMS. COMMON LOGARITHMS. GENERAL PROPERTIES OF LOGARITHMS. 299 commo:n^ logaeithms. n 1 2 3 ^ 5 6 ; 8 9 d 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 62 63 7R'53 79-24 7993 7860 7931 8000 7868 7938 8007 7875 7945 8014 7882 7952 8021 7889 7959 8028 7896 7966 8035 7903 7973 8041 7910 7980 8048 7917 7987 8055 6 7 64 65 66 8062 8129 8195 8069 8180 8202 8075 8142 8209 8082 8149 8215 8089 8156 8222 8096 8162 8228 8102 8169 8235 8109 8176 8241 8116 8182 8248 8122 8189 8254 7 6 7 67 68 69 8261 8325 8388 8267 8331 8395 8274 8338 8401 8280 8344 8407 8287 8351 8414 8293 8357 8420 8299 8363 8426 8306 8370 8432 8312 8376 8439 8319 8382 8445 6 6 6 7 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 72 73 8513 8573 8633 8519 8579 8639 8525 8585 8645 8531 8591 8651 8537 8597 8657 8543 8603 8663 8549 8609 8669 8555 8615 8675 8561 8621 8681 8567 8627 8686 6 6 6 74 7o 76 8692 8751 8808 8698 8756 8814 8704 8762 8820 8710 8768 8825 8716 8774 8831 8722 8779 8837 8727 8785 8842 8733 8791 8848 8739 8797 8854 8745 8802 8859 6 6 6 77 78 79 8865 8921 8976 8871 8927 8982 8876 8932 8987 8882 8938 8993 8887 8943 8998 8893 8949 9004 8899 8954 9009 8904 8960 9015 8910 8965 9020 8915 8971 9025 6 5 6 6 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 82 83 9085 9138 9191 9090 9143 9196 9096 9149 9201 9101 9154 9206 9106 9159 9212 9112 9165 9217 9117 9170 9222 9122 9175 9227 9128 9180 9232 9133 9186 9238 5 5 5 84 85 86 9243 9294 9345 9248 9299 9350 9253 9304 9355 9258 9309 9360 9263 9315 9365 9269 9320 9370 9274 9325 9375 9279 9330 9380 9284 9335 9385 9289 9340 9390 5 I 87 88 89 9395 9445 9494 9400 9450 9499 9405 9455 9504 9410 9460 9509 9415 0465 9513 9420 9469 9518 9425 9474 9523 9430 9479 9528 9435 9484 9533 9440 8489 9538 5 5 4 4 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 92 93 9590 96;38 9685 9595 9643 9689 9600 9647 9694 9605 9652 9699 9609 9657 9703 9614 9661 9708 9619 9666 9713 9624 9671 9717 9628 9675 9722 9633 9680 9727 5 5 4 94 95 96 9731 9777 9823 9736 9782 9827 9741 9786 9832 9745 9791 9836 9750 9795 9841 9754 9800 9845 9759 9805 9850 9763 9809 9854 9768 9814 9859 9773 9818 9863 4 5 5 97 98 99 9868 9912 9956 9872 9917 9961 9877 9921 9965 9S8] 9926 9969 9886 9930 9974 9890 9934 9978 9894 9939 9983 9899 9943 9987 9903 9948 9991 9908 9952 9996 4 4 4 204. Method of Using Logarithmic Tables. — In using a table of logarithms there are two operations, one of which is the inverse of the other: 1°. To find the logarithm of a given number; 2°. To find the number which has a given logarithm. 300 LOGARITHMS. B. TO FIKD THE LOGARITHM OF A NUMBER. 1°. Whe7i the Number has not more than Three Figures. — First determine the characteristic by inspection and write it down. Then look in the column headed n for the first two figures of the number, and at the top of the columns for the third figure. The required mantissa will be in the horizontal line of the first two figures and in the column which has the third figure at the top. This mantissa should be written after the characteristic already found. e.g. Find the logarithm of 687. The characteristic is 2, and the mantissa found in the horizontal line of 68 in the left-hand column and in the column of 7 at the top is 8370. Therefore log 687 = 2.8370. When the characteristic is negative, the minus sign should be written above it, to indicate that it is the charac- teristic alone which is negative. The mantissse of the tables are always positive. Thus log .0687 = 2.8370. When the number consists of two figures only, the man- tissa is found in the column headed 0. Thus, log 68 = 1.8325. When the number consists of one figure only, consider the second figure as zero, and take the mantissa from the column headed 0. Thus the mantissa of 6 is found in the horizontal line of 60 in the column headed 0. 'in |0 lo^ 6 = 0.7782. 2 . When the Numl)er has more than Three Figures. — When a number has more than three figures, use must be made of the principle that when the difference of two num- rO FIND THE LOGARITHM OF A NUMBEB. 301 bers is small compared with either of them, these differ- ences are approximately proportional to the differences of their logarithms. This principle is called the Principle of Proportional Differences. e.g. Find the logarithm of 34567. log 34500 = 4.5378 log 34600 = 4.5391 Difference of the mantissas = 13 Thus a difference of one unit in the third place corre- sponds to a difference of 13 in the logarithms. But the given number differs from 34500, not by a whole unit in the third place, but only by .67 of that unit. Therefore the dif- ference between the logs of 34500 and 34567 will be only .67 of 13 = 8.71, which we take as 9, the nearest integer. Therefore log 34567 = 4.5378 9 4.5387 The difference between one mantissa and the next follow- ing in the tables is called the tabular difference, and the result obtained by multiplying this by the following figures of the number considered as a decimal is called the real difference. It is never necessary to use more than three of the following figures for a multiplier, and seldom more than two. From the above we have the following rule for finding the logarithm of a number of more than three figures : Find the mantissa of the first three figures of the num- ber, and the tabular difference. Multiply this tabular difference by the next two or three figures of the number, considered as a decimal, and add the result to the 7nantissa already found. 302 LOGARITHMS. The tabular difference should be taken from the table at sight. To facilitate this operation, the difference betweeii the last mantissa in one horizontal line and the first of the next is given in the last column, headed D. EXERCISE CXX. Find the logarithms of the following numbers: 1. 956. 2. 58.7. 3. 2.38. 4. .0325. 5. 50 6. .003. 7. 40000. 8. 2. 9. .000007. 10. 28645. 11. 16-. 327. 12. .003579. 13. 2.468. 14. 8.006. C. TO FIND A KUMBER WHICH HAS A GIVEN LOGARITHM. 1°. Whe7i the Exact Mantissa is found in the Tables. — Find the mantissa in the table, and take out as the first two figures of the number tlie two figures of the column headed N which are on the horizontal line of the mantissa, and as the third figure of the number the one at the top of the column in which the mantissa is found, and point off according to the characteristic. e.g. Find the number whose logarithm is 1.9112. Find 9112 in the table and take 81 from the left-hand end of its horizontal line and 5 from the top of its column, and place the decimal point before the 8. log-^ 1.9112 = .815. The symbol log ~ ^ means the 7iumher whose log is. 2°. When the Exact Mantissa is not found in the Table. — Take out from the table the next smaller mantissa, the first three figures of the corresponding number, and the tabular difference, and find tlie real difference between this NUMBER HAVING A GIVEN LOGARITHM. 303 mantissa and the one given. Divide the real difference by the tabular difference to two or, at most, three places in the quotient, annex these figures to the three already taken out, and point off accoi'ding to the characteristic. The result is seldom trustworthy to even two places. It will be seen at once that this process is the reverse of that for finding the correction of the mantissa when the number has more than three figures. EXERCISE CXXI. Find the numbers which have the following logarithms*: 1. 2.9355. 2. f.5635. 3. 2.9948. 4. 3.8845. 5. 0.5982. 6. 3.8340. 7. r.4570. 8. 2.9559. 9. 0.8077. 205. Cologarithms. — The cologarithm of a number is the logarithm of the reciprocal of the number. Thus, colog 987 = log ^ = log 1 - log 987 = 0-2.9943 = -2.9943. To avoid the negative mantissa, the logarithm of the number is usually subtracted from 10 instead of 0. Thus, colog 987 = 10 - log 987, or 10 - 2.9943 = .0057. Of course this logarithm is 10 too large. Such a loga- rithm is called an augmenied logarithm. The colog should be taken from the table at sight. We may begin at the left hand and take each figure from 9 till we come to the last, which should be taken from 10. 304 LOGARITHMS. EXERCISE CXXII. Find the cologarithms of the following numbers : 1. 3784. 2. 3959. 3. 2895. 4. .4265. 5. .078976. 6. .008. 7. 50. 8. .0008. 9. .00009. D. ARITHMETICAL OPERATIONS. 206. Multiplication by Logarithms. — To multiply two or more factors together by means of logarithms, find the logarithm of each factor, add these logarithms and then find the number which corresponds to this resulting loga- rithm. e.g. Find the product of 897, 564, and .0078. log 897 = 2.9528 log 564=2.7513 log .0078 = 3.8921 3.5962 log -13.5962 = 3946.4 207. Division by Logarithms. — To divide one factor by another by means of logarithms, find the logarithm of each factor, subtract the logarithm of the divisor from that of the dividend, and then find the number which cor- responds to the logarithm thus obtained. As in many practical applications it is necessary to per- form both multiplication and division in the same example, it is preferable in all cases to use the cologarithms of the factors of the divisor and add these to the logarithms of the multiplication factors. This method is based upon the principle that to divide by a factor is equivalent to multiplying by its reciprocal. In using cologarithms it must be borne in mind that each ARITHMEriGAL OPERATIONS. 305 colog is augmented, and, tlierefore, that as many lO's must be rejected from the result as there are cologs used. T.. . ,. 1 » 526 X 862 e.g. Fmdthevalueof^3^— . log 526 = 2.7210 log 862 = 2.9355 colog 232 = 7.6345 colog 683 = 7.1656 20.4566 log-^ 0.4566 = 2.8613. 208. Involution by Logarithms. — To raise a number to a power by means of logarithms, find the logarithm of the number, multiply it by the index of the power, and find the number which corresponds to the resulting logarithm. e.g. Raise 249 to the sixth power. log (294)« r= 2.4683 X 6 = 14.8098. log-i 16.8098 = 645330000000000 approximately. 209. Evolution by Logarithms. — To find the root of a number by means of logarithms, take out the logarithm of the number, divide it by the index of the root, and find the number which corresponds to the resulting logarithm. If the characteristic of the logarithm is negative, before dividing by the index, add as many tens to it as there are units in the index of the root, and reject ten from the result- ing logarithm, which would be augmented by 10. For this process consists in adding and subtracting the same multi- ple of 10 and then dividing by the index of the root. e.g. Find the fifth root of .086, 306 LOGARITHMS. log (.086)V5.z. 2.9345-^5 = (48.9345-50) -^ 5 = (48.9345^5)- 10 =1.7869. log~i 1.7869 = .6121, approximately. EXERCISE CXXIII. Note. — A negative quantity has no real logarithm. If such quantities occur in computation, they may be treated as if they were positive and then the sign of the result de- termined by the number of negative factors. If this num- ber be even, the result will be positive, and if odd, negative. In arranging the logarithms and cologarithms for addition, it is best to place an n after each one which has been found for a negative factor, and then a glance will show whether the resulting number should be positive or negative. T.- /. ^1 1 . 23 X - 8 X - 6 ^ e.g. J^md the value of -^ . log 23 = 1.3617 . log 8 = 0.9031/i log 6 = 0.7782^ colog 5 = 9.3010 colog 60 = 8.2218/i 20.5658^ log-i 0.5658w= -3.68. Find by logarithms the values of the following: I. 1. 250.42 X .00687. 2. - 7.8346 X - .086427. 3, - 9.896 X 12.857. 4. .04632 X .008764. ARITHMETICAL OPERATIONS. 307 5. .08 7 6. - 9.876 .0076 • H 18.009 X - .004 8. 27 X - 82 7. .007695 X .004 * 3.8 X - 4.9* 9. (86.42)3. 10. (.0086)3. II. 11. 92/3. 12. H. 13. (- 3.278)^ 14. 192/3. 15. (.12)«/5. (- .000874)5/7. 16. 18. I^To: 17. V'. 0009286. 19. 53/2 X 32/3, 20. 43/8 5275- 21. 5643/5 283 * 22. ^ 5 * 3 210. Theorem. The logarithm of any number to lase h is equal to the product of the logarithm of the number to the base a by logarithm of a to base b. It is required to prove logipi — log„m . logj,a. Let logoW = Xj and log^m = y. Then m = a?", and m — y. Hence, a = J^/*. And . a^/« = b. 308 L00ABITHM8. and, similarly, logaO^ = -. ,\ logbfn = log„m . logi,a, or ^gft^ _ log ^,^, It follows from the above theorem that if the logarithm of any number to base b is known, its logarithm to any other base a may be found by dividing the logarithm of the number to base b by the logarithm of a to base b, e.g. Find log 3 to base 7. Iogio3 = 0.4771. Iogio7 = 0.8451. 4771 EXERCISE CXXIV. Find the following logarithms : 1. logglS. 2. log342. 3. log^S. 4. Iog8.0803. 6. Iogi5.007008. 6. log956.31. When the number can be expressed as an exact power of the base, examples like the above may be solved by in- spection. e.g. Find the value of logiel28. 128 3= 16V*. .-. log,el28 = 7/4. T, log3729. 8. log3,3125. 9. loge,l/4. PART II ELEMENTARY SERIES I CHAPTER XXV. VARIABLES AND LIMITS. 211. Constants and Variables. — A number which, under the conditions of the problem into which it enters, may assume any one of an unlimited number of values is called a variable. A number which, under the conditions t)f the problem into which it enters, has a fixed value is called a constant. Variables are usually represented by the last letters, x, 2/, z, etc., of the alphabet, and constants either by the first letters, a, h, c, etc., or by Arabic numerals. 212. Functions. — Two variables may be so related that a change in the value of one produces a change in the vah.e of the other. In this case one variable is said to be ^func- tion of the other. When one of two variables is a function of the other tho relation between them may be expressed by an equation. Thus, if X and y are functions of each other, we may s: y X X that — = a, or X — aii. or y — —. y '' ^ a , Hence, if the value of one variable be assumed, the corresponding value of the other variable may be computed. The variable for which values are assumed is called Ihe in- dependent variable; and the one whose value is found by computation, the dependent variable. When an equation containing two variables is solved for one of them, the variable involved in the answer is regarded as the independent variable. 811 312 VARIABLES AND LIMITS. Thus, in equation x = ay, y is regarded as tlie inde- pendent variable ; and in the equation y = -, x is regarded as the independent variable. 213. Limit of a Variable. — As a variable changes, its value may approach some constant. If the variable can be made to approach a constant as near as we please without ever becoming absolutely equal to it, the constant is called the limit of the variable. 214. Axioms. — Any quantity, however small, may be taken times enough to exceed any other fixed quantity, however great. Conversely, any quantity, however great, may be divided into so many parts that each part shall be less than any other fixed quantity, however small. 215. Theorem I. If a fraction have a finite numer- ator and an independent variable for its denominator, ive may assign to this denom/hiator a value so great that the value of the fraction shall be less than any assignable value. Let a be the numerator of the fraction, x its denomina- tor, and c any finite value, however small, which we may choose to assign. And let n be the number of times that we must take c to make it greater than a. Then a < nc. a .-. -<6. n Hence, by taking x greater than 7i, we shall have - n> c. X 217. Infinites. — If a variable can become greater than any assigned value, however great that value may be, the variable is said to increase indefinitely, or to increase with- out limit. When a variable is conceived to have a value greater than any assigned value however great, the variable is said to become infinite. Such a variable is called an infinite number, or simply an infinite. An infinite is usually denoted by the symbol qo . It must be borne in mind that this symbol denotes, not a constant, but a variable, which has already increased beyond any assignable limit, but which is still capable of an indefinite increase. 218. Infinitesimals. — If a variable can become less than any assignable value, however small that value may be, the variable is said to decrease indefinitely, or decrease without limit. In this case the variable approaches zero as a limit. When a variable which approaches zero as a limit is conceived to have a value less than any assigned value, 314 VARIABLES AND LIMITS. however small tliis value may 1)e, the variable is said to become infinitesimal. Such a variable is called an infinitesimal number, or simply an infinitesimal. An infinitesimal is often denoted by the symbol 0, which in this case must be understood to represent an exceed- ingly small variable. We often express the relation between finite quantities and infinite and infinitesimal quantities as follows : a ^ n - = 00 , — = 0. 00 The expression tt = Qo cannot be interpreted literally, since we cannot divide by absolute 0; nor can the expres- sion — = be interpreted literally, since we cannot find a number so large that the quotient obtained by dividing a by it shall be absolute zero. The expression - = oo is simply an abbreviated way of writing: when x approaches zero as its limit, then - in- creases without limit. — = is simply an abbreviated way of writing: when X increases without limit, then - approaches zero as its limit. 219. Approach to a Limit. — When a variable ap- proaches a limit, it may approach it in one of three ways : 1°. The variable may be always less tlian its limit; 2°. The variable may be always greater than its limit; 3°. The variable may be alternately greater and less than its limit. VARIABLES AND LIMITS. 315 If X represent the sum of w terms of the series ■ '+\+\+\+---' X is always less than its limit 2. If X represent the sum of n terms of the series ^ 3-4-8 ' X is always greater than its limit 3. If X represent the sum of n terms of the series X is alternately less and greater than its limit 2. 220. Theorem III. If k he any fixed quantity how- ever great, arid x he a variahle which ive may make as small as we please, we may make the product kx less than any assignahle quantity. If there be any smaller value of kx, let it be denoted by s. Since we may make x as small as we please, let us put . *. kx < s, so that s cannot be the smallest value of the product. Hence the product cannot have a smallest value. 221. Theorem IV. If two functions are equal they must have the same limit. Assume it possible for the two functions to have different limits, and denote these limits by L and L '. Put s = \(L-L'), SO that L and L ' differ by 2*\ Now since L is the limit of one function, that function may be made to approach L so as to differ from it by less 316 VARIABLES AND LIMITS. than A', and since L' is the limit of the other function, this function may be made to approach L ' so as to differ from it by less than s. And as the difference between L and L ' = 2s, the functions in the above case must be unequal. But this is contrary to the hypothesis. Hence it is im- possible for the functions to have different limits. 222. Theorem V. The limit of the sum of several functions is equal to the sum of their separate limits. Let the functions be denoted by /(a:), f{x'), f{x"), etc., and their limits by L, L\ L ", etc. ; and let the differences from their limits be denoted by i, i', i'\ etc. Then f{x) = L — i, f{x')=L'-^i\ f{x") = L"-i'\ etc. etc. .-. /(^) +/(,:')+/(:,") + etc. = i: + Z' + Z" + etc. - (i + i' + i" + etc.). We must now prove that i -\- i' -\- i" -[- etc. can be made less than any quantity we can assign. Let h denote this quantity, which may be as small as we please; ^i denote the number of the quantities i, %', i", etc.; and i denote the largest of them. Since the difference between a function and its limit may be made as small as we please, we may make i < -, or ni < h. n But i -{- i' -\- i" -\- etc. < ni^ (/ being the largest.) .*. i -\- i' -\- i" -\- etc. < A. VARIABLES AND LIMITS. 317 Therefore L -\- L' -\- L" -\- etc. is the limit of /W+/(^')-f-/K) + etc. 223. Theoeem VI. The limit of the product of two functions is equal to the product of their separate limits. Using the notation of Theorem V, we have f(x)xf{x') = {L-i){L'-i-) = L. L' - (Li -\- L'i-W). Now as L and L ' are finite, Li' -}- L i can be made as small as we please, and therefore the quantity within the parenthesis may be made as small as we please. Hence L. L' isthe limit of/(.T) X /(«'). Cor. 1. The limit of the product of any numler of functions is equal to the product of their limits. Cor. 2. The limit of any power of a function is equal to the power of its limits lohen these limits are not both zero. 224. Theorem VII. The limit of the quotie^it of two functions is equal to the quotient of their limits when their limits are not both zero. Using the same notation as before, we have f(x) L - i f{x')- L'-i" Now the difference between ^^-7 and -^r-; — r, is L L —t L'i-Li' L\L'-i')' The numerator of this expression evidently approaches zero as its limit, and the denominator approaches L '^ as its limit. 318 VARIABLES AND LIMITS. Hence the expression as a whole has zero for its limit when L ' is not itself zero. 226. Definition. — The expressions •^ ^ -" X — a J at denote the value of these expressions when x becomes equal to a. 226. Theoeem VIII. The formula Lim X r — a J = a"~ is true for all rational values of n. Case I. When n is a positive integer. We have, when x is different from a, T-w //" .M - 1 _J_ n^n - 8 I ^2^n Now suppose X to approach the limit a. Then :c" ~ ^ will approach the limit 6?"~\ x"'"^ tlie limit <3^"~2, etc. Hence «a:"~^ «^^a;"~* etc., will each approach the limit ^n - 1 That is, each term of the second member approaches the limit a/" ~ ^ Because there are n such terms, we have r ' — a' Lim. n - 1 X — a J =a Case II. When n is a positive fractioti. Suppose n = —, p and q being whole numbers. Then a;^ — a"" x'^ — «« X — a X — a Let us put, for convenience in writing, c^ = y, a"^ = Jj VARIABLES AND LIMITS. X = r. a y- h" a;"-«" r- -h" - 1- ■ h X ~ a b" 319 then and X ~ a y* — 0'^ I ~y-o As X approaches indefinitely near to a, and consequently y to 1), the numerator of this fraction (Case I) approaches to pbP~^ as its limit, and the denominator to qb'^'^. Hence the fraction itself approaches to qb^-' q ' Substituting for b its value «^ we have X — a J=a q q (I Hence the same formula holds when n is a positive fraction. Case III. Wlien n is negative. Suppose n — — p, p itself (without the minus sign) being supposed positive. Then x"" — «" X ^ — a p = X- Pa- x—a f a'' - x^ \ \ X — a I fxP - aP\ \ X — a /' — — X ^a When X approaches a, then x'^ approaches a'^, and ' approaches ^«^ "^ Substituting these limiting values, we have I^im. Elzi^ I = _ a-'^^pa^-' = - pa-^-K X — a J=a VARIABLES AND LIMITS. Substituting for — jt> its value % we have Lim. = na' Hence the formula X — a J=a na' is true for all values of n, whether entire or fractional, positive or negative. 227. Definition of Series. — A series is a succession of terms formed in order according to some definite law. 228. Theorem IX. The limit of the series A, + A,x + A^x' + A^x^ + . . . when X is i^idefinitely diminished is A^, provided all the coefficients are finite and the coefficient of the ntlt term approaches a finite limit as n is indefinitely increased. 1°. Suppose the number of terms of the series to be infinite. Let Tc denote the greatest of tlie coefficients A^, A^, etc., and denote the series by Aq + S, Since Ic is the largest of the coefficients A^, A^, etc. . *. hx -\- hx^ + kx? -f- etc. > A^x -\- A^ -\- A^x? -j- etc . •. ^^ < Tex + lex? + hx? + etc. But lex -f- Icx^ + hx^ + etc. may be written in the form lex of the fraction , as may be shown by actual division J. — X of the numerator by the denominator. VARIABLES AND LIMITS. 321 which, when x is indefinitely diminished, can be made as small as we please. Hence by indefinitely diminishing x, Aq can be made to differ from the series by less than any assignable quantity. Hence Aq becomes the limit of the series. 2°. If the number of terms in the series is finite, ;S' must be less than in case 1°; hence, a fortiori, the theorem is true. 229. Theoeem X. In the series Ao + AiX + A^x^ + A^x^ + . . . by taking x small enough ive may make any term as large as ive please compared with the sum. of all that folloiu it, and, hy taking x large enough, we can make any term as large as we please compared with the sum of all that precede it. 1°. The rth term of the series will be A^^, and the ratio of this to the sum of all the terms that follow will be AX A,^,x^^'-^A,^,x^^'-\-... A,^ix-^A,^,x^-\-..: By taking x small enough we can make the denominator of this last fraction as small as we please, and therefore the fraction itself as large as we please. 2°. The ratio of the rth term to the sum of all that precede it will be Ajx'' A„ A,_,x-'^A,_X-' + ... 1 1 • ""^x ""^^ -r • ' ' By taking x large enough we may make the denominator of this last fraction as small as we please, and therefore the fraction as large as we please. 322 VARIABLES AND LIMITS. Cor. In an expression of the form consisting of a finite number of terms in descending powers of X, by taking x small enough we may disregard all the terms but the last, and by taking x large enough we may disregard all the terms but the first. 230. Vanishing Fractions. — A fraction which assumes tlie form — for some particular value of x is called a van- ialiing fractioyi. The fraction, though indeterminate in form when x has this critical value, has a real value. To determine this value is to evaluate the fraction. Sometimes for a particular value of x the fraction assumes the form , which is also indeterminate in form. oo The values of the fractions when they assume these indeterminate forms are really the limiting values of the fractions as x is indefinitely increased or diminished. The limiting value of a fraction when x in both numerator and denominator is indefinitely increased or diminished may be found by Theorem X, cor. e.g. Find the limiting value of ,, ^ , ^\ , when x dx-' -\- 7x^ — 4 is infinite and when x is zero. 1°. When a; = 00 , every term except the first of the numerator may be disregarded, and we have as the limiting value 4:X^ 4 Sx^^S' 2°. When x = 0, every term except the last of the numerator and denominator may be disregarded, and we 7 have as the limiting value — -. VARIABLES AND LIMITS. 323 The limiting value of a fraction which assumes an indeterminate form for a critical value of x may be found by first removing from the numerator and denominator all common factors in x, and substituting the critical value of X in the result. e.g. Find the limiting value of x^ — 4:ax -j- 'Sa^ x^-aJ' 3n X = a. X? - 4:ax + da _ix- a)(x - 3a) _ x^ — a^ ~ {x — a)(x -\- a) ~ x — da X -\- a' Put x = am this result, and we have ^^= 1. 2« EXERCISE CXXV. Find the limiting values of the following: X — al X — a~] 2. « J^ao. X J=o. ax -\-b~\ ax -\-h'l bx -\- aJ = ao' ' bx-^ aj= 0- mx^ ~[ mx^ "I 2)01? — ax J = 00 . ' p^ — ax J _ q. {2x - 3)(S - 6x) -\ (2x - 3) (3 - da;) "] 7x^-Qx-{-4: J =00. ®" 7x^-6x-{-4: J.. x^ — a^~] afi — z^~] 10. X — a J=a' ■ X — z J^ g. 0. 11. + 1 ~] x^-8x-{- 15 -| 324 VABIABLES AND LIMITS. 231. Discussion of Problems. — To discuss the solution of a problem when the answer is literal is to observe between what limiting numerical values of the known elements the problem is possible, and whether any singu- larities or remarkable circumstances occur within these limits. The following discussions will serve to illustrate tlie significance of indeterminate forms of expression, and of and 00 .as limiting values. a. The Product of Two Quantities whose Sum is Constant. Divide a into two parts whose product shall equal h. Let X and y denote the parts. Then, by the con- ditions. x-\- y = a. (1) xy = b. (2) From (1), y — a — X. By substitution in (2), x{a — .t) = l. .-. x^ - ax^h^^', 3nce x=-a±^--l). 1 / a^ and the two parts are o" ^*^ +\/ ^: ^' and VsJt-^- VARIABLES AND LIMITS. ^25 Now these values are imaginary \f h > j- ; that is, if the product of the two parts is greater than the square of half their sum. Cor. Tlie product of tiuo quantities cannot he greater than the square of half their sum. Or, the product of tivo parts of a given quantitij is qreatest when those parts are equal. The two parts will be incommensurable when the differ- ence between their product and the square of half their sum is not a perfect square. b. The General Quadratic Equation. The equation ax^ -^Ix-^- c — has been discussed already in so far as to observe when the values of x become imaginary, when they are real and rational, when real and irrational, and when equal. We will now discuss some peculiarities which may arise by the vanishing of each of the coefficients in turn. Note that c is really the coefficient of x^. If c = 0, then ax^-{-hx = 0', (1) whence x = 0, or . a That is, one of the roots is zero and the other is finite. If ^> = 0, then rf_|_c^O; (2) whence =v-^ 326 VARIABLES AND LIMITS. In this case the roots are equal in vahie and opposite in sign. They will be real or imaginary according as a and c have opposite signs or tlie same sign. It a = 0, then hx + c-Q', (3) and apparently in this case the quadratic has but one root. namely, — — . But every quadratic equation has two roots, and in order to discuss the values of these roots we may proceed as follows : Put - for X in the original equation, and clear of frac- y tions. Then of + % + a = 0. Now put a = 0, and we have oy^ ^hy = 0; mce 2/ = = 0, or _5 c' ••• X - 1 1 c = = 00 , or c Hence, in any quadratic equation one root becomes infinite when the coefficient of x^ becomes zero. This is merely a convenient abbreviation of the follow- ing fuller statement : In the equation ax^ -{- hx -\- c = 0, if a is very small VARIABLES AND LIMITS. 327 one root is very large, and becomes indefinitely great as a is indefinitely diminished. In this case the finite root approaches — — as its limit. c. The ProUem of the Houriers. Two couriers, A and B, are travelling along the same road in the same direction, BR', at the respective rates of m and w miles an hour. At a given hour A is at P, and B is a miles beyond him at Q. After how many hours, and how many miles beyond P, will the couriers be together ? R ^ 9. .R' Let X denote the number of hours after the given time, and y the number of miles beyond P. Then y — a — number of miles beyond §; y=mx', (1) y — a= nx. (2) From (1) and (2), a , am X = — , and y = . 7n — n *^ m — n I. Suppose a to be positive. 1°. Let m > n. In this case both x and y will be positive, and A will overtake B to the right of P. This corresponds with the hypothesis; for since a is positive, B is ahead of A, and since m is greater than w, A is travelling faster than B. 328 VARIABLES AND LIMITS. 2°. Let m = 71. In this case the values of x and ii take — and -— , and each becomes 00 . This result indicates that one never would overtake the other. This interpretation corresponds with the hypothesis made. For B is « miles ahead of A, and both are travelling at the same rate. 3°. Let m. < 71. In this case the values of x and y both become nega- tive. This indicates that che couriers were together before the given time and before they reached the point P. This corresponds with the supposition; for B travels faster and is ahead of A at the given time. He therefore must have overtaken A and have passed him before the given time. II. Suppose « = 0. 1°. Let 771 > n. In this case the values of x and y both assume the form = 0. m — 71 This is as it should be; for since the couriers travel at unequal rates and are together at the given hour, they never could have been together before, nor can they be to- gether again afterward. As A travels faster than B, he must have overtaken B just at the given time. 2°. Let 771 = 71. In this case the values of x and y both assume the form -, and the problem becomes indeterminate. VARIABLES AND LIMITS 329 This corresponds with the given conditions; for the couriers are together and travelling at the same rate. Hence they must have been together during all their past journey, and they must continue together for the future. 3°. Let m < n. This gives the same results as 1°, the only difference being that B must have overtaken A at the given time. III. Suppose a to be negative. 1°. Let m > 71. In this case x and y are both negative, and the couriers must have been together on the road some time before the given hour. This corresponds with the supposition ; for A, being now ahead and travelling faster, must have passed B at some previous point. 2°. Let m = n. This will again give oo for both x and y, and the prob- lem is impossible. These results evidently suit the conditions of the prob- lem; for A is now ahead, and both are travelling at the same rate. Hence the couriers never could have been to- gether in the past, and never can be in the future. 3°. Let m < n. In this case x and y must both be positive, and the couriers must be together at some point farther along the road. This also answers to the given conditions; for B is now behind at the given time, and travelling faster. Hence he must overtake A at some future point. 330 VARIABLES AND LIMITS. d. The ProUem of the Lights. Two lights, A and B, of given intensities, are situated at a given distance apart. Find the point on the line AB where the lights give equal illumination. Let m = illumination of A at a unit's distance, a = distance from A to B, and X — distance from A to P, the point of equal illu- mination. Then a — x will be the distance from B to P. Since the illumination at P varies directly as the inten- sity of the source and inversely as the square of its distance, the illumination of A at P will be -g, and of B at P {a - xy By hypothesis these two illuminations are to be equal. m = n x' {a -xf />• a Via Whence Vm ± Vn The double sign of the denominator gives two values for X, and shows that there must be two points of equal illumi- nation. I. Suppose a to be positive. 1°. Let m > 71. In this case both values of x will be positive, one less and the other greater than a, and the one which is less than a will be greater than — , since the denominator of the VARIABLES AND LIMITS. 331 fraction is less than 2 Vm. Hence the two points of equal illumination will both be on the same side of A, one be- tween A and B and the other beyond B ; and the one be- tween A and B will be nearer to B than to A. Evidently these results are what we ought to expect. The point of equal illumination between the lights ought to be nearer the less intense light, and the second point of illumination ought to be beyond the less intense light, so as to be nearer to it than to the more intense light. 2°. Let m = n. In this case the first value of x will be positive and equal to — , and the second value of x will be oo . That is, one of the points of equal illumination will be midway between the lights, and the other must be at in- finity. The lights being of equal intensity, the points of equal illumination ought to be equally distant from them, and the only such points are the one half way between the two lights and the point at infinity, or nowhere. 3°. Let m < n. In this case the first value of ./• will be positive and less than — , and the second value will be negative and greater Z than a. That is, one of the points of equal illumination will be between A and B and nearer the less intense light, and the other is on. the opposite side of A to B, so as also to be nearer the less intense light, A. II. Suppose a to be zero. 1°. Let m > n. In this case both values of x become zero, and both illuminations become co . 332 VARIABLES AND LIMITS. These results are on the supposition that each light is a mathematical point, which is physically impossible. Mathematical analysis does not concern itself with phy- sical impossibilities. Could each light be reduced to a mathematical point, the intensity of the light would become infinite at that point, and were the two lights together at that point, both illuminations would be equal there and nowhere else. 3°. Let m % + 3^^(^) + . 3^z(MJ_) , 2Mi2 3^^^+^^^+^ = 0^2 H ^ . )l{n + 1)3 37^2 + 5^ + 2 2^3 ^ 3^2 _^ ^ •*• ^^^^- 2 '^ 2 = 2 _ ^^^ + l)(2^ + l) ~ 2 S,= _ 7l{7l + 1)(2?^ + 1) 6 844 THE PR00IIBS8I0N8. 3°. ^3 = ^i'. Here {n + 1)* - n'^ -\- 4w« + 6^^ _|_ 4^ _^ i. Writing 1, 2, 3, etc., in turn for n in this identity, we get 2^ = 1^ + 4 . 13 + 6 . 12 + 4 . 1 + 1 ; 34 = 24 + 4 . 23 + 6 . 22 +.4 . 2 + 1; 44 = 34 + 4 . 33 + 6 . 32 + 4 . 3 + 1; etc. etc. ; i^n-\-iy = 7i^-^^.n^^Q.n^-\-^,n-\- 1. Adding and cancelling as before, we get {n + ly = V + 4.^3 + 6.^% + 4.>S'i + w = 1 + 4. .S3 + n{n + l)(2w + 1) + ^n{n + 1) + w, .-. 4.^%=(n+l)4-[^^(r^+l)(27i+l)+247i+l)+^+l] = ^^4 _^ 4^^3_^ 6^^2 _^ 4,^ _^ 1 __ [2^3_|_3^2_|_,j^27i2+2/i+ w + 1] = n^ + 2^i3 _|_ ^2 ^ ,^2(^^ _|. 1)2 3^ [-^(^ _|. i)-|2, ... ^3=[MM^J=^., B. GEOMETRICAL PROGRESSION". 238. Geometrical Series. — Quantities are said to be in geometrical prog7'essio7i (G.P.) when the ratio of any term to that which immediately precedes it is the same through- out the series. Thus each of the following series forms a geometrical progression : 2, 4, 8, 16, etc. 1, - 1/4, 1/16, - 1/64, etc. a, ar, ar^^ ar^^ etCo OEOMETRtCAL PR00RE88I0N. S45 The constant ratio is called the common ratio, and is found by dividing any term by the one which immediately precedes it. Thus, in the first of the above series, 3 is the common ratio; in the second, — 1/4; and in the third, r. 239. Type Form of the Series. — The type form of a geometric series is a -\- ar -\- ar^ -{- ar^ -\- ar^ . . . -{■ af ~ ^ It will be noticed that in this series the exponent of r in each term is one less than the number of the term. If n denote the number of terms, and I the last or wth term, then / = ar^ ~ ^ 240. Geometrical Means. — When three quantities are in geometrical progression, the middle one is called the geometrical mean between the other two. Let a, b, and c be three quantities in G.P. By definition, — =r — , h^ = aCy and h = Vac. a h That is, the geometrical mean between two quantities is equal to the square root of their product. All the terms in a G.P. between the extremes may be called geometrical means, and any number of such means may be inserted between two terms. Let a and h be the two terms between which 7i geometrical means are to be inserted. The wliole number of terms will be n + 3, and b will be the (w -f 2)th term. Let r be the common ratio. Then ar 346 THE PnOGRESSIONS. h n+ 1 _ _ a n + l r y~a e.g. Insert four geometrical means between 224 and 7. In this case we must find six terms in G.P. of which the first is 224 and the sixth is 7. Therefore 7 =224r^ .-. r^ = l/32, and r = y/xj^'l = 1/2. Hence the means are 112, 56, 28, 14. EXERCISE CXXIX. In finding the common ratio in a G.P. it is often necessary to extract a root of a high index, which is tedious without the use of logarithms. In the following examples it will be easy to extract the required roots by inspection. Kemember that the fourth root is the square root of the square root, that the sixth root is the cube root of the square root, and that the eighth root is the square root of the square root of the square root. 1. Insert two geometrical means between 2 and 250. 2. Insert three geometrical means between — 3 and -768. 3. Insert four geometrical means between 5 and— 1215. 4. Insert five geometrical means between 3 and .000192. 6. Insert four geometrical means between 1/6 and 64/3. GEOMETRICAL PROGRESSION. 347 241. Problem. To find the sum of n terms of a geometrical progression. Let 8 denote the sum, and let the series be a -\- ar -\- ar -\- . . . -\- ar"" \ . Then S = a + ar -{- ar^ ^ . . . + ar'^'K (1) Multiply each side by r : rS = ar -\- ar'^ -\- ar^ -{-... -\- ar^. (2) Subtract (2) from (1), and we get S — rS — a — ar'^, or (1 -r)S= a{l - r"). .-. S= a 1-r e.g. Find the sum of ten terms of the series 2 + 4 -f- 8 + etc. Here a = 2, r = 2, and w = 10. Therefore 1 oio ;S' = 2 A. ^ = 2(210 - 1) = 2(1023) = 2046. 1 — Z 242. Divergent and Convergent Series. — The formula ^ /p'"' ^'* ^ dr^^ ci a—, , or a — , may be written -. I - r r —1 ^ r — 1 r — 1 In the series a -\- ar -\- ar^ + «r^ 4- . . . «r" ~ \. if r be made 1, the series becomes a + a -h a -\- . . . n terms = 7ta. Hence, by sufficiently increasing n, we may cause S to surpass any value however great. When n becomes qo, *S' becomes oo . If r be greater than 1, r"" increases with n, and, by sufficiently increasing n, r'^ may be made as great as we r" — 1 please. When w becomes oo , a— — becomes ao . 348 THE PR0ORE88I0N8. Hence, by sufficiently increasing the value of n, we may cause S to exceed any value however great, and when n = CO , S = cc . In these two cases the geometric series, if supposed continued to an infinite number of terms, is said to be divergent. If r be numerically less than 1, that is a proper frac- tion either positive or negative, r" decreases as n increases. By making n sufficiently large r"^ may be made as small 1 — r" as we please. When n = oo , r"* = 0, and a-- a becomes 1 - r a Hence is the value which 8 approaches as a limit as n is indefinitely increased. In this case the series is said to be convergent. The sum of an infinite series is the limit to which the sum of its first n terms approaches as n is indefinitely in- creased. If r = — 1, the series becomes 8=a — a-\-a — a-\-.,,. In this case the sum of any odd number of terms is «, and of any even number of terms 0. The sum, therefore, does not become infinite when an infinite number of terms are taken, nor does it converge to one definite value. A series which has this property is said to oscillate, and is called an oscillating series. If a series is composed of an infinite number of terms, its sum can be found only when the series is converging. e.g. 1°. Find the sum of the series 1/2 + 1/3 + 2/9 + . . . to six terms. GEOMETRICAL PROOttESSION. 349 1/2(1 - (2/3)^) _ 1/2(1 - 64/729) " ^~ 1-2/3 ~ 1/3 _ 1/2(665/729) 665 ~ 1/3 ~486* EXERCISE CXXX. 1. Find the sum of the G.P. 6 + 18 + 54 + . . . to eight terms. 2. Find the sum of the G.P. 6 - 18 + 54 + . . . to eight terms. 3. Sum — 2 + 2| — 3^ + . . . to six terms. 4. Sum 3/4 + li + 3 + . . . to eight terms. 6. Sum 2 — 4 + 8 — ... to ten terms. 6. Sum 16.2 + 5.4 + 1.8 + . . . to twelve terms. 7. Sum — 1/3 + 1/2 — 3/4 + ... to seven terms. 8. Sum 8/5 - 1 + 5/8 - ... to infinity. 9. Sum .45 + .015 + .0005 + ... to infinity. 10. Sum 1.665 - 1.11 + .74 - ... to infinity. 11. Sum 3-^ + 3-2 + 3-3 -f . . . to infinity. 12. The fifth term of a G.P. is 324 and the eighth term is - 8748. What is the first term ? 13. There are five terms in G. P. The sum of the first and second is 30, and the sum of the fourth and fifth is 1920. What are the numbers ? 14. There are three numbers in G. P. The sum of the first and second is 24, and of the second and third is — 72. What are the numbers ? 15. There are three numbers in G.P. Tlie second minus the first equals 36, and the third plus the second equals 210. What are the numbers ? 350 THE PROGRESSIONS. 243. The Value of Repeating Decimals. — The value of a repeating or a recurring decimal may be found by sum- ming a G.P. to infinity. e.g. 1°. Find the value of the repeating decimal .333-|-. 3 3 3 •^^^ "^ 10 + 10^ + 10^ + • • • *^ '""^^^^^ 3 1 3 10 1 10* 1 - 1/10 "10* 9 ~3' Find the value of the circulating decimal .24L _^ 41 ~ 10 + 102 • 1 _ W "I ..1 - 1/102 ~ 99 J 41 102^ _ ^ . 41 103 X 99 - 10 2 X 99 -f 41 239 10 + 103 X 99 - 10 + 990 990 990* 244. Rule for Values of Recurring Decimals. — Note 241 2 that the last answer = — — - — . y yu Hence we obtain the following arithmetical rule for finding the value of a mixed circulating decimal : Sithtract the non-repeating figures from all the digits down to the end of the first period, and write as a denomi- nator as many 9'^ as there are digits i7i the repeating part, folloived hy as many ciphers as there are digits in the non- repeating part. Note also that the answer to the previous example = 3/9. Hence we obtain the following rule for finding the value of a pure recurring decimal : COMPOUND INTEREST AND ANNUITIES. 351 Write as a doiominator to the recurring digits as many 9's as there are digits in the period. EXERCISE CXXXI. Sum the following recurring decimals as geometrical progressions, and show in each case- that the result is in agreement with the rules just given: 1. .15. 2. .185. 8. .396. 4. .428571. 5. .012987. 6. .79. 7. .315. 8. .116.. 9. .19324. C. COMPOUND INTEREST AND ANNUITIES. 245. Compound Interest. — There are many problems in Geometrical Progression of which an approximate solution can be obtained readily by means of logarithms. Among these the different cases of compound interest and annuities are of especial importance. Money is said to be invested at compound interest when at stated intervals the interest which has accrued is added to the principal, so as itself to draw interest. These addi- tions are made usually annually, semi-annually, or quarterly. 246. Problem I. To find the amotmt at the end of a given time of a sum of 7noney invested at compound ititerest at a given rate. Let F denote the given sum, 71 denote the number of years, r denote the interest of one dollar for one year, and A denote the required amount. 1°. Suppose the interest to be computed annually. At the end of the first year the amount will be P + riP = P(l + r); 352 THE PROGRESSIONS. at the end of the second year the amount will be P(l + ^) -!_ rP(l + r) = P{\ + r)(l + r) = P(l + r)^; at the end of the third year the amount will be P(l + rf + rP{l + rf = P(l + r)2(l + r) = P(l + r)^; and at the end of the 7ith year the amount will be P(l + r)^-'-}-rP(l + r)'^-^ = P(l + r)"-i(l + r) = P(l + r)". The amounts P(l + r), P(l + rf, P(l + r)^ . . . . P(l + r)% are in geometrical progression, the first term being P(l+r), the last term P(l + ry\ and the common ratio 1 + r. ^ = P(l + r)^ (1) To solve this by logarithms, it is necessary to take out log P, log (1 + r)% and the antilog of the sum of these two logs. 2°. If the interest be computed semi-annually, the for- mula for the amount becomes A = p{l + If; (3) and if the interest be computed quarterly, the formula be- comes A = P {'+iT' (^) 247. Present Worth. — The present ivortli of a sum of money due at some future time without interest is the principal which put at interest for the given time would amount to the given sum. 248. Problem II. To find tlie present imrtli, at com- pound interest, of a fixed sum due at a future date. In formula (1), if A denotes the given sum, r the cur- COMPOUND INTEREST AND ANNUITIES. 353 rent rate of interest, and n the given number of years, then P will evidently denote the present worth. Hence ^ = (]-^» = ^(i + '•)-"• w To solve this by logarithms, it is necessary to take out the log of A, the colog of (1 + ^)^ and the antilog of their sum. Of course, if the interest is to be computed semi-annu- ally or quarterly, P must be found from formula (2) or (3). 249. Problem III. To find the amount at a given time of a fixed sum invested at stated intervals at compound interest. Let P denote the fixed sum, and use A, r, and n as be- fore. Then the amounts of the stated investments, on the supposition that they are made annually, will be as follow : A, = P(l + r)% A, = P(l + rr-\ A, = P(l^ry-^ A^ = P(l + r)"-("-« = P(l + r). The sum of these amounts is -P(l + r)-{- P(l + ry -h ^"(1 + ^)' + ^(1 + ry. This is a geometrical progression, of which the first term is P(l + ^)? the common ratio (1 + r), and the number of terms 7i. Hence ^ ' 1 + r — 1 r To solve this by means of logarithms, first find by loga- 354 THE PIIOGEESSIONS . rithms the value of (1 + r)" + S from this subtract 1 -|- r, find the logarithm of the result, of P, and the colog of r, and, finally, the antilog of the sum of the three. 250. Annuities. — An annuity is a fixed sum of money payable at equal intervals of time. If the payment continue for a definite time, the annuity is called q. fixed annuity ; if only during a person's life, a life an7iuity ; and if for all time, a perpetuity. Annuities may pay annually, semi-annually, quarterly, or at any other stated times, but the principles of dealing with all the cases being the same, we shall consider only the case of annual annuities. 251. Problem IV. To find tlie present value of an annuity of a given amount payable at the end of each of n successive years. Let A denote the amount of each payment, P the pres- ent worth of the whole annuity, and P, , P^, etc., the present worth of the successive payments, beginning with the first. Then P, = J(l + r)-\ P, = A{1 + rY\ P^=A{l^rf\ mn,^P=A[^^.^^^,-^.....^^J^ 1 L_ In case of a perpetuity, n becomes oo , and y- — ; — r- be- (1 -f- ry A comes 0. Therefore P = — . r COMPOUND INTEREST AND ANNUITIES. 355 That is, the present ivorth of a perpetuity is the qttotient obtained by dividinff the amount of the anmml papnent by the interest of one dollar for one year. 252. Problem V. To find the amount of an annuity to run for n years which can be purchased for a given sum of money, the rate of compound interest being known. In formula (6), P denotes the present value or the pur- chase-money, and A the amount of the annuity. From (6), we obtain rP _ rP(l-fr)" 1- (1 -f rf Formula (7) is also the formula for finding by what fixed annual payment of A dollars an obligation of P dollars may be cancelled in a given number of years, r being the interest of one dollar for one year. 253. Problem VI. To find the present ivorth of an annuity to begin after m years and to continue for n years, allotving compound interest. By (6), the value of the annuity at the expiration of m years is (1 + and by (4), the present worth of this sum due in m years is A^ _ ^ rV (l + rY _ A{(l + rY-l) 356 THE P1100RES8I0N8. EXERCISE CXXXII. 1. What will be the amount of 2000 dollars for 15 years at 5 per cent, the interest being compounded annually ? 2. What will be the amount of 800 dollars for 9 years 3 months at 4 per cent, the interest being compounded quarterly ? 3. What sum of money will amount to 11240.60 in 5 years 6 months at 6 per cent, the interest being com- pounded semi-annually ? 4. In how many years will 968 dollars amount to 11269.40 at 5 per cent, the interest being compounded semi-annually ? 6. What is the present worth of a note for 600 dollars due 9 years hence, allowing 4-^ per cent compound in- terest ? 6. At what rate per annum will 2600 dollars give $416.40 in 3 years and 9 months, the interest being com- pounded quarterly ? 7. In how many years will 500 dollars double itself at 5 per cent, the interest being compounded annually ? 8. In how many years will a sum of money double itself at 4 per cent, the interest being compounded quar- terly? 9. What is the present value of an annuity of 500 dollars to continue for 20 years, allowing 4 per cent com- pound interest ? 10. What is the present value of a perpetuity of 300 dollars, allowing 5 per cent compound interest ? 11. What is the present value of an annuity of 400 HARMONIC .PROGRESSION. 357 dollars to begin 8 years hence and to run for 15 years, allowing 4 per cent compound interest ? 12. What fixed annual payment must be made to can- cel an obligation of 3000 dollars in 8 years, allowing 3^ per cent interest ? 13. What annuity to continue 12 years can be pur- chased for 4000 dollars, allowing 5 per cent compound interest ? D. HARMONIC PROGRESSION". 254. Harmonic Progression. — Three quantities are said to be in harmonic progression when the first is to the third as the difference between the first and second is to the difference between the second and third. An harmonic progression is denoted by the abbreviation H.P. a, J), and c are in H.P. when a-.l = a — h:h — c. A series is said to be harmonic when every three con- secutive terms are in H.P. 255. Theorem I. If three quantities are in har- monic progression.) their reciprocals are in arithmetical progression. Let a, h, and c be three quantities in harmonic pro- gression. Then a'.c ^= a — h'.l) — c. Whence «(J — c) = c{a — V), or ah — ac = ac — he. Dividing each term by adc, we have c h~ 1) a' 358 THE PROGRESSIONS. Harmonical properties are interesting because of their importance in geometry and in the theory of sound. In algebra, the theorem just proved is the only one of any im- portance. There is no general formula for the sum of any number of terms in H.P. Questions in H.P. are solved usually by taking the reciprocals of their tejms, and making use of the properties of the resulting A. P. 256. Theorem II. The liarmonic mean of two quan- tities is equal to ticice their ])roduct divided by tlieir sum. If a, b, and c are in H.P., — , ^, and — are in A. P. a b' c .: ^+1 = ^. a c b a -\- c 267. Theorem III. The geometric mean of two quan- tities is also the geometric mean of the arith7netic and har- monic mea7is of the quantities. Denote the arithmetic, geometric, and harmonic means of a and ^ by J, 0, and H, respectively. Then A = 2 • G = Vab. H = Ub .'. A . H = ab = 0\ 258. Problem. To insert n harmonic means betivem a and b. Insert n arithmetical means between - and -=-, and the a b reciprocals of these will be the required harmonic means. BARMONIC PROGRESSION. 359 EXERCISE CXXXIII. 1. Insert two harmonic means between 3 and 12. 2. Insert two liarmonic means between 2 and 1/5. 3. Find the fifth term of the H.P. 1/2, 1/4, 1/6. 4. Insert three harmonic means between 5 and 25. 5. If a, h, c, are in A. P., and i, c, d, are in H.P., prove that a \h — c -. d. 6. Show that if a, h, c, d, be in H.P., then will ?>{h - a){d - c) = {c - i){d - a). 7. Show that if .a, h, c, be in A. P., h, c, d, in G.P., and c, d, e, in H.P., then will a, c, e, be in G.P. CHAPTER XXVII. BINOMIAL THEOREM. 259. Theorem. — When n is a positive integer, (a + xY = a" + na"" -^x-{- ^^^^a" " V + 1.2.3 "^ ^ -\- . . . to n -\- 1 terms. 1°. When n = 1, we have {a -{- xY — a -\- X = a"" -{- na"" ~ % since a^"^ = «o = 1. By actual multiplication, when ?^ = 2, we have {a-\-xy=a^-\-2ax-{-x^=a''+na'' -^x-{- ^^^^~^V " V, since a^'^ — aP = 1. When w = 3, we have (a-\-xf = «3 + 3 A + 3«a;2 + a;^ = «" + na"" -^x-\- \ _ \ 1 . z When w = 4, we have (a + a;)^= a^ + 4 A + Ga^a^^ + ^ao? -|_ a;4 = ^n _|_ ^^n- 1^ y^(^-l) 2 o , n{n-l){n-^) + 1.2 "^ ^+ 17273 "" ^* ^^ - 1)(^ - 2)(^ - 3) + 1.2.3.4 "^ ''• 360 BINOMIAL THEOREM. 361 We thus see that the theorem holds true when n ^ \, 2, 3, or 4. 2°. Now multiply each member of the expression (a + xf = fl" 4- na^'-'^x + \ ~ ^U ^'-^x^ 1 . Z + 1.2.3 "^ ^ 4" . • . to (w + 1) terms, which we have found to hold true when n = l,2, 3, and 4, hy a -{- X, and we obtain (a + a:)'* + i = «« + !+ [«"a; + na^'x] -f- • • • to 7i + 2 terms. Note that the second term of the last aggregate is ob- tained by multiplying the fifth term of the expression of (a + ^)" hy a. Note also that each aggregate contains two terms in ax with identical exponents, and that, if we let r + 1 denote the number of the aggregate, the coefficient of these two terms of each aggregate after the first will be respectively n{n-l), . .(n-(r- 1)) n{n - 1) . . . (?i - r) 1 . 2 . . . . r ^""^ 1.2. ...r+1 • ^(,, _ 1) . . . (^ _ (^ _ 1)) n(n-l)..,(n-r) 1.2. ...r "^ 1.2. ...r + 1 n{n-l). . .(n-(r- 1)) T n - r n 1.2. ...r L ■^r+lj 362 BINOMIAL THEOREM. ^ n{n-l)...(7i-{r-l)) n^l 1.2 r ^ r + 1 _{n + l)n{n - 1) ... (w - (r - 1)) ~ 1. 2. . . . r(r+l) ' whatever r may be, and this is the general expression for the sum of the coefficients of the term in ax in each bracket after the first. Therefore we have (a + xf^^ :=«" + ! + {n + l)a''x + i^±li)^«~ - 1^^^ 1 . Z {n^l)n{n-l) _ (n ^ l)n(n - l)(n - ^) + 1.2.3 "" ^ "^ 1.2.3.4 "^ ^ + . . . to (« + 2) terms. If we put n -\- 1 = n' , WQ will have 1 . /O + >»'(«' -W-V -%^ + ... to (»' + 1) terms, which agrees with the theorem. We therefore conclude that the theorem will be true for the next higher value of n if it be true for any one value of n. But by actual multiplication the theorem has been shown to hold true when ?i = 1, 2, 3, and 4. It therefore niust hold true when n = 5, 6, 7, or any positive integer. 260. The Binomial Coefficients. — The quantities n{n-J) n{n - l){n - 2) ""' 1.2 ' 1.2.3 '^^''" are known as the binomial coefficients. Note that the factors in the numerators begin with n BINOMIAL THEOREM. Z^^ and decrease by 1, and that their number is one less than the number of the term in which it occurs; also that the factors in the denominators begin with one and increase by unity, and that the number of factors in the denominator is the same as in the numerator. e.g. The coefficient of the fifth term of the develop- meut of (. + x)» is >K >' -!)(>» -2)(« - 3) _ ^ ^ 1.2.3.4 Note carefully that the binomial coefficient of the next term in the development of a binomial expression can be obtained by multiplying the coefficient of the last by the exponent of a in that term and dividing by the number of the term. Thus the binomial coefficient of the third term is -^ — --^, and the exponent of a is n — 2. The binomial 1 . Z coefficient of the fourth term is — ^^ — - — ^^r -. This is the coefficient of the third multiplied by (w — 2) and di- vided by 3. 261. Developments. — AVhen a single algebraic expression is changed into the sum of a series of terms, it is said to be developed, and the series is called its development. A development may be true in form, yet may equal the func- tion only for certain special values of x. No development can equal the function except for the values of x which make it convergent. EXERCISE CXXXIV. Find the binomial coefficients of the development of the following expressions: 1. {a^xf. 2. {a-^xf. 3. (a-[-xy. 4. (a^xf. 5 {a^xY. 6. {a^x)". 7. {a-\-x)\ 8. {a-^xf. 9. {a ^ x)\ 364 BINOMIAL THEOREM. 262. Coefficients. — Note in the above examples that after the middle of the development, the coefficients of the first half are repeated in the reverse order. When n is odd, the number of terms in the development will be even. There will be no middle term, and the coefficients of the terms each side of the middle of the series will be the same. When n is even, the number of terms in the development will be odd, and there will be a middle term whose coefficient will be the largest of all. 263. Exponents. — Note also that the sum of the expo- nents of the two terms of the binomial in each term of the development is equal to 7i, and that the exponent of the second term of the binomial is always one less than the number of the term in which it occurs in the develop- ment. The exponent of the first term will be n, minus the exponent of the second term. e.g. In the sixth term of the development of (a + xy we have a'^x^. 264. Signs. — When both terms of the binomial to be developed are positive, all the terms of the development are positive, since all powers of positive quantities are positive. When the first term of the binomial is positive and the second term negative, every other term of the development beginning with the second is negative. e.g. Write the product of the powers of the first and second terms of the binomial (c — 2x^y in the fourth term of its development. c%- 2xY =c^X -Sx'= - 8(^x\ EXERCISE CXXXV. Write the product of the powers of the two terms of the following binomials in the given term of their develop- ment. N.B. — When the terms of the binomial expression to be developed are complex, they should in all cases be thrown BINOMIAL THEOREM. 365 with their signs within parentheses, the powers to which these are to be raised should be indicated, and the binomial co'efficient should be written before and then the indicated operation should be performed. 1. In the fifth term of (a + 2x^y\ 2. In the fourteenth term of (3 — aY^. 3. In the fourth term of (5^^ — Ix^)"*. 4. In the eighth term of (6a — x/by^. I 1 V^ 5. In the seventh term of (2a; — — 1 . / 1 Y' 6. In the eleventh term of \\.x . \ 2 ^/xl 265. Practical Rules. — The work of developing a power of a binomial is facilitated by the following arrange- ment: 1°. In one line write all the powers of the first term beginning with the ^^th and ending with the 0th, or unity. 2°. Under these write the corresponding powers of the second term, beginning with the 0th, or unity, and ending with the wth. 3°. Under these, in a third line, write the binomial coefficients. 4°. Form the continued product of each column of three factors, and connect these products with the proper signs. The result will be the required development. e.g. Develop (2« — Zx^y. Powers of 2a, 32as + 16a4 +8a3 + 4«5 -f2« +1. Powers of - Soj^ 1 - S** + Oa;* — 27aj8 _|- 81*8 _ 243«io. Binom. Coef. 1 -f- 5 +10 +10 +5 +1. (2a - Zx'f = S2a' - UOa*x'-\- T20a^x*~ 1080aV+ SlOax^- 24'dx^^ Perhaps the easiest way to write out a binomial expres- 366 BINOMIAL THEOREM. sion is first to throw tlie complex terms with their signs within parentheses, indicate the powers to which these are to be raised, and then find the binomial coefiicients by successive applications of the rule already given for finding the coefficient of the next term to the one already ob- tained. EXERCISE CXXXVI. Develop the following expressions : 1. {a + x)\ 2. {a -xy. 3. {i-^xy. 4. (^-3)^ 5. (3. :+2#. 6. {"^x-yy. 7. (1 - da'Y. 8. (1 -xyy. 9. (3«-2/3)«. 10. p 3y V 3 ^ 2^-y . 11. (c^/3 + f^ 3/4)4 12. (m- V2 _ n^y. 13. {x^^ - 2y^"y. 14. [a^ + 5 Vxf. 15. ( Va^ + -i ^ay. 16. (xy^-^dy-'^r-y. 17. («V2^- 2/3 4-^,-1/2^2/3)7^ 18. (1 - l/xr. - 266. The General Term. — The general term of the development of {a -\- xy is usually designated the rth term, r standing for the number of the term. In any term of the development of {a + x)'^ : 1°. The exponent of x is one less than the number of the term. 2°. The exponent of a is n minus the exponent of x. 3°. The last factor of the numerator is one greater than the exponent of a. 4°. The last factor of the denominator is the same as the exponent of x. Therefore, in the rth term, The exponent of x will be r — 1 ; BINOMIAL THEOREM. 367 The exponent of a will be y^ — (r — 1) or 7^ — r + 1 ; The last factor of the numerator will be ';^ — r -}- ^5 The last factor of the denominator will be r — 1. Hence the formula for the rth term is n{n - \){ii - 2) . . . (^ - r + 2) 1.2.3 r:": V^^W ' e.g. The seventh term of (2aV2 _ I- 2)12. In this case n — 12 and r = 7; hence the seventh term will be 12 . 11 . 10 . 9 . 8 . 7 = 924 . (64«3Z'-i2) ^ 59136«3Z>-i2. EXERCISE CXXXVII. 1. Find the fourth term of {x — 5)^^. 2. Find the tenth term of (1 + 1x)^. 3. Find the twelfth term of (2a; - \y^. 4. Find the fourth term of («/3) + Uy^. 5. Find the fifth term of (2« - 1/^)^. 6. Find the seventh term of {— —1 . -2\6 7. Find the fifth term /^3/2 y5/2\8 8.- Find the value of (x + V2y -\- {x - V2Y. 9. Find the value oi (V2 -]- ly - { V2 - If. 10 Find the value of [2 - ^{1 -x)f+[2-\- ^(l-x)Y, 11. Find the middle term of {a/x -f- x/aY^. 12. Find the two middle terms of ida ^ j . (o 1 \ 9 — X^ — —- ^ dx 368 BINOMIAL THEOREM. 267. Binomial Theorem for any Rational Index. — We have seen that when n is a positive integer, the binomial function develops into a finite series, the number of whose terms is w -f 1. This is because the factor n — r -\- 1 vanishes when r — n -\- 1. Now as r is necessarily integral, n — r -{- 1 cannot vanish for any fractional or negative value of n. Hence when n is negative or fractional, a function when devel- oped by the binomial theorem must produce an infinite series of terms. It is shown in Higher Algebra that the development is true in form for all rational values of n. It must, however, be borne in mind that the series is in reality an expansion of the function only for those values of x which render the series convergent. EXERCISE CXXXVIII. Develop each of the following binomials to five terms: 1. (a - x)y\ 2. {a + x)y\ 3. (1 - xY\ 4. (1 -f xy^ 5. (3 - ^x)y\ 6. 1/ n - x. 7. 1/ v'lT^. 8. y{x^ + %). 9. («' - ^x- V2) - y\ CHAPTER XXVIII. PERMUTATIONS AND COMBINATIONS. 268. Permutation. — To permute a group of things is to arrange them in a different order, and the various different orders in which the things in a group may be arranged are called the im'mutations of the group. Thus I permute the group formed by the three letters abc when I change their order into acb, and the six differ- ent orders in which the letters of this group may be written are called the permutations of this group. These permuta- tions are abc, ach, hca, hac, cab, cba. 269. Combination. — To combine a given number of things into groups each of which shall contain the same number of things is to select from the whole the requisite number of things and put them together without regard to the order in which they are placed, and the various groups that may be formed in this way out of the whole number are called the combwations of the things. Thus the four letters a, b, c, d, may be combined two at a time, or by twos, in six different ways, namely, ab, ac, ad, be, bd, cd. If the letters were taken three at a time, or by threes, it would be possible to make only four combinations, namely, abc, aM, acd, bed. 370 PERMUTATIONS AND COMBINATIONS. 270. Symbols of Combination and Permutation. — If the whole number of things at our disposal be denoted by w, and the number to be put into each group be denoted by r, then the number of possible combinations will be de- noted by the symbol ^C^. This symbol is read, oi things combined by r's. Thus in the above example and 'C, = 4. When things are combined by 2's there are two possible permutations for each group. Thus we may write ab, or ba. Of the four letters a, b, c, d, the possible combinations by 2's are ab, ac, ad, be, bd, cd. Of each of these groups there are two possible permu- tations. Hence the possible permutations of the four letters by 2's are ab, ac, ad, be, bd, cd, ba, ca, da, cb, db, dc — 12. Of the same four letters the possible combinations by 3's are abc, abd, acd, bed. Of each of these groups there are six possible permuta- tions. Hence the possible permutations of the four letters by 3's are abc, abd, acd. bed. acb. adb. adc, bde. bea, bda, cda. cdb, bac, bad, cad. cbd. cab. dab. dac, dbc, cba, dha^ dca. deb ^ 24. PERMUTATIONS AND COMBINATIONS. 37l In any case, the number of permutations is equal to the product of the number of combinations and the number of permutations of each combination. Using n and r as above, the number of permutations that are possible is denoted by the symbol "P^. Thus, ^P, ^ 12 and ^P, = 24. 271. Number of Permutations. — The important fact to which attention was called a short time since may be sym- bolized thus: - «P^ =r "6; X ^P,. This is a special case of the following general principle : If one operation can be j)erformed in m ways, and if after it has been performed in any one of these ways a second operation can be performed in n ways, the number of ways of performing the two operations will be m X n. The truth of this statement is evident. For there will be n ways of performing the second operation for each way of performing the first; that is, n ways of performing the two for each way of performing the first ; and as there are m ways of performing the first, there must be m X n ways of performing the two. e.g. There are ten steamers plying between Liverpool and Dublin. In how many ways can a man go from Liver- pool to Dublin and return by a different steamer ? There are ten ways of making the first passage, and with each of these is a choice of nine ways of returning. Hence the number of possible ways of making the two journeys is 10 X 9 == 90. This principle applies also to the case in which there are more than two operations each of which may be per- formed in a given number of ways. e.g. Three travellers arrive at a town in which there are four hotels. In how many ways can they find accommo- dation, each at a different hotel ? 372 PERMUTATIONS AND COMBINATIONS. The first tniveller has a choice of four hotels, and after he has made his selection in any one way, the second has a choice of three. Hence the first two can make their choice in 4 X 3 == 12 ways. With any one of these selections, the third can select his hotel in two ways. Hence the possible number of ways is 4 X 3 X 2 = 24. 272. Peoblem I. To find the number of permutations of n dissimilar things taken r at a time. This is equivalent to finding in how many different ways we may put one thing in each of r places when we have n different things at our disposal. Evidently we may select any one of the n objects for the first place; hence we may fill that place in n different ways. After any object has been selected for the first place there remain n — 1 objects, any one of which may be se- lected for the second place. Hence the first two places may • be filled in 7i{n — 1) different ways. After any selection has been made for the first two places there remain n — 2 objects, any one of which may be selected for the third place. Hence the first three places can be filled in n{n — l)(vi — 2) different ways. And so on. Notice that a new factor is introduced for each place that is filled, so that the number of factors will be equal always to the number of places filled. Notice also that the first factor is the number of objects at our disposal, and that each subsequent factor is dimin- ished by unity, so that each factor is the number of things at our disposal diminished by a number which is one less than that of the corresponding place. Hence the rth fac- tor will hQ 71 — {r — 1) = 71 — r -\- 1. Hence the number of permutations of w things taken r at a time, or "P^ = 7i{7i — l){^n — 2) ... r factors, or "P^ = 7l{7l - 1){71 - 2) ... (71 - 7- + 1). When r in the above formula for the number of per- PERMUTATIONS AND COMBINATIONS. 373 mutations equals n^ the last factor becomes 1, and the for- mula becomes -P,^ = n{n - l)(?^ - 2) ... 3 . 2 . 1. This product is (idXlQdi factorial n. It is usually denoted by the symbol \n, or n\ e.g. 1°. Six persons enter a room in which there are six chairs. In how many ways may they be seated ? Here we have «Pe=|G = 6X5X4X3X2X1 = 720. e.g. 2"". Five persons enter a room where there are eight chairs. In how many ways may they be seated ? Here we have sp, = 8X7X6X5X4 = 6720. e.g. 3°. How many different numbers of six digits may be formed out of the nine digits 1, 2, 3, ... 9 ? Here we have 9Pg = 9X8X7X6X5X4 = 60480. 273. Problem II. To find lioiv many of the permuta- tio?is ^Pr contain a particular ohject. Denote the objects by the letters of the alphabet. Find first how many permutations there are of all the letters b4it a when taken r — 1 at a time. Then associate a with each of these in every possible way. The result of these two operations must be all the permutations of the n letters taken r at a time which contain the letter a. The permutations oi n — 1 things taken r — 1 at a time are "-ip^_i = {n - l){n - 2) . . . {n - r -\- 1). In each of these groups a can have r positions, since it may occur first, or last, or in every intermediate position between the letters of each group. 874 PERMUTATIONS AND COMBINATIONS. Hence the number of permutations which contain the letter a is r{n - l){n - 2) . . . (w - r + 1). In a similar way we may find that the number of per- mutations which contain two objects or letters is r{r - l){n- 2) . . . {n - r + 1). For if the two letters a and b be left out and the re- maining letters are arranged in groups of r — 2 letters, the number of permutations would be (n- 2){n-d). , . (n - r + 1). Since each of these groups contains r — 2 letters, i may be associated with each in r — 1 different ways. Hence the number of permutations which contain b would be (r - l){7i - 2){?i - 3) . . . {71 - r 4- 1). As each of these groups contains r — 1 letters, a may be associated with it in r different ways. Hence the num- ber of permutations which contain a and b would be r{r - l){n - 2){7i - 3) . . . (w - r + 1). In a similar way, the number of permutations contain- ing three objects or letters would be r{r - !)(/• - 2)(7i - S) . . . (n — r -{- 1), etc. etc, e.g. How many numbers of four digits can be formed out of the six digits 1, 2, 3, 4, 5, 6 ? How many of these will contain 1 ? How many will contain 1 and 2 ? How many will contain 1, 2, and 3 ? 1°. ep, = C) X 5 X 4 X 3 = 360. 2°. r{7i - l){7i - 2)(7i -3) = 4X5X4X3 = 240. 3°. r{r - l)(7i - 2){7i -3) = 4x3x4x3 = 144. 4°. r{r - l)(r - 2)(w - 3) = 4 x 3 X 2 X 3 = 72. PERMUTATIONS AND COMBINATiONS. 375 274. Peoblem. To find the number of permutations of n tilings all together, ivhen u of the things are alike. Denote the required number of permutations by x. Now if the u things were all unlike they would give rise to "Pu, or u\, permutations, each one of which might be com- bined with the X permutations, and thus give rise to "P„ , or n !, permutations. Hence ^up — np n\ or X— —,-. ui Similarly, if among the n objects there were u alike of one kind and v of another, then ^ up vp — np •^ ' -L u ' -^ V — -'■ nf n\ , or X =^ —. — r, etc. u\ v\ e.g. How many permutations can be made from the letters in the word Mississippi ? Here there are 11 letters in all, and among them 4 s's, 4 i's, and 2ys. _ 11! 11.10.9.8.7.6.5.4.3.2.1 ^ ~ 4! 4! 2l ~ 4.3.2.1.4.3.2.1.2.1 = 34650. If the permutations were to contain no repeated letters, the number of different letters being 4, the permutations would be *P, = 4 . 3 . 2 . 1 --= 24. EXERCISE CXXXIX. Find the value of : 1. '^Pr . 2. i^Pg. 3. 'Pr 4. How many permutations can be made of the letters in the word number ? 376 PERMUTATIONS AND COMBINATIONS. 6. How many permutations can be made of the letters in the word q^iadruple ? 6. How many permutations can be made of the letters in the word priiiciple ? 7. In how many ways may 4 red, 3 blue, and 5 white cubes be arranged in a pile ? 8. In how many ways can 7 cards each of a different prismatic color be arranged in piles of 4 cards each ? 9. How many of these piles would contain red ? 10. How many of them would contain red and green ? 11. How many of them would contain red, green, and blue? 12. A pack consists of 8 white, 6 red, and 4 blue cards. In how many ways may they be arranged ? 275. Problem. To find the number of combinations of n things tahen r at a time. As we have already seen, nr - !^ - ^(^^ - l){n - 2) . . . (^ - r + 1) \l~ ' t e.g. How many different committees of 8 persons each can be formed out of a board of 16 men ? Here .^^^16.15.14.13.12.11.10.9 8.7.6.5.4.3.2.1 = 12870. 276. Problem. To find the number of ti^nes any 'particular object, a^ will be present in ** 6^. If we form "~ ^6V_i combinations from all the objects except a taken r — 1 togetlier we can place a with each of these groups, and thus form all the combinations of the PERMUTATIONS AND COMBINATIONS. 377 n objects taken r together which contain a. Hence a occurs in '*"^6'^_, of the combinations. Similarly, two particular objects will occur in ""^(7^_2 of the combina- tions; etc. e.g. Out of a guard of 14 men, how many different squads of 6 men can be drafted for duty each night ? In how many of these squads would any one particular man be ? In how many of these squads would any two given men be ? 1°. ^^C'e = 3003. 2°. '^C, = 1287. 3°. 126; = 495. e.g. From 10 books in how many ways can a selection of 4 books be made, 1° when a specified book is included, 2° when a specified book is excluded ? 1°. Since one book is to be included in each selection, we have only to choose 3 out of the remaining 9. 'C, = 84. 2°. Since one book is always to be excluded, we must select the 4 books out of the remaining 9. »C; = 126. EXERCISE CXL. 1. In a certain district 4 representatives are to be elected, and there are 8 candidates. In how many differ- ent ways may a ticket be made up, each ticket to contain four names ? 2. Out of 9 red balls, 4 white balls, and 6 black balls, how many different, combinations may be formed each con- sisting of 5 red balls, 1 white ball, and 3 black balls ? Out of the 9 red balls 126 combinations may be formed 378 ' PERMUTATIONS AND COMBINATIONS. each containing 5 balls. Each of these may contain one of the 4 white balls, and there may be formed 20 combinations out of 6 black balls taken 2 at a time. As each of these may be combined with the 126 previous groups, hence the combinations will equal 126 X 4 X 20 = 10080. 3. How many combinations can be formed out of 5 red, 7 white, and 6 blue objects, each combination to consist of 3 red, 4 white, and 2 blue objects ? 4. On the supposition that the colored objects of each set are all of different shape, how many permutations of these objects could be formed with 3 red, 4 white, and 2 blue in each resulting set ? 6. Out of 12 doctors, 15 teachers, and 10 lawyers, how many different committees can be formed, each containing 4 doctors, 5 teachers, and 3 lawyers ? 6. There are fifteen points in a plane no three of which are in a line. How many ^triangles can be formed by join- ing them in threes ? 277. Meaning of the Binomial Coefficients. — {a + xy = [a -\- x)(a -\- x) = a^ -\- 2ax + x^; {a + xY ={a-]-x){a + x){a -^x) = a^-{- da^x+dax^ + x^; {a -j- xy z=z (a -\- x){a + ^)(« + ^)(« + x) = «^ 4- 4 A +6ftV _^ 4^^3 _^ ^4. (a -\- xy = (a -\- x){a -\~ x) . . . n factors = a^+ na^-'x + ^^^^^~ "^V " V . . . to w + 1 terms. These products are formed by taking a letter from each of the 7i factors and combining them in every possible way. We may take an a from each and combine these n a's PERMUTATIONS AND COMBINATIONS. 379 into a product in every possible way. As the letters are all alike, there is only one way of combining them. Hence «" is one term of the product. The letter x can be taken once, and a the remaining {n — 1) times, and the number of combinations of «"~^ and X will be the number of ways in which x may be taken out of the n factors, and this is the number of ways of taking n things 1 at a time, or ^C\ = n. Hence the term «" " ^x will occur "Ci times and we have Again, the letter x can be taken twice, and a the re- maining {n — 2) times, and the number of ways in which 2 x's can be taken is the number of ways of taking n things 2 at a time, or "62 = — — — ^ — - . Hence the term «" ~ "^x^ will occur ^6^2 times, and we have "6'2««-V. And, in general, x can be taken r times (r being a positive integer not greater than n), and a the remaining {n — r) times, and the number of ways in which r x'% can be taken is the number of ways of taking n things r at a time, or _ n{n - l){n - 2) . . . (n - (r - 1)) ^^ - 1 . 2 . 3 . . . . r Hence we shall have ^C^a ™ ~ ^x^. Hence (« + a:)" = «" -f ^ C^a'' " ^a; + " C^a'' - ^x^ + . . . wCVa"- ^a;^ + ... to [~Cna"-"2:^ = x""]. We thus see that the binomial coefficients are simply •the number of different ways in which 71 things can be taken 1, 2, 3, . . . up to w at a time. They are 1, ~(7i, ^C^, "C'a, ... ~6; ... up to "C„. 380 PERMUTATIONS AND COMBINATIONS. They are often written C^, Ci, C\, C^, . . .Cr,..Cnj Cq being understood to be 1. If we make both a and x equal to 1, the formula becomes (1 + 1)- = 1 + 6; + c; + c; . . . + a . . . + c'^, or 2*^ rr 1 + 6\ + ^2 + Cg . . . + C; . . . + Chi. That is, the sum of the binomial coefficients in any expression io n -{- 1 terms is equal to 2^ — 1. Or the sum of all the possible ways of taking n things 1, 2, 3, up to n at a time is equal to 2" — 1. CHAPTER XXIX. DEPRESSION OP EQUATIONS. 278. General Equation of wth Degree in x. — The most general form of an integral equation of the nth. degree in X is in which n is a positive integer. If we divide this equation through by Jo 7 and put —^ =z ai, -J — a^, etc., we obtain xn + a.x''-'' + «,x"-2 + . . . an._,x + «^ = 0, (1) which we will consider as the general form of an integral equation of the nth degree in x. The coefficients «i, a.^, etc., may be integral, fractional, or surd, but we shall consider only the cases in which these coefficients are rational. If none of the coefficients a^, a.^, etc., are zero, the equation is said to be complete ; and if one or more of them are zeros, incomiMe. Any value of x which causes the first member of (1) to vanish, or become zero, is called a root of the equation. It is proved in Higher Algebra that every equation of the above form has at least one root, and we shall assume this to be true in the present chapter. 279. Theorem 1. If a is a root of the equation x"" + a^x''-'^. + «2^"~^ + . . . «„-i^ + rt'n = 0, the first rneinber of the equation is divisible hy x — a. 382 DEPRESSION OF EQUATIONS The division of the first member hy x — a may be con- tinued until the remainder does not contain x. Denote this remainder by R and the quotient obtained by Q. Then we have {x-a)Q^R = 0, as a form which the general equation may be made to as- sume. But a is assumed to be a root of the equation. Hence if we put X — a, the first member must vanish. ... o.e + ^ = o, or i? = 0. Therefore x — a \^ contained in the first member with- out a remainder. 280. Theorem II. Conversely, if the first me7)iber of the equation x^ + a^x'^'^ + a.iX^~'^ + • . . ctfi-i^ + «„ = is divisible hy x— a, then a is a root of the equation. In this case the equation may be made to take the form {x — a)Q— 0, the first member of which vanishes when x = a. Therefore a must be a root of the equation. Cor. If the first member of the equation be divisible by ax -\- h, then is a root of the equation. 281. Theorem III. An equation of the nth degree has n roots. We have assumed what may be proved in more advanced algebra that the equation x"" + a^x'^''^ -\- «2^""^ + • . . an-iX + «„ = has at least one root. DEPRESSION OF EQUATIONS. 383 Denote this root by a. Tlien the first member is divisi- ble by X — a, aud the equation may be written {x - «)(:r"-i + h^x""-^ + . . . K-yX-^ hn) = 0, of which re = rt is a solution, and of which a farther solu- tion may be obtained by putting a;"-i + b.x''-' + . . . b^.^x + ^^ = 0. This division lowers, or depresses, the degree of the equation by unity. The new equation is the same in form as (1), and therefore may be assumed to have at least one root. Denote this root by h. Then the first member is divisi- ble hy X — h, and the equation may be written {x - h){x--' + c,x^-' + . . . Cn-,x + c^) = 0, of which a; = ^ is a solution, and of which a further solu- tion may be obtained by putting Jb I v^JC' I • « • (yyi — i^ l~ t/^ — \y» The degree of this equation has been depressed two units from that of (1). It is still of the same general form as (1), and may be assumed to have at least one root. Denote this root by c. As the first member is divisible hy X — c, the equation may be written {X - C)(X^-' + chx^-'-i- . . . dn-iX + dn) = 0, and may be solved by putting X — c = 0, and x""-^ + d.x''-'' + . . . dn-iX -{-dn = 0. The degree of our original equation has been depressed now by three units. This process may be continued till the degree of the original equation has been depressed n — 1 units, and we reach an equation of the first degree of the form x — k = 0, of which k is the root. 384 DEPRESSION OF EQUATIONS. As each division by a linear factor depresses the degree of the equation by unity, it must be divided by ^^ — 1 fac- tors to depress it to the first degree. This implies n — 1 roots, which together with the root of the resulting linear equation make n roots. Cor. 1. The equation (a) ^•n _^ ^^^n-l _^ ^^^n-2 _|_ ^ . . a^_,x + a„ = may be written {x — a){x — b){x — c) . . . to n factors = 0; and the equation AoX'' + A,x''-'-{-A,x^-^-\- . . . A^.,x + A, = may be written Aq{x — a)(x — b){x — c) . . . to 72 factors = 0. (3) Cor. 2. The substitution of any oth§r than one of the n values a, d, c, etc., for x in the first member of (2) or (3) would not cause it to vanish. Hence an equation of the ni\\ degree has only 7i roots. Of these 7i roots some may be rational, some may be surd, and some may be imaginary. Also some of the n roots may be equal. . Cor. 3. The solution of an equation of the ni\\ degree consists merely in resolving it into its linear factors, and equating each of these factors to zero. Cor. 4. The degree of an equation in x may be de- pressed by unity by dividing it through by x minus one of its roots. Cor. 5. An equation in x may be tested for a suspected root by dividing it through by x minus the suspected root. Cor. 6. When all the roots but two of an equation in X are known, the equation may be depressed to a quadratic equation, which may then be solved by the rule already given. DEPRESSION OF EQUATIONS 385 EXERCISE CXLI. Form the equations which have the following roots : 1. I, 2, and 3. 2. — 2, — 3, 4, and 5. 3. 1, - 2, - 3, and 0. 4. 4, - 1, - 3/2, ^d 1/3. 5. -3, -3, 4/3, and 4/3. 6. 3, - 4, - 1/4, and 1/5. Prove that the numbers given are roots of the equation and find the other roots. In testing for suspected roots, use method of synthetic division : Equation. 7. x^ - ^Ix + 84 = 0. 8. 2a;3 + bx^ - 4:3x - 90 = 0. 9. x^-\-2x^ -nx-i-Q = 0. 10. 4:X^ - 4:X^ -7x^-4.x-{-4: = 0. 11. 9x^ - Ux^ - 2a;2 _ 24:c + 9 = 0. 12. Sx^ - Ux^ + 20a; - 8 = 0. 13. x^ - 15.^2 + 10:r + 24 = 0. 14. x^-4:X^-5x^+20x'^+4:X-li)=0. 15. a^-74:X^-24:X^-{-937x-S^0=0, Number. 4. - 5. 2. 1/2, 2. 1/3, 3. 2/3. -1,2. 1, - 1, 2. 1, 3, - 5. CHAPTER XXX. UNDETERMINED COEFFICIENTS. A. FUNCTIOI^S OF FINITE DIMENSION'S. 282. Theorem I. A71 integral expression of the ntJi in X cannot vanish for more than n values of x, ex- cept the coefficie7its of all the poivers of x are zero. Let Ax"" + J5a:"-i + Cx''-^ + . . . vanish for the n values of x, a, h, c, . . . It must then be equivalent to A{x — a){x — h){x — c) . , . If now we substitute for x any value k different from each of the n values a, b, c, . . . , we have A{k - a)(k - b){k - c) . . . Now as k is different from a, b, c, . . . , the expression cannot vanish for the value x = k, except A itself is zero. If A be 2ero, the original expression reduces to which is of the (fi — l)th degree, and as before can vanish for only 71 — 1 values of x, except B = 0. And so on. Hence an expression of the nth. degree in x catmot van- ish for more tha7i 71 values of x, except the coefficie7its of all the powers of x are zero; and when all these coefficients are zero, it is evident that the expression must vanish for all the powers of x. 283. Theorem II. If ttvo integral expressions of the nth degree in x be equal to one another for more than 7i PARTIAL FRACTIONS. 387 values of x, tliey ivill be equal for all values of x, and all the coefficients of the saine powers of x in the two expres- sions mMst he equal. Let Ax^^Bx"" - ^^Cx^-^-{- . . . =A'x''-\-B'x''-'^^C'x''-^-\- . . . Then must A = A\ B = B', C^C . . . By transposition, we have {A - yl>'" + (^ - B')x"-' ^{C- 6'>«-2 . . . = 0, and this must be true for all values of x for which the two original expressions are equal, and therefore for more than n values of x. Hence by Theorem I, A - A' = 0, B - B' =0, C- C = 0, . . . or A = A', B = B', C = C, . ., When two integral expressions in x of finite dimensions are equal for all values of x, all the coefficients of the same power of X in the two expressions must be equal to each other. For in this case n is finite, and the possible values of X infinite, and therefore > n. B. PARTIAL FEACTIONS. 284. Definition of Partial Fractions.— The sum of the two fractions ^ and - — ■ — is 1 — X l-\- X 1 — x^ With reference to the last fraction, the parts which make it up by addition are called its partial fractions. It is often necessary to separate a fraction into its partials. In this separation it is understood that the denominators of the partials shall be of the first degree when practicable, but at any rate of a lower degree than that of the original fraction. e.g. 1. Separate ^ into partial fractions. x X 388 UNDETERMINED COEFFICIENTS. Since the denominator = {1 — x){l -\- x), assume 2 + 8a; A ^ B + l-x^ ~ 1 - x^ 1 -\- x' in which A and B are coefficients to be determined. Clearing of fractions, we have 2 + 8a; = ^(1 + a;) + B{1 - x) = (^ + B)x'^ + (^ - B)x. And as this is to be true for all values of x, we may apply Theorem II, which gives ^ + ^ = 2, and A- B^S. .'. 2.4 = 10, and A = b. Also, 2^ == — 6, and ^ = — 3. Hence the partials are and — :; — ■ — . 1 — X 1 -\-x From the above example we may derive the following rule for separating a prope'r fraction into its partials: Resolve the denominator, if possible, into real linear factors, and form fractions icith undetermined numerators, and put their sum equal to the original fraction. Clear of fractions, and equate the coefficiefits of the like powers of x. EXERCISE CXLII. Separate the following fractions into partials with linear denominators : 1. 7x +17 34 - 2x x^-\-5x-j- 6* x^ + 2a; - 8 25 - a; 13a; - 26 3. -0 T^- 4. x^ - X - 12' a;2 _ 3a; - 40 PARTIAL FRACTIONS. 389 llx - 7 '■^x' -lx-16' c c. . a;^ + 3a; + 2 e.g. 2. Separate ^(^ _ ^^^^ fractious. 10 - 15a; 6a:2-26a;+24' into partial Assume a;2 + 3^ + 2 A 2)(x-3) 5 a;- 2 3* 6(a; - l)(a; - 2)(a: - 3) ~ Q{x - 1) Theu, cleariug of fractions, we have a;2 + 3a; + 2 = ^(x-2)(a;-3)+65(a;-l)(a;-3)+6(7(a;-l)(a;-2) = Ax^-bAx^QA-^QBx'-UBx-^rl^B^QCx^-l^Cx-{-l'ZG x^ = A ^ - 6A x+ QA -\-QB -24:B + 18^ +6(7 -ISC + 126' Therefore, equating coefficients, we have A-\- 6B-{- 6(7=1, 5A + 24^ + 18(7= - 3, and 6.4 + 18i? + 126'=2. Whence A = 3, B = - 2, and C= 5/3. a;2 + 3a; + 2 1 2 5 • * 6(a;-l)(a;-2)(a;-3) 2(a;-l) a; - 2 ^ 3(a; - 3) * There is, however, a shorter way of solving this ex- ample. Since in the expression a;2 + dx -\-2 = A{x- 2){x - 3) + QB{x - l)(a; - 3) + 6(7(a; - l)(a; - 2) X may have any value whatever, we may put a; = 1. 390 UNDETERMINED COEFFICIENTS. Then we shall have 6 = 2.4, and ^ = 3. If we put X = ^, we shall have 12 = - Q>B, and B= ~ 2. If we put X = 3, we shall have 20 = 12(7, and C = 5/3. It is much shorter to use this method when by inspec- tion we can find values of x which will cause all the terms except one of the right-hand member of the identity to vanish. EXERCISE CXLIII. Separate the following fractions into their partials : ^2 - Ux + 37 9^2 _ 3g^ _ g9 2. "• (x-3){x^-9x-\-20Y {2x-j-2)(x^-9y 2dx - Ux^ 3x - 2 3. 77i TwT^ 57- 4. (2x-l)(9-x'y "• (x-l){x^- 5x-{-Qy X 2;^ J- X + 1 6. ' *" (x + l)(x + 3){x -j- 5y "• (x-]-l)(x^-5x-^Qy 7a;2 _|_ 7^^ _ 6 e.g. 3. Separate -, — r~rW o\ ^^^^ ^^^ partials. (.T+ 1) (a; — Z) In forming this fraction by addition there may have A been a fraction in the form of -, — t^tto, one in the form of (x-j-iy 7? n and one in the form of -. Hence in our as- x+r x-2 sumption we must make provision for all these. 7a;2 + 7a;-6 A ^ B ^ C Assume -. — r^rY27 ^ — i — r--i \2 H r^ + {x^\)\x-%) {x^Vf' x-\-\' x-^^ PARTIAL FRACTIONS. 801 Clearing of fractions, we have Ix'^lx - 6 = A{x - 2) + ^(x+ l)(a:-2)+ C(^+l)l - Putting X — — I, we have - 6 = — 3^, and A = 2. Putting :c — 2, we have 36 = 9C, and (7=4. Equating coefficients of x^, we have ^+ (7=7. .-. ^= 7 - C=3. 7a:^ + 7a; - 6 _ 2 3 4 ^®^^® (2: + 1)2(:?; - 2) ~ {x^lf +a; + l + 2;-2- e.g. 4. Separate yg — into partials. X J. The denominator = {x — l){x^ -\- x-\- 1)^ and the qua- dratic factor is not separable into real factors. But a proper fraction which has a quadratic factor for its denominator may have a linear factor for its numerator. We must make provision for this by assuming that bx^ + l A Bx-\-C x^ — 1 x — 1 x^ -\- x-\- 1' 6x^-^1^ A{x^ + x-\-l)-^{Bx-\- C){x Putting ic = 1, we have 6 = 3A, and ^ = 2. Equating the coefficients of x^, we have A-\-B = 6. ... 5 = 5-^ = 3. Equating the constant terms, we have A+ C=l. 392 UNDETERMINED COEFFICIENTS. Whence C = 1 -- 2 = - 1. bx' 4-1 2 , 'dx - \ Therefore x^ — 1 X — 1 x^ -{- X -\- 1' Observe that each of the separations into partial fractions given is characterized by this : that it introduces just as many undetermined coefficients as equations for them to satisfy. This is characteristic of any proper application of the method of undetermined coefficients in which the number of coefficients is finite. EXERCISE CXLIV. Separate the following fractions into partials: 12a;2 - a: + 10 1. 6. 1 x^ + r 2a;3 + 2x^ -f 10 x'-i-: X^ - -X+1 4. x^-1 x^ - d (^ + m^'' + 1)' (x^ -]-i){x - ly G. FUNCTIONS OF INFINITE DIMENSIONS. 285. Theorem II. If Uvo integral functions of x of infinite dimeyisions, and arranged in asce^iding order, are equal to one another for all values of x ivUich malce the series convergent, the coefficients of the like powers of x in the two series will he equal. JjQt A^hx-{- Cx^^ . . , = A' ^ B'x + C'x^ + . . . be true for all values of x which render both convergent. Then will A=: A', B = B', C — C , etc. For if the series are both convergent their difference will be convergent, and we shall have A- A ^{B- B')x + (C - C')x^ . . . = for all values of x for which the series is convergent. EXPANSION OF FUNCTIONS. 393 But when x is sufficiently small, the series is convergent and A — J' is greater than all that follows, and its sign must control that of the series; that is, the A — A' will be >, =, or < zero according as the series is >, =, or < zero. But the whole series = 0. .-. A-A' = 0, or A = A'. By striking out A and A' as equal, we may in like manner prove B — B' ; and then C = C, etc. For since {B - B')x + {C - C')x 4- . . . r= for all values of x which make the original series convergent, and therefore for other values of x than zero, both members of the equation may be divided by x and the conclusion be drawn that B - B' -^{C- C')x^. . . =:0 for values of :r which make the original equation convergent. D. EXPANSION OF FUNCTIONS. A function may be developed into an infinite series in various ways; and whenever the series is convergent, the function is equal to its development, which is then called its expansion. It is important to bear in mind that when the series into which a finite function is developed becomes divergent for any value of x the function cannot equal its development. A proper fraction may be developed into an infinite series in ascending powers of x by division. The four following expansions by division are im- portant : 1. -^— = 1 + a; + 2^2 + ic3 + a;^ -f . . . 1 — x 2. -4— =1-^ + ^^-^^+^;^+. . . l-\-x 894 UNDETEBMINED COEFFICIENTS. 3. 4. (1 — a;) 1 5x^ {1 + xy l-2x-\-3x^- 4:x^ + 5a;^ + . . . A function which is not a perfect power may be devel- oped into an infinite series in ascending order by evolution. e.g. yi - :c = 1 16 6x^ 128' • • If a function of x which has but one value for each value of X be expanded in ascending powers of x, the powers must all be integral. For were the exponent of any term to become fractional, that term would be many-valued for eacli value of x, which contradicts the hypothesis. The following example illustrates the expansion of a fraction by the method of undetermined coefficients. Expand — — — — -^ to five terms in ascending powers JL ""j~ X ~j~ X of X. Assume 1 2 ~^~^2 = A-i-Bx+ Ct? 4- B^x^^Ex^'-^Fx^^ Qx^-^.., 1 - re - a;2 = ^(1 + a; + a;2) + B{x ■\- x^ ^ x^) -f C{x^ + a;3 + x"-) + D{x? + rc^ + x"") + E{x^ + ^' + ^') + Fix'' + a:« + x') + G{x'> + a;"^ + a:8) + . . . A^ A x+ A 7?+B x'+C vf' ^D ^+E -\-B + + D ■\-E + F + c + D + E ^F + G X^-\- Whence EXPANSION OF FUNCTIONS. 395 A-\-B= -1, and B=-2, A+B+C=-h and C = 0, B + C-\- D ^0, and i) = 2, (7 + i> + ^ = 0, and E = -2, D + E-\- F =0, and F=0, E-{. F-\- G = 0, and G = 2. ;+:+:.-! ^-+^-^- - 2x^ + 2x^ + and In certain cases the operation of expanding fractions into series may be abridged. 1°. If the numerator and denominator of the fraction contain only even powers of x, we may assume a series con- taining only even powers, as ^ + Bx^ -\- Cx'^ + . • . 2°. If the numerator of the fraction contains only odd powers of x and the denominator only even powers, we may assume a series containing only odd powers of x. 3°. If every term in the numerator contains x, but not every term in the denominator, we may assume a series beginning with the lowest power of x in the numerator. 4°. If the numerator does not contain x, we may find by actual division what power of x will occur in the first term of the expansion. e.g. ^ _ — 3 gives by division l/3a;~* as the first term of the quotient. Hence we may assume 396 UNDETERMINED COEFFICIENTS. EXERCISE CXLV. Expand each of the following fractions to five terms in ascending powers of x : l-%x-\- Zx^ 3. 1 + 32; - ^X^' 3 - 4:0? 4. dx ^- 4:-dx'' 2 - 3a: + 4a;2 1 + 2x - - 5x' 2 - ^x? 1 + 4:X^' 2x 3 - 2x^' The following example illustrates the method of develop- ing a radical by the method of undetermined coefficients. To expand Vl + ^• Assume Vl -\- X = A-^Bx + Cx^-i- Dx^ + Ex^ + Then, squaring each member, we have l-\-x=:A'' + 2ABx -i-2AC -\-B^ Whence x^-\-2AD + 2BC x^-{-2AB -{-2BD x' + A^ = 1, A = 1, 2AB = 1, B = 1/2, 2AC+B^=:0, 6'= -1/8. 2AD-\-2BC=:0, D = l/16, 2AE + 2BD + C2 = 0, B=- 5/138. Therefore ^(l-\-x) = l + \/%x - l/W 4- l/16a;3 - 5/128:^4+ . . EXPANSION OF FUNCTIONS. EXERCISE CXLVI. 1. Expand 1/(1 + x + x^) to x\ 2. Expand y (———) to x^. 3. Expand \/{l + x) to x^. Ex. Let ij = dx-2x^-\- 3x^ - 4:X^ -{- . , , Express x in ascending powers of 2/ to ?/*. Assume x = Ay -{- By'^ -\- Cy^ + Dy^ -f • • • = A{^x- 2x^ + 3.?;3 - 4:X* + . . .^ + ^(9^2 ~ 12x^ + 22:^^ + . • •) + C{27x^- 54^-4 -f. ..) + i>(81a:* + ..'.)• 397 .-. x = 3Ax-2A x^ + 3.4 x^ - 4.4 ^^ + .., + 95 -125 + 225 + 27C - 54C + S1D Whence SA = 1 -2^ + 95 = 3J -125 + 27(7=0 - 4^ + 22 5 - 54C + 81i> = 0. Whence A = 1/3, ^ = 2/27, C*= - 1/243, D=- 14/2187. Therefore x = l/'dy + 2/2? f-1/2 43?/3-l4 /21873/^+.. 398 UNDETERMINED COEFFICIENTS. EXERCISE CXLVII. 1. It y = 2x -\- x'^ — 2a^ — 3x^ -{-..., find x in terms of y to y^. 2. It y := X -\- x^ -\- x^ -\- x^ -\- , . . , find x in terms of y to y\ 3. lty = x — a^-{-x^ — x'^-\-,.., find x in terms of y to y\ CHAPTEK XXXI. CONTINUED FRACTIONS. 286. Definition of a Continued Fraction. — An expres- sion of the form a± o± — g ± etc. is called a contmued fraction. For convenience, continued fractions usually are written in the form a ± - - — etc. c ±e ±g ± In this chapter we shall consider only the simpler form «, H , — , etc., in which the numerators are each unity and «, , a^, a^, etc., are positive integers. The fractions a, , ^, — , etc., are called the first, sec- ond, third, etc., elements of the continued fraction. 287. The Convergents. — The fraction obtained by stop- ping at any element is called a convergent of the continued fraction. Thus a^ , a. -\ , and a,-\ , — are the first, «, a^ +«3 400 CONTINUED FRACTIONS. second, and third convergents of the continued fraction given above. These convergents may be reduced to the forms f', «-^^-±i, and <"■"' + ^j"' + "■ . 1 rt?-^{a/ quotients, «5„ being the nth partial quotient. When the number of partial quotients is finite the continued fraction is said to be termi7iating . If the number of these quotients is un- limited the fraction is called an infinite continued fractio7i. Since a^, a^, a^, etc., are positive integers, a continued fraction of the form a, -\ , 1- etc. must be greater «^, + «, ^ than unity; while a continued fraction of the form of — -L — — must be less than unity. «, ^ «, + ^3 + . . . ^ ^ The complete quotient at any stage is the quotient from that point on to the end. Thus an is the nth partial quotient, and a„ -\ is the correspond- ^n + 1 I ^n + 2 I • • • ing complete quotient. The complete quotient at any stage may be denoted by K. As we have seen, the ^th convergent is Qn a„qn - 1 + 5'n - 2 ' This value evidently may be converted into that of the whole continued fraction by substituting K in the place of cin- Denote the value of the entire fraction by x. Then will Kq„ _ 1 + (7„ _ 2 * 290. Theorem II. The difference between tivo suc- cessive convergents is a fraction whose numerator is imity and whose denomi^iator is the product of the denominators of the convergents, and this difference taken in regular order is alternately positive and negative. Pn^ Pn-l _ (fnP„-l+P,,-2 Pn - 1 qn qn-i «uqn-i-\- qn-2 qn-i CONTINUED FRACTIONS. 403 ~ {Clnqn-l+ qn-'z)gn-l Pn_ Pn-l _ Pn-2qn-l—Pn- l^n - 2 qn qn-l~ qnqn-1 .*. Pnqn-l-Pn-iqn = " (i?n - l5'n - 2 " i?n- 25'n - l)- So also in succession Pn-iqn-2 - Pn-2qn-l= - i?n - 25'n - 3 + i?n - sS'n - 2- p.q.-p.qz = -p.q.+p^q.' But p,q, - p,q, = (a^a, + 1) - a^if, = 1 = (" 1)^- Also, since the successive convergents, beginning with the first, are alternately less and greater than the fraction, the successive convergents are alternately greater and less than the preceding. Therefore the successive difference will be alternately positive and negative, so that the numer- ator of the fraction will be (— 1)", in which n is the num- ber of the convergent used as a subtrahend. Hence Pnqn-i -Pn-iqn = (- 1)". (1) Hence, also, ^ -^^^ = till!. (2) q,t qn-i qnqn-i ^ CoK. 1. All convergents are in their lowest terms. For every common measure of jt7„ and q^ must also be a measure of Pnqn-\ — Pn-\qn and, from (1), of ± 1. Hence pn and q,^ can have no common measure. Cor. 2. In the continued fraction _i- -i- _1_ «, + a, -f ^3+ ' 404 CONTINUED FRACTIONS. wliicli is less than unity, ^, /y _^ n—( — ^Y-'^ Olid -^" _ Pn-\ _ {— 1)" Yn Y« - 1 7nYn - 1 since the first convergent will be too large, the next too small, etc. 291. Theorem III. Each co^ivergent is nearer in value to the continued fraction than any preceding con- Let X denote the continued fraction, and — , ^C!L±i qn qn+i and i-!?-±l denote three consecutive convergents. Then x differs from ^"^ ^ only in taking the complete 5'ji+ 2 {n -\- '^) quotient in place of «„ + ^. Hence X = Kqn^x-^ qn ' Pn K{p^^^ q^ ~ p,,q^ -f 1 ) _ K qn{ Kqn + 1 + 1 and q,, < <7„+i; hence on both accounts K . 1 qnJi^qn + 1 + 5'n ^'n + i A^„ +!+$'«* Combining the result of this article with that of article 290, it follows that CONTINUED FRACTIONS. 405 The co7ivergents of an odd order continually increase, hut are akuays less than the continued fraction j The convergents of an even order continually decrease, but are always greater than the continued fract\on. 292. Theorem IV. The value of x differs from — 1 , , ., 1 by less than — ^ and by more than ^ Let — , ^^"^ S ^ ^ be three consecutive conver- gents, and let K denote the {n + 2)th complete quotient. Then ^ = pL±l±^n^ qn {Kqn + 1 + qn)qn qn(Kqn + 1 + ^n) _ Kp^ ^ iqn + Pnqn - ^^Mn + 1 - Pnqn qn{Kqn^X-\-qn) - -^(Pn + iqn - Paqn.l) _ ^^ qn{Kq^ f 1 + ^n) qn{Kqn + 1 + ?n) 1 4^„ + l + |) Now K is greater than 1, therefore — differs from 2: qn by less than and by more than — -5. Mn+. qnqn-tl + qn And since q^ < §'„ + 1 , the difference between — and x qn must be less than -^ and greater than ^-^ — . qn <'q n + l 293. Theorem V. The last convergent preceding a large partial quotient is a close approximation to the value of the fraction. 406 CONTINUED FRACTIONS. By the last theorem, the error in taking -— instead of fin the whole continued fraction is less than , or, since (7n + i = «n + ign + ^n-i, less than —y- -— — ^, or less than j. Hence the larger a^ +i is, the nearer does — approximate to the continued fraction. Therefore when a^ + x is relatively large, the value of x differs but little from that of ^. qn 294. Theorem VI. Every fraction whose numerator and denominator are positive integers can he converted into a terminating continued fraction. m Let — be a fraction whose numerator and denominator 71 are positive integers. Divide mhj n and let a^ be the integral quotient and p the remainder. Then m . p ,1 n n ^ n P Divide n hy p and let a„ be the integral quotient and g be the remainder. Then n q 1 P ^ p P q Divide p^y q and let ^g be the integral quotient and r be the remainder, and so on. Therefore — = «, H , — ^ ^«a 4- «3 + • • • CONTINUED m ACTIONS. 407 If 7)1 < 71, the first integral quotient will be zero. w 1 Put — = and proceed as before. 71 n m The above process is the same as that of finding the greatest common measure of m and n, a^, a^, a^ being the successive quotients. As m and }i, being positive integers, are commensurable, the process must terminate after a finite number of divisions. K7n Cor. Evidently — and -^r^- will give the same contin- •^ 71 Rn * ued fraction. e.g. 251 1. Reduce — — to a continued fraction. oO/C Find the greatest common divisor of 251 and 802 by the usual method. quotients. 251 1 1 1 1 •'• 802 "3+5+8 +6* e.g. 2. Reduce 3.1416 to a continued fraction. 1416 251 802 3 ] 6 49 5 1 8 6 3.1416 = 3 + 10000 1416 8 10000 88 7 16 11 and 3.1416 = 3 + 1416 10000 1 2 i 7 + 16 + 11* 1 L i 7 + 16 + ri' 408 CONTINUED FRACTIONS. 355 e.g. 3. Show that -— ^ is a close approximation to 3.14159, differing from it by less than .000004. 3.14159 = 3 + 159 100000 7 854 887 15 29 33 1 1 4 25 1 7 4 1 1 1 1 1 7 + 15 -hl + 25 + 1 + 7 +4 The successive convergents are 3 1' 22 r 333 355 106' lis' The last convergent precedes the large quotient 25, and hence is a close approximation to x. It differs from it by less than — . .„ , and there- fore bv less than 25 X (100)' 25 X (113)2 , or .000004. EXERCISE CXLVIII. Express the following as continued fractions; 3. 3.61. 144 53 72 59* ' 91- 112 749 153' ^' 326- 436 3015 345* *• 6961 6. 89 CONTINUED FRACTIONS. 409 Calculate the successive convergents to the following continued fraction : 24-- - - — - ^' "^6+1+1 + 11+2' 1 L 1 ?L 1 1 ^^' 2+2 +3 +1 +2 + 6* "• "^3 + 1+2+2+1+9* 1111 12 -_ 2 + 3 + 1 + 4 * 13. Find a series of fractions converging to .24226, the excess in days of the tropical year over 365 days. 14. A metre is 39.37079 inches; show by the theory of continued fractions that 32 metres are nearly equal to 35 yards. 16. A kilometre is very nearly equal to .62138 mile. Qi, .1. ^ XT- i. .. 5 18 23 64 bnow that the tractions — , — , ^r^, -p— are successive ap- proximations to the ratio of a kilometre to a mile. 16. Two scales of equal lengths are divided into 162 and 209 equal parts respectively. If their zero points are coincident, show that the thirty-first division of one nearly coincides with the fortieth of the other. 17. The modulus of the common system of logarithms is approximately equal to .43429. Express this decimal as a continued fraction, find its sixth convergent, and deter- mine the limits to the error made in taking this convergent for the fraction itself. 18. The base of the Napierean system of logarithms is 2.7183 approximately. Express this decimal as a continued fraction, find its eighth convergent, and determine the limits to the error made in taking this convergent for the fraction itself. 410 CONTINUED FRACTIONS. 295. Periodic Continued Fractions. — AVlien the partial quotients of a continued fraction continually recur in the same order, the fraction is called 2i periodic continued fraction. A periodic continued fraction is said to be simple or mixed according as the recurrence begins at the beginning or not. Thus, 1 1, 1 1 1 ^"^^>+c+a + i^c +. .. is a simple periodic fraction. L 1 1 L is a mixed periodic fraction. 296. Theorem VII. A quadratic surd can he ex- pressed as an infinite periodic continued fractio7i. e.g. Eeduce V% to a continued fraction. The integer next below VSi^ 2. Hence 1^8 — 2 expressed as an equivalent fraction with a rational numerator is (|/8-2)(l^+2)_ 4 y8 + 2 i/8 + 2 V8 = 2 + -— ^ = 2+ ^ 4^+2 V84-2 The integer next below ^^— is 1. V8 + 2 _ i^8- 2 Hence 1 1 i^S = 2 + — — — = 2 + 1 ^8-^ 1+4/8 CONTINUED FRACTIONS. 411 1 = 2 + i ( V8 - 2)( V8 + 2) !■ + |/8 + 2 The integer next below V8 + ^ is 4. Hence (l/8-2)( V8 +3) 4/8 + 2=4+4/8-2 = 4-1- ^8 + 2 4 4+ ^ ^ =4 + V¥+2 4^8 + 2 * 4 At this point the steps begin to recur : ^=^ + r+4 + r+r+... Thus 4^8 is seen to be equivalent to a periodic fraction with one non-periodic element, which is half the last partial quotient of the recurring portion. This law holds good for every quadratic suixl. Note in the above example that the last partial quotient in the recurring portion is an integer + the given surd. The following is a very compact and convenient form for working such examples : 4/8 = 2 + 4/8-2 = 2+ --i , i^+2 4/8 + 2 i/8 - 2 , , 1 -A = -^H ~A = 1 + 4/8+2' i/8 + 2 = 4 + ^/8-2 = 4+ -— ^ , 4/8 + 2 412 CONTINUED FRACTIONS. i^ = i+i:«^^=i + V8 + 3' ^+3 = 4 + 4^-8 = 4 + -^^, ••• ^=« + r+r+r+ !'«*''• 296. Theokem VIII. An infinite periodic fraction may he expressed as a quadratic surd. Let the partial quotient be 1, 2, 3, 1, 2, 3, etc. Then ^-i L 1 _1±^ ±nen a:- i _^ 2 -f 3 + ^~ 10 + Sa:* .-. 10a; + 3a;^ = 7+ 2a:. .-. 3a;2 + 8a: - 7 = 0, .-. a: = l/3( 1^ - 4). EXERCISE CXLIX. Express the following as periodic continued fractions : 1. Vl. 2. '/13. 3. V2. 4. V6^ 6. Vl7. 6. 4/19. Express the following continued fractions as quadratic surds : 111 14.2.111 "^ 2 + 2 + 2+... *■ +2 + 3 + 2+3 + ... 11111 9- 1 + 2 + 3 + 4+1 + .. . ANSWERS. EXERCISE 1. 1. 105. 2. 525. 3. 2625. 4. 26460. 5. 85050. 6. 396900. 7. 91875. 8. 1701000. • 9. 165375. 10. 1181250. 11. 5a cts. 12. 120a sq. in 13. Wmn, 14. 25a^c. EXERCISE II. 1. 67f. 2. 35i. 3. 1130. 4. Same. 5. Same. 6. Same. EXERCISE III. 1. 2aW + lOa^^ + 12. 2. a - I2b^ + 3. 3. Qx^y + 5 - 5b\ 4. Sa'y + 9«/ - 7. 6. 7a3a;-5aV + 6. 1, :?; = 12. 6. X = 2S4r^. EXERCISE IV. 2. «/ = 7. s. z = 7^. a; = 4. X c - 15. 1. x = 9a EXERCISE V. 1. 27 and 36. 2. 45 and 58. 3. 30 and 120. 4. 17 and 85. 5. 75, 150, and 225. 6. 72, 36, and 12. 7. 525, 175, and 35. 8. Harness $45 ; horse $135 ; carriage $270. 9. History $1.38 ; arithmetic 69 cts. ; speller 23 cts. 10. Sister's age 10 ; boy's 13 ; brother's 18. 2 ANSWIJRS TO QILLETS ALGEBRA. EXERCISE VI. 1. 70 and 105. 2. 27 and 45. 3. $1.20, $1.80, $1.35, and 10.54. 4. $45000, $30000, 124000, and $18000. 5. $100.00, $25.00, and $200.00. EXERCISE VII. 1. 3« - 4Z> - ^ah + ^ad + 6. 2. 3m -\- Aifi — 20cx -\- 26cy — 5c^. 3. 7 + 24c - 32^ - 122;. 4. 5x ~ ab - ac + la. 5. 18m + 16^ - 24^> + 32c. 6. 2a;-f 62:+21, or 8a:+21. 7. 52;-3(« + 2^»-3c) + 9; 5a; + 3(- <*- 2^» + 3c)+ 9. 8. 7«^-4c(2^>-4f?-6c)+3; lah+4.c{-U-^4.d-\-^c)-\-'d. 9. 27-2«2(- 3c + 5Z»-6); 27 + 2«2(3c - 5^ + 6). 10. 10a; - 5( - 42;2 - 5 A + 7) ; lOa; + 5(42;^ + 5 A - 7). EXERCISE VIII. 1. 8 and 12. 2. 3 and 9. 3. Harness = $60 ; horse = $180 ; carriage = $480. 4. $2000 the first month, $5920 the second month, and $23720 the third month. 6. 20. 6. 36. 7. 18. 8. 132. 9. 99. 10. 25, 48, and 46. EXERCISE IX. 1. 245 bushels in all, 98 bushels of rye, and 70 bushels of barley. 2. 231 in all, 154 baldwins, and 42 greenings. 3. First and second 30 miles, second and third 32 miles, and first and fourth 80 miles. 4. Louis had 320, and Howard 80. 5. First 77, second 81, and third 68. 6. Winning candidate 18156. Losing candidates 17344, 17624, and 17400, respectively. 7. M to N 21 miles, N io 6' 6 miles, and 6^ to ^81 miles. ANSWEE^S TO OILLET'S ALGEBRA. 1. 4. 7. 10. 12. 14. 16. 1&. 20. 22. 23. 24. 6. 5. - 6. 18. 10. - 6. 0. 15. - 18. EXERCISE X. 1. 18. 2. 6. 3. 18. 4. 6. - 6. 7. - 18. 8. - 18. 9. .1. 0. 12. 0. 13. 0. 14. .6. 6. 17. '^a. 18. — 2«. .9. 12 + (+6), 6 + (+12), and 12-(-6); 12 + (- 6), - 6 + (+ 12), and - 6 - (- 12); a -{-{-a), a + (-a); -G+(-12), -12+ (-6), -6- (+12). EXERCISE XI. U B.C. 3. 40 A.D. 30 B.C. 6. b B.C. C A.D. 9. 20° below zero. 11. Has risen 16°. 13. Has fallen 8°. 15. 17° warmer. 17. 12° warmer. 19. 5 miles south. 21. 6 years older. 6 B.C. 2. a A.D. 5. 50 A.D. 8. Has fallen 12°. Has fallen 7°. Has risen a°. 8° colder. 3 miles west. 4 years younger. 2 years younger. The grocer owes Hermon 3 dollars. 7 pounds less. 25. 20000 dollars poorer. 1. c a — b, EXERCISE XII. -{a + b). 2. m a + b 3. In 6 hours. First will have travelled 24 miles, and second 18 miles. 4. a m -\- n miles. 6, 50 and 58. hours. First 7na m-{- n miles, second 7ia 7W + ANSWERS TO GILLET'S ALGEBRA. EXERCISE XIII. 1. 25«. 2. ^loj^x ;. 3. - 36«52. 4. - 56rc. 6. - Qx^. 6. 2ac^x. 7, 2y^ - 2ac - 5. 8. 4.a^x — ax^ — ^ab — 8. 9. — dx-{- Qab + c. 10. 5a;2 - aW - c + 7. 11. 19/12;?; = lj\x. 12. - V12y. 13. 16(« + b). 14. - ft - (x + y). 15. 5(a + b) -5(m + ?i). 16. 4:a(b-\-x). 17. c{a^ - h^). 18. - 2az - 4. 19. 0. 20. 2a- b-\-5c-}-3d. 21. 4:^ + 3y + 2 + 5^. 22. c? — xy. 23. «2^,3 _p ^2^^ 24. 3ft + lOc -\-^d-x. 25. X -\- b — c -\- d. 26. First 7500, second 7000, 27. (4:X - 50) dollars. third 6500, and fourth 28. 8000 dollars. 6000. 29. {a + Z')a; — mq 5 EXERCISE XIV. 1. (« + m)x + (* + ^)2/- 2. {mn + pq)x — 2by, 3. (3 + 6^ + la)x - 6y -\- m H-^. 4. 8(a-{-b-\- l)x + (5 - ■ 10)^. 5. x-}-8 and 2a; + 8. 6. Albert is 12 and Howard 24, 7. In 9 hours. 72 miles and 54 miles. 2 ' .2 9. (ft — m)x -\- {b — 7i)y -\- {c — p)z. 10. 2(d - f)x ^^e-d)y^ 4(/ + e)z. 11. l/12(8ft + 9% -- 2(1 - 3ft)^. 12. (2ft - U)x — (4ft + b)y. 13. Herbert is l/2ft, and Horace a. T ct , ab ., , ac ., 14. In 7 hours, 7 miles, and ^^ miles, J — (J ' b — b — c ANSWBB8 TO GILLET'S ALGEBRA. EXERCISE XV. 1. — ^x — y + 14:Z. 2. 4« — ^ + 2c. 3. 8a' - 2a^ + 4«2 _ I5a + 14. 4. 6. 8. 10. 12. 13. 15. 17. 19. 21. 20A2 + IQa'x. 5. 4«3 - 2. 4/3a:2 - l/2x - 1/2. 7. « — 5 + c. - 2a - 95 - 8. 9. X - 8«2a;2 -f 12. 2^« - 7^* - 3. 11. 9 and 18. A has 172.50, and B $77.50, - X + tj. 14. 2x - 11. 5:^: + 4^ + 7«. - 11. 16. a^ + ^ + ^ 4. 1, 4(« + h). 18. 2a{c - x). 2«2(J_a;)+4«J(«-5). 20. x-8. 6fl + m , bm — 6fl^ 6 EXERCISE XVI. 1. — Sab — m — 2ax, 2. ox — 2a. 3. 2b — 4:C. 4. lOa^ — '^y -\- 5^. 5. — 9ax — 2by. 6. 0. 7. 0. 8. 3m. 1. X — {a -{■ b). 3. a; — (— a + 3a^ — 5. a; — ( — a; -|- 2« - 7. ic— (— «-|-^— ^— ^ 9. x—{—2x-\-2m- 11. ic— (2m + 3a — 13. x — (a -\- b -\- p EXERCISE XVIII. 1. m^p-\-q-\-a — b — c-\-d. 2. m-{-a — b-{-p-\-q — n-\-k, 3. 15«a; — i:by. 4. 0. 5. p -\- b -\- s -^ t -\- m -\- n. 6. $8360, $16120, $23880, and $31640. EXERCISE XVII. 2. a; — (m + ^^)' %). 4. a; _ (35 - 2c - 5^). - 2^^). 6. ^_(_3+« + 5). n^7i). 8. a; — (— a; — rt + 5). -2m). 10. :^-— (—2x -{-2ab —m) 2b). 12. x—(2am-{-b-{-p—q—7i) - q -\- m -»). ANSWEES TO GILLET'S ALGEBRA. a-\- b -\- c 3b -\- c and a -\- b — c 4 - 4 ^ 2 8. llax. 9. — 2ax — 6by 10. — 2x -\- 2y. 11. — 4:bz. 12. - (m + 6)^ + 2x -\- iab - 5. 13. 6, 18, 36, 54, and 72. C2!. 1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 1. 3. 5. 7. 9. 10. 12. 14. 16. 17. 21ab. — a^b^xy^. a^bcdm. 30a%hnx^. IQSabkni^x^. — SOagxYz. A:bc^gnx?z^. — 2ia'^xY' a^bx^y^. 2. 5. 8. 11. 14. 17. 20. 23. 26. 29. EXERCISE XIX. 3Wb. ?>a%^x^z^. . — abcdx^. 10 ba^7n^xy^. , bg?)} ny'^. . Iba^bhix^yz. . - ^alrc'j^y. — '^((^xhi^. — apqx^y'^. — 2/bacmhiH^. 3. 6. 9. 12. 15. 18. 21. 24. 27. 30. — 32AY. — 4:2mVxY. — d^b'^cxK ^Labmn. 4:axY. — 4:abgxyz*. 4:abexy. — m^n^x^. Za%cdh^K Zo^bcxy'^, EXERCISE XX, 2. X - 40. \. X = 20. 3. $19000, $9000, $12000, and $7000. 4. 9, 10, 17, 19, and 2G. EXERCISE XXIII. - 3G.t2//2^ - \:%xy'^zK 2. a%^c^ - 3x^ + Zxy + Zxz. 4. a^bc — ab^c -\- ab(^. a%^c - id)^(^ + a^bc'. 6. lia'b^ + 2MbK 15a;y-18.i;y+24;z;y. 8. 56a:y + 40a;y. - bx^z^ -j- Sx^z^ — Sx^y^z^. - 4:Sx^yh^ + 9QxYz'- U. 91a;y + 106x^y^ - 8xYz^ + lOxH/z'. 13. - «2Z>V + a^JV _^ ^2J3^2, a^'^c - a%h + aWcK 15. - W + 9/^a^ - Uq, - 5/22^2 _^ 5/3^_y _^ 10/3^-. ANSWERS TO GILLET'S ALGEBRA. 7 18. - 2a'x' + 7/2aV. 19. 5/2^4^2 _ 5/3^3^3 _^ ^2^4^ 20. 21/2xhj - xy. 21. l/2a:y - 3^y. 22. — x^ -\- l6/4:9xy. EXERCISE XXIV. 1. x^ — 1. 2. x^y -f a^y + xy^. 3. - 3:^:5 _^ dx^-dx^-{-12x^ ^. x^-\-x^—x-l. 5. 2:4 + a:^ + 2,^2 - x -{- 3. 6. x* - ISa;^ + 36. 7. a:^ — ?/^. 8. x^ + y^ 9. ^'C^ + x^y^ -\- y^. 10. a;54-5ic4+10x3+10a;2 + 5a:+l. n. a; = 11. 12. $20000, $52000, and $48000. 13. 5. 14. x^ - 5ax^ + lO^V - 13«V + ISff^a; - Qa\ 15. x^ — 4«V + Stt^a; — a^ 16. x^-\-2x^y—xy—4:a^y^—xy-\-2xy^-{-y^. 17. ^* — a'^. 18. •'c^ — (^* — d + c)a;2 — (be — ca -{- ab)x -\- abc. 19. \ — X -^ x^ — X? -\- 2x^ — a^ -\- x^ -\- a^, 20. a^ — b^. 21. 27^^ — 64«/^. 22. 125aV + 27%«. 23. 64aV _ i25^9a;3^ 24. $300, $550, and $350. 26. 8. EXERCISE XXV. 1. 9x^ + dx^ - 2x^ + 62; - 4. 2. a;« + ^^ - 2a;« - 2x^ - 5x^ - x^ -}- 5x^ + 9. 3. 2x^ - lOa:^ + 5x^ - 222;^ - 52;2 -^ 5a; + 1. 4. 2x^ - 7a^ +6a^ + da^ - 3x^ -f 4a: - 4. 5. 1 - 6x^ + 5x\ 6. 1 - 7x^ + Qx\ 7. 1 + :6- - 8a;2 + 19x^ - I6x\ 8. 4 - 9x2 _j_ X2a;3 - 4:f4. 9. x^ + .-c^ - 2x^ - x^ -\- x^ + ^ -f 1. 10. 2x^ - bx^ 4- 2x^ + 6x^ - 3x\ EXERCISE XXVI. 1. 5xy. 2. 'Sa\ 3. 9«2. 4 7^2^. 6. — 17:r. 6. — ll.r^^. 7. 5z^. 8. 9«c2. 9. 2xy. 10. — 3«2^. 11. i/ba^y. 12. — 9a;2?/2^;3, 13, 15(x + y)V. AJVSWEBS TO GILLETS ALGEBRA. 14. - 39(« - hfxK 15. - 'dOcd(a + bfxy. 16. bQa%\c - ciyxyK 17. b(a + bfx. 18. - ^ac{h -dfy. 19. Ud(b + c)x. 20. 2a^c\ 21. - 2Wy. 22. X = 60. 23. 48 and 132. EXERCISE XXVil. 1. x^ + xy + y\ 2. a^ - ab -\- b. 3. a^ - ?>a% + 1)\ 4. 8a;3 + mx^y + 21y\ 6. 6/6a*-l/5a'b-l/da^^ . 6. - 2a% - 4:ab\ 7. 6x^y - 6xy^ + 8xy. 8. 2a - Sb -\- 46'. 9. dx — 2y — 4. 10. 2/3« - l/6^> - c. EXERCISE XXVIII. 1. X - S. 2. x-\- 3. 3. X — 7. 4. X - 2. 5. 2x - - 3. 6. Sx + 8. 7. 4:x - 3. 8. 5x + 4. 9. 7x + 5. 10. x^ + xy + y^. 11. ^ + y- 12. 9 A2 4- 12abx + 16^^ 13. 2aV - 3(^b\ 14. 7a:2 + 5xy + 2f, 15. ^3 _ 2a;2 -I- a; + 1. 16. f - 3x^ -h 2:?; - 1. 17. .^2 - a;y + y^. 18. x^ + X - y. 19. x^ - 2x -h 3. 20. x^ ^ 5x-\- G. 21. 7«2 - SrtZ* + 2bK 22. 8. 23. - 8. 24. 5. 25. — 5. 26. - 18. 27. 5. 28. - 10. 29. 10. 30. -5. 31. 7a- -45. 32. 0. 33. -392-+2 84. X + 1/3. 35. x^ - \/2x -f 3/4. 36. \ — x-\-^ — Qi?-\-x'^~ etc. 37. \\2x^2x^ ^ 27? + etc. 88. 2{x - yf - 4.{x ~ yf - {x - y), EXERCISE XXIX. 1. X = 5^.. 2. X = — 2. 8. X — 3. 4. a: = 20. 5. a; =.11. e. 31 doz. 7, 8 sheep. ANSWEBS TO GILLETS ALGEBRA. - 27/64a;2^ EXERCISE XXXII. 2. x^if. 3. 6. — 125a;y. 6. 8. dh^. 9. 11. 49ai»Z>V. 12. 14. - 27a».r^ 15. - 32a:iy 4/9a^^io. EXERCISE XXXIII. «2 + ^ab + 9Z>2. a;^ — Vdxy ■\- 25«/^. 9a;^ — %xy + «/^. 81a:2 - 36a^y + 42/2 16 ^2 4" ^ahxy -{- ^Wy'^. - 82; + .t2. - '2/Ux + 1/9^*2. a;'^ -\- lax -\- (^ 2. 4. 6. 8. 10. 12. 14. 16. 18. «2 - 6a5 + 952. A,x^ -f 12 2;?/ + 9?/^. ^x^ + 30a;?/ + 25f/2. "IWW - lOabc + c2 x^ — 2abcx + c^lt^^, x^ - 2.^2 + 1. x^ + 4/3«2; + 4/9«2. ^2 _ 3^a; + 9/4a'2. 16 + 8a; + a;?. EXERCISE XXXIV. 1 _j_ 4:^:2 _^ 9^4 + 4a; + 6a;2 + 12a;3. 1 + 4a; + 10a:2 _|_ 00.^3 _|_ 253^4 j^ 24a;5 + 16a;^ \j^^x-\- 10a;2 + 20a;3 + 25a:4 + 34a;5 + 36a:« + 30a;^ 4- 40a;8 4- 2e5a:io. 4. a^^h^^c^^^^- lab + %ac + 40a;« + 25a;io. Uc - 2ad + Ud — 2cd. 9^2 4- 452 ^ ^2 _|_ ^2 _^ i2ab - Qac - Uc + 6ad. 4- 4Z»^ - 2cd. EXERCISE XXXV. x^ + 2ax^ 4- 3 A 4- a^. 2. a;^ — 3«a;2 4- 3 A — «^. a;^— 6a;2|/4-12a;^2_g^3^ ^ 3^3 _|_ i2a;22/ 4- Qxy-^ -{- y^. ^ 27a''5 - U6x^y + 226xy^ - 12byK a^b^ 4- 3a2^,2c 4- 3«5c2 4- c^ 8^3^,3 _ 36«2^,2^ _|_ 54^^,c2 - 27c3. 125^3 _ 75«2^6' 4- 15a^>V _ js^. 10 ANSWEMS TO QILLETB ALGEBRA. 10. Ux^ - 240a;'*3/2 + 300a;y - 125?/«. EXERCISE XXXVI. 1. a^-2x^ + x+l. 2. l/2a;2 - 1/3^2/ + 1/4/. 8. 4 and 9. 4. 5 and 8. 5. 3 and 5. 6. 42. EXERCISE XXXVII. o -4- ft'>'3^y9 6. x^y^z. — x^y^. EXERCISE XXXVIII. a^-\-2a- 1. 2. 2a^ — 3 A — fl^a:^. 4. •2a4 + 4«V - 4c4. 6. 4a;2 - 2^2; -h 2*2. g a;^ — 2a;2?/ + '^xy^ — y^. 10. 52;^ — ?tx^y — 4x^/2 + y^. 12. ^^ - ^«/ + y^' dx^ — 4:Xy^ — 2/. 2x^ - 5a; + 3. ^ - 2x^ + 3a; - 4. 2 - 3« - a2 -j. 2^3. a;2 — l/2xy — y\ ic^ — 2xy -\- y''^ EXERCISE XXXIX. 1. 106929. 2. 14356521. 3. 714025. 4. 25060036. 5. 387420489. e. 25836889. 7. .00092416. 8. .00000784. 9. 4816.36. 10. 1867.1041. 11. 1435.6521. 12. 64.128064. 3789. 2.1319. EXERCISE XL. 2. 5006. 3. 5083. 4. 129.63. 6. ,9486+. 7. 2.4919+. 8. .0923+. EXERCISE XL!. 1. «3 _^ 3^2 _^ 3^ _|. 1, 2. x^ + Qx^ + 12a; + 8. 3. «V — TiaH^y^ + 3aa;/ — /. 4. 8m3 - 12m2 + 6m - 1. 6. 64^3 - 144^2* + 108aZ>2 _ 27*^^ ANSWERS TO OILLET'S ALGEBRA. ll 6. 1 + 3a; H- Qx^ + Ta;^ + Qx^ + 'ix^ + a:^ 7. 1 - C^x + 21a:2 - 44?;=* + Q>^x^ - 54^^ + 27:?;«. 8. «^ + 6r?2/; - 3r^2^ + 12rtS2 _ i^ahc + Sr/c^ + 8^^ - I2}pc + 6^>6'2 - c^. 9. 8r^6 - 36«5 -f 66^4 - G3rr^ + 33^2 _ g^^ _^ ]^ 10. 1 - 3a: + 6a;2 - \0x^ + 12.^'^ - 12a:5 + lO.c" - 6a;' + 3a;8 - a;^ EXERCISE XLil. 1. 1 — x. 2. 1 H- 2a:. 3. 2a: — 3^. 4. 3a;^ — z^. 5. a-\- 8b. 6. 4a: — 3a:2. 7. 1 + ^ + .... 8. l---_-_--etc. EXERCISE XLIil. 1. 2460375. 2. 11089567. 3.- 1191016. 4. 17173512. 5. 109215352. 6. 102503.232. 7. 820.025856. 8. 20910.518875. 9. 056623104. EXERCISE XLIV. 1. 478. 2. 3.84. 3. 4.68. 4. 9.36. 5. 27.55. 6. 1.357 + . 7. .5848+. 8. .2154+. 9. 1.587+. 10. .7368 + . ll. 3.045+. 12. 2.502 + . 13. 9a\x^-\-6aWxy+4.bY. 14. x = 2. 15. 15 ft. by 12 ft. 16. 2«3 + 4c2. 17. X = 1. 18. 48 ft. by 40 ft. 19. 90 of port and 150 of claret. 20. 44. EXERCISE XLV. 1. 14a:2 - 43a: + 20. 2. 20a:2 + 62a: + 48. 3. 28 - 47a: + 15a:2. 4. 30 - 20a: - 80a;2. 5. x^ + 16a: + 63. 6. a:2 - 8a: + 15. 7. x^ + 3a: - 54. 8. a:2 - 4a: - 77. 9. x^-x-30. ■ 10. a;2 + 3x - 28. 11. x^ + 6a: + 9. 12. a:2 - 8a: + 16. 13. a;2 - 64. 14. z'^ - 36. 12 ANSWERS TO OILLET'S ALGEBRA. 15. d6x^ + 39.i; - 108. 16. 72a;2 + 12a; - 24. 17. 24:X^ - 19x - 175. 18. 18a;2 ~^x- 180. 19. 4:ax^-(5a-lr^b)x-\-bb. 20. 18aa;2+(24«+6c)a;+8c. 21. ba^x^ + (^^ — 5«c)a; — be. 22. {2a^ + 2a*)a;2 - («J + V- 2ac)a; — 5c. 23. 24 - 36a; - 108^2, 24. 63 - 44a; - 32a;2. EXERCISE XLVI. 1. 20^:4 - 47^2 _^ 21. 2. 21a;8^47^4^20. 3. 30:?:« - IQx^ - 32. 4. 42a;io + 4a;5 _ g^ 5. 2^4 _ 20^2 _ 96^ 6. 54^2 -|. 3^6 _ rj^j.^ 7. x-\- 12Vx-\-35. 8. 6.a; — 2 |/^ — 48. 9. a; -49. 10. 9a; H- 24 V^4- 16. 11. c^-5. 12. m — 5. 13. 6m^ - 18m^ + 16. 14. 3n^ H- 21?i3 - 180. 15. 510 + 55 _ 5g^ 16. «i4 _ 2aT _ 99, 17. a:8 - 49. 18. m« - 36. 19. 4:X^ - 16. 20. 26a^x^ - 9. 21. 9x - 175. 22. 36a; - 147. 23. a;2 + 4. 24. 4a;4 4- 45. 3. EXERCISE XLVII. {a + a;)2-3(« + x)-2% = a;^ + (2«-3)a;+a2_3«_28. (m + 2;)2-f m+a;— 72 = a;^— (2m — l)a; + m2+ m —72. \x-bf ^ ^{x-b)-^h = x^-{2b-\)x -^b^-^b-^:b. 4. (a;-m)2-5(a;-m)-84=a;2-(2m-f 5)a;+m2+5m-84. 5. a;2 — m 4- 5. g, a;2 _ 3 _^ ^^ 7. (a; - 4)2 - {x - ay = (2a - 8)x - d^ + 16. 8. a;3 + a;2 + a;+l. 9. 1 - 3a;2 + 2a;4. 10. X — 5^. 11. 5| days. \%, 30 min. 13. {x - 5)2 - (a; + 6)2 = - 22a; - 11. 14. \x + 7)2 -\x- 5)2 = 24a; + 24. 15. X - 23. 16. X - 23. 17. a;2 - a; + 5. 18. 11. 19. 3. 20. 220 - 16a;. 21. 7a; + 148. 22. a; = - If. 23. 4i liours. ANSWERS TO OILLETS ALGEBRA. 13 EXERCISE XLVIII. 1. x^ -h aK 2. x^ + 27. 3. x^ - 343. 4. a;3 - (^. 6. 8a;<' - 27a^ 6. a;2 _|. 4^ _|_ lt5. 7. 4«V _ ;^4^^2 _|_ 49, 8. ■ a:« 4- ^x' + 3a:4 _^ 4^3 _|_ 3^,2 _|_ 2a; + 1. 9_ 1 _ 9.^2 _^ 33^4 _ (.3^0 _^ 66^ _ 36^10 _^ s.'cil 10 a^j^\/%lb\ 11. l/8rtV - 8/27^'V. 12. l/125«V2+l/216^>%i^ 13. 9«V + «V + 1/9 A2. 14. l/16««a'i'^ - l/Ua^h^x''^ + 1/36Z»V^ EXERCISE XLIX. 1. x^ + 8.r + 16. 2. m^ - 10m + 25. 3, x^ -'dx + 9/4. 4. w2 - 5w + 25/4. 6. x' + 72; + 49/4. 6. if - ^y + 81/4. ^ ^2 _ 3/4a; _|_ 9/04. ^ ^} + 5/6^ + 25/144. 9. x" + Z'.r + ^^74. 10. a;2 - 5Z^:c + 25^>V4. 11. x^-^x-^ 1/4. 12. ^^ — ?/ + 1/4. EXERCISE L. 1. x^ + 6a;3 4- 9. 2. m^ - Vlw? + 36. 3 ^4 _ 5^2 _^ 25/4. 4. «» + 7^4 + 49/4. 5. x'' + ^>a:3 + 574. 6. 2;* - 2;2 + 1/4. 7 ,^10 _ 2/3x5 _|_ 1/9. g^ ^^6 _ 3/4^3 _|_ 9/64^ 9. (^+2)H6(^+2)+9. 10. (:^-5)2-3(x-5)+9/4. EXERCISE LI. 1 ^2 _ 8a; + 16 - 18. 2. a;2 - 12a; +36-6. 3_ r^i + 7a: + 49/4-52/4. 4. a:2_7^+49/4_233/20. 5^ 1/1 6a:* + 1/2^;^ + \x^ + 32a: + 256. 6. 27 days. 7. 3^ days. 8^ ^2 _ 9^ + 81/4 - 69/4. 9. 22+ii^+i2i/4_i49/4. 10. x^+bx+b'/4:--j^. 11. y'-bij+b'/4.-'^^. 12. 16/81a;4 _ 4o/27a:3 + 100/9a:2 _ 250/3a: + 625. 13. 72 miles. 14. 5j\ hours. 14 ANSWERS TO OILLETS ALGEBRA. EXERCISE Lll. \. x^-{- '^x + 9/4. 2. x} - 5x -f 25/4. 3. x^-3x+ 9/4. 4. x^ + 9.T + 81/4. 6. {x + ay - 6/d(x + rt) + 25/36. 6. x = 2f . 7. .^2 + -.^• + by4:a^. 8. ?/2 — n/my + 7^V4m2. 9. x^ + 3/2:»2 _|_ 9/16. 10. z^ - 3z^ + 9/4. 11. {z - 5)4 + 3/7(z - 5)2+9/196. 12. a: = 4:-^. EXERCISE Llll. 1. 2{x^ + 3/2a; + 9/16 -f 39/16). 2. 3(:?;2 _ 6a; + 9 - 13). 3. 4:(x^ - 'S/2x + 9/16 + 19/16). 4. 5(a;2+5a;4-25/4-41/4). 5. Q{x^-\-7x-\-49/4-^7/12). 6. 1.4142+. 7. 1.442+. 8. 7(2;2-9:z;+81/4-53/4). 9. 8(x2-52;H- 25/4-31/4). 10. 9(.'c2-9:c+81/4-53/4). n. 10(a;2+7a;H-49/4-81/4). 12. n(x^ - %l\\x + 1/121 + 32/121). 14. m(.3-Vm. + — ,--^^^^j. EXERCISE LIV. 1. lai^U + c). 2. 2«2^(.t2 - 4a; + a^). 3. 5Z»V(J:r -^(?y - !)• 4. 7«(1 - «2 4. 2a3). 5. 2a;3(3 + a; + 2a;2). 6. 15«2(i _ 15^*2). 7. 5:^3(^2 _. 2«2 _ 3^3)^ g^ 19^2(2^3 + 3a). 9. (3a:2 — X — \)x. 10. xif^'lxy — Zx -\- "ly). EXERCISE LV. 1. {x^a)(x- a), 2. (a: + 3)(a;-3). 3. 4(a + 4)(« - 4). 4. \zax + 5Z')(3aa; - 5J). 5. (9 + 4«a;2)(9 - ^ax^). 6. (7«2a; + 4«V)(7a2a; - 4aV). ANSWERS TO OILLET'S ALGEBRA. 15 7. (x+13){x-l). 8. 0/+5)(^-13). 9. {a + 2)(« - 6). 10. {b + 23)(b + 1). 11. 6 ft. 12. 86. 13. 3(2 + a){2 - a). u. 3«(4« + 6b){^a - 6b). 15. 3«(3rt2 -f 5a:2)(3«2 - 5x^). 16. bx{5ax^ + 3.^7/)(5^.T^ — 3:?:?/^). 17. (:?: + 10)(a: + 4). 18. {x -j- l){x - 17). 19. (a:+ l)(a:- 11). 20. (.T + 29)(^' + 1). 21. lOJ^ hours; 21y\ hours. 22. 67. EXERCISE LVI. 1. (x-\-b)(x+7). 2. {x-3){x-9). 3. (x-\-i){x-8). 4. (.T - 3)(a; + 10). 6. (x-7){x + 6). 6. (:^:+5)(.'r-4). 7. 2(a; - 8)(^ + 3). 8. (.tH-5)(3x+ 11). 9. (2rc-l)(3a:-7). lo. (4^; + 1)(5.t + 8). 11. (bx - 3){7x -^ 12). 12. 4(7:c + 5)(2a;-5). 13. A can do it in 17| days, B in 14|| days, and C in 13^ days. 14. 2(2 - x){3 + 4.T). 15. 4(6 - 7x)(2 - 3x). 16. (5 4- 3.t)(7 + 4:7:). n. {2x + 7){3x - a). 18. {ax — 6){bx + 7). 19. {ax + b){cx — d). 20. {x — {a — b)) (x + {a + ^')). 21. l{a-}-b)x + 2)({a-b)x-4:). 22. 3(a;+ 6)(a: - 3). 23. 7{x - Q){x + 5). 24. 10(a: - 2){x + 7). 25. 5^2(3^; - 2)(5a; + 3). 26. 93. EXERCISE LVII. 0. 1. 44. 2. - 10. 3. x^-x. 4. x^-x + 1. 5. 28. 6. C3 + ^3 _ J3 _ ^3 7. if — if = 0, ■ 8. .^6 _ y6 —Q^ 9. if - ?/" = 0. 10. -if + .v' = 0. LI. f + y' = 2y\ 12. y' + y'=.2yK 16 ANSWERS TO OILLET'S ALGEBRA. 13. y*-\-y' = 2f. 14. -y-+if = 0, 15. y"" -\-y'' = 2^". 16. 2/" + ^" = ^^Z""- EXERCISE LVIIi. 1. It is. 2. It is. 3. It is. 4. It is not. 6. It is. 6. It is. 7. It is not. 8. It is not. 9. It is. 10. It is. 11. It is not. 12. It is not. 13. It is not. 14. It is not. is. It is. 16. It is not. 17. It is. 18. It is. 19. It is not. 20. It is. 21. x^ + hT" + Z>V + h^x^ + ¥x^ + ¥x + h\ 22. x'^ — h'j(? + y^^ — b^x -\- h^ with — 2^^ as remainder. 23. x:' -\- bx* -\- b^x^ + b^^'^ -\- b'^x-\-b^ with 2b^ as remainder. 24. X' ~ bx^ + b^x^ - Z/V + b^x^ - Wx^ + b^x - b~ with 2^^ as remainder. 25. X?' -\- bx^ -\- h'^x^ + V^x + b^ with 2b^ as remainder. 26. x^ — bx" + Wx^ - IH^ + b^x^ - b^x + b\ 27. x^ + bx^ -\- b^x + b\ 28. (^ - bx^ + b^x^ - ^V _^ ^4^ _ j5^ EXERCISE LIX. 1. 2:'^ + a:^^ + x^y^ -\- xy^ -\- y^. 2 x"^ -\- x^y -\- x'y'^ + x!^y^ + a;^«/* + x^y^ + a;;^^ -f 2/'^- 3. X? — x^y + x^y'^ — 2:y + ^^'^ — y^' 4. 2;^ — x^y + .^**?/^ — 0:^2/^ + ^2/* ~ ^^'^ + y^' 5. a;2 4- 3a: + 9. 6. x^ + 3a:2+ 9^ _|_ 27. 7. a;3 - 2a;2 + 4a; - 8. 8. a;^ - 2a;3_j_4^_8^_|_ 16^ 9. 0. 10. V'-aK 11. 0. 12. 0. 13. 0. 14. - 12^2^. 16. W - '^V'c - Ui^ + 1c\ 16. {x -\- 2)(3a; + 4). 17. (.T-l)(.T+5). 18. (a: + 4)(a; - 3). 19. (.T - 2)(3a: - 2). 20. \x - l)(4.r - 3). ANSWERS TO GILLETS ALGEBRA. IT EXERCISE LX. 1. 13i hours. 2. 120 hours. 3. 50 days, 214 V 21a;2 - 11a; 40 3 - ^iCiWx^ ' 15a;2 - 9a; + 42 7a; 4- 8 15a;2 + 13a; 9- hx 9a; + 20 11 loa^x a;2- 16 Their sum = 30 and 9a^x 2 + 10;r + 24 6 -. 3a; and r^'a; 9a=^^a;' x^ 16 x^ x 2a;2 + a; + 44. X? - 16. • 14a; + 48 x" 6a 4rt^ a; - 30 ' a^ - a; - 30 _ a;2 + 25x- - 43 ~ Tc^"- a; - 30 • oa; -|- 6 6rt — oa; — 6 30 4«:^ 45x^ +20a;2-3a;+18 15.t2 + 14a; - 8 * 25^>2 - 84rt6' 36^2 4^2 16. — b'^ — 4ac 18. 4«2 4a^> * 20 ANSWERS TO GILLET'S ALGEBRA. 19 iC^*" ^2rn-\-n _r_ ^TO-f-2n ^n^ 20. a;^"* + aj^"""^" + a^^'^+^n _|_ ^m+sn _|_ ^«^ EXERCISE LXXI. 1. 1. 2. --^— . 3. 1/x, 4. — ^ -^^ 5. a X X? - 13a; + 42 15.^:2 - 26a; + 8 a: + 7 10. a'^ — a;'^ 2a;- 1 EXERCISE LXXII. ab 2. 7/12. «- 11 a~% * a; + l a; + 5* . • 14 - 17a: - - 6a;2 2« - - 1* 2a;- - 1 2a;- - 3 * 6a;2 - 23a; + 20 3a;- 2 2a;2. - 21a; + 27 6. 4a; + 6 ' ' a; — 5 EXERCISE LXXIII. 1. x^ -X- 12, a;2 - 15a; + 56, ^x + 48. 2. 9a;2 - 9a; - 28, lOa;^ — 43a; -f 28, 5a;2 -f 51a; - 44. 3. -40a;2 + 94a;-48, -35a;^-19a;-42, -3(a;2-6a; + 9). 4. a;2 - 64, a;^ - 64, 7a;2 + 28a; - 224. 5. x'^x- 42, a;^ -f 216, 5a;2 - 30a; + 180. 6. i/'^"' -4- o;S»»+'» _L_ '7;*"' + ^" -4- -|.3'« + 3n _j_ ™2»i+4n _j_ ^m+5» _j_ ^6»» EXERCISE LXXIV. 1. A is 48 and B is 12. 2. 49^ minutes after 3 o'clock. a. 17 and 28. 4. 35 dimes, 5 cents, and 10 dollars. 5. 98 and 215, . , cmim — 1) _,, cin — 1) 6. A's age = — ^^ '- ; B's age = -^ . m — n ® m — n AI^SWEBS TO GILLpT'S ALGEBRA. 21 100(^ - a) 11 7. • 8. TT"* ac 3 9. 27y\ minutes after 5 o'clock. n — r , nq 4- r 10. ;— r and ^ ,' . S' + l ^ + 1 EXERCISE LXXV. 1. -3f 2. 11. 3. 7. 4. -8. 6. 1. 6. - If. 7. - 10. 8. - 1. 9. 7. 10. 3. 11. 1. 12. — 2. 13. 1/4. 14. -3^- 15. If. EXERCISE LXXVI. 1. 5^. 2. 6. 3. 9-1^. 4. 1. 6. 1*. EXERCISE LXXVII. 1. 1/2. 2. 4f. 3. 4i. 4. -4/7. 5. Hf. 6. 3. 7. 3i. 8. -4i. 9. 2^. . ab — cd 10. 0. 11. -^^ j. 12. 2yV«. EXERCISE LXXVIII. 1. 55 minutes. 2. 37^ min. and 25 min. 3. 130000. 4. $84000. 6. A 39 miles and B 27 miles. 6. 283. 7. 536. 8. Of the first -^^ ^, and of the second -^^ z-^. a — a — i 9. $750 and $500. 10. 192 miles. 11. $15.36 and $4.56. 12. Hound 72 and fox 108. 13, 300. 14. Man 84 cents, boy 42 cents, 15. 85 gallons of spirits and 35 of water. 16. 1500. 17. 28. 18. 3 shillings. EXERCISE LXXIX. 1. Vn^. 2. V^i:^ 3. V^^. 4. V'2ba%\ 6. V^aF. 6. Vl^\ ANSWEBS TO GILLET'S ALGEBRA. 8. yl/9«y or — d-. VU V9 >. V -77^2- or —=.—. 10. VflJ^ + 2«^> + b\ 11. Va;2 - 2xy + 2/2. 12. V^a"" + 426^^ _^ 49^ 13. fe 14. '^STaV. „ l//Y9iy3 15. hi 16. \/x^^lbx'^lbx^l2b. 17. \^a^ - 9^2 _^ 27^ _ 27. 18. .727«V |/27«V ^ ^^0^ " f 64.3 • EXERCISE LXXX. 1. 24/3. 2. 5 1^3. 3. 64/5: 4. 7 1/15. 6. 16 V2. 6. 9 V7. 7. 3 1^5". 8. 4 V'y. 9. 8 1^11. 10. 4.aV¥l). 11. 5«:c2 4/5«. 12. la^x^V^ax. 13. 2a{a + J) 1^6?. 14. 2x^y{x - y) V^xy, EXERCISE LXXXI. 1. i/99. 2. 1^208. 3. V252. 4. ^^72. 5. >^^320. 6. ^864. 7. VSia^ - 9«2j. 8. V^x^ + Qx^y + Zxy^, EXERCISE LXXXII. 1. 1/2 V2. 2. 1/5 i/5. 3. 1/3 VQ, 4. 1/6 l/fK 5. i- f^2l^. 6. . — ^ V¥-b\ ' Ix ^ a — h 1 ./^^-TT.— 7-7^ . 1 7. -t-t; ^^^ + 10:c + 24. 8. — ri^ i^^^ + 2a; -~ 35. ANSW£JMS TO GILLET'S ALGEBEA, 23 9. ^r-V-r VlOx^ + x-2. 10. 5-^^ 4/12^21146^+42. 2x -{-1 dx — i 11. — ^ ^20 + 7x-Qx\ 12. T y 12a;2 + 7:c - 12. OX ~j~ 4: 'lit/ O EXERCISE LXXXIII. 1. 18^2". 2. 37 V2. 3. ^^15". 4. 2/5 V'e". 6. 25a2a; 4/3^^ 6. -^ V^. w(?i + Vns) 7. ISft^* V'2«2^2, 8^ n — s EXERCISE LXXXIV. 1. 4 V'5. 2. - 3«2^ f ^. 3. 2^> Vb. 31 3_ 4. UV2a, 5. ^VQ. 6. -l9aVab. y u 7. (13c - d5cd) V2i. 8. (6- - X -~-^ V^^ EXERCISE LXXXV. 1. 96 Vd. 2. 1^ H. 3. 24 y^. 4, 1/2 4/6". 6. ~4/^. 6. 4^% 7. 64/10 + 7 VT5 + 84/6 + 24. 8. 6 + V16. 9. 6 4/2r- 46. 10. 2 l^e. 11. 6ff - 6a; + 5 Vox, 12. 3 4/7 - 47. 13. 64/5 + 14. 14. 53-144/5. 15^ 32-104/7. . 16. a; + 7 4/^+ 13. 17. a; + 18 4/^+ 81. 18. x + Vx-dO, 19. a; -2 4/3^+3. 20. -3. 0^2. 24 AlfSWEIiS TO GILLErS ALGEBRA. 21. 15 + 4^11: 22. V^^ - 3x - 40. 23. 2a; + 2 + 2 i/a;2 -f '2^ - 24. 24. V^f^^^. "•^ 2a; -f 2 |/a;2 - 9. 26. ISa;^ |/(:j''^ - 13« + 42. 24a% + 8« t^6^+ 4^. 28. 35«Z>a; + 245«&. 25a; - 58 - 24 Vx^ - x - 42. 30. 63^3 V^~rT{j. 34«3 _ 98^2 + 30«2 4/«2 _ 2« - 15. 32. x - 29. 33. x + 2. 34. - 13. 35. - 7a; - 26. 36. 9«V _ 72«V _ 25a;5 - 175a;^ 27. 29. 31. EXERCISE LXXXVI. 1. 113. 2. - 166. 3. 172. 4. - 6. 5. a — 4^. 6. 9c2 — 4a;. 7. X. 8. 2j» — q, 9. 2a;. 10. 25a;2+75?/2_49«2. n. - 2ax. 12. 2x^ -\- 6x. EXERCISE LXXXVII. 1. 44. 2. 59. 3. «2 + J2 + ^2 _ - 2«J - 2ac - 2hc. 4. 64. EXERCISE LXXXVIII. 1. - 64/3 + 8 V 104-134/42" 64/7-1/2" ^- 26 • ^- 25 • 4. 2. 5. ^ - i^a^ - l^. 3 _ i/9 _ «4 6. 7a; + 3 + 8 Vi 3a; + 15 8. 1 1 + a;2 11. Vs. aVx. 12. '^'S. 13. 44/5. 14. 15. Vx - 7. 16. f :z^^ + 2a; + 4. 17. V^x-^3. Vx - 9. 18. \^x - 3. 19. Va; - 7. 20. 21. 4/3a;-2. 22. Vbx-1. 23. 11 - 3 Vf 3 3 4/f - 2 1^3" 19 - 6 i^ 24. 3 . 25. ^^ . ANSWBBS TO GILLErS . AIGEBMA. 26. 2+^6. ^/X'U 27. ^. y ». f. 29. a — X : 30. 4 + 4/15; EXERCISE LXXXIX. 1. VlOO, ^^125, and ^^11/2. 'i^(a - by, and 2. '\/{a + hy, y'Ca^ f x^y. 4. 1/2 |/^. 3. -♦/10125. 6. t^a^ 6. 7. 3/2^8/3. EXERCISE XC. 1. 14. 2. 8. 3. 20. 4. 2|. 5. 13. 6. 6/5. 7. 144. 8. 2. 9. 13. 4|^. 10. ItV 11- 5. 14 (^-*)' 12. 12. 15. 1/6. 16. 19. 3«/4. 23. 20. 17. (v;^-i)2. a - 1. 21. 42i. EXERCISE XCI. 18. 2/5. 22. 9«/10. 1. 4. 24/17. 12^3_ 2. Hh a{c - 1)2 ^- 4. • 3. 3|. 6. 8/45. 7. 25/168. («-&)2 25 EXERCISE XCII. 1. 16. 2. 1/64. 3 1/5. 4. 1/6. 6. 1/1000. 6. 36. 7. a^VK 8. a-^/'^W^'^. - 9. «V3Z,i2. 10. a-%-y'^. 11^ «i/5 _|_ ji/2 ^ ^4/3^ j2^ ^^3/2 _|_ ^1/3^2^ 13. a^/V/s + a^/^^^, 14. a;2/32/0V3 4- ^s/v/s^ 26 AJ)f^8WEBS TO GILLETS ALGEBkA. 3-1 1 16. VX^ 3-. 16. T"vf* 18. — T- X S- - + -r— . 19. tC^/^ — ^*/^ 20. 1 - ^'/^ 21. «'/' + 2'^/^ 22. X^-1, 23. :r3 + 2a:V2 -|- 3 + 2:?;-^/2 + a;-3. 4n 8m 2n 4h 25. ic5/2 + //2. 26. i«^^ + ^V + 2/'- 27. a:«/3 _ ^4/3^4/3 _^ ^8/3^ 28. iiJ^/^ - 22;«/5i/V4 4- 4a;V5^V2 _ SrrVy/* + \^. 29. ic2/3-^-2/3. 30. «4/10 _|_ ^3/10^1/5 _|_ ^2/10^2/5 _|_ ^1/10^3/5 _^ ^.4/5^ 31. ^ + 2/- EXERCISE XCIII. 1. x = 2, y = ^. 2. X = ^, y = b. 3. a; = 2, 2/ = 1. 4. a:=:4, i/ = — 1. 5. ic = 1, «/ = 2. 6. ^= — 3, ?/=:4. 7. X = 6^ y — — Q, 8. a^ = — 1, «/ = — 2. 9. 2; = 3, _^ = — 1. \Q^ X — 1 , y = 5. 11. a; — 3, ?/ = 8. 12. x — 2, y — 3. 'T* '7*^ or 13. 1 - 2 ~ "8~ ~ 16"' ^*' ''^^ -2^+1. 15. a; = 15, 2/ = 16. 16. x = ^, y = 2. 17. »: = 2, y = - 1/2. 18. x = \,y = l, he ac 19. X — b, y = b 20. .T = — r-7' ^ = — ^^^^t.* ^ a-\- h ^ a-^h h — c a — c _, 07 21. X — 7 , y = ,. 22. X = 2h — a, y = 2a — u. b — a ^ a — ^ ac he 23. x:=:a, y = h, 24. X = -2x12' y «24_^2' ^-«2_^ AirSWEnS TO GILLET'S ALOEBBA. 2? 7« + 8Z/ 8« -h n EXERCISE XCIV. 1. 7andl. 2. 8/15. 3. 45. 4. 54. 5. 58 years and 18 years. 6. Each would do it in 50 days. 7. Tea 28 cents a pound, and sugar 3 cents. 8. 4 gals, from the first and 3 gals, from the second. 9. 2 gals, from the first and 10 gals, from the second. 10. Tea 30 cents a pound, and sugar 3^ cents. EXERCISE XCV. 1. x = 2, 2. x=\, 8. a; = 3. ^ = 3, 2/=-2, 2/ = 5, z =4. ^ = 3. = -3 4. x = l, 5. a; = 4, 6. a; = 3/2, 2/ = 3, 2/ =-3, 2/ = 2/3, z= -b. ^=2. ;2? = 5/6. 7. x = 15, 8. i. = 3. 9. a; = 9, y = lS, 2/ = 6, ^=18, z = 20. z = 9. ^ = 6. 10. X = S, y = Q, z ^=- 5. 1. 5m 2m 3a;' + 4a;' - 7n - V^x^^ 3n « - 6a:* + lO/. 3» + »i n + 3»n 12. a;2" + a; ' +»:" + '" + a; ' + ic^"*. 13. 4a:2/3 _^ 25a;V3 _|- iGa:^ - I2:r - 24a;S/3. EXERCISE XCVI. 1. 9, 11, and 18. 2. 37, 25, and 16. 3. 124, $32, and 116. 4. A, 1420; B, $640; C, $1040. 6. A in 40 days, B in 120 days, and C in 60 days. 6. A in 10 days, B in 15 days, and C in 12 days. 7. 234. 8. 253. 9. 428. 10. A, _ , ■ . ; B,—— —-———; C, rs -\-st—rt —7's -\~st^rt rs—st-{-rt 28 ANSWERS TO OILLET'8 ALGEBRA. 11. Rate of stream, 2 miles per hour; rate rowing in still water, 10 miles per hour. 12. Rate of the current, 3 miles per hour; rate of crew in still water, 12 miles per hour. 13. Rates 36 and 27 miles per hour respectively, and distance 75 G miles. 14. Rates 25 and 30 miles per hour respectively, and distance 330 miles. 15. 15 persons, and 5 dollars a piece. 16. Number of persons ^, ^ ; each received bm — an om — an EXERCISE XCVII. 1. {x-\-V^^){x- V^^).2. {x ^ Vf){x - Vl). 3. (^+4 V~i)(x-4: V^). 4. 3(^ + V^)(x - 1/3). 6. b{x + VE){x - Vb). 6. 7(a;4- V'^)(x- V^^). 7. 2{x + 1/2 Vq){x - 1/2 |/6). 8. 3(ic + 1/3 V- lb)(x - 1/3 V- 15). 9. 5(a; + l/5|/l0)(:r- 1/5 VIO). 10. ^{x + 1/4 V- U){x - 1/4 ^"-^nj) = 4(2: + 1/2 V'^^)(x - 1/2 \r^), 11. ^x + 2/3 \^){:x - 2/3 VI). 12. 7(a: + 1/7 V- 'db){x - 1/7 V^^^), EXERCISE XCVIII. 1. ^[x^——-)[x^ — —y 2. 2{x^\/2){x+^ 3. 5(.+ IlA4_^)(, .-^-^^ •^•+ 10 / 3 + 2|/ir\/ . _3-24^> / _3 + 2|/ll\/ , AWSWEBS TO OILLET'S ALGEBRA. 29 6. 8. 10. 11. 12. 13. 14. 1. 4. 7. 10. 13. ^8> J , -4 + l/88\/ , -4- V88\ 6 a; + 4 b: + 3 + i/- ^% + ! - V- 87> i^2> 8 2{x^4.){x-\-l). 7. 661 at 125, and 108i at 140. 7(a; + l)(a:H-2/7). 9. 1l{x-\-'^ ^ Vb){x^2-- Vb). d{x^^){x- 2/3). 4(a: - 3 + t'+'6)(a; - 3 - i^+~6). 15(a:- 3/5)(a: + 2/3). S + ^/yN/ 6-Vf s — bm , ^ — «^» and ■)• 16. X a — h " b — a a{c - If 4c • acres. 16. a: = 4/9. - 3, 6. 5,7. 3, -3. — a, b. 2/5, - 4/3. EXERCISE XCIX. 2. 5, - 9. 5. - 4, - 4. 8. a, — «. 11. - 3/2, 1. 14. - 8/7, - 1. 3. - 5, - 8, 6. 5, 5. 9. — a, — b. 12. 4/3, - 3. 15. 3/2, 4. 16. 5/3, 4. 1. x^ - lOx + 21 EXERCISE C. 0. 3. a:2 + 8a; + 7 = 0. 5. x^ 4- 9a: = 0. 7. x^ + 16a: + 64 = 0. 9. 4a:2 - 15x + 9 = 0. 11. 16a:2- 28a: +.11 = 0. 13. 18a;2+27a:+ 10 = 0. 15. 5a;2 - 33a: - 14 = 0. 4. 6. 8. 10. 12. 14. 16. a;2 + 2a: - 24 = 0. 0:2 — 2a; = q^ a;2 - 49 = 0. x^ - 22a: +121 = 0. 18a:2 _ 18a: + 1 = 0. ^.2 2a:2 8a: + 22 = 0. - a: - 3 = 0. 6a: + 4 = 0. so ANSWERS TO GILLET'S ALGBBM. 17. x^ - ix — ^ = 0. 18. x^ — lOo; 4- 22 = 0. 19. x^ - I82; + 85 = 0. 20. 25x^ - 35.T + 13 = 0. IP- — 4«c 21. 242;2 - 44a; + 21 = 0. 22. a;^-2a; ^^' a; + 2 • ^^- 2 - a;' EXERCISE CI. 1. 1, - 1/3. 2. 2, - 3. 3. 2, 3. 4. 4,1/4. 5. -1,2. 6. -3/4,-9/4. 7. 5, - 6yV 8. 1, - 7/32. 9. a, 1/a. 10. 3, 13/11. 11. 2, 1/2. 12. 1/2, - 3. 13. 5, - 1/6. EXERCISE CM. 2. 5/7, 3/4. 3. - a, b. 4. - 3/4, -2. 5. 2/3, - 5/4. 6. ± 6, ± 9. 7. ± 6, ± 10. 8. ±2/3, ±3/4. 9. ±^,±^:^. EXERCISE cm. 1. 15 and 8, or - 23/2 and - 37/2. 2. 3, 4, and 5, or — 1, 0, and 1. 3. 20 and 8, or — 14 and — 26. 4. 5, 6, and 7, or — 1, 0, and 1. 6. 4 and 2. 6. 1, 2, 3, 4; or 5, 6, 7, 8. 7. 3, 4, 5, 6, or - 4/3, - 1/3, 2/3, 5/3. 8. 20 barrels; 6 dollars a barrel. 9. $80 or $20. 10. $60. 11. 8 miles an hour. 12. 7 miles an hour. EXERCISE CIV. 1. -2, -4. 2. 20, "6. 3. 5, -5/2. 4. 1, 4|. 5. 1, 2f 6. 3, 1/2. 7. 4, -4^. 8. 1, -3/4. 9. 2, -2/9. ^ , /o -6± VS ,77 10. 7,-1/3. 11. . 12. —a-\-I),—a — I?. AJSfSWEBS TO OILLErS ALGEBRA. 31 13. —a, —3ak 14. , ab, is. a, b. (t 16. 0, — — Y. 17. 2ft— 6, db—2a. 18. «, 1/a. a -\- b ' 19. ^', y. 20. |(-3±^). 21. ± V^H^^, 22. 1/8 (-25 ±4/33). 23. 3/5, -2/3. 24. 3, + 1/6. EXERCISE CV. 1. 30 and 40 miles per hour. 2. 40 and 45 miles per hour. 3. 2 1 hours. 4. 2^^^ hours. 2,,, 1 2|/ft3 5. a'^^/^, 6. — i— + ~^^- a" Vb^ Vb^ 1 ' ^ {m — n)bd ~ cb-\- dx C2 - ^>2' 10. (^ - 2?/)(7a; - 11). EXERCISE CVI. 1. 5/2, 3/2. 2. ± 2/3 1/3, ± 4/5". 3. 6, 5f. 4. ±V^, ±l/2f6. 6. 5,6. 6. ±2/3 1^, ±1/3 t/2r 7. ± 1/2 V^, ± 1/3 |/6. 8. ± 1/6 V6, ± 1/3 V2, 9. 3/2, - 2. 10. 3/5, - 4/7. 11 13. - 1, - 1/2. 14. t^m, 4. EXERCISE evil. 1. 3 miles an hour, 3^^ hours. 2. 5 miles an hour, 5| hours. db^ - 75ftV 31 4/5 + 85 ^' 25^2 • 11 5. 8 days. 6. 16 days. 82 ANSWERS TO QILLET'S ALGEBHA. EXERCISE CVIII. I. a: 1= 3, y = ±b, 2. x = 7/2, y=± 5/2, x= -3, y = ±5. x= - 7/2, y=± 5/2. 3. x = ZV%^ y—±2V5, 4. x='dm.—n,y=±{m-\-3n), x = ~3 i^2, y=±2V5. x=n—dm,y=±{m-\-^7i). 5. 25, 9/16. 6. - 243, '^'26^ 7. {a^ + l)'fa~^-^ - ). EXERCISE ex. I, X = 7, y = Q, 2. X = 8, y = 3, X = 6, y = 7. x = 3, y = 8. Z, X = 5, y = 2, 4. rr =r 3, 2/ = 7, x= 2, y = 5. X = 7, y = 3. 6. X = 1, y = ^> Q, X = '2, y = S, X = — b, y = — 1. X = — 8, y = -- 2. 1^ X — 2, y — —9, %, x= — Q, y =zl2f X = 9, y — — 2. x= — 12, ?/ = 6. 9. a; = 7, 2/ — 4, 10. a: = 5, ?/ = 3, a; — — 4, ?/ = — 7. ^ =: 3, 2/ = 5. II. X = Q, «/ = 4, 12. X = 5, y = 8, x= — 4, «/ = — 6. ic = 8, ^ = 5. AN8WEBS TO GILLET'S ALGEBRA. 33 13. X = Qi, y = 3, 14. ^ r= 9, 2/ = "^j :c= 3, ?/=.6. :k = _ 7, 2/- -9. 15. x = h -\-a, y=a—h, 16. x=±(2a—b), y=±{a—2b), x=b—ay y=—a—b. x=±{a—2b),y=±(2a—b). EXERCISE CXI. 1. x= ± 4:, y = ±1, 2. X = ± 8, y =^ ^^ 5, x=^ ± 14:, y= ^4:. x= ±3, y= ± 5, 3. x= ±Q, y= ±2. 4. x= :i^9, y=^ ±4. 6. X = ± 4, y — ± b, Q. X = ±2, y — ±4, x = ±^V^,y=±V^. x= ±V2, y = ±3V2. EXERCISE CXIlo 1. 3 and 5. 2, 4 and 7. 3. 5 and 9, 4. 4 and 10. 5. 3 and 4. 6. 3 and 7. 7. 2 and 3. 8. 1 and 2. 9. Cows 30 dollars apiece and sheep 3 dollars apiece. 10. 13. 11. ^^:rj2' 12. {4x - dy'^)(dx^ - 2y). is. - 14. 25. 15. 4 4/2". 16. ^ 26b^ - 84ac 3Qd' ' b 17. 4+|/2. III. 3. 4, 8; 13, 1. 6. 9, 8,3. EXERCISE CXIII 1. 2, 3. 2. 1, 10; 14, 2. 4. 1, 11. 5. 7. 7. 5, 6, 7. 8. 4, 2, 7.- 9. 3, 11, 1; 7, 4, 1; 2, 8, 2; 6, 1, 2; 1, 5, 3. 10. 1, 5, 2; 3, 1, 4; 2, 3, 3. 11. a; = 4 + 13i?, 2/ = 1 + '^i>- 12. a; = lli> — 2, y = ^p — 2. 13. 8, 7. 14. 64, 44. 15. liy using the 7-inch five times and the 13-inch once. 34 ANSWERS TO GILLETS ALGEBRA. 16. By using 6 four-pound weights and 3 seven-pound weights. 17. By using the fifty- and twenty-cent pieces respec- tively 1, 17; 3, 12; 5, 7; or 7, 2. 18. By using the half-dollars, quarter - dollars, and dimes respectively 1, 18, 1; 4, 10, 6; or 7, 2, 11. 19. 5 pigs, 10 sheep, and 15 calves. 20. 92, 90. 21. 19/9, 2/5; 10/9, 7/5; or 1/9, 12/5. EXERCISE CXIV. Z. X> 2i. 4. x>m. 5. ^>4|. 6. X > 3.9, y > 4.9. EXERCISE CXVII. 1. 151 : 208. 2. 6 : 11. 3. 589 : 1008. 4. x^ -y^'.x — y. 6. x^ — y^ \x — ■y- 6. x"" -%f\x - y. 7. 144 : 125. 8 . 15 : 8. 9. 0, 4, 16, oc ) , - 32. 10. - If 11 . 18. 12. 385, 660. 13. 11. 14 . 5 : 37. 15. 5 : 6 or - : 3 : 5. 16. 9 : 7, or - 8 : 3. EXERCISE CXVill. 17. 5. 1. ah c ' bb 1 2. y. 3. -. 9 4. -. c 5. 3i. 4.a n 6. O*. 7. /7 ■ \« ^ m{b—a) *• 'd{c-h)' 9. 2. 10. -4. 11. -2i. 12. 6. 13. 2i. 14. 1/2. 15. -3/14. EXERCISE CXIX. 1. 13i 2. 2|. 3. 3.6. 4. 16. 5. 1256.64. 6. 523.5 cu. ft. 7. 4752 cu. ft. 8. 2^ cu. ft. 9. 18 miles. lo. EXERCISE CXX. 15/32. 1. 2.9805. 2 1.7686. 3. 0.3766. 4. 2.5119. 5. 1.6990. 6, 3.4771. 7. 4.6021. 8. 0.3010. AN8WEBS TO GILLET'S ALGEBRA. 35 9. 13. 6.8451. 10. 0.3923. 14. 4.4571. 11. 1.2121 0.9034. EXERCISE CXXI. ). 12. 3.5538 1. 862. 4. 7665. 7. .2864. 2. .366. 6. 3.9645. 8. .09034. EXERCISE CXXII. 3. 6. 9. .0988. .006823. 6.42285. 1. 6.42221. 4. 10.3701. 7. 8.3010. 2. 6.4024. 5. 11.1025. 8. 13.0969. EXERCISE CXXIII. 3. 6.5383. 6. 12.0969. 9. 14.0458. 1. 4. 7. 10 13. 16. 19. 172. .000406. - 2340.52. .000000636. - 378.45. 2.3388. 23.2578. 20. 2. .677. 3. 5. .0114289. 6. 8. 118.916. 9. 11. 4.326. 12 14. 7.12. 15 17. — .006535. 18 .8834. 21. .15811. EXERCISE CXXIV. 22 - 127.205. - 1299.39. 645300. 1.71. .07852. .2475. 5. - .70214 1. 3.9073. 2. 3.4022. 3. 1.4999. 4. 7. 2.7871. 6. 5. 2.1683. 8. 5/2. 6. 9. 18346. -1/3. 1. 6. 9. 1. 2. m/p. 6. 2«. 10. EXERCISE CXXV. 00 . 3. a/h. 0. 7. - 10/7. 5z*. 11. - 3/2. EXERCISE CXXVI. 4. h/a. 8. -9/4. 12. -2. 1. 3. 7. 11. 13. 64; 78; - 75 11. 4. 3i. 8. 9th. 12. 8/9, 7/9, 6/9. ;8. 2. 52; 83; - 7. 5. 2h 0. 9. 19 th. 10, 12, 14, . . . 52. , . . . 1/9. -14; 55; -19f. 6. 1/6. 10. 16th. 36 ANSWERS TO QILLET'S ALGEBRA. 14. 4a — bh, 3a — 4:b, 2a — 'db , . . — 5a + ib. 15. d = 4:, a = 2. 16. d= — 3, a = 21. 17. - 28f. EXERCISE CXXVII. 1. 624. 2. 187. 3. 255. 4. 810. 5. 0. 6. 357. 7. 1/2 {n^ + 37^2). 8. n{a + ^')2 - oi(n - l)ab, 9. 80. 10. 1941. 11. 1080. 12. 1160. 13. 8 + 12 + 16 + . . . + 76. /S' = 680. 14. 12j\ + 14if + 16f f + . . . 97|f . ^ = 2200. 15. 8729. 16. 41832. EXERCISE CXXVIII. 1. 603. 2. 3375. 3. 13. 4. 33. 5. 10 days. 6 8 clays. 7. ±5. 8. ± 2^. 9. 9 days. 10. 50500 yards. 11. $5195. 12. ± 20, ± 30, ± 40. 13. ± 8, ± 12, ± 16, ± 20. 14. =F 4, ± 2, ± 8, ± 14. EXERCISE CXXIX. 1. 10, 50. 2. ± 12, -48, ± 192. 3. - 15, 45, - 135, 405. 4. ± .6, .12, ± .024, .0048, ± 00096 6. 1/3, 2/3, 4/3, 8/3, 16/3, 32/3. EXERCISE CXXX. 1. 19680. 2. -9840. 3. 1281/512. 4. 191i. 5. -682. 6. 53144/2187. 7. -463/192. 8. 64/65. 9. 27/58. 10. .999. 11. 1/2. 12. 4. 13. 6, 24, 96, 384, 1536. 14. - 12, 36, - 108. 15. 24, 60, 150; or 27, 63, 147. EXERCISE CXXXI. 1. 5/33. 2. 5/27. 3. 44/111. 4. 3/7. 5. 1/77. 6. 4/5. 7. 52/165. 8. 7/60. 9. 143/740. ANSWERS TO OILLETS ALGEBRA 37 EXERCISE CXXXII. 1. $4159.09. 2. i?1153.94. 3. $897.00. 4. 5? yrs. 5. $403.90. 6. .04. 7. 14 yrs. 2 mo. 12 da. 8. 17^ yrs. 9. $6785.71 10. $6000. 11. $3246.42. 12. $437.50. 13, 451.33. EXERCISE CXXXIII. 1. 4 and 6. 2. 1/2 and 2/7. 3. 1/10. 4. 6i 8i, m. EXERCISE CXXXIV. 1. 1.2.1. 2. 1.3.3.1 .. 3. 1.4.6.4.1. 4. 1.5.10. 10.5.: 1. 5. 1.6.15.20.15.6.1. 6. 1.7.21. 35.35 .21.7.1. 7. 1.8.28.56.70.56.28.8.1. 8. 1.9.36.84.126.126.84.36.9.1. 9. 1.10.45.120.210.252.210.120.45.10.1. EXERCISE CXXXV. 1. a^^ X {:ix'y = 16«iV2, 2. 32 X (- ay^ ~ - S)a^\ 3. (5«3)4(_ 7^^3)3 ^ _ 214375^%^ 4. 5Vx-J--«V. 5. (2^)^(-^)-l^ 6. 4^X^=1. EXERCISE CXXXVI. 2. a^ - 8t«^a; + 28««a;2-56«5:x;3 _^ TO^?*^*- 56A-5 + 28a2a;6 — Sax^ -f a;^. 3. 1 + 9:?; + 36:?;2 _^ 34^3 _|_ 126a:* + 126^;^ + Mx^ + Ux'^ + 9^'S 4- x\ 4. :r^ - 15a;4 + 90.r3 - 270a;2 + 405a; - 243. 5. 81a;4 + 'ZUx^y +■ 216a;y + %Qxif + 16?/^ 6. 32a;5 - SOx^ij + 80^:3^2 _ 49^2^3 _|_ i()^^4 _ ^5^ 7. 1 -18a2 + I35a^ - 540rt« + 1215a^ - 1458^1^ + 729a^l 38 ANSWERS TO GILLET'S ALGEBRA. 8. l-7a;?/+21a;y-35^-y+35:z;y-21a.-y+ "ixSj^-x^if. 9. 729«« - 972a^ + 540«^ - IQW + ?|^' _ ^^ + ^^. ^°- 729 + 27 "^ 3 "^ " "^ 4a:2 + 8^^* + g4^,6- 12. w?~^ — 6m~^/W + 15?/?r2'M'* — 207n~^/hi^ + 15wr^;i^ 14. a^2 _^ 20a9a;V2 + 150A + 500A3/2 ^ Q2bx^ 15. «^ + 16«29/6 -^ 96ftii/3 -I- 25Gft5/2 4_ 256^^3. 16. x^ + 15a;i2/5«/-2/5-j-90a:V5;i/-4/5-f 270x6/5^-«/5+405:?;3/5^-8/3 + 243?/-l 17. ttV2^-lV3_|-7a5/2^-10/3_|_21^3/2^-2_^ 35«V2^-2/3_|_35^-l/2^2/3 + 21a-3/2^2 _|_ 7^-5/2^10/3 _^ a-y^^y. 10 45 120 210 252 210 120 45 10 1 3o' ic^ a;^ EXERCISE CXXXVII. 1. - 35750a;^«. 2. - 112G402;9. 3 - 312a;2. .n '^'^ ll'^O ,,, 10500 81 x^ 70:ry» ^ ^^ 2a;4 + 24a;2 + 8. 9. 140 V^. 10. 2(3G5 - 'SQ3x + 63.^2 - x^). li. 252. 189^^^ _ 21a^ J_ 12. g , ^g . 13. ^g. EXERCISE CXXXVIII. 1. aV4 _ l/4«-3/4^ _ 3/32«-V4a;2 _ r/128«-"/V - 77/2048rt-iVV. 2. a^/2 + 3/2aV2a; + 3/8«-V2a;2 _ 1/16^,-3/2^^3 -f 3/128a-V2a;4. ANSWERS TO GILLET'S ALOEBIiA. 39 3. 1 + 4.« + 10.f2 -f mx^ 4- 'dbx\ 4. 1 - Ix + 28.^2 _ 84a;3 + 210a;^ 6. 3V4 _ _J_ X l-x^ \—x^ '^-l :^, 2V27" 8^3'^ l^h'^ 128^3^ 6. 1 + 1/3:^; + 2/9:^2 4- 14/81a;3 + 35/243^;^ 1 1 . 3 o 11 3 , 44 , '■ ^-r + ^^^-T2^^ +-625^- 8. x-^ - ^x-'^ij + IQx-hf - Ux-y + 256a;-iy. 9. «-i + 1/2^-5:^-1/2 _j_ b/8a-^x'^ + 15/16«-%-3/2 + 195/128«-%-2 EXERCISE CXXXIX. 1. 4. 7. 10. 8648640. 720. 27720. 240. 2. 259459200. 3. 5. 181440. 6. 8. 840. 9. 11. 96. 12. EXERCISE CXL. 5040. 90720. 480. 9^89180. 1. 4. 70. 1512000. 2. 10080. 5. 178378200. EXERCISE CXLI. 3. 5250. 6. 455. 1. 2. 3. 4. 5. 6. 7. a;3 _ Qx^ _|_ x^ - 4.^•3 - 6a;^ - 1L^•3 9.T^ 4- 30a;3 - 20:^^4-21a;3 3, -r. 11a; - 6 == 0. 19a;2 + 46:^ + 120 = 0. x^ - 6x = 0. - 48x2 — 19x + 12 = 0. -47:^2- 120.r+ 144 = 0. - 240^:2 - Vdx + 12 = 0. 8. - 2, |. 9. - 2± i^. 10. ^(-3± V-7). 11. 13. ^(-1±V 3, -4. - 8). 12. 2, 2. 14. - 2, 4. 15. -7,8, 40 ANSWERS TO GILLETS ALGEBRA. EXERCISE CXLII. and , „ . 2. ^ and— 3. • z and r— r. 4. ^ and re — 4 :»+3 ' X — 8 X -}- 6' 6. ^ — r-p> and -. 6. ^ 1 and — 2a; + 3 a; — 5 * 3a; — 4 2a; — 6' EXERCISE CXLIII. 3 4 1. a; — 3 'a; — 4 a; — 5 3 2 , 5 3. 2a; + 2 a;-3'a; + 3 1.4 1 2a; -1 '3 + a; 3-a; 1 4,7 4. irrz T^-z -0 + 2(a; - 1) a; - 2 ^ 2(a; - 3) • 3 5 1 5. 4(a; + 3) 8{x + 5) 8(x + 1)' 1 7 13 6- 1o/^_J_1^ -^ ^^4- 12(a; + 1) 3(a; - 2) ^ 4(a; - 3)* EXERCISE CXLIV. a;- 2 3. *• ' 3(a; + 1) 3(a;^ - a; + 1) 7 5a; - 3 ^' x — l+a;2_|_ x-\-l' 5a; +6 _ 3a; - 4 a;'^ + a;+l a;^ — a;-|-l* 1 4a; -8 *• 5(a;+2) +5(a;2+ 1)* 11 ^- 2(a;2 + 1) "^ H^ ~ 1)' ' ANSWERS TO OILLETS ALGEBRA. 41 EXERCISE CXLV. 1. 1 - bx + nx" - 86.^3 + M^xK 2. 2 - Ix + 28a;2 - ^Ix^ + 322^;^ 3. 3 - 19:^2 + 95a;4 - 475^6 + 2375^8. 4. 2 - 11:?;2 _^ 44^4 _ 1^(5^6 _|_ 704^:8. 3 , 9 , , 27 , , 81 . , 243 9 2 , 4 „ , 8 , , 16 „ , 32 3 «• 3-"+r" +^^+ 81^+243^ • EXERCISE CXLVI. . , 1 > 3 2 3 , , 3 , 1. l + 2^+8^'^-r6^+l28^- 2. 1 - a; 4- -a;2 - -2;3 + -x^. 3. 1+3^-^-^ + 81^'^ -^3^^. EXERCISE CXLVII. 1 12.^3 13 4 1. ^=2^-8^^+16^ -128^- 2. X = y — y'^ -^ 1/ — y^. 3. a; = 2/+2/' + 2/+8/ + ..- EXERCISE CXLVIII. 1111 111111 1- 1 + 8+1+5* ^- 1+3+1+3+1+3 3 3 + 1 1 i i i ^ i ^- ^1+1+1+1+3+2+2 1^111111 ^•.1 +2+1 + 2 +1 +2 +1+2* 5 2+1 1 1 1 i i 1 ^- ^3 +2 +1 +3 +2 +1 +2 i_j_l 11^ 11111 6- i + i+i+i+r+i+i+i+1+3- . 42 ANSWERS TO GILLETS ALGEBliA. 1 4.L 1 L L 1 1 1 ^- +3" + l + a +r + 3 +1+3* L 1 L 1 L i *■ 2+3+4+5+6+7* 9. 2/1, 13/6, 15/7, 28/13, 323/150, 674/313. 10. 1/2, 2/5, 7/17, 9/22, 25/61, 159/388. 11. 3/1, 10/3, 13/4, 36/11, 85/26, 121/37, 1174/359. 12. 1/2, 3/7, 4/9, 19/43. 13. 1/4, 7/29, 8/33, 39/161, 47/194. 1 1 L 1 1 ^.lL .76 1 I*'- 2 + 3 + 3 + 3 + 1 + 1 "^ 7 + . . . ' 175 ' 262325' _\ 231700* 18. - + 2^ _^ 2 + 1 + 1 + 4 + 1 + 1 + 19 + . . . ' 71 ' 1__ _1_ 103589' 98548* EXERCISE CXLIX. 1 2^1 i 1 1 1 1 1 1 '• -^1+1 + 1+4+1 + 1 + 1 + 4+. ,, ^' "^1+1 + 1+1 + 6 + 1+1+1+1+6+.., i 1 „ . 1 1 1 1 2+2 1 1 ^' "^"^2+2... *• ^ + 2 + 4+2+4+.. ^- ^+8+8+... ,1111_11 ®- + 2 + i_^3_j_i4_2 + 8 1 1_ 1 1 1 1 + 2 + 1 + 3 + 1 + 2 + 8 + .• 7. 1^2-1. 8. 1^6-1. 9. 1/5(2 1^39-9). COMPLETE LIST OF HENRY HOLT & CO.'S EDUCATIONAL PUBLICATIONS. All prices art net except those marked withan asterisk (*), which are retail. All books bound in cloth, unless otherwise indicated. SCIENCE. CATALOGUE PRICE PAGE Allen's Laboratory Physics, PupiVs Edition % 80 2 T^& s,2i.xn^. Teacher'' s Edition 100 2 Arthur, Barnes, and Coulter''s Plant Dissection 120 3 Barker's Physics. A dvanced Course 3 50 4. 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(Harper.) 70 234 Janon's Recueil de Poesies 80 144 Labiche and Delacour's La Cagnotte. Play. Paper 20 158 Les Petits Oiseaux. Play. Paper 20 158 Labiche et Martin's La Poudreaux Yeux. Play. Paper 20 158 Lacombe's Petite Histoire du Peuple Franfais 60 161 La Fontaine's Fables Choisies. (Delbos.) Boards 40 146 Leclerq's Trois Proverbes. Plays. Paper 20 158 Mace's Bouchee de Pain. Vocab 100 152 The same. Vocab. Paper , 52 152 Madame de M.'s La Petite Maman. With Mme. de Gaulle's Le Bracelet. Paper 20 162 Mazeres' Le Collier de Perles. Play. Paper 20 158 de Maistre's Voyage autour de ma Chambre. Paper 28 152 Merimee's Colomba. (Cameron.) 60 237 The same. Boards 36 237 Moli^re's L'Avare. Play. (Joynes.) Boards 20 146 Le Bourgeois Gentilhomme. Play. (Delbos.) Paper 20 146 Le Misanthrope. Play. (Joynes.) Boards 20 147 Musiciens C^l^bres i 00 153 The same. Paper 52 153 Musset's Un Caprice. Play. Paper 20 158 Porchat's Trois Mois sous la Neige 70 153 The same. Paper 32 153 Pressense's Rosa. Vocab. (Pylodet.) 100 154 The same. Paper 52 154 Pylodet's Gouttes de Ros^e 50 144 Le9ons de Litterature Fran^atse Classique 130 144 Theatre Fran9ais Classique. Paper 20 144 La Litterature Fran9aise Contemporaine i 10 144 La M^re I'Oie. Boards 40 163 Racine's Athalie. Play. (Joynes.) Boards 20 147, Esther. Play. (Joynes.) Boards 20 147 Les Plaideurs. Play. (Delbos.) 20 147 Regent's French and German Poems. Boards 20 St. Germain's Pour une :fipingle. Vocab 75 163 The same. Paper 36 163 Sand's La Petite Fadette. (BScher.) i 00 154 The same. Boards 5a 154 Marianne. Paper 30 154 Sandeau's Mademoiselle de la Seiglifere. Play. Boards 20 159 La Maison de Penarvan. Play. Boards 20 159 Scribe et Legouve. La Bataille de Dames. Play. Boards 20 159 Les Doigts de Fee. Play. Boards ... 20 159 Scribe et Melesville's Valerie. Play Paper 20 159 Segur's Les Petites Filles Modeles. Paper 24 163 Siraudin et Thiboust's Les Femmes qui Pleurent. Play. Paper 20 159 Souvestre's Un Philosophe sous les Toits 60 154 The same. Paper 28 154 La Vieille Cousine, with Les Ricochets. Plays. Paper 20 160 La Loterie de Francfort, with Curo's La Jeune Savante. Plays. Boards 20 160 Le Testament de Mme. Patural, with Drohojowska's Demoi- selle de Saint-Cyr. Plays. Boards 20 160 Tvine's Les Origines de la France Coiiiemporaine. (Edgren ) Boards. 50 237 Educational Publications CATALOGUE PRICE i'AGE Thiers'" Expedition de Bonaparte en figypte. (Rdgren.) Boards $ 35 238 ToepfiFer's Bibliothfeque de mon Oncie. (Marcou.) 238 Vacquerie's Jean Baud ry, Piay. Paper 20 160 Verconsin's C'fiiait Gertrude. En Wagon. (.Together.) Plays. Boards. 30 23S Verne's Michel Strogoff. (.Lewis.) 70 155 Walter's Classic French Letters 75 239 GREEK AND LATIN. Brooks's Introduction to Attic Greek i 10 164 Goodell's The Greek in English 60 165 Greek Lessons. Part I. The Greek in English. Part IL The Greek of Xenophon 125 166 Judson's The Latin in English 243 Peck's Gai Suetoni Tranquilli De Vila Caesarum Libri Duo i 20 167 Lati n Pronunciation ... 40 167 Preparatory Latin and Greek Texts 120 168 Latin part separate 80 168 Greek part separate 60 168 Richardson's Six Months' Preparation for Caesar 90 245 Scrivener's Greek Testament 200 168 Williams's Extracts from Various Greek Authors i cxj 169 ITALIAN AND SPANISH. ITALIAN Montague's Manual of Italian Grammar. Half roan 100 171 Nota's La Fiera. Paper 60 173 Ongaro's Rosa deir Alpi. Paper. 60 173 Parlate Italiano ? Boards 40 173 Pellico's Francesca da Rimini. Paper 60 173 SPANISH. Caballero''s La Familia de Alvareda. Paper -5 173 i Habla vd. Espanol ? Boards ^^ 40 172 £ Habla V. Ingles ? Boards . .. 40 172 Lope de Vega's Obras Maestras. Burnished buckram 100 173 Manning's Practical Spanish Grammar. (Revised Ed.) 100 170 Ramsey's Text-book of Modern Spanish 180 172 Saies's Spanish Hive , , i c» 172 UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. \ Lir21-100TO-9,'47 (A5702sl6)476 ! i 800537 ^gj- UNIVERSITY OF CALIFORNIA LIBRARY '!! nHiiiiii KJ^^ 'li '11! ! „:.;! illlil ■ :,!! I III li llili m .iiill'll r lil li i iiiiiiiiililllliliiiii'""'' '"'i I mm