GIFT or Dr. Horace Ivie Digitized by tine Internet Arciiive in 2008 witii funding from IVIicrosoft Corporation http://www.arcliive.org/details/elementsofalgebrOOrayjrich ECLECTIC EDUCATIONAL SERIES. BATS NEW HIGHER ALGEBRA. ELEMENTS OF ALGEBRA, FOR ^ . ^ COLLEGES, SCHOOLS, AND PRIVATE STUDENTS. By JOSEPH RAY, M. D., LATE PROFESSOR OF MATHEMATICS IN WOODWARD COLLEGE. Edited by DEL. KEMPER, A. M., Prof, op Mathematics, Hamden Sidney College. NEW-YORK •:. CINCINNATI •:. CHICAGO AMERICAN BOOK COMPANY FROM THE PBE88 OF VAN ANTWERP, BRAGG, A, CO. Ray's Mathematical Series. ^ S^t^^ 111) r, \\ 6 )r «uc^ Hlv \tZv\ O -^ ARITHMETIC. FVL ^ Ray's New Primary Arithmetic. Ray's New Intellectual Arithmetic. Ray's New Practical Arithmetic. Ray's New Higher Arithmetic. TWO-BOOK SERIES. Ray's New Rlementary Arithmetic. Ray's kW Practical Arithmetic. AI.GEBRA. Ray's New Elementary Algebra. Ray's New Higher Algebra. HIGHER MATHEMATICS. Ray's Plane and Solid Geometry. Ray's Geometry and Trigonometry. Ray's Analytic Geometry. Ray's Elements of Astronomy. Ray's Surveying and Navigation. Ray's Differential and Integral Calculus. eour.ATJQN pei^ Entered according to Act of Congress, in the year 1852, by W, B. Smith, in the Clerk's Office of the District Court of the United States, for the District of Ohio. Entered according to Act of Congress, in the year 1866, bv Sabgent, Wilson & HiNKLE, in the Clerlj's Office of the District Court o'f the United States for the Southern District of Ohio. PREFACE, Algebra is justly regarded one of the most interesting and useful branches of education, and an acquaintance with it is now sought by all who advance beyond the more common elements. To those who would know Mathematics, a knowledge not merely of its elementary principles, but also of its higher parts, is essen-- tial; while no one can lay claim to that discipline of mind which education confers, who is not familiar with the logic of Algebra. It is both a demonstrative and a practical science— a system of truths and reasoning, from which is derived a collection of Kulee that may be used in the solution of an endless variety of problems, not only interesting to the student, but many of which are of the highest possible utility in the arts of life. The object of the present treatise is to present an outline of this science in a brief, clear, and practical form. The aim throughout has been to demonstrate every principle, and to fur- nish the student the means of understanding clearly the rationale of every process he is required to perform. No effort has been made to simplify subjects by omitting that which is difficult, but rather to present them in such a light as to render their acquisi- tion within the reach of all who will take the pains to study. To fix the principles in the mind of the student, and to show their bearing and utility, great attention has been paid to the preparation of practical exercises. These are intended rather to illustrate the principles of the science, than as difficult problems to torture the ingenuity of the learner, or amuse the already skillful Algebraist. An effort has been made throughout the work to observe a natural and strictly logical connection between the different parts, so that the learner may not be required to rely on a prin- 924223 (") iv PREFACE. ciple, or employ a process, with the rationale of which he is not already acquainted. The reference by Articles will always en- able him to trace any subject back to its first principles. The limits of a preface will not permit a statement of the peculiarities of the work, nor is it necessary, as those who are interested to know will examine it for themselves. It is, however, proper to remark, that Quadratic Equations have received more than usual attention. The same may be said of lladicals, of the Binomial Theorem, and of Logarithms, all of which are so useful in other branches of Mathematics. On some subjects it was necessary to be brief, to bring the work within suitable limits. For example, what is here given of the Theory of Equations, is to be regarded merely as an outline of the more practical and interesting parts of the subject, which alone is sufficient for a distinct treatise, as may be seen by reference to the works of Young or Hymers in English, or of DeFourcy or Reynaud in French. Some topics and exercises, deemed both useful and interesting, will be found here, not hitherto presented to the notice of stu- dents. But these, as well as the general manner of treating the subject, are submitted, with deference, to the intelligent educa- tional public, to whom the author is already greatly indebted for the favor with which his previous works have been received. Woodward College, May, 1852. Publishers' Notice. — This work, originally published as Ray's Algebra, Part II., w^as revi.sed, in 1867, by Dr. L. D. Potter. Portions of the work were revised in 1875, by Prof. Del. Kemper. CONTENTS I.— FUNDAMENTAL RULES. ARTICLES. Definitions and Notation 1 — 36 Exercises on the Definitions and Notation 36 Examples to be written in Algebraic Symbols 36 Addition — General Rule. — Subtraction — Rule 37 — 45 Bracket, or Vinculum 46 Observations on Addition and Subtraction 47 — 61 Multiplication — Preliminary principle 52 — 53 Rule of Coefficients — of Exponents 55 — 56 Rule of the Signs— General Rule 60— 61 Multiplication by Detached Coefficients .... 62 Remarks on Algebraic Multiplication .... 63 — 66 Division — Rule of Signs — Coefficients — Exponents .... 67 — 70 Division of a Monomial by a Monomial .... 71 Division of Polynomials by Monomials .... 74 Division of one Polynomial by another .... 75 — 76 Division by Detached Coefficients 77 II.— THEOREMS, FACTORING, Etc. Algebraic Thkorems — Square of the sum of two quantities . 78 Square of the difference of two quantities ... 79 Product of Sum and Difference 80 Transfer of factors in a fraction 81 Any quantity whose exponent is is =1 ... 82 a"»— 6"* is divisible by « — b 83 a2"i_j2r« is divisible by a-f 6 85 a2™+i-f62m+i ig divisible by a+6 86 Factoring — Of Numbers — Of Algebraic quantities .... 87 — 95 Greatest Common Divisor 96 — 108 liKAST Common Multiple 109 — 113 (v) vi CONTENTS. III.— ALGEBEAIC FEACTIONS. ARTICLES. Definitions — Proposition — Lowest terms 114 — 119 To reduce a Fraction to an Entire or Mixed Quantity ... 121 To reduce a Mixed Quantity to the form of a Fraction . . 122 Signs of Fractions 123 To reduce Fractions to a Common Denominator 125 — 126 To reduce a Quantity to a Fraction with a given Denominator 127 — 128 Addition and Subtraction of Fractions 129 — 130 Multiplication and Division of Fractions 131 — 132 Eeduction of Complex Fractions 133 Eesolution of Fractions into Series 134 Miscellaneous Propositions in Fractions 135 — 137 Theorems in Fractions — Miscellaneous exercises 138 — 139 IV.— SIMPLE EQUATIONS. Definitions and Elementary principles 140 — 149 Transposition — Clearing of Fractions 150 — 151 Solution of Simple Equations — Eule 152 — 153 Questions involving Simple Equations 154 Simple Equations with two unknown quantities 155 Elimination by Substitution — Comparison — Addition, etc. . 156 — 158 Problems producing Simple Equations containing two un- known quantities 159 Simple Equations involving three or more unknown quan- tities 160 Problems producing Simple Equations containing three or more unknown quantities 161 Y.— SUPPLEMENT TO SIMPLE EQUATIONS. Generalization — Formation of Eules — Examples .... 162 — 163 Negative Solutions — Discussion of Problems — Couriers . . 164 — 166 Cases of Indetermination and Impossible Problems . . . 167 — 169 A Simple Equation bos but One Eoot 170 CONTENTS. * vii VI.— FORMATION OF POWERS— EXTRACTION OF ROOTS- RADICALS— INEQUALITIES. ARTICLES. Involution or Formation of Powers — Newton's Method . . 172 Square Root of Numbers — Of Fractions — Theorem . . . 173 — 179 Approximate Square Roots 180 Square Root of Monomials — Of Polynomials 182 — 184 Cube Root of Numbers — Approximate Cube Roots .... 185 — 189 Cube Root of Monomials— Of Polynomials 190—191 Fourth Root— Sixth Root— Nth Root, etc 192 Signs of the Roots-Nth Root of Monomials 193—194 Radical Quantities — Definitions — Reduction of Radicals . 195 — 203 Addition and Subtraction of Radicals 204 Multiplication and Division of Radicals 205* To render Rational the Denominator of a Fraction .... 206 Powers and Roots of Radicals — Imaginary Quantities , . . 207 — 210 Theory of Fractional Exj)onents 211 Multiplication and Division in Fractional Exponents . . . 212 — 213 Powers and Roots of Quantities with Fractional Exponents . 214 Simple Equations containing Radicals 216 Inequalities — Propositions I to V — Examples 217 — 2^:4 VII.— QUADRATIC EQUATIONS. Definitions — Pure Quadratic Equations — Problems .... 224 — 229 Affected Quadratic Equations 230 Completing the Square — General Rule — Hindoo Method , . 2:]1 — 232 Problems producing Affected Equations 233 Discussion of General Equation — Problem of Lights . . . 234 — 239 Trinomial Equations — Definitions 240 Binomial Surds — Theorems — Square Root of A±|/B ... 241 Varieties of Trinomials — Form of Fourth Degree .... 242 — 243 Simultaneous Quadratic Equations 244 Pure Equations — Affected Equations 245 — 250 Questions producing Simultaneous Quadratic Equations . . 251 Formulse — General Solutions 262 Special Artifices and Examples 253 viii CONTENTS. VIII.— RATIO— PROPORTION— PROGRESSIONS. ARTICLES. Ratio — Kinds — Antecedent and Consequent 254 — 256 Ratio — Multiplication and Division of 259 Ratio of Equality — Of greater and less Inequality .... 260 Ratio— Compound — Duplicate — Triplicate ....... 261 Ratios — Comparison of 262 Proportion — Definitions 263 — 266 Product of Means equal to Product of Extremes 267 Proportion from two Equal Products 268 Product of the Extremes equal to the Square of the Mean . 269 Proportion by Alternation — By Inversion 270 — 271 Proportion from equality of Antecedents and Consequents . 272 Proportion by Composition — By Division 273 — 274 Proportion by Composition and Division 275 Like Powers or Roots of Proportionals are in Proportion . . 276 Products of Proportionals are in Proportion 277 Continued Proportion — Exercises — Problems 278—279 IIarmonical Proportion 280 Variation — Propositions — Exercises 281 — 290 Arithmetical Progression — Increasing and Decreasing . . 291 Last Term— Rule— Sum of Series— Rule— Table 292—294 To insert m Arithmetical Means between two numbers, Ex. 16 294 Geometrical Progression — Increasing and Decreasing . . 295 Last Term— How to find it— Sum of Series— Rule .... 296—297 Sura of Decreasing Infinite Geometrical Series 299 Table of General Formulae 300 To find a Geometric Mean between two numbers, Ex. 18 . . 300 To insert m Geometrical Means between two numbers, Ex. 19 300 Circulating Decimals — To find the value of 301 IIarmonical Progression — Proposition 302 — 303 Problems in Arithmetical and Geometrical Progression . . 304 IX.— PERMUTATIONS— COMBINATIONS— BINOMIAL THEOREM. Permutations 305 — 307 Combinations 308 — 309 CONTENTS. ix ARTU'LFS. Binomial Theorem when the Exponent is a Positive Integer :^I0 Binomial Theorem applied to Polynomials ....... 311 X.— INDETERMINATE COEFFICIENTS— BINOMIAL THEOREM —GENERAL DEMONSTRATION— SUMMATION AND INTERPOLATION OF SERIES. . Indeterminate Coefficients — Theorem — Evolution . . . 314 — 317 Decomposition of Rational Fractions 318 IkNOMiAL Theorem for any Exponent — Application of . . . 319 — 321 Extraction of Roots by the Binomial Theorem 322 Limit of Error in a Converging Series 323 Differential Method op Series — Orders of Differences . . 324 To find the nth term of a Series — The sum of n terms . . . 326 — 327 Piling of Cannon Balls and Shells 328 — 331 Interpolation of Series 333 — 335 Summation of Infinite Series 336 — 338 Recurring Series 339—343 Reversion of Series 344 — 346 XI.-CONTINUEDFRACTIONS— LOGARITHMS— EXPONENTIAL EQUATIONS— INTEREST AND ANNUITIES. Continued Fractions 347 — 356 Logarithms — Definitions — Characteristic Table 357 — 359 Properties of Logarithms — Multiplication — Division . . . 360 — 361 Formation of Powers — Extraction of Roots 362 — 363 Logarithms of Decimals— Of Base— Of 364—368 Computation of Logarithms — Logarithmic Series .... 370—373 Naperian Logarithms — Computation of 375 Common Logarithms — Computation of by Series .... 377 Single Position 380 Double Position 381 Exponential Equations . . . ' 382 — 383 Interest and Annuities — Simj)le Interest 384 — 385 Compound Interest — Increase of Population 386 — 387 Compound Discount — Formulee — Annuities 388 — 391 X CONTENDS. XIL— GENERAL THEORY OF EQUATIONS. ARTICLES. Definitions — General Form of Equations 393 — 394 An Equation whose Root is a is divisible by x — a 395 An Equation of the nth Degree has n and only n roots . . 396 — 397 Relations of the Roots and Coefficients of an Equation . . 398 What Equations have no Fractional Roots 399 To change the Signs of the Roots of an Equation .... 400 Number of Imaginary Roots of an Equation must be even . 401 Descartes' Rule of the Signs 402 Limits of a Root — A method of finding 403 Transformation of Equations . . 404 — 408 Synthetic Division — Transformation of Equations by. . . 409 — 410 Derived Polynomials — Law of — Transformation by .... 411 — 413 Equal Roots 414 Limits of the Roots of Equations 415 Limit of the greatest Root— Of the Negative Roots .... 416—418 Sturm's Theorem 420 — 427 XIII.— RESOLUTION OF NUMERICAL EQUATIONS. Rational Roots — Rule for finding 429 Horner's Method op Approximation 430 — 434 Approximation by Double Position 436 Newton's Method op Approximation 437 Cardan's Rule for Solving Cubic Equations 438 — 441 Reciprocal or Recurring Equations 442 Binomial Equations 443 — 444 HIGHER ALGEBRA. I. DEFINITIONS. Article 1. Mathematics is the science of the exaot relations of Quantity as to Magnitude or Form. S. duantity, as the subject of mathematical investiga- tions, is any thing capable of being measured, or about which the question How much? may be asked. It may be, 1. Geometric, involving Form ; 2. Number. 3. Number is quantity considered as composed of equal parts of the same kind, each called the unit; and the magni- tude of the quantity is indicated by its ratio to the unit. 4. Numbers are represented by conventional symbols. When the symbols used arc general, as distinguished from the arithmetical symbols, viz., the Arabic numerals, the process of investigation is called Algebraic. Hence, we have the following definitions : 5. Algebra is the method of investigating the relations of numbers by means of general symbols. Remark. — It should be remembered that the word ^^ quantiti/,^^ whenever used ia algebra, is synonymous with '^'•numhery G. The algebraic symbols are of two kinds: 1. Symbols of numbers; 2. Symbols of relation. Numbers are usually represented by letters ; as, a, 6, x, y : sometimes, of course, when known, by the Arabic numerals. T« The symbols of relation, usually called Signs, are the representatives of certain phrases, and are used to express operations with precision and brevity. The principal alge- braic signs are : = -|- — X -^ \^' 11 12 RAY S ALGEBRA, SECOND BOOK. 8. The Sign of Equality, =, is read equal to. It de- notes that the quantities between which it is placed are equal. Thus, £C=5, denotes that the quantity represented by x. equals 5,<. ^ ;.• ' ■^;^,\.!]?hefSij5n .({f A4^i^ +, is read pZ?fs. It denotes thai the quantity to which it is prefixed is to be added. Thus, a-\-h denotes that h is to be added to a. 10. The Sign of Subtraction, — , is read minm. It denotes that the quantity to which it is prefixed is to be subtracted. Thus, a — h denotes that h is to be subtracted from a. 11. The signs -\- and — are called (lie signs. The former is called the lyositioe^ the latter the negative sign ; they are said to be contrary^ or opposite. 12. Every quantity is supposed to have either the posi- tive or negative sign. When a quantity has no sign pre- fixed to it, -f- is understood. Thus, a=-\-a. Quantities having the positive sign are called positive; those having the negative sign, negative. 13. Quantities having the same sign are said to have like signs ; those having different signs, unlike signs. Thus, -\-a and -{-&, or — a and — 6, have like signs ; while +c and — d have unlike sio;ns. o 14. The Sign of Multiplication, X, is read into., or multiplied hij. It denotes that the quantities between ■which it is placed are to be multiplied together. The product of two or more letters is also expressed by a dot or period, or by writing the letters in close succes- sion. The last method is generally to be preferred. Thus, the continued product of the numbers designated by a, 6, and c, is denoted by ayjjy^c., or a.hx^ or abc. DEFINITIONS. 13 15. Factors are quantities that are to be multiplied together. Thus, in the product a6, there are two fac- tors, a and h] in the product 3x5X7, there are three factors, 3, 5, and 7. 16, The Sign of Division, -r-, is read divided hy. It denotes that the quantity preceding it is to be divided by that following it. Division is also expressed by placing the dividend as the numerator, and the divisor as the denominator of a frac- tion. Thus, a-^?>, or -, signifies that a is to be divided by h. IT. The Sign of Inequality, >, denotes that one of the two quantities between which it is placed is greater than the other. The opening of the sign is toward the greater quantity. Thus, a^6, denotes that a is greater than h. It is read a greater than h. Also, c^-f-c — 2ah. 20. The Degree of any term is equal to the number of literal factors which it contains. 16 RAY'S ALGEBRA, SECOND BOOK. Thus, 5a is of the first degree; it contains one literal factor. ax is of the second degree; it contains two literal factors. Sa^b^c='Saaabbc, is of the sixth degree. 30. A polynomial is said to be homogeneous when each of its terms is of the same degree. Thus, a-{-b — 3c is homogeneous; each term being of the first degree. x^ — 7x1/- is homogeneous ; each term being of the third degree. X- — Sx^^ is not homogeneous. 31. An algebraic quantity is said to be arranged ac- cording to the dimensions of any letter it contains, when the exponents of that letter occur in the order of their magnitudes, either increasing or decreasing. Thus, ax~-\-a'^X-~a^x^% is arranged according to the ascending powers of a; and bx"^ — b''''x'^-{-b'^x, is arranged according to the de- scending powers of X. 32. A Parenthesis, ( ), is used to show that all the terms of a polynomial which it incloses are to be consid- ered together, as a single term. Thus, 10 — [a — b) means that a — b is to be subtracted from 10. h{a-\-b — c) means that a-\-b — e is to be multiplied by 5. 5, c, are 3, 5, and 6 respectively. Ans. 492. 13. Find the value of 7/^ a^^W when a=5 and 6=3. Ans. -,^. Verify the following, by giving to each letter any value whatever : 14. a{mA^ii){m — n)=om^ — an^. TO BE EXPRESSED IN ALGEBRAIC SYMBOLS. 1. Five times a, plus the second power of b. 2. x^ plus y divided by Sz. 3. X plus y, divided by 3;^. 4. 3 into X minus n times y, divided by m minus n. 5. a third power minus x third power, divided by a sec- ond power minus x second power. 6. The square root of m minus the square root of n. 7. The square root of m minus n. 1. 5a+Z>2 1. x+. Zz ADDITION. ANSWERS. 3. "+^. Zz 5. 4 3x— Tiy 6. 7. 19 ADDITION. ST. Addition, in Algebra, is the process of finding the simplest expression for the sum of two or more algebraic quantities. There are three cases of algebraic addition : 1st. "When the quantities are similar, and have like signs. 2d. When the quantities are similar, but the signs unlike. 8d. When the quantities are dissimilar, or part similar and part dissimilar. 38. First Case. — Let it be required to find the sura of Sx^y, bx^y^ and ^x'^y. Here, x'^y is taken, in the first term, 3 times; in the second, 5 times; and in the third, 7 times; hence, in all, it is taken 15 times. Since adding the quantities can not change their character, and since each term is positive, their sum is positive. Find the sura of — Scc^y, — 5x^y, and — ^x'^y. Here, X^y is taken, in the first term, — 3 times ; in the second, — 5 times ; and in the third, — 7 times ; hence, in all, it is taken — 15 times. Therefore, to add similar quantities having the same sign. OPERATION. -f Ix'^y -f-] ^x^ OPERATIOX. — ^x'^^y — Ix'^y — 15a:-?/ Rule. — Add the coefficients, and prefix the sum, with the common sign, to the literal pari. 20 RAY'S ALGEBRA, SECOND BOOK. 39. Second Case. — Let it be required to find the sum of -(-9a, — 5«, -i-4a, and — 2a. Here, -\-9a-{~4a is -f 13a; and —5a— 2a is —7a. operation. Now, since the sum of two equal quantities, of -\-9a 'which one is positive and the other negative, is — 5a evidently 0, — 7a will cancel -l-7a in the quan- -|-4a tity -(-13a, and leave -(-6a for the aggregate, or re- — 2a suit of the four quantities. ~ In like manner, to obtain the sum of — 9a, -(-Sa, — 4a, and -(-2a, we find the sum of — 9a and —4a is — 13a, and the sum of -f 5a and -(-2a is -f-7a. operation. Now, -(-7a will cancel — 7a in the quantity — 13a; — 9a which leaves — 6a for the aggregate. Therefore, -f 5a — 4a 4- 2a —6a TO ADD SIMILAR QUANTITIES HAVING DIFFERENT SIGNS, Rule. — 1. Add the positive and negative coefficients sep- arately. 2. Subtract the less sum from the greater ^ and give to the difference the sign of the greater. 3. Prefix this difference to the literal pai-t. 40. Third Case. — Let it be required to find the sum of 5a2— 8^>+c, -\-h—a\ and 56+3a^ In writing the quantities, we place, for con- operation. venience, those which are similar under each Sa^ — 8b-\-C other. — a--\- b The sum in the first column is -|-7a-, and in 3a2-|-56 the second, — 26; there being no term similar to C, it is annexed, with its proper sign. 7a^— 26-l-c 41« From the preceding, we derive the following GENERAL RULE FOR ADDITION OF ALGEBRAIC QUANTITIES. 1. Write the quantities to he added^ placing those that are umiiar under each other^ SUBTRACTION. 21 2. Add the similar quantities by the rules already given. 3. Annex the other quantities with their proper signs. Remark. — In algebraic Addition, Subtraction, and Multiplica- tion, it is best to begin the operation at the left hand. 1. Find the sum of 4aa:-}-36^, hax-\-^hy^ ^ax-\-Qhy, and 20ax-\-hy. Ans. 37a£c-j-18%. 2. Of 10cz—2ax\ lhcz—Zax\ 24cz—9ax\ and Scz lax^. Ans. h2cz — 22ax 3. Of 3rcy— 10/, — a;y-f 5/, 8xy— 6/, and 4xy -1-2/. Ans. 14a:y— 9/. 4. Of a-fi-^c-ft?, a-^t-{-c— J, «-[-&— c-fcZ, a—h^c -f (Z, and _a+6-f-c-|-(Z. Ans. 3a-|-36 + 3c+3(^. 5. Of 3(a;2— /), 8(a;2— /), and —h(x'~y''). Ans. 6(a;^ — y"^). 6. Of 10«2& — 12a36c — 156V-I-10, — 4a26 + 8a36c _1062c*— 4, — 3a26_3a36c-f 20Z>V— 3, and 2a'^6-fl2a36c -f 55V-J-2. Ans. ba'h-\-Mhc-{-h. 7. Of a"— Z;"-|-3x^, 2a'^—ZV—xP, and rt"*-!- 46'*— a;?. Ans. 4a"'-f 2ic^— a;«. SUBTRACTION. 42. Subtraction, in Algebra, is the process of finding the difference between two algebraic quantities. The quantity to be subtracted is called the subtrahend; that from which the subtraction is to be made, the minuend; the quantity left, the difference or remainder. The explanation of the principles on which the opera- tions depend, may be divided into two cases. 1st. Where all the terms are positive. 2d. Where the terms are either partly or wholly negative. 22 RAY'S ALGEBRA, SECOND BOOK. 43. First Case. — Let it be required to subtract 4a from 7a. It is evident that 7 times any quantity, operation. less 4 times that quantity, is equal to 3 times la Minuend, the quantity; therefore, la less 4a is equal 4a Subtrahend, to 3a. 3a Remainder. If it be required to subtract b from a, we operation. can only indicate the operation, by placing a Minuend, the sign minus before the quantity to be sub- b Subtrahend, tracted. c^_6 Remainder. 44. Second Case. — Let it be required to subtract h — c from a. If we subtract b from a, the result, a — 6, operation. is obviously too little, for the quantity b a Minuend, ought to be diminished by C before it is taken b — e Subtrahend. from a. We have, in fact, subtracted a quan- q^ b-\-c Remainder. tity too great by c, and to obtain a true re- sult, the difference, a — b, must be increased by C; this gives, for the true remainder, a — b-\-C. To illustrate the above example by figures, let a=9, 6=5, and C=S ; and let it be required to subtract 5 — 3 from 9. The operation and illustration may be compared, thus: From a Take b—c From 9 . . Take 5— 3 . . . . r=9 =2 Rem. a—b^c Rem. 9-5-1-3 . . . . =7 In the examples already explained, the same result would have been obtained by changing the signs of the quantity to be sub- tracted, and then adding it. 45. From the preceding, we derive the following RULE FOR SUBTRACTION OF ALGEBRAIC QUANTITIES. 1. Write the quantities^ placing similar terms under each other. SUBTRACTION. 23 2. Conceive the signs of all the terms of the subtrahend to he changed, from, -j- to — , or from — to -\-^ and then pro- ceed hy the rule for algebraic addition. Remark. — 1. Beginners should solve a few examples hy actually changing the signs of the subtrahend. 2. Proof. — Add the remainder and the subtrahend, as in arithmetic. (1) (1) From Sa^b-Scx- .H The ^ame, with f Sa^b-Bcx- z^ Take 3a2Z>4-4ca;— Sz^ V the signs of the ^ —^a''b—4tcx-\-Zz' ■ I subtrahend j — — Kern, ba'b—l cx-^-^z" ) changed. VRem. ba'b~lcx-[-2z'' (2) (3) From 5a' — ^mz-\-hy*' From ax^ — 3c'/ — z^ Take — 2a^-|-3mg-i-6/ Take fca;^— 3r/-fy» Rem, la^—6mz — y ' Kern, (a— 6)x"^ — f—z^ 4. From 4a— 26+3c take 3a+4&— c. Ans. a — 65+4c. 5. From 9a-,^— 4y+9 take Trc^-f 5?/— 14. Ads. 2x^— 9y+23. 6. From 23a;/— 7y+lla;' take 11a;/— 5^— Qx^. Ans. 12ic/— 2y-f 20x^ 7. From 12x+18 take 12a:— 18+^^. Ans. 36—^. 8. From x'—f take —4:—f-^4x\ Ans. 4— 3x^ 9. From 4ax^-\-hx-\-c take Sx^~2x-{-b. Ans. (4a— 3)a;3-f (2H-6)a:+c— 5. 10. From — IT^c^+Qax^— ^a'ic+lSa' take — 19x'+9ax^ — 9a2a;+l7a^ Ans. 2x^-\-2a'x—2a\ 11. From a^-\-Sx'-JrSx-\-l take x'^—Sx'^Sx—l. Ans. 6a;2-f 2. 12. From 9a^x'—lS+20ab'x—Wcx' take Sb^'cx'' -\-9a"'x'—6-\-3ab'x. Ans. 11ab^x—1b"'cx'—^. 13. From 4a'^+2x^— »« take a*"— i"-l-3xP and 2a"» ^2b"—xP. Ans. a"^-f 46"— x«. 24 RAY S ALGEBRA, SECOND BOOK. The Bracket, or Vinculum. — As the Bracket, or Vin- culum, is frequently employed in relation to Addition and Subtraction, it is important that the rules, which govern its use, should be well understood. 46. 1st. Where the sign phis precedes a parenthesis, or vinculum, it may be omitted without affecting the expression. This is self-evident, as is also the converse, viz. : Any number of terms may be inclosed within a parenthesis preceded by the sign plus, without affecting the value of the expression. Thus. a+(6— c)=a+6— c; 6+(5— 3)=6-|-5— 3, and a-f6— c -fd=a-f(6— c+d); 5+4—3+2=5+ (4—3+2). 2d. Where the sign minus precedes a vinculum, it may be omitted if the signs of all the terms within it be changed. For the minus indicates subtraction, which is effected by changing the signs of all the terms of the quantity to be subtracted. Thus, a—{b~c)—a — 6+c. a—{x—y^z)=a—x-^y-z. Sometimes several brackets, or vinculums, are employed in the same expression, all of which may be removed. Thus, a— {a+6— [a+6— c— (a— 6+c)]|, =a— {a+6— [a+6— c— a+6— c]}, =:a — \a-\-h — a — 6+c+a— 6+c}, =a — a — 6+a-j-6— c— a+6— c=6 — 2c. 3d. Any quantity may be inclosed in a parenthesis preceded by the sign minus, provided the signs of all the inclosed terms be changed. This is evident from the preceding principle. Thus, a— 6+c=a— (6— c)=c— (6— a). OBSERVATIONS. 25 This principle often enables us to express the same quan' tity under several different forms. Thus, a — 6-)-c-f c/=ia— [6— c — d}, Simplify, as much as possible, the following expressions; 1. (1— 2.T+3a:0 + (3+2a:— a:^). Ans. 4+2x1 2. ^a-h-c)-\-{h^c—d)-\-{d-e-^f)-^(e—f-^g). Ans a — g. 3. 3(:r^+/)- { {x'-\-2xy-^if)—(2xij—x^~f) } . Ans. x^-\-y^^ 4. a — (x — «) — [x — (« — x)]. Ans. 3a — 3a;. 5. 1— {1— [1— (1— a:)]}. Ans. X. OBSERVATIONS ON ADDITION AND SUBTRACTION. 47. All quantities are to be regarded as positive, unless, for Bome special reason, they are otherwise designated. Negative quan- tities are always, in some particular respect, the opposite of positive quantities. Thus : If a merchant's ffains are positive, his losses are negative; if lati- tude north of the equator is -|-j that south is — •, if distance to the right of a certain line is -f , that to the left is — ; if time after a certain hour is -f? time be/ore that hour is — ; if motion in one direction be -f » naotion in an opposite direction is — ; and so on. 48. This relation of the signs gives rise to some important particulars. 1st. TVie addition, to any quantity, of a negative number, produces a LESS result than adding zero. Thus, 10 10 10 10 10 10 10 3 2 1 -1 2 -3 13 12 11 10 9 8 7 It will also be seen, from this illustration, that adding a negative number produces the same result as subtracting an equal positive number. 2d Bk. 3* 26" RAY'S ALGEBRA, SECOND BOOK. 2(1 The subtraction of a negative quantiti/ produces a greater result than subtracting zero. Thus, 10 ]0 10 10 10 10 10 3 2 1 —1 —2 —3 7 ~8 "9 10 11 12 13 Here, subtracting a negative number produces the same result as adding an equal positive number. 49* When two negative quantities are considered algebraically, that is called the least which contains the greatest number of units; thus, — 3 is said to be less than — 2, But, that which contains the greatest number of units is said to be nuvierically the greatest ; thus, — 3 is numerically greater than — 2. SO* The sum of two positive quantities is always greater than either of them. Thus, -|-5-|-3=-f8. The sum of two negative quantities, algebraically considered, is less than either of them. Thus, — 5 — 3= — 8. The sum of a positive and negative quantity is always less than the positive quantity. Thus, -f 5 — 3=-[-2. 51. The difference of two positive quantities, as in arithmetic, is always less than the greater quantity. Thus, 2a from ba leaves 3a, or 5a— (+2a)=+3a. The difference of two negative quantities is always greater, alge- braically considered, than the minuend. Thus, — 2a from — 5a leaves —3a, or — 5a— (— 2a)=— 3a. The difference between a positive and a negative quantity, found by subtracting the latter from the former, is always greater than either of them. Thus, 2a— (— a)=:3a 1. The latitude of A is 10° N. (+) ; the latitude of B is 5° S. ( — ) ; what their difference of latitude? Ans. 15°. 2. At T A. M., a thermometer stood at —9° ; at 2 P. M., at 4-15° V what was the change of temperature ? Ans. 24°. MULTIPLICATION. 27 MULTIPLICATION. 52. Multiplication, in Algebra, is the process of tak- ing one algebraic quantity as many times as there are units in another. The quantity to be multiplied is called the multiplicand; the quantity by which we multiply, the multiplier; and the result, the product. The multiplicand and multiplier are called factors. 53. In explanation of the subject of algebraic multipli- cation, we begin with the following Preliminary Principle.— T'/ie product of two factors is the same, whichever he made the midtiplier. To prove this, suppose we have a sash containing a vertical and b horizontal rows. Since there are a vertical rows and b panes in each row, the whole number of panes will be represented by b taken a times; that is, by ab, or by a taken b times ; that is, by ba. Hence, ab is equal to ba. In a similar manner, it may be shown that The product of three, or of any number of factors, is the same, in whatever order they are taken. Thus, a'Xh'Xc=ahc, cab, hac, or cha, and 2x3x4=4 X2x 3=3x2x4=4x3x2; the product in each case being 24. Also, acy^6=^6ac, or 6ca; and so on. It also follows from this principle, that When either of the factors of a product is multiplied, the product itself is multiplied. Thus, 2x3, multiplied by 5, may be written 5x2x3, or 5x3x2 ; that is, 10x3, or 15x2, either of which is equal to 30. Remark. — The distinction between the multiplication of num- bers and of /actors should be carefully noticed. Thus, 32X2=^4, but 3X2 multiplied by 2 equals 6x2, or 3x4. 28 RAY'S ALGEBRA, SECOND BOOK. 54. In multiplication there are four things to be con- sidered in relation to each term, viz. : the coefficient; the literal part; the exponent; and the sign. 55. Of the Coefficient and Literal Part. — 1. Let it be required to find the product of 2ac by 8^. To indicate the multiplication, we may write operation. the product thus, 2acX36. But, by Art. 53, this 2ac is the same as 2x3X«^c, and 2x3=6; therefore, 36 the product is 6a6c. Hence, ~ ~ 6a0c product. Rule of the Coefficients. — Multiply together the coefficients of the factors for the coefficient of the product. Rule for the Literal Part. — Annex to the coefficient all the letters of the factors in alphabetical order. 2. 3acX56= l^ahc. 3. 2amy^cn= 2acmn. 4. 5aX4o'ic= 20aax. 5. 7c3/x3y2;=: 21cy7/z. 56. Of the Exponent. — To determine the rule of the exponents, 1. Let it be required to find the product of 2a'^ by 3a'. Since 2a^=z2aa, and 3a^=3aaa^ the product operation. will be 2aaX3««a, or Gaaaaa, which, for the 2a^=2aa sake of brevity, is written 6a'*. Hence, we Sa^=Saaa have the following 6a^=6aaaaa Rule of the Exponents. — Add the exponents of any letter in the factors for its exponent in the product. 2. ahXci= a""}) 3. x^yXxy= ir»/ 4. rj?x^zy^ax^=. a*j?z^ 5. a'"Xa"= «'"■'"'*. 7. x"'+^XaJ"~^= a;'«+\ 5'7. From the two preceding articles, we derive the fol- lowing MULTIPLICATION. 29 GENERAL RULE FOR MULTIPLYING ONE POSITIVE MONOMIAL BY ANOTHER. 1. Multiply the coefficients for the coefficient of the product. 2. Annex all the letters found in both factors. 3. When the same letter occurs in both, add its exponents. 1. Multiply be hj z Ans. bcz. 2. Multiply Sax hj by Ans. Sabxy. 3. Multiply 4am by Sbn Ans. 12abmn. 4. Multiply Mx by ^ax^y Ans. 85aVy. 5. Multiply 3tt'"ic" by 9a"a;"*. . . Ans. 27a"'+"a;"'+'*. 58. — 1. Required to find the product of a-\-b by m. Here, the sum of the units in a and b is to be operation. taken m times. The units in a taken in times a-f 6 :=.m,a^ and the units in h taken m times =^mb\ m hence, a-\-b taken m times r^ma-^mb. Hence, , when the signs are positive, we have the fol- lowing Rule for Multiplying a Polynomial by a Monomial. — Multiply each term of the multiplicand by the multiplier. 2. Multiply x-{-y by w Ans. wx-f-^y. 3. ax^-^-cz by 3ar Ans. Sa'^cx^-\Sac^z. 4. 2a2+362 by ^ab Ans. lOa^fe+lSaR 5. mx-\-ny-\-vz by m^n. Ans. mhix-\-in^f^^y-\-'^^^^'^z. 59. — 1. Required to find the product of a+Z> by m-\-n. Here, a-(-6 is to be taken as many times as there are units in W-fn, which is evidently as many times as there are units in m, plus as many times as there are units in n. Thus, a+6 ma-\-mb= multiplicand taken m times. na-\-nbz= multiplicand taken n times. »ia-f 77i6-j-wa4-n6= multiplicand taken {m-{-n) times. 30 RAY'S ALGEBRA, SECOND BOOK. Hence, when the signs are positive, we have the following Rule for Multiplying one Polynomial by Another. — Multiply each term of the multiplicand hy each term of the multiplier^ and add the products. 2. Multiply x-\-y by a+c. . . Ans. ax-\-ay-\-cx-^cy. 8. 2x-\-Sz by Sx-{-2z. . . . Ans. 6x'-\-lSxz+6z\ 4. 2aH-c by a+2c Ans. 2a'-^bac-\-2cK 5. x^-\-xy-{-y^ by x-j-y. . . Ans. x^-^2x'^y-\-2xy^-\-y^. 6. a^+2ai+&' by a+b. . Ans. a^+Sa'b+Sa¥+h\ 60« Of the Signs. — In the preceding article it was as- sumed that the product of two positive quantities is posi- tive. The general rule for this, and the other cases which may arise in algebraic multiplication, may be deduced, as follows : 1st. Let it be required to find the product of +^ by a. The quantity 6, taken once, is -{-b; taken twice, is -|-26; taken 3 times, is -{-Sb; and hence, taken a times, it is -\-ab. Hence, the product of two positive quantities is positive; or, more briefly expressed, plus multiplied by plus gives plus. 2d. Let it be required to find the product of — b by a. The quantity — 6, taken once, is — 6; taken twice, is — 26; taken 3 times, is — 36; and hence, taken a times, is — ab. Hence, a negative quantity, multiplied by a positive quantity, gives a negative product; or, more briefly, minus multiplied by plus gives 3d. Let it be required to multiply b by — a. Since by^-^a implies that b is to be added a times to 0, 6X — ct must indicate (Art. 47) that 6 is to be subtracted a times from 0. Sub- tracted once, it is — 6; subtracted twice, — 26; and so on. Hence, subtracted a times, it is — ab. Therefore, a positive multiplied by a negative quantity, gives a negative product; or, plus multiplied by minus gives minus. MULTIPLICATION. 31 4th. Let it be required to multiply — h by — a. Reasoning as above, — 6 subtracted a times from 0, gives -\-ab. Hence, the product of two negative quantities is positive; or, more briefly, minus multiplied by minus gives plus. Note. — The following proof of the 3d and 4th cases is generally regarded as more satisfactory than the preceding. Let it be required to find the product of c — d by a — h. Here it is required to take e — d as many times as there are units in a — b. This will be done by taking C — d as many times as there are units in a, and then subtracting, from this product, G — d taken as many times as there are units in 6. Thus, c — d a—b ac — ad=c—d taken a times, be — bdz=c — d taken b times. ac—ad—bc-\-bd. By subtraction, =c — d taken a — 6 times. The final result, in the terms, — 6c and -\-bd, is what it would have been if we had added the partial products, assuming that -\-C multiplied by — b gives — 6c, and that — d multiplied by — 6 gives -\-bd. As we know the result to be correct, we infer that the assumption would be correct, viz.: that plus by minus gives minus^ and minus by minus gives plus. From the above, we derive the following General Rule for the Signs. — The product of like signs gives plus, and of unlike signs, minus. GENERAL RULE FOR THE MULTIPLICATION OF ALGEBRAIC QUANTITIES. 61. — 1. Multiply every term of the multiplicand hy each term of the multiplier, observing the rules for the coefficients, the exponents, and the signs. 2. Add the several partial products together. 32 RAY'S ALGEBRA, SECOND BOOK. NUMERICAL EXAMPLES TO VERIFY THE RULE OF THE SIGNS. 1. Multiply V-4 by 5. Ans. 35-20=.15=:3x5. 2. 84-3 by 6-4. Ans. 48-14— 12=22==llx2. GENERAL EXAMPLES. 1. Multiply 4a2_3ac-f 2 by ^ax. Ans. 20a'x-lbah'x-\-10ax. 2. ba—2ah-\~10 by —9ah. Ans. ~4ba'h-{-lSa'h'—90ah. 3. 2x-{-Sz by 2x—Sz. Ans. ^x'-9z\ 4. 4a2_6a-f9 by 2a+3. Ans. Sa'+2l. 5 a — h-^c — d by a-\-h — c — d. Ans. a'—h^—c^^d'—2ad'{-2bc. 6. a^-\-f-\-z' by x'+f. Ans. a:^4-a-.y+.TV4-a:y+/+/;22^ 7. a'-\-Sa'h-\-Sah''-\-h' by a3_3a2^^3a^,2_2,3^ Ans. a^— 3a*i2+3a2i*_Z,6. 8 12x'—Sx'i/-\-Uxf—10f by 3a:H-2y. Ans. 36a:*+29a;y— 20/. 9. a^+a.'K+ic^ by a/^ — ax-{-x'^. Ans. a*-|-a^a;^+a;*. 10. a'-i-2ah-{'2h^ by a2_2a^;+262. Ans. a*+4?>*. 11. l-\-x-^x'^-{-a^'{-x* by 1 — x. Ans. 1 — a;^. 12. 27a^-\-9x'7/-hSxf-]-f by 3a;— 3/. Ans. 81a^*— /. 13. a'-\-2a'b-{-2ab''^h' by a"— 2a2& + 2a7>2_Z,3^ Ans. a^ — 6*. 14. x'—a^-\-x^—x+l by ir'+a^— 1. Ans. rr" — x*-{-a^ — x'^-}-2x — 1. 15. l-i-x-{-x*-\-x^ by 1 — x-\-x^ — ar*. Ans. 1 — a^. 16. Multiply together x — 3, x-\-4, x — 5, and x-\-6. Ans. a;*+2a;»— 41x2— 42a:+360. 17. a+t, a—h, a^-\-ah+h\ and a'—ah+h-. Ans. tt'' — 6^ MULTIPLICATION. 33 62. Multiplication by Detached Coefficients. — In tlie multiplicatiou of polyooiiiials, it is evident that the coeffi- cients of the product depend on the coefficients of the fac- tors, and not upon the literal parts of the terms. Hence, by detaching the coefficients of the factors from, the literal parts, and multiplying them together, we shall obtain the coefficients of the product. If to these coeffi- cients, the proper letters are then annexed, the whole prod- uct will be obtained. This method is applicable where the powers of the same letter increase or decrease regularly. 1. Multiply a'— 2a6-(-Z>2 by a-\-h. orERATioN. 1-2-1-1 After finding the coeflBcients, it is obvious ji j that a^ will be the first term, and b^ the last -, f, , ^ term; hence, the entire product is a^ — a^b , , ^ . ^ — ab^-\-b^. 1_1_1_^1 2. Multiply a'—Sa'h-j-h^ by a'—h\ In this example, supposing the powers operation of a to decrease regularly toward the 1 — 3-|-0-f 1 left, it is obvious that there is a terra 1-j-O — 1 wanting in each factor. These must be -[ 3 i oij supplied by 0. The entire product is l-f3 1 a^—Sa^ b -a^b^-] Aa-b^—b->. i_3_i 1 4_o_i 3. Multiply m^ -\-mhi-\-'mn'^ -\-7i^ by m — n. Ans. m* — n^. 4. Multiply l-]-2z-\-Sz'^4z'-\-b^ by 1—z. Ans. l-i^z-{-z^-\-^-^z'—^J. By this method, let the general examples. Art. 61, from 7 to 14 in- clusive, be solved. REiMARKS ON ALGEBRAIC MULTIPLICATION. OS« The degree of the product of any two monomials is equal to the sum of the degrees of the multiplicand and multiplier. Thus, 2a-b, which is of the 3d degree, multiplied by oab^ of the 4th degree, gives ^W'b^^ which is of the 7th degree. 34 RAY'S ALGEBRA, SECOND BOOK. This is also true of two polynomials; as an illustration of which, see Example 7, Art. 61. 04:« In the multiplication of two polynomials, when the partial products do not contain similar terms, if there be m terms in the multiplicand, and n terms in the multiplier, the number of terms in the product will be nvyn. Thus, in Example 6, Art. 61, there are 3 terms in the multiplicand, 2 in the multiplier, and 3x2=6 ia the product. 6S« If the partial products contain similar terms, the number of terms in the reduced product will evidently be less than my^^n; see Examples 7 to 18 inclusive. Art. 61. 06* When the multiplication of two polynomials, indicated by a parenthesis, as (m-\-n)[p — g), is actually performed, the expres- sion is said to be expanded, or developed. DIVISION OT. Division, in Algebra, is the process of finding how many times one algebraic quantity is contained in another. Or, having the product of two factors, and one of them given, Division teaches the method of finding the other. The quantity by which we divide is called the divisor; the quantity to be divided, the dividend; the result of the operation, the quotient. 68. In division, as in multiplication, there are four things to be considered, viz. : the sign; the coefficient; the exponent; and the literal part. 60. To ascertain the rule of the siprns. Since, +«X+6=+a6 ^ -aX+6=:-a6 I therefore, -f ax— 6=— a6 I — ax— 6=+a6 J )4-a6H-H-6^+a — a6-^+6=— a + a6-= — br=—a —ab-i — 6=4-a DIVISION. 35 From the foregoing illustration, we derive the following Rule of the Signs. — Like signs in the divisor and divi- dend give plus in the quotient; unlike signs give minus. TO. The rule of the coefficients, the rule of the exponents^ and the i-ide of the literal part, may all be derived from the solution of a single example. Required to find how often 2a^ is contained in 6a^h, 2a^ 2 Since division is the reverse of multiplication, the quotient mul- tiplied by the divisor, must produce the dividend; hence, to obtain this quotient, it is obvious, 1st. That the coefficient of the quotient must be such a number, that when multiplied by 2 the product shall be 6 ; therefore, to obtain it, we divide 6 by 2. Hence, the Rule of the Coefficients. — Divide the coefficient of the dividend by the coefficient of the divisor. 2d. The exponent of a in the quotient must be such a number, that when 2, the exponent of a in the divisor, is added to it, the sum shall be 5; that is, it must be 3, or 5 — 2. Hence, the Rule of the Exponents. — Suhtract the exponent of any letter in the divisor from the exponent of the same letter in the dividend for its exponent in the quotient. 3d. The letter b, which is a factor of the dividend, but not of the divisor, must be in the quotient. Hence, the Rule of the Literal Part. — Write, in the quotient, every letter found in the dividend, and not in the divisor. Tl. The preceding rules, taken together, give the fol- lowing 36 RAYS ALGEBRA, SECOND BOOK. GENERAL RULE FOR DIVIDING ONE MONOMIAL BY ANOTHER. 1. Prefix the proper sign, on the ptinciple that like signs give plus, and unlike signs give minus. 2. Divide the coefficient of the dividend hy that of the divisor. 3. Subtract the exponent of the divisor from that of the dividend, when the same letter or letters occur in both. 4. Annex any letter found in the dividend but not in the divisor. 1. Divide 4a^ by 2a'' and by —2a' 2. SOa*b' by ^a'b. . . . 3. — 28xy2* by —Ixyh. 4. —Sba^b^c by bab\ . . 5. S2xyz by — Sxy. . . 6. 42c^m^7i by — Scmn. 7. £c'"+" and ic"»-" each by x 8. 1;'"+" by v'^-^p. . . . Ans. 2a' and — 2a^ . . Ans. 6a^b' . Ans. 4xYz\ . Ans. — 7abc. . . Ans. — 4:z. Ans. — 14c^m. Ans. x"" and x'"~'^\ . . . Ans. v""-^. Note. — In the following examples, the quantities included within tLe parenthesis are to be considered together, as a single quantity 9. Divide (a-f 6)' by (a-\-by. . . -. . Ans. (a+6). 10. (m — ny by (m — ny Ans. (m — w)'. 11. S(a—byx'y by 2(a—b)xy. , . Ans. 4(a—byx. 12. (a-{-bx'y+^ by (ia+bx')P-\ . . Ans. (^a+bx'y. •72. It is evident that one monomial can not be divided by another in the following cases : 1st. When the coefficient of the dividend is not exactly divisible by the coefficient of the divisor. 2d. When the same literal factor has a greater exponent in the divisor than in the dividend. 3d. When the divisor contains one or more literal fac tors not -^ound in the dividend. DIVISION. 37 In each of these cases the division is to be indicated by a fraction. See Art. 119. 73. It has been shown, Art. 53, that any product is multiplied by multiplying either of its factors ; hence, con- versely, any dividend will he divided hy dividing either of its factors. Thus, 6x9^3r=2x9; or, 6x3=18. T4. Division of Polynomials by Monomials. — In mul- tiplying a polynomial by a monomial, we multiply each term of the multiplicand by the multiplier. Hence, conversely, we have the following RULE FOR DIVIDING A POLYNOMIAL BY A MONOMIAL. Divide each term of the dividend hy the divisor^ accord- ing to the rule for the division of monomials. Note. — Place the divisor on the left, as in arithmetic. 1. Divide a}-\-ah by a Ans. a -{-h. 2. '6xy-\-2x^y by — xy Ans. — 3 — 2x. 8. 10ah—lbz'—2bz by 5^. . . . Ans. 2a'—Sz—6. 4. Sah-]-12ahx—9a''h by —Sah. Ans. —l—4x-^Sa. 5. 5x^y — 40aV3/^-|-25a*icy by — ^xy. Ans. — x^y'^-\-Sa'^xy — 5a*. 6. 4:ahc—2iah^—S2ahd by — 4a6. Ans. —c-{-6b-i-Sd. 7. a'"Z>3+a'"+iZ>2+a"-26 by ah. Ans. a'«-^6'+a'"6+a"-3. 8. Sa(^x-\-y)-]-c\x-]-yy by x-\-y. Ans. Sa+cXx-^y). 9. (h-\-c)(h—c^'—{h—c)(h+cy by (6-|-c)(&— c). Ans. (fe— c)— (6-f c)=— 2c. 10. h^c(m-{-n)—hc\m-{-n) by 5c(m-f ?i) Ans. b—c. 38 RAY'S ALGEBRA, SECOl^D BOOK. DIVISION OF ONE POLYNOMIAL BY ANOTHER. •75. To deduce a rule for the division of polynomials, we shall first form a product, and then reverse the operation. Multiplication, or formation of a product. a3_5cx26 a^—ba^b -o-'b^^^a^b^ a^— 3a46_i 1 aW-\-ba^b'^ a^^2ab~b'^ Quotient. Division, or decomposition of a product a5_3a46-lla362-j-5a263 a3_5a26 i.stii+2a<6^ 1 Ia362_^5a2^)3 2d Remainder, — tt^ft^-f-Sa^ft^ — a362^5a2^>3 '.'A Remainder, The dividend, or product, and the divisor, being given, (Art. 67), it is now required to find the quotient, or the other factor. This dividend has been formed by multiplying the divisor by the several terms of the quotient, and adding the partial products to- gether. These several unknown terms, constituting the quotient, we are now to find. Arranging the dividend and divisor according to the decreasing powers of the letter a, it is plain that the division of a'', the first term of the dividend, by a^, the first term of the divisor, will give a^, the first term of the quotient. If we subtract from the dividend a''— 5a^6, which is the product of the divisor a^—5a^b by a^, the first term of the quotient, the remainder -\-2a^b—lla^b^-\-5a-b^, will be the product of the divisor by the other terms of the quotient. The first term -fSa^^ of the Ist remainder, is the product of the 1st term a^ of the divisor by the 1st of the remaining unknown terms of the quotient; therefore, we shall obtain the 2d term of the required quotient, by dividing -f2a'*6 by a^; this gives -f2a6. Multiplying the divisor by -\-2ab, and subtracting the product, we have a 2d remainder, which is the product of the divisor by the remaining term or terms of the quotient; hence, the division of the 1st term —a^b^ of this 2d remainder, by the 1st term a^ of the divisor, must give the 3d term of the quotient, which is found to be —62. The remainder zero, shows that the quotient a2-f2a6— 62 is exact, since the subtraction of the three partial products has exhausted the dividend. DIVISION. 39 It is immaterial whether the divisor be placed on the right or left of the dividend ; by placing it on the right, it is more easily multi- plied by the respective terms of the quotient. TO. From the above, we derive the following RULE FOR THE DIVISION OF ONE POLYNOMIAL BY ANOTHER. 1. Arrange the dividend and divisor with reference to a certain letter. 2. Divide the first term of the dividend hy the first term of the divisor^ for the first term of the quotient. Multiply the divisor hy this term, and subtract the product from the dividend. 3. Divide the first term of the remainder hy the first term of the divisor, for the second term of the quotient. Multiply the divisor hy this term, and subtract the product from the last remainder. 4. Proceed in the same mariner, and if the final remainder is 0, the division is said to he exact. 1. Divide 15x'+16x^— 15/ by ^x—Sy. OPERATION. 15x^-]-16xy-lby^\ 5x-Sy Ibx^— 9xy ^x\-by, Quotient. 4-25a:.v— 15?/2 2. Divide m'^ — n'^ by m-^-n. OPERATION. rri'—n^ |m-fn ^2_j_77in m— ^ Quotient. — mn — 7i2 — Tnn— ^2 3. Divide a;'-fy by x-\-y. OPERATION. x^-^y^ \ x^y X'^ArX^y x^—xy\y', Quot. — a;22/+ 2/3 —x^y—xy"^ J^xy^^y^ xy'^^y^ 40 RAYS ALGEBRA, SECOND BOOK. 4. Divide 1x'y-\-bxy''-\-2x^-^y^ by Zxy-\-x'^y\ Arranging the divisor and dividend with reference to a;, we have the following: OPERATION. 2x^^lx^y^bxy'^^y^ \x'^j^^xy^y'^ 2x'^-\-Qx'^y^2xy'^ '•2x-\y, Quotient. 5. Divide x^-\-x^ — ^x^-\hx^ by x — x^. Division performed, by arranging both quantities according to the ascending powers of x. Division performed, by arranging both quantities according to the descendini/ powers of x. x^-\-x^—7x^-\-dx-'\x--x^ Bx'^—7x*^x^-\-x2\ —x^^x x'^—x^ x-y2x^—dx^, hx'^-hx^ —hx^y2x^-^x, 2a;3— 7a:* Quotient. — 2xt-f x?- Quotient. 2a:3_2a:4 -2a:4+2a:3 — 5a:4+5a:5 — a:3-f a;2 —5x*-^dx-> -x^^x^ The two quotients above are the same, but differently arranged. 6. Divide ^x^-\-bxy — 4j/^ by Zx-\-^y. Ans. 2x — y. 7. x^— 40a:— 63 by a:— 7. Ans. a:'^-|-7a;-|-9. 8. 3/i^H-16/i*A:-33/tVi:^-f-14/iV^3 ^y A2_|_7/j,. Ans. Zh^—bK'h-^2hlc\ 9. a^— 243 by a— 3. A. «*+3a»-f-9a2-f 27a-|-81. 10. a:«— 2a'a;''-j-«« by x'—2ax-\-a} Ans. a;*+2aa:^-f 3«V4-2a'a:+a*. 11. 1— 6a:5-f-5x« by l_2a:+a:^ Ans. l-j-2.'r+3a:H-4a;3-h5^*- 12. P^-\-p^l-\-'^P^' — 2q'^-\-7qr — 3/'^ by p — ^-f 3r. Ans. p-\-2q — r. 13. 4a:54-4a:— a:^ by 3.^+23:^+2. Ans. 2x'—Sx'-^2x. 14. ic*— a« by a^-^2ax''-j-2a'x-{-a\ Ans. a^—2ax''-j-2a'x—a\ DIVISION. 41 15. Divide m^-(-2mp — n"^ — 2/«g-f-i^^ — 2^ ^y '^^ — n-\-p — q. Ans. m-\-n-\-j)-\-q. 16. a^^b'-\~c^—Sabc by a+6+c. Ans. a"^.-[-6"^-|-c2 — ab — ac — be. 17. a;"'+"-f-.x"yM-*'^"'3/"''-fy"''"" by a;"-f^"*. A. a;"'-]-^". 18. aa;^ — (a'^-\-b)x^-\-b^ by ao; — Z>. Ans. x'^ — ax — b. 19. a2m_3^»«^«_|.2c2« by a"*— c". Ans. a"* — 2c». 20. x*-[-^ * — a:^ — a:"^ by x — x'^. Ans. x^ — x'^. 21. a«4-a«6--f a*6*-fa2^>6_|_58 ^y a'^a'b^a'b'-\^ah'-}~b\ Ans. a*— a^t-fa'-^/;-^— aZ/3-fZ,*. 22. a^-f (a— l)x-+(a— l>3 + (rt — l)a;* — .T.s by a — a;. Ans. a-\-x-{-x'^-\-x^-\-x*. 23. 1— 9a;8— 8.^» by 1+2.^+.^'^ Ans. 1 — 2x-\- Sx'—4x^-{- ^x*—6x'-{- 1x^—Sx\ 24. l-|-2x by 1 — Sx to 5 terms in the quotient. Ans. l+5a;+15a;^+45a;3-|-135a;*-|- etc. 77. Division by Detached Coefficients. — Division may sometimes be conveniently performed by detaching the coefficients, as explained in Art. 62. Thus, 1. Let it be required to divide x'^-{-2xi/-^i/^ by x-\-i/. l-fS-fljl+l Hence, the coefficients of the quotient 1-fl 1+1 are 1 and 1. A\so,x^-ir'X=x,a.ndy^-^y—y; -|-l-fl therefore, the quotient is Ix^ly, or x-\^y. 2. Divide 12a'—2Qa'b—Sa'b'-{-10ab'—Sb' by Sa^— 2a& 12—26—8+10—813—2+1 Hence, the coefficients of the 12— 8+4 4—6—8 quotient are 4—6—8. Also, —18-12+10 a^--a2=a2^ and b*^b^=b^) —18+12— 6 therefore, the quotient is ia^ —24+16—8 _6a6— 862. —26+16-8 2d Bk. 42 RAY'S ALGEBRA, SECOND BOOK. 1. Divide a^+a:^ by a+a ;. 1+0+0+1 1 + 1 11+1 1-1+1 —1 +1+1 +1+1 a2_aa:+x2, Quotient. 4. Divide w^ — 6m*n-\-10mbi^ — 10mV-j-5m7i* — 7i* by m' — 2mn-\-n^. Ans. 77i^ — 8??i^n+3m»^ — «^. 5. ©ivide a^—Za'h'-\-2>a'h'—h^ by a^— 3a^Z>+3a6'^— //. Ans. a?-\-^d'h^^al'-^h\ Most of the examples in Art. 76 may be solved by this method. II. ALGEBRAIC THEOREMS, DERIVED FROM MULTIPLICATION AND DIVISION. Remark — One of the chief objects of Algebra is to establish certain general truths. The following theorems serve to show some of its most simple applications. TS. Theorem I. — The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first hy the second, plus the square of the second. Let a represent one of the quantities and 6 a +5 the other. a +6 Then,a+6= their sum; and (a+6)X(«+6), «^+ «& or (a+6)2= the square of their sum. By mul- + a6+62 tipjying, we obtain a^-\-2ab-\-b-, which proves a~^2ab^i^ the theorem. APPLICATION. i: (2+5)2=4+20+25:=49. 2. (2w+3»)2=:4//i2+i2wn+9?i2. ALGEBRAIC THEOREMS. 43 4. (aa;2-}-3a;2;3)2==a2a;*-t- 6aa;32;3_|_9a;22;6. TO. Theorem II. — Hie square of the difference of two quantities is equal to the square of the first, minus twice the product of the first hy the second, plus the square of the second. Let a represent one of the quantities, and b a — b the other. a — b Then, a — 6= their difference; and (a — 6)X <^^ — «^ {a — 6), or [a—bY= the square of their dif- — ab-^b"^ ference. By multiplying, we obtain a^ — 2ab (jfi 2a6+62 -f-62, which proves the theorem. APPLICATION. 1. (5_3)2:^25— 304-9=4. 2. (2a;— 2/)2i=4a;2— 4a:2/+2/^. 3. (3a;— 52;)2z=9x2_30a:^-f 25^2. 4. (a^— 3ea;)2=a222_6aca;2r+9c2a;2. 80. Theorem III. — The product of the sum and differ- ence of two quantities, is equal to the difference of their squares. Let a represent one of the quantities, and b a -\^b the other. a —b Then, a^bz= their sum, and a~b= their a^-\- ab dilference. Multiplying, we obtain a^ — b^, — ab—b which proves the theorem. a^—b^ APPLICATION. •1. (7+4)(7-4)=49-16==z33=llx3. 2. (2a;-f2/)(2a;-2/)=4a;2— 2/2. 3. (3a2+462)(3a2_452)^9a4_l664. 4. (3aa;+562/)(3aa;— 56?/) =9a22;2_ 2562^2. 81. Theorem IV. — Any factor may he transferred from one term of a fraction to another, if, at the same time, the sign of its exponent he changed. 44 RAY'S ALGEBRA, SECOND BOOK. Take the fraction -7—3. Since we may divide both terms by the same quantity without changing the value of the fraction, (Ray's Arithmetic, 3d Book, Art. 136), divide first by ic^, and then by a:\ (Art. 70). Thus, ax^ ax"^ ax^ a a ax^ a bx^ ~~ h ' hx^ ~ bx^-^ ~ 6x-2 * * "TT " bx-'' a ax- In a similar manner, it may be shown that t-t, =^ — ? — . ' ^ bx- b 1 ar-2 1 Also, — = = — - =x-^, and 0:"'= — -,, from which it follows that, Tlie reciprocal of a quantity is equal to the same quantify with the sign of its exponent changed. EXAMPLES. a^b b S. ^,,^=ab-r^='-; b'" a-' 2. a'"- \„. a-'"- 4. a"*-":^ ^',^. a"-'" 82. Theorem V. — Any quantity, whose exponent -is 0, is equal to unity. If we divide a- by a^, and apply the rule for the exponents 0,2 (Art. 70), we find — ^=a2-2_(;(0. y^y^.^ since any quantity is con- a ^2 tained in itself once, — =:rl ; therefore, a"=^l. x"^ x'"' Similarly, — z=^x"'-^—X^. But-— =:1; therefore, x^=l, which proves the theorem. By this notation, we may preserve the trace of a letter, which has a^b disappeared in division. Thus, —j-=za^-^b^-^—a^b^=.a. SS* Theorem VI. — jThe difference of the same power of two quantities is always divisible by the difference of the quantities. ALGEBRAIC THEOREMS. 45 If we divide a---b~^ a^—b^; etc., successively by a b, the quo- tients will be found, by trial, to follow a simple law, both as to the exponents and the signs. Thus, (^a^^b-)-^{a—b)^a+b ; (a^^—b"')-^{a—b)=a-^ab-\-b~; la^-bi)-^[a—b)=a^^a^b^ab'-irb"; \a^—b'')-^[CL—b)—a^^a^b-\-a-b-^ab^'-^b^^ etc. The general and direct proof of this theorem is as follows : Let us divide a'^—b'^ by a—b. a'^^b'^la-b a'^^a^'-^b a'"-i b—b"'=b{a'"-^ — fi'"-' ) .-.+*J&^),Q„„, In performing this division, we see that the first term of the quo- tient is a'^-\ and the first remainder, 6(a'"-' — 6"*-^). The remainder consists of two factors, b and a"*"^ — b^-K Now, if the second of these factors, viz., a"'-' — 6"'-i, is divisible by a — 6, then will the quantity a'^—b*^ be divisible by a—b. That is. If the differcjice of (he same powers of two quant{tie& is divisible hy the difference of the quantities themselves^ then will the difference of the next higher powers of the same quan- tities he divisihlc hy the difference of the quantities. But we have seen that a^ — b"^ is divisible by a—b ; hence, a^ — b^ is also divisible by a—b. Again, since a^ — 6^ is divisible by a— by it follows that a"*— 6* is divisible Tsy it, and so on; which proves the theorem generally. 84. Lemma. — In proving the next two theorems, it is necessary to notice, that the even powers of a negative quantity are positive, and the odd powers negative. Thus, — a, the 1st power of — a, is negative. — aX — a:=a^, the 2d power, is positive — aX— <^X — ^= — ^^ *^i6 3d power, is negative. — aX~«X-"<^X— ^=^^ ^^e 4th power, is positive ; and so on. 46 KAY'S ALGEBRA, SECOND BOOK. 85. Theorem VII. — TJie difference of the even powers of the same degree of two quantities^ is always divisible hy the sum of the quantities. If we take the quantities a — b and a"* — 6*", and put — c instead of 6, a — b will become a — ( — c)=(X-|-C; and when m is even, b^ will become c"^, and a^ — 6"* will become a"*— (-f c^^j^a^i — c'" : but a"* — b^ is always divisible by a — b ; Therefore, a"^ — c"* is always divisible by a-{-C when 7)1 is even, which is the theorem. EXAMPLES. 1. (^a'^—b'^)^{a^b)=a—b. 2. \a'^—b^)^{a-Yb)=a^—a^b+ab'^—b\ 3. \a^—b^)^{a-]~b)=za''—a^b-]-a^b'^—a^b^-\-a¥—b\ 86. Theorem VIII. — Tlie sum of the odd powers of the same degree of two quantities^ is always divisible by the sum of the quantities. If we take the quantities a — 6 and a"^—b"^, and put — c instead of 6, a—b will become a — (— c)=a-f c; and when m is odd, 6"* will become —c"", (Art. 84), and a'^—b'^ will become a"*— (— c'") =:a"^-\-d^: but am—b"^ is always divisible by a—b\ Therefore, a^-\-d^ is always divisible by a+C when m is odd, which is the theorem. EXAMPLES. 1. (a3+63)_^(ct4-6)=ra2— a6-f 62. 3. \a^j^b~)-^{a-]rb)=af>—a^b-\-a^b'^~a^ty^^a^b^—ab^-\-b^. By a method of proof similar to that employed in Theorem VI., it may be shown that the sum of two quantities of the same degree can never be divided by the difference of the quantities. Thus, a+6, a2_|.ft2 a3-|-6^, a^-\-b^, etc., are not divisible by a — b. When, in either of the last three theorems, a or 6 becomes unity, the form of the quotient will be obvious. Thus, («••■»— l)-^(a-l)=aH«^+a24-a+l. (1-f aS)-=-(l-fa}=:l— a-f a2— a3-f «S etc. FACTORING. 47 FACTORING. 8T- The following summary of the principles of arith- metic should be remembered : Proposition I. — A factor of any number is a factor of any multiple of that number. Proposition 11. — A factor of two numbers is a factor of their sum. From these are inferred the following, and the converse of each : 1. Every number ending in 0, 2, 4, 6, or 8, is divisible by 2. 2. Every number is divisible by 4, when the number denoted by its two right hand digits is divisible by 4. 3. Every number ending in or 5, is divisible by 5. 4. Every number ending with 0, 00, etc., is divisible by 10, 100, etc. 88. A Divisor or Factor of a quantity, is a quantity that will exactly divide it without a remainder. Thus, a is a factor or divisor of a6, and a-\-x is a divisor or fac- tor of a^ — x^ 89. A Prime Quantity is one which is exactly divisible, only by itself and unity. Thus, x, y, and x-{-z, are prime quantities ; while xy, and ax-\-az, are not prime. 90. Two quantities are said to be prime to each other, or relatively prime, when no quantity except unity will exactly divide them both. Thus, ab and cd are prime to each other. Ol. A Composite Quantity is one which is the prod- uct of two or more factors, neither of which is unity. Thus, a^ — x"^ is a composite quantity, the factors being a-{-x and a — x. 48 RAY'S ALGEBRA. SECOND BOOK. 02. To separate a monomial into its prime factors, Rule. — Resolve the coefficient into its prime factors; then, these with the literal factors of the monomials, will he the prime factors of the given quantity. 1. Find the prime factors of ISah^. Ans. 2x3x3x<^-?>-&. 2. Of 28.x V Ans. 2X^X1 XX.X.1/.Z.Z.Z, 3. Of 210ax'i/z\ . , Ans. 2x^X^X1 -a.x.x.x.i/.z.z. 03. To separate a polynomial into its factors, when one of them is a monomial, Kule. — Divide the given quantify hy the greatest monomial (hat will exactly divide each of its terms. The divisor will be one factor, and the quotient the other, 1. Separate into factors, a-\-ax. . . Ans. a(l-{-ic). 2. xz-\-yz Ans. z(x-\-yy 3. x^y-\-xy^ Ans. xy{x-\-y'). 4. ^aV-^^a^hc Ans. 3a6(26+3ac.) 5. a^bx^y — ab'^xy'^-\-abcxyz^. Ans. abxy(ax'^ — hy-\-cz^'). 04« To separate any binomial or trinomial which is the product of two or more polynomials, into its prime factors. 1st. Any trinomial can be separated into two binomial factors, when the extremes are squares and positive, and the middle term is twice the product of the square roots of the extreme terms. The factors will be the sum or difference of the square roots of the extreme terms, according as the sign of the middle term is plus or minus. (See Arts. 78, 79.) Thus, a^+2a6+62z=(a4-6)(a+6); a^—2ab+b^={a—b){a—b). 2d. Any binomial, which is the diflFerence of two squares, can be separated into factors, one of which is the sum and the other the diflFerence of their roots. (See Art. 80.) Thus, a^—b-=(a-]-b){a—b). FACTORING. 49 3d. Any binomial which is the difFerenc^^ of the same powers of two quantities, can be separated into at least two factors, one of which is the difference of the two quantities. (See Art. 83.) x^—xf^{x—y){x^-\-xy-^y^). Thus, Similarly, x^ — i/^^=(x — y){x'^-{--x,^y-\-^'^y'^-{-xy^-\-y^). 4th. Any binomial which is the difference of the even powers of two quantities, higher than the second degree, can be separated into at least three factors. (See Art. 85.) Thus, a4— 64=(a2+62)(a2— 62)=(a2^62)(«^6)(a-6). 5th. Any binomial which is the sum of the odd powers of two quantities, can be separated into at least two factors, one of which is the sum of the quantities. (Art. 86.) Thus, a3+63=(a+6)(a2— a6+62j 6th. The following examples of the factoring of binomials com- posed of the sum of like even powers of quantities may be verified either by multiplication or by (iivision : a2+62^(a+v/'2^+6)(a— 1/2^+6). a4+64=(a2+i/2.o6+62)(a2— v^2.a6+62). a6+66^(a2+62)(a4_«262_^64)^(«2_|_J2)(^J^_^3,„^,_|_52) (a2_v/3:a6+62). a8+68==(a*+l/2:a262-f&*)(a*-V^-a'6'+6*). aio+6io=(a24-&2)(a8_a662_|,a464— a266-f68). al 2_|-61 2=..(a4_|.64)(tt8_a464_^58). mm mm a-^rnJ^h^^={a''^V2.a ^ b^ +6'")(a'»— V 2^a^ 6 2" -f6"*). a3'«+63'»=(a'"+6'")(a2"»— a'"6'«-|-62"'). Separate the following into their simplest factors; 1. c^-\-2cd+d^. 2. a^x^-{-2axhj+y^. 3. 2bx^y^-\-20xyH-riz\ 4. 9Z^— 6z222_j.^4, 5. 4)712x2 — imn^x-\-n^. 6. x^—z\ 7. 9a2x*— 25. 2d Bk. 5^^ 8. U—a^b^z\ 9. a^—xK 10. 2^4-1. 11. t/3-1. 12. a^x^—b^yK 13. x^-jry^. 14. x^ — y^. 50 RAY S ALGEBRA, SECOND BOOK. 94:. To separate a quadratic trinomial into its factors. A Quadratic Trinomial is of the form x^-\-ax-\-b, in which the sign of the second term may be either plus or minus. Such a quantity may be resolved into factors by inspection. Ob- serve carefully the product resulting from the multiplication of two factors of the form x-\-a, and a;+6. Thus, a;2— 5a;+6=:(a:— 2)(a:— 3), since the first term of each factor must be X, and the other terms, — 2 and — 3, must be such that their sum will be — 5, and their product -j-6. Trinomials to be decomposed into binomial factors. 1. .'c2+3x+2 Ans. (x-\-l)(x-{-2) Ans. (x — 3) (a: — 5) x ' SxJrU. x-^—x—2. . x-^-\-x—12. x'—x-12. x^j^2x—Zh Ans. (ic4-l)(x— 2) Ans. (a^— 3)(x+4) Ans. (x+3)(.x— 4) Ans. (x— 5)(.x+7) 95. Examples to be resolved into factors, by first sep- arating the monomial factor, and then applying Arts. 93 and 94. Ex. 1. a7?y — ax}^^axrj{x'^—y'^)=^axy(x-\-y)(x—y). ^ax'^-\-Qaxy-\-^ay^. 2cx'—\2cx^\^c. Ziii^n — 3??i?i'. . 2d(?y — 2xif'. . . 2x'-^6x—S. . . . . . Ans. Sa(x-\-y){x^y). . . .Ans. 2<8a3^3 all the quantities. The least powers of Iba^x^y^ ^SX^Ci'^^^y^ « and x, are a^ andiC; hence, the G.C.D. is 3a-x. Find the G.C.D. of the following quantities : 8. lbahc\ and 21h'cd Ans. Sbc. 4. 4a^b, lOtt-'c, and l^a^'hc Ans. 2a\ 6. 4:axy, 20xyz, and 12xyz\ . . . Ans. 4^;^^ 6. 12aV2^, 18aa;V, SOd^x^z, and 6ax^zK Ans. 6ax^z. lOO. Previous to investigating the rule for finding the G.C.D. of two polynomials, it is necessary to introduce the following propositions : Proposition I. — A divisor of any quantity is also a divisor of any midtiple of that quantity. Thus, if A will divide B, it will divide 2B, 3B, etc. Proposition II. — A divisor of two quantities is also a divi- sor of their sum or their difference. Thus, if A will divide B and C, it will divide B-f C, or B— C. ThiB is evident from Art. 74. GREATEST COMMON DIVISOR. 53 lOl. Let it be required to find the G.C.D. of two poly- nomials, A and B, of wliich A is tlie greater. If we divide A by B, and there 16 no remainder, B is evidently the B)A(Q G.C.D., since it can have no divisor BQ greater than itself. A— BQ^R, 1st. Rem. . Divide A by B, and call the quo- tient Q; then if there is a remainder R)B(Q'' R, it is evidently equal to A— BQ. RQ^ If, now, there is any common divisor B— RQ^=R^ 2d Rem. of A and B, it will also divide BQ (Prop. 1st) and A— BQ or R (Prop. A=BQ +R Since the 2d); or the common divisor must B=RQ^4-R^ eiuaUoV^o divide A, B, and R, and can not be product of the divisor by tho greater than R. quotient, plue the remainder. Now, if R will exactly divide B, it will also exactly divide BQ (Prop. 1st) and BQ+R (Prop. 2(1). Consequently, it will divide A, since A=BQ-)-R, and will be tho common divisor of the two polynomials A and B. It will also bo the greatest common divisor, since no divisor of A, B, and R can bo greater than R. Suppose, however, that when we undertake to divide R into B, to ascertain if it will exactly divide it, we find that the quotient is Q^, with a remainder R''. Now, reasoning as before, if R'' exactly divides R, it will also divide RQ^ (Prop. 1st) and also B (Prop. 2d), since B=RQ^+R^; and whatever exactly divides B and R, will also exactly divide A, since A=BQ-|-R; therefore, if R^ exactly divides R, it will ex- actly divide both A and B, and will be their common divisor. It will also be the greatest common divisor, since the greatest divisor of R^ is R^ itself. By continuing to divide the last divisor by the last remainder, we may apply the same reasoning to every successive divisor and re- mainder; and when any division becomes exact, the last divisor will be the greatest common measure of A and B. The same method of proof may be applied to numbers ; for ex- ample, let A=T=120, and B=r35. lOS. When a remainder becomes unity, or does not contain the letter of arrangement, it is evident that there is no common divisor of the two quantities. 54 RAY'S ALGEBRA, SECOND BOOK. 103* If either quantity contains a factor not found in the other, that factor may be canceled without affecting the common divisor. Thus, a is the G.C.D. of ax and ay, and will be, if we cancel X in ax^ or y in ay. 1.04* We may multiply either quantity by a factor not found . in the other, without changing the G.C.D. Thus, in the two quan- tities, ax and ay, if we multiply ax by m, or ay by n, the G.C.D. will still be a. 103* But if we multiply either quantity by a factor found in the other, we change the G.C.D. Thus, in the two quantities, ax and ay, if we multiply ay by x, or ax by y the G.C.D. becomes ax or ay. lOO* From Art. 101, it is evident that the three preceding articles apply also to the successive remainders. lOT- It is evident that any common factor of two quantities, must also be a factor of their G.C.D. Where such common factor is easily seen, we may set it aside, and find the G.C.D. of what re- mains. Thus, take 55a: and 15a:. Setting aside a;, we find the greatest common measure of 55 and 15 to be 5. Annexing ar, we have 5a:. Remark. — The illustrative examples, in the five articles above, are monomials, but the same principles obviously apply to poly- nomials. We shall now show the application of these principles. 1. Find the G.C.D. of x^—z" and a:*— rrV. Here the second quantity contains x^ as a operation. factor, but it is not a factor of the first; we x^ — z^\x^ — z^ may, therefore, cancel it (Art. 103), and the a? — xz"^ [^ second quantity becomes x"^ — z^. Then divide xz^ — z^ the first by it. or [x — z)z'^ After dividing, we find that z"^ is a factor of the remainder, but not of X^ — z^, the next divi- X^ — 0^ ^x — z dend. We, therefore, cancel it (Art. 103), and a:^ — xz\x-\-z the second divisor becomes x — z. Then, divid- xz — z^ ing by this, we find there is no remainder; there- xz — z^ fore, x—z is the G.C.D. GREATEST COMMON DIVISOR. 55 2. Find the G.C.D. of o^^-f a;V and x^—x^z\ The factor X- is common to both quantities ; it is, therefore, a factor of the greatest divisor (Art. 107), and may be taken out and reserved. Doing this, the quantities become x^'-{-z^ and x^ — xz^. The second quantity still contains a common factor, X, which the first does not; canceling this, it becomes x^ — 0-. Then, pro- ceeding as in the first example, we find that X-^-Z divides without a remainder; therefore, X^{x-\-z) is the required G.C.D. OPERATION. x^ — xz"^ \x or (x\z)z'^ a;2_^2 \x-\-Z x^-\-xz\x — z — xz — z'^ 3. Find the G.C.D. of 1 Oa^o^^— 4(71t,— Ga^, and bhx:'—lllx By separating the monomial factors, we find 1 0a%2_4a2a;_6a2^2a2( 5.^2— 2a:— 3), and 56:c2— 116a:+66=6(5x2—lla:-)-6). The factors 2a^ and b have no common measure, and hence are not factors of the common divisor. We may, therefore, suppress them (Art. 103), and proceed to find the G.C.D. of the remaining quantities, which is found to be x — 1. OPERATION. 5a:2_l la:+6 15a;2— 2a:— 3 5a:2- 2a:— 3 11 —to 4- 9 or _9(a:-l) 5a:2— 2a:— 3|a:— 1 5a:2— 5a: & 3a:— 3 3a:— 3 4. Find the G.C.D. of ^a''—^ay^y\ and Za^—^a''y In solving this exam- ple, it is necessary, in two instances, to multi- ply the dividend, that the coefficient of the first term may be divisible by the first term of the di- visor (Art. 104,). OPERATION. 3a'>— 3a22/^02/2— ?/3|4a2._5cr2/_|_2/2 4 ""^ 1 2a^ — 1 2a^y ^\ay^ — ^y"^ 12a^—15a2y^Say^ 3a^y-{- ay2—iy^ 4 \Sa-^Sy 12a^y\- 4a?/2— 16?/3 [over.] 5G RAY'S ALGEBRA, SECOND BOOK. We find 19?/2 is a 12a'^y-\- ^ay- — IG?/^ [brought over.] factor of the first re- I2a'^y—lbay-+ Sy^ mainder, but not of the Iday'^ — 19y'' first divisor, and hence or 19y-(^a — y) can not be a factor of the G.C.D. ; it must, 4:a-—5ay-^y-\a-~y g.c.d. therefore, be suppressed. 4a~—4ay \4a~y Hence, — ay^y"^ —ay^y^ TO FIND THE GREATEST COMMON DIVISOR OF TWO POLYNOMIALS, 108. Rule. — 1. Divide the greater polynomial hy the less, and if there is no remainder, the less quantity will he the divisor sought. 2. If there he a remainder, divide the first divisor hy it, and continue to divide the last divisor hy the last remainder, until a divisor is ohtained which leaves no remainder; this ivill he the G. CD. of the two given polynomials. Notes. — 1. When the highest power of the Zca2-|-8a26 and 9a'—Sab'-\-Sd'b'—9a'b. Ans. a — b. 10. ic*-|-^'^^*4-<^* ^o^ ic*-f-aic' — a^x — a*. Ans. x'^^ax-\-(.r. 11. a;*— pa:^-j-(2 — l)a;'^+i^^ — 2 aiitl ^* — qx^-]-(p — l)x^ -\-qx^p. Ans. ^2— 1. Ans. a+36. Ans. a^— a;^ Ans . a:— 1. Ans. 3x-2. LEAST COMMON MULTIPLE. 109. A Multiple of a quantity is any quantity thiit contains it exactly. Thus, 6 is a multiple of 2 or of 3 ; and ab is a multiple of a or of Z>; also, a(b — c) is a multiple of a or (b — c). no. A Common Multiple of two or more quantities, is a quantity that contains either of them exactly. Thus, 12 is a common multiple of 2 and 3 ; and 20a:3/, of 2x and 5^. 111. The Least Common Multiple of two or more quantities, is the least quantity that will contain them ex- actly. Thus, 6 is the least common multiple of 2 and 3 y lOa;^, of 2x and 5y. Note. — L.C.M. stands for least common multiple. 112. To find the L.C.M. of two or more quantities. 58 RAY'S ALGEBRA, SECOND BOOK, It is evident that the L.C.M. of two or more quantitiea contains all the prime /actors of each of the quantities oncCy and does not contain any prime factor besides. Thus, the L.C.M. of ah and be must contain the factors a, 6, c, and no other factor. Assuming the principle above stated, let us find the L.C.M. of mx, nx, and m'^nz. Arranging the quantities as in the OPERATION. margin, we see that m is a prime factor m\mx nx ni^nz common to two of them. It must, there- r nx mnz fore, even if found in only one of the X X mz quantities, be a factor of the L.C.M.; and I 1 1 mz as it can occur but once in the L.C.M., myny^xy^inz=m~nxz we cancel m in each of the quantities in which it is found, which is done by dividing by it. For the same reason we divide by n and by x. We thus find that the L.C.M. must contain the factors m, n, and x\ also, me, otherwise it would not contain all the prime factors found in one of the quantities. Hence, m'Xn'X.x'Xmz=m?nxz, contains all the prime factors of the quantities once, and contains no other factor ; it is, therefore, the required L.C.M. Hence, TO FIND THE LEAST COMMON MULTIPLE OF TWO OR MORE QUANTITIES, Rule. — 1. Arrange the quantities in a horizontal line, divide by any prime factor that will exactly divide two or more of them, and set the quotients and the undivided quan- tities in a line beneath. 2. Continue dividing as before, until no prime factor, ex- cept unity, will exactly divide two or more of the quantities. 3. Multiply the divisors and the quantities in the last line together, and the product will be the L.C.M. required. Or, Separate the quantities into their prime factors ; then, to form a product, 1st, take each factor once; 2d, if any factor occurs more than once, fake it the greatest number of times it occurs in either of the quantities. ALGEBRAIC FRACTIONS. 59 113. Since the G.C.D. of two quantities contains all the factors common to both, if we divide the product of two quantities hy their G.C.D., the quotient will he their L.C.M. 1. Find the L.C.M. of ^a\ 9ax^ and 24x^ Ans. 720^ 2. 32xy, 'iQax'y, bd'x(x—y). Ans. 160aV/(a:— 3/) 3. Sx-{-6y and 2x^—8/ Ans. 6x^—24/ 4. a^-\-x^ and a^ — x^. . . Ans. a* — a^x-\-aa? — x* 5. x~l, x'—l, rr— 2, and x"^ — 4. Ans. .r,* — 5a;2-j-4 6. x'—l, x'^1, (x—ly, (x-]-iy, x^—l, and x^-\-l Ans. x^^ — X® — a:*-f 1 7. Sx'~llx-^6, 2.x2— 7a^+3, and 6x'—1x-\-2. (See Art. 113.) Ans. 6a^—2bx'-\-2Bx—6. III. ALGEBRAIC FRACTIONS. DEFINITIONS. 114. Algebraic Fractions are represented in the same manner as common fractions in arithmetic. The quantity below the line is called the denominator^ because it denominates, or shows the number of parts into which the unit is divided ; the quantity above the line is called the numerator, because it numbers, or shows how many parts are taken. Thus, in the fraction -, a unit is supposed to be di- c-\-d vided into c-\-d equal parts, and a — b of those parts are taken. 113. The terms proper, improper, simple, compound, and complex, have the same meaning when applied to alg'^braio fractions, as to common numerical fractions. 60 RAY S ALGEBRA, SECOND BOOK. 116. An Entire Algebraic Quantity is one not ex- pressed under the form of a fraction. IIT. A Mixed Quantity is one composed of an entire quantity and a fraction. lis. Proposition. — The value of a fraction is not altered, when both terms are multiplied or divided by the same quantity. A 77lA Let ^=Q- Then, will — ^=Q- For, since the numerator of a fraction may always be considered a dividend, and the denominator a divisor, if we multiply the numerator or dividend by any quan- tity, as m, the quotient will be increased m times; if we multiply the denominator or divisor by m, the quotient will be diminished as much, or it will be divided by m. Therefore, the value of the frac- tion is not changed. Or, the Proposition may be proved thus: --^ (Art. 81), ^^— = -^= (Art. 82), -. A similar method of reasoning may be applied to the division of the terms of a fraction. Case I. — To reduce a Fraction to its Lowest Terms. 119. From Art. 118, we have the following Kule. — Divide both terms of the fraction by any quantity that will exactly divide them, and continue this process as long as possible. Or, Divide both terms by their greatest common divisor. Or, Resolve both terms into their prime factors^ and then cancel those factors which are common. In algebraic fractions, the last is generally the best method. ALGEBRAIC FRACTIONS. 61 1. Reduce jj-j — ^3 to its lowest terms. 10aex^_ 2acx^ _2ax^_2a 15bot^~3bc^~3bxS~3dx' ' ^ ^ , lOacx^ 2a Or, dividing by bcx\ j^^-.^^-^. Or, 2. 3. 4. 5. 6. 12. 10acrc2_ 2aX5ca:2 _ 2a 156c^ " 3tox5c^2 — 35^- Za^hx ax-\-x'^ Sbx — ex Sa^ — Sab' 1-x ^"^- 3^- 1—x' a"" Ans. Ans. Ans, a-\-x a — b' 1 1-j-x 1 a n+l' Ans a* f_ mnp — my m^p-\-mp^ ' Q 2ax — 4ax^ o. Ans. . Ans. n — m m-\-p' l-2x ^ax 3 ^ ha^-\-hax . 5a y. — :r^ — ^— . Ans. 10 ^''+2.x-3 ^^^ ' x:^-\-bx-\-6' ' xm\-2' ^- a^-4^-f5 ^a:'^— 5ar+5 a — .X x-1 X ic^H-l 15ry^-f-35.x^H-3x+7 27a:*+63a;-^— 12x'^— 28;c' * * Ans. x'—x-i-l' 9x=*— 4x' The following examples are to be solved by factoring, but the process requires care and practice. -I o T. T x^-l-Ca-l-c^x-T-ac . , Id. lieduce ., ;, — ( — -—r- to its lowest terms. X -{-{o-\-c)x-}-bc x--\-{a^c)x-\-ac—x^-{-ax^cx-\-aG Also, x^^{b-^c)x^bc={x-\-c){x-\~b)] .'. the fraction becomes (x-^c){x^a) __x-{-a (x+c) {x'^F) ~ x^b , Ans. ^ . ac-\-bi/-\-aTi/-\-bc a/-j-2bx-[-2ax-{-b/' Ans. 62 RAY S ALGEBHA, SECOND BOOK. 10. — —. T Ans. -, x'—y*' x^—f it>- T~m. -^°s. ~-j-. a* — b^x^ a^ — bx -H ax'^—bx'^^'^ , X- J-i- -TT. T^-^ Ans. rn~\ a^bx — b^x? ' ' ' b{a-\-bxy 120. Exercises in Division, in which the quotient is a fraction, and capable of being reduced : 1. Divide 2aV by 5a V6 Ans. -^ 56* 2. ax-\-x^ by obx — ex Ans. ,, ob — c a-'-^ab-^b^ a-\-b • 3. a^ — b^ by a^ — b^ Ans. 4. a'—b' by (a— by Ans. ^!±^^H^' a — 6 Case II. — To reduce a Fraction to an Entire or Mixed Quantity. IISI. Since the numerator of the fraction may be re- garded as a dividend, and the denominator as the divisor, this is merely a ease of division. Hence, Kule. — Divide the numerator by the denominator, for the entire part. If there be a remainder, place it over the de- nominator, for the fractional part, and reduce it to its lowest terms. 1 T> 1 a^4-a^ — ax^ . . . X. Keduce to an entire or mixed quantity. a^ — ax ^ ^ - '.. — =a^x^ — 5 —a-\-x-\ , Ans. ALGEBRAIC FRACTIONS. 63 Reduce the following to entire or mixed quantities : „ ax — x^ . x^ A. Ans. X . a a D. — ^ — i— Ans. a-\~h-\ r. a — h a — h ^- i=3x ^°^' ^+'^+r=:3x- - ^-\^hx^ , , "Ihx o- ., , Ans. icH — x^ — bx X — b ^ aV — ^-\-xz — z — x-\-\ . „ , z — 1 X^ — 1 x-\-l Case III. — To reduce a Mixed Quantity to the form OF A Fraction. ISS. This is, obviously, the reverse of Case II. Hence, we have the following Rule. — 1. Multiply the entire part by the denominator of the fraction. 2. Add the numerator to the product^ if the sign of the fraction be plus^ or subtract it, if the sign be minus. 3. Place the result over the denominator. Before applying this rule, it is necessary to consider 12S. The Signs of Fractions.— Each of the several terms of the numerator and denominator of a fraction is preceded by the sign plus or minus, expressed or under- stood ; and the fraction, taken as a whole, is also preceded by the sign plus or minus, expressed or understood. Thus, in the fraction ; — , the sign of a^ig plus; of 6^ minus: ' x^y ' ^ r ) ' while the sign of each term of the denominator is plus; but the sign of the fraction, taken as a whole, is minus. 64 RAY'S ALGEBRA, SECOND BOOK. 1S4. It is often convenient to change the signs of the numerator or denominator of a fraction, or both. By the rule for the signs, in Division (Art. 69), we have, =+&; or, changing the signs of both terms, z=-j-6. If we change the sign of the jinmerator, we have = — b. If we change the sign of the denominator, we have — — =—b. Hence, 1. l%e signs of both terms of a fraction may he changed,, Without altering its value or changing its sign, as a whole. 2. If the sign of either term be changed, the sign of the fraction will be changed. Hence, also, 3. The signs of either term of a fraction may be changed, without altering its value, if the sign of the fraction be changed at the same time. Thus, — \ — — — — (— a— a;)=:a-f cc. a—x a—x —a-lx ^ ' ' And, a —aA "— =a4 ■ — =— x. ' u—x a—x ' —a-\-x Applying the above principles, the sign of the fraction may be ^ade plus, in all cases, if desired. Heduce the following quantities to a fractional form : 1. 2-1-1 and 2—3 Ans. K} and |. o ■ , ^^ — ^^ A a^-\-x^ ^. a-\-x-\ Ans. — - — . X X o. a^ — ax-\~x^ — — ; — . Ans. a-\-x a-{-x 4. 2a-;.+(^=^'. Ans."- X x' - a' . ab o. a —J Ans. — -y. ALGEBRAIC FRACTIONS. 65 0. a — X — - — — - , . . Ans. , a-j-x a-\-x 1. 1-^^^' Ans. ^, Case IV. — To reduce Fractions of Different Denom iNATORs TO Equivalent Fractions having A Common Denominator. a b \25» — 1. Let it be required to reduce — , -, and -, to a common denominator. If we multiply both terms of the first fraction by nr, of the sec- ond by Tnr, and of the third by mn, we have anr bmr , emn , , and • . Tiinr unnr mnr As the terms of each fraction have thus been multiplied by the same quantity, the value of the fractions has not been changed. (Art, 118.) Hence, TO REDUCE fractions TO A COMMON DENOMINATOR, Rule. — Multiply both terms of each fraction by the prod- uct of all the denominators^ except its own. Or, 1. Multiply each numerator by the product of all the denominators except its own., for the new numerators. 2. Multiply all the denominators together for the common denominator. Reduce the fractions in each of the following to a com- mon denominator: 12,3 .^^yz 2xz Zxy 2. -, -, and - Ans. — , — , — x' y z xyz xyz xyz' 3. r- and - Ans. — and -y. b a ab ab' . x'^A-ax J ox — CL^ . . . Ans. -7-^^ and ,, x^ — a^ X' — or 4. and X — a a x^a 2d Bk. 6 66 RAY'S ALGEBRA, SECOND BOOK. 1!30. It frequently happens, that the denominators of the fractions to be reduced contain a common factor. In such cases the preceding rule does not give the least com- mon denominator. 1X-1 •! 1 ah c 1, Let it be required to reduce — , — , and — , to their - , ^ . m mn nr least common denominator. Since the denominators of these fractions contain only three prime factors, m, n, and r, it is evident that the least common denomina- tor will contain these three factors, and no others; that is, it will be mnr^ the L.C.M. of m, mn, and nr. It now remains to reduce each fraction, without altering its value, to another whose denominator shall be mnr. To effect this, we must multiply both terms, of the first fraction by nr, of the second by r, and of the third by m. But these multi- pliers will evidently be obtained by dividing mnr by m, m,n, and nr\ that is, by dividing the L.C.M. of the given denominators by the several denominators. Hence, TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO EQUIVALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR, Rule. — 1. Find the L.C.M. of all the denominators; this will he the common denominator. 2. Divide the L.C.M. hy the first of the given denominators, and midtiply the quotient hy the first of the given numerators; the product will he the first of the required numerators. 3. Proceed thus to find each of the other numerators. Reduce the fractions, in each of the following, to equiv- alent fractions having the least common denominator : g a h c a 2hy Sex 6xy^ 3x' 2y * 6xy^ 6xy' 6xy' Q _^ l_ ^ A„, K ^—h) y(a±h) z ^- a_|_6' a—h' a'—h'^' a'—h' ' d'—h' ' a'-^h'^' ALGEBRAIC FRACTIONS. 67 . m — n m-\-n mW (m — ny (^m,-\-ny m^n^ tn-^ii 111 — n m^ — n^ iiv — iv m^ — iv w* — nr Other exercises will be found in Addition of Fractions. Note. — The two following Articles may be of frequent use. 12T. To reduce an entire quantity to the form of a fraction having a given denominator, Hule, — Multiply the entire quantity hy the given denomina- tor^ and write the product over it. 1. Reduce a; to a fraction whose denominator is a. ax Ans. — . a 2. Reduce ^az to a fraction whose denominator is z^. 2a2» Ans. — -. z^ 3. Reduce x-\-y to a fraction whose denominator is x — y. . x^ — y''- Ans. ^. x—y l!38. To convert a fraction to an equivalent one hav- ing a given denominator, Rule. — Divide the given denominator hy the denominator of the given fraction^ and midtiply loth terms hy the quo- tient. 1. Convert | to an equivalent fraction, having 49 for its denominator. Ans. |i. a 5 2. Convert -^ and — to equivalent fractions having the denominator 9c^. Ans. -p^-— and ^ — 3. Convert and — -^ to equivalent fractions hav- oc— 6 a+6 (a-\-hy (a—hy ing the denominator a^ — h'\ Ans. ., A -, ~ — A-, 68 RAYS ALGEBRA, SECOND BOOK. Case v.— ADDITION AND SUBTRACTION OF FRACTIONS. l!39. — 1. Required to find the value of -, -, and -. (1 (.1 (X Since in each of these fractions the unit is supposed to be divided into d parts, it is evident that their sum will be expressed by the „ . a-f6+c „ iraction , . Hence, d ' Rule for the Addition of Fractions.— 1 . Reduce the fractions^ if necessary^ to a common denominator. 2. Add the numerators^ and. write their sum over the com- mon denominator. 130. — 2. Let it be required to subtract — from -. d a The unit being, in each case, divided into the same parts, the difference will evidently be expressed by — ^ — . Hence, Rule for the Subtraction of Fractions. — 1. Reduce the fractions, if necessary, to a common denonfilnator. 2. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the remainder over the common denominator. EXAMPLES IN ADDITION OF FRACTIONS. 1. Add - and jv together Ans. ^j, 2. Add -r and - toj^ether Ans. - — '-.—, h a ^ ah 11 2 3. Add = and _ together. . . . Ans. .j ^. * ^- ^ ^ , o c , Z> , « A a-\-hx^-\-c:x^ 4. Find the value of — f- — -j- -. . Ans. X x^ a? x^ r -r.. 1 .1 1 o^ . ad— he . a-\-hx 5. Find the value of -,-f ,, . , ^ . . . Ans. , . d ' d{c-\-dx) c-^dx ALGEBRAIC FRACTIONS. 69 6. Find the value of 4+ 1+ f. Ans. i!^±9^«. aO ac be abc 7. Of -4- ^J-. . 8. Of -^ ■ ^ Ans. x'-{-f 4(1 -1-a:) ' 4(1— ar) ' 2(H-ar^)" Ans. 9. Of-?=?+^^4-^''. 10. Of -^ ■ -^ 1 1— a:*- Ans. 0. 4a3(a+.'c) ' 4a'(a— a:) * 2a''=(a^4-a:'^)" Ans. a'—3^' 11. Of a(a — i)(a — c) 6(6 — a){h — c) c(c — d){c — 6)* ahc EXAMPLES IN SUBTRACTION OF FRACTIONS. on, in tlie following, from the Subtract the second fract first : - 5.^ - 3y 1. w- and —. la 7 9 1 ^ 1 2. y and -j~r. a — a-f-o 3. and P—9. p+q 4. and - — ^. n n — 1 . bx — Sory Ans. — u — ^. 7a . . . Ans. . . . Ans. 26 a'—h^' 4pq Ans. 1—271 1 A ^-1 6. and (2r+lX^+2) (a:-f-l)(^-|-2)(x+3)- Ans. (2r+l)(a:+3/ 70 RATS ALGEBRA, SECOND BOOK. 7. - and ^ , . , V - Ans c c(c-j-c?x) * * c-f-c/a:' 1 3m-f2n, 1 3m— 2n 12m-t ^' 2-3;7i— 271 ^"^ 2*37714:271 • 9^^=:47/;^' 9. . !^+^ . and ^—t±^-^. A. ^+^ (a — 6)(x — a) (a — b)(x — 6) * (x — a)(x — by Find the value m in r^£. 4771 — 371 m+37i 2n 10. Of Q7i ^ — qTT ^^-1 ^- • • • ^^^' 1 0(1 — n) 3(1 — 7i) 1 — u 1 — n 11. Of ^ h-i — Aiis. 0. at) ac be 12. Of^±^ ^ ^=:^ Ans.l. ..Q 1 1 a;+3 . x-\-S Case VI.— MULTIPLICATION OF FRACTIONS. 131. — 1. Required to find the product of 7 by -^. a Here, as in arithmetic, we take the part of ^, which is expressed 1 1 ^ a a by -^, and then multiply by C. Thus, the -^ part of r- is v-^, and c Or thus, J- and -j^=:ab-^ and CO?-* (Art. 81). Multiplying, we have a&-^C(i-^=^. Hence, IRiUle. — Multiply the numerators together for a new numer- ator^ and the denominators together for a new denominator Remarks. — 1st. To multiply a fraction by an integral quantity, reduce the latter to the form of a fraction, by writing unity beneath it J or, multiply the numerator by the integer. ALGEBRAIC FRACTIONS. 71 2d. If either of the factors is a mixed quantity, reduce it to an improper fraction, 3d. When the numerators and denominators have common factors^ let such factors be first separated, and then canceled. a+5 a^—b^^ ~ioW~ {a+b){a—by^^b ~2b{a~b)' 2a2 (a4_6)2_2a2x(^+&)(«+&) Find the products of the fractions in the following : 1. ^ by 3^ and — by ^,. X' a X a , -+-. a X a •^ and 2-\ x^y x—y 1+6- and a;2-|-5x-l-4 ^ax a} — x^ hc-{-hx x-\-y ' a— &' (y^—yf ' 8. x4-l+- by cc— 1-1--. X '^ X 9 l^^by?^-!-^ . x"^ , bed Ans. — and . y a . a* — X* . Ans. — - — . a^x Ans. Ans. Ans Ans. Ans. xy^' 4a:(a-f-ic) '■6y{c~x)' a\a-\-b') x—y Ans. Ans. x^+H— ,. xf 9£ ;+2+ 8a6* p — qx p-^qx Ans. rs-l-(H-|-2s)ic-l-g^a::^, Find the value »-'t^^)(^^')-(^')(^')■ atZ be ' 72 RAY S ALGEBRA, SECOND BOOK. Case VII.— DIVISION of fractions. 13I3. — 1. Required to find the quotient of - by -7. Here, as in arithmetic, the quotient of y by -^ is -x-,andthequo- ^. , „ « ^ . 1 c . a(i ,. . , , , ad tient or -j- by G times — „ or -^, is -y- diviaed by C, or -z — . b •' d d b ^ ' be a c Or thus, J- and -^= (Art. 81) ab-^ and cd-^. Dividing, we have afx-'^ ad „ — T-, =^v— . Hence, cd- 1 6c ' Rule. — Invert the divisor^ and proceed as in multiplication of fractions. Remark. — To divide a fraction by an integral quantity, reduce the latter to the form of a fraction, by writing unity beneath it ; or, multiply the denominator by the integer. Remarks 2 and 3, Art. 131, apply equally well to division of frac- tions. Required, in their simplest forms, the quotients 1. Of — j Ans. ~. xy xy^ a^x 2. Of — i =-— J Ans.-- a-\- c a — o a^ — c^ „ „„ .r' — d^x ax — a^ . x'-fax^ 3. Of \ Ans. ^~ . a^ X (T 5. Of 't^fj^-^y^t ■. Ans. 1. ^ — y ^ — y a f\o «* — ^ a^xAr^ K «+^/ 21 I 2\ 6. Of , o . 2-^ 3 ^.3 - • ^^^- {ce-^ax-^x'). a^ — Zflrar+j^ a^ — ar^ x 8. Of 3ar ALGEBRAIC FRACTIONS. 2x 2 • x—1 Ans 73 3 4* x—1 .0. Of ( x'—\ \-^lx—~\, . . Ans. x'-\-\-\-x-{-~. \ x* I \ X I ' X^ X To REDUCE A Complex Fraction to a Simple one. 133. This is merely a case of division, in which the dividend and divisor are either fractions or mixed quan- tities. c b n Thus, — ^ is the same as to divide a-\ — by m . ' n c r m r acr^hr ac-\-b mr—n ac-{-b X mr—n cmr—cn Or, the following method, obviously true, will generally be found more convenient. Multiply both terms of the complex fraction by the product of the denominators, or by their L. CM. acr^ br Thus, in the above, multiplying by cr, we have, at once, -— — — — , Solve the following examples by both methods : 2:r— 2 Ans. :~1 a c ^- e __g' ^'''- M{eh-fgy 7 ^^ 2d Bk. 7* a-^\ a—\ CT— l"^aH-l a — 1 a-)-l a^bJr- Ans. a-h6H-- . Ans. 74 RAY'S ALGEBRA, SECOND BOOK. Resolution of Fractions into Series. 134* An Infinite Series consists of an unlimited num- ber of terms which observe the same law. The Law of a Series is a relation existing between its terms, such as that when some of them are known the others may be found. Thus, in the infinite series 1 1 — ^ 3+) ^^c., any term may be found by multiplying the preceding term by . Any proper algebraic fraction, whose denominator is a polynomial, may, by division, be resolved into an infinite series. 1 X 1. Convert the fraction = into an infinite series. l-x\l-\-x 1-^x 1— 2a;-l-2a:2_2a;3+, etc. It is evident that the law of — 2x ' this series is, that each term, — 2a;— 2.t2 after the second, is equal to -j-2a:2 the preceding term, multiplied 42a;24-2a:3 by ~x. — 2a;3 Resolve the following fractions into infinite series : 2. .j ^=1 — r^-j-r* — r^-j-r® — ^ etc., to infinity. 1 -r-i-v l_|_^_y^_r*-fr«-f r^— r»— r^^+j etc. 4. —7-7=1 h-i 3+, etc. a-\-b a a^ or 135. Miscellaneous Propositions in Fractions. — The answer to some general question, that is, the solution to a literal equation (cf. Arts. 162-165), may happen to be a fraction: e. g., we may have x = -j-- When the two terms of the fraction are finite numbers, the fraction, being the ratio of two finite numbers, has a determinate value. But ALGEBRAIC FRACTIONS. 75 the values of the numerator and denominator may be changed, by reason of some suppositions as to the values of the known numbers involved in the question, thus giving rise to anomalous results requiring explanation. 130. — 1. Thus, suppose a; :=-r- If, while the denomina- tor remains constant, the numerator changes, the value of the fraction varies directly with the numerator : if a de- creases, the fraction decreases ; if a becomes 0, the fraction likewise becomes 0, that is, -7- = 0. Also, if a increases, b being constant, the fraction increases ; if a becomes 00, the fraction becomes 00 ; that is, -j- = co. 2. If the denominator changes while the numerator re- mains constant, the value of the fraction varies inversely; that is, if b decreases, the value of the fraction iiicreases, and, vice versa, if b increases, the value of the fraction de- creases ; if b becomes 0, the fraction becomes 00, or -tt = 00 ; if, on the other hand, b becomes 00, the fraction becomes 0, or - = 0. 137. If the numerator and denominator are both ren- dered zero simultaneously, the solution assumes the form x = jr. In this case, the unknown number x is said to be indeterminate, inasmuch as it may evidently, at this stage of the investigation, have any value whatever ; since the only condition imposed upon it is, that it shall give, when multiplied by zero, a product equal to zero. Hence, the form -rr has been called the symbol of indetermination. Nev- ertheless, it may, and indeed generally does, happen that the indetermination is only apparent, being due to the pres- ence, in numerator and denominator, of a common factor which the particular hypothesis reduces to zero, and which 76 RAY'S ALGEBRA, SECOND BOOK, if suppressed before making the hypothesis, will leave the result in a determinate form. Thus, suppose some equation has given us a; = j- : if we make b=a, we have x = -?r: if, however, we cancel the common factor a — b, and Vien (i^ 1 make b=a, we have x^=2a. So, if re = -^. « '• for a=l, a -\-a — z we have x=-7r: but cancelling the common factor a — 1 2 before making a=l, gives x = -^. These considerations show that it is not safe to assunne the symbol ■Q- as indicating absolute indetermination, until we have ascertained whether the result has not been caused, as in the examples cited, by the presence of a common factor which becomes zero under the par- ticular supposition imposed. Finally, it should be remembered that all of these symbols, discussed in this and the preceding article, as well as some others of the same character, omitted as unsuited to an elementary work, are to be interpreted as mere abbreviations ; other- wise, they are without meaning. 13S. Theorem. — If the same quantity he added to both terms of a proper fraction^ the new fraction resulting will be greater than the first; hut if the same quantity he added to both terms of an improper fraction^ the new fraction result- ing will be less than the first. Let m represent the quantity to be added to each term of any fraction, as j ; then, the resulting traction is . Reducing * and = to a common denominator, we have ^ h b^7n ' ah \-am ah-\-bm Since the denominators are the same, that fraction which has the greater numerator is the greater. Now, if * is a proper fraction, or if a is less than 6, the second fraction is obviously greater; but if it is improper, and a greater than 6, the second is less than the first; which proves the theorem. ALGEBRAIC FRACTIONS. 77 1.39. Theorem. — If tlm same quantity he subtracted from hoth terms of a proper fraction, the new fraction resulting will be less than the first; but if the same quantity he sub- tracted from, both terms of an improper fraction, the new fraction resulting will be greater than the first. Let m represent the quantity to be subtracted from each term of any fraction, as y ; then, the resulting fraction is b — m Reducing these fractions to a common denominatoi', we have ab—am ah — hin b'^—bm b^—brn Reasoning as in the preceding theorem, when the original frac- tion is proper, the second fraction is evidently less than the first ; when improper, it is greater. MISCELLANEOUS EXERCISES. X X — 3 X x-\-Z 18 1. Prove that X — 3 X x-\-Z X x^ — 9* ^- ^a—b){a—c) ~^ {b—a){b—c) "^ {(•—a){c—b) ' 3. Find the value of j x-\ —^ I —- I x — ^ j, when x=o}i. Ans. 9. . „ _ x-\-2a , x-\-2h , 4ab a o 4. Of — L^-p—" , when .-^=—77. Ans. 2. X — Za X — 2b a-\-b 5. Prove that the sum or difference of any two quanti- ties, divided by their product, is equal to the sum or dif- ference of their reciprocals. fi Tf ^'+^^' . ^'+^' . __f!±^L__l ''• "•' la—b){a—c) "^ (b—a)(ib—c) "^ (^c—a)(c—b) ' prove that when the terms are multiplied respectively by 6-f c, a-f-c, and a-\-b, the sum =^0 ; and that when mul- tiplied respectively by be, ac, and ah, it is =M 78 KAYS ALGEBRA, SECOND BOOK. lY. SIMPLE EQUATIONS. DEFINITIONS AND ELEMENTARY PRINCIPLES. 140* An Equation is an algebraic expression, stating t"he equality between two quantities. Thus, X — 5=3 is an equation, stating that if 5 be subtracted from cc, the remainder will be 3. 141. Every equation is composed of two parts, sepa- rated from each other by the sign of equality. The First Member of an equation is the quantity on the left of the sign of equality. The Second Member is the quantity on the right of the sign of equality. Each member of an equation is composed of one or more terms. 142. There are generally two classes of quantities in an equation, the known and the nnhnown. The Known Quantities are represented either by num- bers or the first letters of the alphabet ; as, a, &, c, etc. The Unknown Quantities are represented by the last letters of the alphabet ; as, cc, y^ z^ etc. 143* Equations are divided into degrees, called firsf^ second, tliird, and so on. The Degree of an equation depends on the highest power of the unknown quantity which it contains. A Simple Equation, or an cqiiation of the first degree, is one that contains no power of the unknown quantity higher than the first. SIMPLE EQUATIONS. 79 A Quadratic Equation, or an equation of the second degree, is one in which the highest power of the unknown quantity is a square. Similarly, we have equations of the third degree, fourth degree, and so on. Those of the third degree are generally called cubic equations ; and those of the fourth degree, hiquadratic equations. Thus, ax — 6==rC, is an equation of the 1st degree. x--\^2px—q^ " " " 2d " or quadratic equation. x^—px^zzq^ " " " 3d " or cubic " X^-\^ax^-\^px^q, " " 4tli " or biquadratic " x^-{^ax^-^^bx"-'^—c, " nth degree. When any equation contains more than one unknown quantity, its degree is equal to the greatest sum of the exponents of the unknown quantity, in any of its terms. Thus, xy-f ax-— 5?/=C, is an equation of the 2d degree. X-y-\x^- — cx—a^ is an equation of the 3d degree. 144. A Complete Equation of any degree is one that contains all the powers of the unknown quantity, from O^up to the given degree. An Incomplete Equation is an equation in which one or more terms are wanting. Thus, X^-j-px-j-grzrO, is a complete equation of the second degree, the term q being equivalent to qx^\ since x"=rl. Art. 82. x3-fpx--|-gx-|-r=i0, is a complete equation of the third degree. ax^=zq^ is an incomplete equation of the second degree. x3-[-^X=g, is an incomplete equation of the third degree. 145. An Identical Equation is one in which the two members are identical ; or, one in which one of the r/iem- hers is the result of the operations indicated in the other. Thus, ax—b=ax—b, 8x— 3x=ox, (x-f-3)(x— 3)=x2 — 9, are identical equations. 80 RAY'S ALGEBRA, SECOND BOOK. Equations are also distinguished as mcmerical and literal. A Numerical Equation is one in which all the known quantities are expressed by numbers; as, 2.x^-\-'^x=^0x +15. A Literal Equation is one in which the known quan- tities are represented by letters, or by letters and numbers ; as, ax-[^h=^cx-\-d^ and ax-\-h=^^x-\-b. 140* Every equation may be regarded as the statement, in algebraic language, of a particular question. Thus, a;— 5=9, may be regarded as the statement of the follow- ing question: To find a number from which, if 5 be subtracted, the remainder shall be 9. To Solve an Equation is to find the value of the unknoivn quantity. An equation is said to be verified when the value of the unknown quantity, being substituted for it, the two mem- bers are rendered equal to each other. Thus, in the equation X — 5=9, if 14, which is the true value of X, be substituted instead of it, We have, 14—5=9; Or, 9=9. 14*7. The value of the unknown quantity, in any equa- tion, is called the root of that equation. SIMPLE EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 148. All the rules employed in the solution of equa- tions are founded on this evident principle : If we perform the same operation on two equal quantities^ the results will he equal. This principle may be otherwise expressed in the follow- ing self-evident propositions, or SIMPLE EQUATIONS. 81 AXIOMS. 1. Jf^to two equal quantities^ the same quantity he added., the sums will he equal. 2. If, from two equal quantities, (he same quantity he suh~ traded^ the remainders will he equal. 3. If two equal quantities he multiplied hy the same quan- tity, the products will he equal. 4. If two equal quantities he divided hy the same quantity ^ the quotients will he equal. 5. If two equal quantities he raised, to the same power, the results will he equal. 6. If the same root of two equal quantities he extracted, the results will he equal. 140. There are two operatioDS of constant use in the solution of equations. These are Transposition, and Clear- ing an Equation of Fractions^ TRANSPOSITION. ISO. Suppose we have the equation x — 6=c. By Axiom 1, Art. 148, we may add any quantity to both members of this equation without destroying the equality. Adding b to both sides, We have, X — 6-| 6=re-|-6; Or, a;=c-f6, since — 6-|-6"0. Comparing this result with the original equation, we find that it is the same as if we had removed the term 6 to the other side of the equation, with its sign changed. Again, take the equation a;-j-6=:C. Subtracting b from both sides, Ax. 2, ic-f 6 — 6=:C — 6; Or, a:=c — b. Here again we have the same result as if we had transposed b to the other side with its sign changed. The same method may be employed in removing a term from the second member of the equa- tion to the first. Hence, Rule of Transposition. — Any quantity may he transposed from one side of an equation to the other , if at the same time, its sign he changed. 151. To Clear an Equation of Fractions. — 1. Let it be required to clear the following equation of fractions : ah he Since, by Ax. 3, Art. 148, we may multiply both members of this equation by any quantity without destroying the equality, we first multiply by a6, the denominator of the first fraction. This gives, .... X ^—=^aoa. ^ ' be Multiplying both members again by be, We have, .... bex—abx=ab^cd. (1) Dividing both members by 6, Ax. 4, Art. 148, AVe have, .... ex—ax=abcd. (2) If. instead of multiplying successively by ah and 6c, we had, at once, multiplied by aby^bc, or a62c, we would have obtained the form (1) by one operation. By multiplying both members by abe, the L.C.M. of the denominators, we would have obtained the reduced form (2). Of these three methods, the third is the most simple. Hence, Eule for Clearing an Equation of Fractions.— KwcZ the L.C.M. of all the denominators^ and multiply each tarn of the equation hy it. Clear the following equations of fractions : 2.%—%=\ Ans. 4x— 3a:=12. 3 4 3. ^ + 5=5 Ans. 3a:+2x=6a 4 6 4. |_|+^__3i Ans. 6x-3x+2x:r.84. SIMPLE EQUATIONS. 83 5. 2x-^=='^^. . Ans. 20x-2rr+6==5a:— 15. When a fraction, whose denominator is to be removed, is preceded by a minus sign, the signs of all the terms in the numerator must be changed. See Art. 46, 2d. Thus, in the above example, wt, have 20x—{2x—6)=:5x—lb, or 20a;— 2a:-t-6=^5a;— 15. 6. a:— '^^S— ^. A.12a:— 3a;+6=60— 2a;— 4. ax hx he ac X — a X — a 2nh ^ X ^ ax bx ,0 7, J 7. -T^ -f 1 =171. . . Ans. cx-4-a^x — b^x=aocm. ab be ac 8. a-|-6 a — h d^ — h'^' Ans. ax — a^ — hx-\-ab — ax-^a^ — hx-\-ah=i2nh. SOLUTION OF SIMPLE EQUATIONS CONTAINING ONLY ONE UNKNOWN QUANTITY. 15S. The unknown quantity in an equation may be combined with the known quantities, either by addition, subtraction, multiplication, or division; or by two or more of these different methods. 1. Let it be required to find the value of a;, in the equation a-\-x=b, where the unknown quantity is connected by addition. By subtracting a from each side (Art. 148), we have x=zb — a. 2. Let it be required to find the value of x, in the equation X — a=6, where the unknown quantity is connected by subtraction. By adding a to each side (Art. 148), we have x=zh-\-a. 3. Let it be required to find the value of a;, in the equation ax^:=h, where the unknown quantity is connected by multiplication. 84 RAY'S ALGEBRA, SECOND BOOK. By dividing each side by a, we have h a 4. Let it be required to find the value of .-r, in the equation a where the unknown quantity is connected by division. By multiplying each side by or, we have x=^hy^a=ab. From the solution of these examples, we see that When the nnJcnown quantify is connected hy addition^ it is to he separated hy suhtraction. When connected hy suhtraction, it is separated hy addition. When connected hy multiplication, it is separated hy division. When connected hy division, it is separated hy multiplication. 5. Let it be required to find the value of x, in the equation „ 24-2.r ^, OX — - — ^ — ^=x-\- o. Clearing of fractions, 21a:— (24— 2a:)=7x+56, Or, 21a:-24-|-2a:=7a;-l-56. Transposing, 2LT-f 2a:— 7a;=56+24; Reducing, 16a:=80; Dividing by 16, a:=^|g=5. In this solution there are three steps, viz. : 1st. Clearing the equation of fractions ; 2d. Transposition; and 3d. Reduc- ing like terms, and dividing hy the coefficient of x. Let the value of x be substituted instead of x in the original equation, and, if it is the true value, the two members will be equal to each other. This is called veri- fication. 24— 2a: Original equation, 3a: _ — =a:-f 8. 24 2v5 Substituting 5 for X, 3X5 _-i-^=5-f 8; Or, 15—2=5+8; or, 13=13. SIMPLE EQUATIONS. 85 6. Find the value of a:, in the equation X ^=^4- !-• ab be 1st step, abcx—cx—ac=abcd-\-ax. 2d step, abcx—cx—ax=abcd^ac. Factoring, {abe — c—a) a:=ac(6d-f 1). 3d step, x—~^ — ^^^—L. abc—G—a 1S3. From the solution of the preceding examples, we derive the following Rule for the Solution of a Simple Equation. — 1. If necessary, clear the equation of fractions, and perform all the operations indicated. 2. Transpose all the terms containing the unknown quan- tity to one side, and the known quantities to the other. 3. Reduce each member to its simplest form. 4. Divide both sides by the coefficient of the unknown quantity. Find the value of the unknown quantity in the following* ar^Y 2x-^ x-4. '• 14 21 ~^^^~ 4 • 1st step, 18:K-f 42— 8a:+28-(-231=21a:— 84-, 2d step, 18a:— 8x-21a:=— 231— 42— 28— 84; Sd step, — lla;=— 385, a;=^35. 8-3+2f=7f, 7|=7t. 8. 5(cc-f-l)— 2=8(a:+5) Ans. x=Q. 9. 3(a:— 2)+4=4(3— cc) Ans. x=2. 10. 5^3(4— a-)+4(3—2x)=0 Ans. x=^\. 86 11. RAY'S ALGEBRA, SECOND BOOK . , Aus. a;=:12. . . Ans. a;=10. - — -4-7 12- i+l-i+H 13.^ + ^- a; Zx 14. -2 ^=10+ -6"- ,, a;— 7i 3a;— 9 . 27— 5a; 16. 5x-?^^ + l=3x+i^+?. 3 r. Ans. ic=:A. ^ Ans, cc=14. . Ans. x=:7-^^. Ans. a;=r8. ._ 7a:+9 3a^+l 9.T-13 249— 9x , 17. — ^ ^= ^ j-^— . Ans.:rz=9. 18. i(2x—10)—j\{Sx-40)=^U—l(b1—x). Ans. cc=:l7. 19. i(4+|x)-K2x— l)=j| Ans. x=i. Ans. x=4:. 20. 3ix{28-(|+24)}=3ix{2>+|} Ai 21. K=^-y)-A(l-3.^)=x-3'g( 5x- ^^-^-^ ). Ans. a;=rll. When one or more of the denominators is a compound quantity, as in the two following examples, it is generally best to multiply all the terms by the L.C.M. of the other denominators, collect the terms, and proceed as before. ^^ 9.r-f3 ^ 3x-6 . , 3a:-f 22 9 Ans. x=S. 23. ?^+^ ic-f2 3.T,— 9 ^, , 3.r+9 . . . -12-^2| + ^^. Ans. 0.^5. -9^ 24. hx-{-2x—a=Sx—2c Ans. x 25. a'x-]-h^=h^x-\-a^ Ans. x= b—V 26. a-\-b ' :^b^=a''-j-bx Ans. x^a-\-b. SIMPLE EQUATIOXS. 87 ^u. hx d a ex . ad ^t' =T 7 Ans. a;=-7- a c b d he »rt a — h a-\-h . c ^ 29. 5-l-^+3a6=0.. . . Ans. x=ex Subtracting and adding 3, -^-}- ^^^ l^^d after 1st game. X \ of the above, or, . . -^-\- 1= second loss. X Subtracting and adding 2, ^A^ 4= had after 2d game. X 4 1 of the above, or, . . =-;+ ■=.■= third loss. 7 14 7 Zx 24 Subtracting, .... ---(-"^= had after 3d game. Zx 24 Then, —+-y =12; from which we find a;=S;20. 6. Out of a cask of wine which had leaked away i, 35 gallons were drawn, and then, being gauged, it was -J full ; how much did it hold? Let a:-- the number of gallons it held; Then, f= « " " leaked out. X There had been taken away c+^^ gallons. 2d Bk. 8 90 RAY'S ALGEBRA, SECOND BOOK. There remained x— ( ^+35 \ gal.; .-. x— I f+35 W--. From which the answer is readily found. Y. A laborer was engaged for 20 days. For each da;y that he worked, he received 50 cents and his boarding ; and for each day that he was idle, he paid 25 cents for his boarding. At the expiration of the time, he received $4 ; how many days did he work, and how many days was he idle? Let , . . x=: the number of days he worked ; Then, . . 20— rr=3 " " " " was idle. Also, . . . 60a:z=r wages due for work. And . . . 25(20—0:)= the amount to be deducted for boarding. .-. 50a:— 25(20 -ic) =400. From which the answer is readily found. 8. What two numbers are as 3 to 5, to each of which, if 9 be added, the sums shall be to each other as 6 to 7? Let 3a:= the first, and 5a:= the second number. Then, 3a:+9 : 5:c+9 : : 6 : 7. But in every proportion, the product of the means is equal to the product of the extremes. (Ray's Arith., 3d Book, Art. 200.) Hence, 6(5a:4-9)=7(3a:-f 9). From which the answer is readily found. AVhen, as in the above example, two or more unknown quantities have to each other a given ratio, A&&uyne each of them a multiple of some other unknown quantity^ so that they shall have to each other the given ratio. 9. A courier, who traveled at the rate of 31^^ miles in 5 hours, was dispatched from a certain city ; 8 hours after his departure, another courier was sent to overtake him, who traveled at the rate of 22^ miles in 3 hours. In what time did he overtake the first, and at what distance from the place of departure? Let X^= the number of hours that the second courier trnvols. Then, since the first courier travels at the rate of 3n miles iu SmrLE EQUATIONS. 91 GHx b hours; that is, -fg- miles in 1 hour, he will travel -y^ miles in X hours ; and since he started 8 hours before the second courier, the whole distance traveled by him will be (8-f rr)63. Again, since the second courier travels at the rate of 22^ miles in 3 hours, that is, 4^5 miles in 1 hour, he will travel 45^ miles in X hours. But the couriers are together at the end of the time X] therefore, the distance traveled by each must be the same. Hence, 45a: — -:=(8-fa:)|3| from which the answer is readily found. 10. A smuggler had a quantity of brandy, which he ex- pected would sell for 198 shillings ; after he had sold 10 gallons, a revenue officer seized one third of the remainder, in consequence of which, what he sold brought him only 162 shillings. Required the number of gallons he had, and the price per gallon. Let x= the number of gallons ; Then, — is the price per gallon, in shillings; and '—^ — is the X o quantity seized, the value of which is 198 — 162=36 shillings. /p 20 198 X — =36; from which the answer is readily found. 2 ^^ X 11. There are three numbers whose sum is 133 ; the sec- ond is twice the first, and the third twice the second. Re- quired the numbers. Ans. 19, 38, and 76. 12. There are three numbers whose sum is 187 ; the sec- ond is 3 times, and the third 4^ times, the first. Required the numbers. Ans. 22, 66, and 99. 13. There are two numbers, of which the first is 3| times the second, and their difference is 100. Required the numbers. Ans. 40 and 140. 14. Two numbers are to each other as 3 to 7 ; if 16 be added to the first and subtracted from the second, the sum will be to the difference as 7 to 3. What are the numbers ? Ans. 12 and 28. 92 RAY'S ALGEBRA, SECOND BOOK. 15. What two numbers are to each other as 2 to 3, to each of which if 6 be added the sums will be as 4 to 5 ? Ans. 6 and 9. 16. A person, at the time of his marriage, was three times as old as his wife, but 15 years after he was only twice as old. What were their ages on their wedding day? Ans. Man 45, and wife 15. 17. A bill of $34 was paid in half dollars and dimes, and the number of pieces of both sorts was 100 ; how many were there of each ? Ans 60 half dollars, 40 dimes. 18. There are three numbers whose sum is 156 ; the sec- ond is 3} times the first, and the third is equal to the re- mainder left, after subtracting the diiference of the first and second from 100. Required the numbers. Ans. 28, 98, and 30. 19. What number is that, whose half, third, and fourth parts, taken together, are equal to 52? Ans. 48. 20. What number is that, which being increased by its six sevenths, and diminished by 20, shall be equal to 45 ? Ans. 35. 21. What nurdber is that, to which if its third and fourth parts be added, the sum will exceed its sixth part by 51 ? Ans. 36. 22. Find a number which, being multiplied by 4, be- comes as much above 40 as it is now below it, Ans. 16. 23. What number is that, to which if 1 6 be added, 4 times the sum will be equal to 10 times the number in- creased by 1 ? Ans. 9. 24. If a certain number be multiplied by 4, and 20 be added to the product the sum will be 32. What is the number? Ans. 3. 25. If 5 be subtracted from three fourths of a certain number the remainder will be equal to the number divided by 3. Required the number. Ans. 12. SIMPLE EQUATIONS. 93 26. The rent of an estate is greater by 8 % than it was last year. The rent this year is $1890. What was it last year? Ans. $1750. Observe that the interest on any sum of money is found by mul- tiplying the principal by the rate per cent., and dividing by 100. 27. An estate is divided as follows : The eldest child receives one fourth, the second 20 %, and the third 15 % of the whole. The remainder, which is $2168, is given to the widow. Required the value of the estate, and the share of each child. Ans. Estate $5420; shares $1355, $1084, and $813. 28. The sum of two numbers is 30 ; and if the less be subtracted from the greater, one fourth of the remainder will be 3. Required the numbers. Ans. 9 and 21. 29. A laborer was engaged for 28 days, upon the con- dition that for every day he worked he was to receive 75 cents, and for every day he was absent, he was to forfeit 25 cents. At the end of his time he received $12. How many days did he work? Ans. 19. 30. At what time between two and three o'clock will the hour and minute hands of a watch be together? Ans. 2h. 10m. 54/y sec. The face of a watch is divided into 60 minute spaces, and the minute hand moves twelve times as fast as the hour hand. Let x= distance from XII to the point of meeting; it will also express the number of min. after 2 when the hands are together. Let X= No. min. after 2 o'clock, or distance min. hand has gone. Then, .t— 10=iz distance hour hand has gone after 2 o'clock. x^ 12(a:~10)==12a;— 120; lla:=:120; and x=.\Ol^ min., or the hands are together 10 min. 54-fi- sec. after 2 o'clock. 31. The hour and minute hand of a clock are together at noon ; when are they next together ? Ans. Ih. 5/j min. 94 RAY'S ALGEBRA, SECOND BOOK. 32. At what time between 8 and 9 o'clock are the hour and minute hands of a watch opposite to each other ? Ans. 8 h. lOlJ min. 33. A has three times as much money as B, but if B give A $50, then A will have four times as much as B. Find the money of each. Ans. A, $750 ; B, $250. 34. From a bag of money which contained a certain sum, there was taken $20 more than its half j from the re- mainder, $30 more than its third part ; and from the re- mainder, $40 more than its fourth part, and then there was nothing left. What sum did it contain ? Ans. $290. 35. A merchant gains the first year, 15 % on his capi- tal ; the second year, 20 % on the capital at the close of the first ; and the third year, 25 ^ on the capital at the close of the second ; when he finds that he has cleared $1000.50. Required his capital. Ans. $1380. 36. A is twice as old as B ; 22 years ago, he was three times as old. What is A's age ? Ans. 88. 37. A person buys 4 houses ; for the second, he gives half as much again as for the first ; for the third, half as much again as for the second ; and for the fourth, as much as for the first and third together : he pays $8000 for them all. Required the cost of each. Ans. $1000, $1500, $2250, and $3250. 38. A cistern is filled in 24 minutes by 3 pipes, the first of which conveys 8 gallons more, and the second 7 gallons less, than the third every 3 minutes. The cistern holds 1050 gallons. How much flows through each pipe in a minute? Ans. 17/^, 12/^, 14l|. 39. A can do a piece of work in 3 da3"s, B in 6 days, and C in 9 days. Find the time in which all together can perform it. Ans. l-^j days. SIMPLE EQUATIONS. 95 Let a;= the required number of days. Then, in one day, A can do I, B 1, and C i, and all three - of the whole work. 3' b' y X Hence, 1+Hi=?- 40. If A does a piece of work in 10 days, whicli A and B can do together in 7 days, how long would it take B to do it alone? Ans. 23i days. 41. A performs § of a piece of work in 4 days ; he then receives the assistance of B, and the two together finish it in 6 days. Required the time in which each can do it alone. Ans. A, 14 days ; B, 21 days. 42. A person bought an equal number of sheep, cows, and oxen, for |330 ; each sheep cost $3, each cow $12, and each ox §18. Required the number of each. Ans. 10. 43. A sum of money is to be divided among five per- sons — A, B, C, D, and E. B received $10 less than A ; C, |16 more than B ; D, $5 less than C; E, $15 more than D ; and the shares of the last two are equal to the sum of the shares of the other three. Required the share of each. Ans. A, $21 ; B, $11 ; C, $27 ; D, $22 ; E, $37. 44. A bought eggs at 18 cts. a dozen, but had he bought 5 more for the same money, they would have cost him 2^ cts. a dozen less. How many did he buy? Ans. 31. 45. A person bought a number of sheep for $94 ; having lost 7 of them, he sold i of the remainder at prime cost, for $20. How many had he at first? Ans. 47. 46. There are two places, 154 miles distant from each other, from which two persons, A and B, set out at the same instant, to meet on the road. A travels at the rate of 3 mi. in 2 hr., and B at the rate of 5 mi. in 4 hr. How long, and how far, did each travel before they met? Ans. 56 hr. ; A traveled 84, B, 70 mi. 47. A person bought a chaise, horse, and harness, for $450 ; the horse came to twice the price of the harness, 96 RAY S ALGEBRA, SECOND BOOK. and the chaise to twice the price of the horse and harness. What was the cost of each ? Ans. Chaise $300, horse $100, harness $50. 48. There is a fish whose tail weighs 9 lbs. ; his head weighs as much as his tail and half his body, and his body weighs as much as his head and his tail. What is his whole weight? Ans. 72 lbs. 49. Find that number, which, multiplied by 5, and 24 taken from the product, the remainder divided by 6, and 13 added to the quotient, will still give the same number. Ans. 54. 50. In a bag containing eagles and dollars, there are three times as many eagles as dollars ; but if 8 eagles and as many dollars be taken away, there will be left five times as many eagles as dollars. How many were there of each ? Ans. 48 eagles, 16 dollars. 51. If 10 apples cost a cent, and 25 pears cost 2 cents, and you buy 100 apples and pears for 9^ cents, how many of each will you have? Ans. 75 apples and 25 pears. 52. Suppose that for every 8 sheep a farmer keeps, he should plow an acre of land, and allow one acre of pasture for every 5 sheep, how many sheep may he keep on 325 acres? Ans. 1000. 53. A person has just 2 hours spare time ; how far may he ride in a stage which travels 12 miles an hour, so as to return home in time, walking back at the rate of 4 miles an hour? Ans. 6 miles. 54. If 65 lbs. of sea-water contain 2 lbs. of salt, how- much fresh water must be added to these 65 lbs., in order that the quantity of salt contained in 25 lbs. of the new mixture shall be reduced to i of a lb.? Ans. 135 lbs. 55. A mass of copper and tin weighs 80 lbs. ; and for every 7 lbs. of copper, there are 3 lbs of tin. How much copper must be added to the mass, that for every 11 lbs. ol' copper there may be 4 lbs. of tin? Ans. 10 lbs. SIMPLE EQUATIONS. 97 56. A merchant maintained himself for three years, at a cost of $250 a year ; and in each of those years augmented that part of his stock which was not so expended, by J there- of At the end of the third year his original stock was doubled. What was that stock ? Ans. $3700. SIMPLE EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. ISS. When an equation contains two or more unknown quantities, the value of any one of them is entirely depend- ent on the rest, and can become known only when the values of the rest are given, or known. Thus, in the equation the value of x depends on the values of y and a, and cau only become known when they are known ; therefore, To find the value of any unknown quantity, we must obtain a single equation containing it and known quantities. The method' of doing this is termed elimination, which may be defined briefly, thus : Elimination is the process of deducing, from two or more equations containing two or more unknown quantities, a single equation containing only one unknown quantity. There are three principal methods of elimination : 1st. Elimination hy Suhs^fitution. 2d. Elimination hy Comparison. 3d. Elimination hy Addition and Siihtraction. 150. Elimination by Substitution consists in finding the value of one of the unknown quantities in one of the equations, and substituting this value in the other equa- tion. 2a Bk. 9* 98 RAY'S ALGEBRA, SECOND BOOK. To explain this method, let it be required to find the values of X and y^ in the following equations: 2a:+3?/=33, (1) 4a:+5?/=59. (2) From (1), by transposing Zy and dividing by 2, we have 33-3y Substituting this value of a:, instead of x in (2), we have 66—6^+^=59; -2/=-7; 2/=7; a;^3 3-3X7^6 The following is the general form to which two equations of the first degree, containing two unknown quantities, may always be reduced. The signs of the known quantities, a, 6, c, etc., may be either plus or minus. axA-hyz^c, (1) a'x-^b'y=&. (2) From (1), by transposing 6y, and dividing by a, we have a Substituting this value of x in (2), we have a'c—a'by -f- ab'y=a&\ {ah' — a'b)y=a& — a'c ; _ ac'—a'G y-ab'—a'b' I a&—a'c \ ^_€-by_<^-f>\ ab'-a'b } _ ab'c-a'bc-ab& -\-a'bG jjut ^— ^— _ — a{ab'—a'b) h b'c—bc' „ = -1.7 >j,' Hence, Rule for Elimination ty Substitution. — Find an expres- sion for the value of one of the unknown quantities in either SIMPLE EQUATIONS. 99 equation^ and substitute this value, instead of the same un- known quantity, in the other equation; there will thus he formed a new equation, containing only one unknown quantity. 1. Zx—by— 2, \ Ans. cc=4, 2a:+7i/=r22. J y=2. 2. 5a;— 3(a;— ?/)=13, | A. x=b, x—y—A. i y=l- 3. 4x=G^—3y, ^ Ans. a:=rlO, 2a; +3^=44. / y= 8. 4x-^Sy=8y+3. 5. ax^by=c — d, i _n{c—d) _m{G--d) \ . n(c—d) ) Ans. x=z^ ^, y- ) an^brrv ^ imx=ny. ) an^brrv ^ an-\-brrb Remark. — This method is always to be preferred where the value of one of the unknown quantities may be found in terms of the other, as in examples 4 and 5 above. 157. Elimination by Comparison consists in finding the value of the same unknown quantity in two different equations, and then placing these values equal to each other. To illustrate this method, we will take the same equations as in the preceding article. 2a;+3y=33, (1) 4a;+5y^59. (2) From (1), by transposing and dividing, we have a;= — ^ — . KQ tyj. From (2), by transposing and dividing, we have a;= — ^ — . Placing these values of x equal to each other, 59— 5j/ _33— 31/ 4 "~~2~' 59—5^ = 66 — 63/, by clearing of fractions; y = 7, by transposition. The value of x may be found similarly, by first finding the values of y, and placing them equal to each other. Or, it may generally be found most readily by substitution. Thus, 4a;+5X7=59; Whence, x=^^^^=.Q. 100 RAY'S ALGEBRA, SECOND BOOK. General equations, ax-{-by=iCj (1) a'x-\-l/y=&. (2) From (1), by transposing and dividing, Xz= -. Prom (2), by transposing and dividing, a;= ^' . Equating these values of x^ c—hy_&—h'y a^c—a^by=a& — Clb^y, by clearing of fraction ; {ah^—a'b)y—a&—a'G^ by transposing; a& — a^G „ ,^, c—ax ^ ,„, &—a^x From (1), y^—^- ; from (2), y^ ^, . Equating these values of 2/, &—a^xG — ax ~~¥~""~b~' b&—a^bx= b^c-ab^x ; {ab'—a'b)x=b'c~b&\ b'c-b& „ iC— -XT Til- Hence, Rule for Elimination by Comparison. — Find an expres- sion for the value of the same unknown quantity in each of the given equations^ and place these values equal to each other; there will thus he formed a new equation^ containing only one vnknown quantity. 1. 3a:— 2?/= 9, ^ Ans. ic=5, 2. 7a;4-y=102/+7, ^ Ans. x^\0, x^y= 22/+3. / y^ 7. 3. 4c-f- ^=51, y Ans. x=.Q, %x—\^y=z 9. J y=Z. 4. ma:=7?y, ) an ^ ^ Ans. x= — — . a. f x-\-y=a. J m-f ri am y-= m-\-n 5. ax-\-by—p^'\ bq—dp cx-\-dy^q. J ^~bc-ad} aq-cp^ ^ ad— be Remark.— This method is generally to be preferred v^'here the equations are literal, and sometimes in other cases. SIMPLE EQUATIONS. 101 158. Elimination by Addition and Subtraction con- sists iu multiplying or dividing two equations, so as to render the coefficient of one of the unknown quantities the same in both ; and then, by addition or subtraction, caus- ing the terms containing it to disappear. Taking the same equations as in the preceding articles, 2a;+3^=33, (1) 4xi-5y^69. (2) It is evident if we multiply (1) by 2, that' the ^coefficient of a; will be the same in the two equations. I ; i ',J I ', ,* >^ 'j l^ 4x-\-6i/=:66 (3), by multit)iyiog.(l),biy',2;, , ,., .^ , , 4x~\-by=iz59, (2) brought. davvH:. • , ; ' ' \^,' J ','^ ]\l ^ , 2/:=:: 7, by subtractiou. If the signs of the coefficients of x had been different^ the terms in X would have been canceled by adding. Having obtained the value of y^ that of X may be obtained in the same way, or by substitution. Thus, Multiply (1) by 5, and (2) by 3, and the coefficients of y will be the same in both. 10a:+15i/=.lG5, (4) by multiplying (1) by 5. ]2a:+157/=177, (5) by multiplying (2) by 3. '2xzzzl2, by subtracting (4) from (5). x= 6. Or, by substitution, from either of the original equations. Thus, From (1) 2rc-f-3x7=33 ; 2a:=33-21=12; x=6. General equations, ax-\-by=zC, (1) a'x-\-b'y=^c^. (2) It is evident that we shall render the coefficients of X the same in both equations, by multiplying (1) by a^, and (2) by a. aa^x-\-a^by=:za^c, (3), by multiplying (1) by a^; aa^x-\-ab^y=a&, (4), by multiplying (2) by a; {ab^~a^b)y=a&—a^c, by subtracting; ac^—a^G 102 RAY'S ALGEBRA, SECOND BOOK. The coefficients of y in the two equations will evidently become equal by multiplying (1) by 6'', and (2) by b. ' ah^xA^bh^y—l/c, (5) by multiplying (1) by b^; a'bx^bb'y=bG% (6) by multiplying (2) by 6; {ab'—a^b)x—b'G—b&^ by subtracting; b'c-b& „ x^=^—j~, jy. Hence, Rule for Elimination by Addition and Subtraction. — 1. Multiply or divide fjie equations, if necessary, so that one of the unknown c^Hxntities will have the same coefficient in hoth. _ , . -- ' Yy^u^l^a'he^lTi^ d^ffe^^^ the sum, of the equations, accord- ing as the signs of the equal terms are alike or unlike, and the resulting equation will contain only one unknown quantity. Remark. — When the coefficients of the quantity to be eliminated are prime to each other, multiply each by the other. When the coefficients are not prime, multiply by such numbers as will produce their L.C.M. If the equations have fractional coefficients, they ought to be cleared, before applying the rule. 1. a:+3?/.=]0,-) Sx-\-2y^ 9. J 2. 2rr+3?/=18, ^ Sx-2y--= 1.) 3. 2x~ %=n Sx-12y=l^ :} Ans. a:=:l, y=S. Ans. x=zS, Ans. x=zl, 2/=-l. 40 " 2x—y a^-2/, +^y=h 5. x-{-ay=b ax —C.) A. x= y= A. x=.^, ac-\-b2 a^^bi ab — c a^b' The following examples may be solved by either of the three methods of elimination : 1. ^x-Ay^ 8,-) 13a;-f7y=101.J 2. x-:^{y- 2)^5, Ay-l{x^\Q)=.Z. Ans. X=A, A. a;=5, y-2. q ^^y ft 10 ^1. Ans. iC— 18, y-10. SIMPLE EQUATIONS. 103 4. 5^1/ 1 Ans. a:=:2. 8a;-4zz.9?/, ;} 6. K^+^)+K^-2/)=59, 6a;— 33z/=0. 3a:+4v+3 2x+l~y y-8 %+5x— 8 _ x-f^ 7a:4 6 12 ~T~ Ans. X- -6, y -4. Ans. X— 99, y^ 15. Ans. a:; =7, y -9. 8. ax -by x\y 1} 9. Sax -2by=c, l 10. — [--=«, ^3/ 11 be , ac Ans. a:—- y, nxid y— -.. Ube , c/ 156— a \ 2a-^3ab' ^ b\ 2a-f36 / X y 11. (a2_&2)(^5a;_^3y)^(4o^_5)2a6, «62C Ans. X- y ma—nb^ mb—nd' a-y j^-^{a-\-b^c)bx=b^y-{-{a']~2b)ab. Ans. Xz y^ a^b' ab a -b' Rb:m ARK. —Transpose b-y \xi (2), multiply by 3, and subtract; there will then result an equation involving X. PROBLEMS PRODUCING SIMPLE EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. ISO. The questions contained in Art. 154, may all be solved by using one unknown symbol, although, in some cases, there were two or more unknown quantities. It frequently happens, however, that the conditions of a problem are such as to require the use of two or more sym- bols for the unknown quantities. In this case, the number of equations must be equal to the number of symbols, and 104 RAYS ALGEBRA, SECOND BOOK. the value of the unknown quantities may be found by either of the three methods of elimination. A problem may often be solved by using either one or more unknown quantities. In illustration, take the fol- lowing : 1. The difference of two numbers is a, and the less is to the greater as m to n ; required the numbers. Solution by using one unknown quantity. Let mx=z the less number, and nx= the greater. Then, nx—mx=a. a ma ^ na .'. mx=z , and nxz n-^m n—m n — m Solution by using two unknown quantities. Let x= the less number, and ?/=: the greater. Then, y — x=a, (1) And x\y\\in\n\ or, my—nx. (2) nx Since myzr^nx^ we have y^^- — -; Substituting this value of y in (1), we find as before, ma , na X= , and ?/= . n—m ^ n — di 2. The hour and minute hands of a watch are opposite at 6 o'clock ; when are they next opposite ? Let X^= minute spaces moved over by the hour hand, and 2/~ min- ute spaces moved over by the minute hand. Then, since the minute hand moves 12 times as fast as the hour hand, x:y'.'.l\\2, or y=^l2x. (1) But the minute hand must evidently pass over 60 minutes moro than the hour hand ; hence, i/^x+60. (2) Substituting, 12a:=a:-f 60, llrc=60. Xz=b^~ min. y=i65JL min. =1 h., 5A. m. Hence, the hands are next opposite at 5^^ m. past 7. In a similar manner the period of coincidence of tlie hands may be found. SIMPLE EQUATIONS. 105 3. There is a number consisting of two digits, which divided by the sum of its digits, gives a quotient 7 ; but if the digits be written in an inverse order, and the number thence arising be divided by the sum of the digits -{-4, the quotient =3. Required the number. Ans. 84. In solving questions of this kind, observe that any number con- sisting of two places of figures, is equal to 10 times the figure in the ten's place plus the figure in the unit's place. Thus, 35 is equal to 10x3+5; 456^100X4+10X54 6. Let a:= the tens' digit, and ^= the units' digit. Then, 10x+2/= the number. And 102/+a:i=: the number when the digits are reversed. Also, ^.^^^1. i^,=3. From these equations we readily find a:=:8, and 2/=4. 4. A farmer sells to one man 5 sheep and 7 cows for fill, and to another, at the same rate, 7 sheep and 5 cows for $93. Required the price of a sheep and of a cow. Ans. Sheep, $4; cow, $13. 5. If 7 lbs. of tea and 9 lbs. of coffee cost $5.20, and, at the same rate, 4 lbs. of tea and 11 lbs. of coffee cost $8.85, what is the price of a lb. of each? Ans. Tea, 55c. ; coffee, 15c. G. A and B are in trade together with different sums ; if $50 be added to A's money, and $20 be taken from B's, they will have the same sum; but if A's money were 3 times, and B's 5 times as great as each really is, they would to- gether have $2350. How much has each ? Ans. A, $250; B, $320. 7. A and B together have $9800 ; A invests the sixth- part of his money in business, and B the fifth part, and then each has the same sum remaining. How much has each? Ans. A, $4800 ; B, $5000. Suggestion. — Let 6a;== A's money, and by^^ B's. lOG RAY S ALGEBRA, SECOND BOOK. 8. Find a fraction, such that if the numerator and de- nominator be each increased by 1, the value is l; but if Bach be diminished by 1, the value is |. Ans. ^. 9. Find two numbers, such that ^ of the first exceeds I of the second by 3, and | of the first and i of the sec- ond are together equal to 10. Ans. 24 and 20. ^ ^ 10. A grocer knows neither the weight nor the first cost of a box of tea he had purchased. He only recollects that if he had sold the whole at 30 cts. per lb., he would have gained $1, but if he had sold it at 22 cts. per lb., he would have lost $3. Required the number of lbs. in the box, and the first cost per lb. Ans. 50 lbs. at 28 cts. 11. The rent of a field is a certain fixed number of bu. of wheat, and a fixed number of bu. of corn. AVhen wheat is 55 cts., and corn 33 cts. per bu., the portions of rent by wheat and corn are equal ; but when wheat is 65 cts. and corn 41 cts., the rent is increased by $1.40. What is the grain rent? Ans. 6 bu. of wheat, 10 of corn. 12. The quantity of water which flows from an orifice is proportional to the area of the orifice, and the velocity of the water. Now, there are two orifices, the areas being as 5 to 13, and the velocities as 8 to 7; and from one there issued in a certain time 561 cubic feet more than from the other. How much water did each discharge ? Ans. 440 and 1001 cubic feet. 13. Find two numbers in the ratio of 5 to 7, to which two other required numbers, in the ratio of 3 to 5, being respectively added, the sums shall be in the ratio of 9 to 13 ; and the difibrence of those sums :=16. Ans. 30 and 42, 6 \nd 10. * 14. A boy spends 30 cts. in apples and pears, buying his apples at 4 and his pears at 5 for a ct. ; he then finds that half his apples and ] of his pears cost 13 cts. How many of each did he buy ? Ans. 72 apples, GO pears- SIMPLE EQUATIONS. 107 15. A farmer rents a farm for $245 per year ; the till- able land being valued at $2 per acre, and the pasture at $1.40 ; now the number of acres of tillable, is to half the excess of the tillable above the pasture, as 28 to 9. How many acres are there of each? Ans. 98 acres tillable, 35 of pasture. 16. Find that number of 2 figures to which, if the num- ber formed by changing the places of the digits be added, the sura is 121 ; and if the less of the same two numbers be taken from the greater, the remainder is 9. Ans. 65 or 56. lY. To determine three numbers such that if 6 be added to the first and second, the sums will be in the ratio of 2 to 3 ; if 5 be added to the first and third, the sums will be in the ratio of 7 to 11 ; but if 36 be subtracted from the second and third, the remainders will be as 6 to 7. Ans. 30, 48, 50. Suggestion.— Let 2a:— 6, Sx—6, and 2/ be the numbers. 18. Two persons, A and B, can perform a piece of work in 16 days. They work together for 4 days, when A, being called off, B is left to finish it, which he does in 36 days more. In what time could each do it separately ? Ans. A in 24, B in 48 days. 19. A and B drink from a cask of beer for 2 hr., after which A falls asleep, and B drinks the remainder in 2 hr. and 48 min. ; but if B had fallen asleep and A had con- tinued to drink, it would have taken him 4 hr. and 40 min. to finish the cask. In what time could each singly drink the whole? Ans. A in 10, B in 6 hrs. 20. Di^^ide the fraction | into two parts, so that the numerators of the two parts taken together shall be equal to their denominators taken together. Ans. i and j^. 21. A purse holds 19 crowns and 6 guineas. Now 4 crowns and 5 guineas fill i| of it. How many of each will it hold? Ans. 21 crowns or 63 guineas. 108 RAY S ALGEBRA, SECOND BOOK. 22. When wheat was 5 shillings a bu. and rye 3 shil- lings, a man wanted to fill his sack with a mixture of rye and wheat for the money he had in his purse. If he bought 7 bu. of rye and laid out the rest of his money in wheat, he would want 2 bu. to fill his sack ; but if he bought 6 bu. of wheat, and filled his sack with rye, he would have 6 shillings left. How must he lay out his money, and fill his sack ? Ans. Buy 9 bu. of wheat, and 12 rye. SIMPLE EQUATIONS, INVOLVING THREE OR MORE UN- KNOWN QUANTITIES. lOO. Simple equations, involving three or more un- known quantities, may be solved by either of the three methods of elimination, explained in Arts. 155 to 159; but the third method is generally to be preferred. 1. Given 5cc— %+2z=r.48, (1) Sx-]-Sy—^z=24, (2) 2x — bi/-^Sz^l9, (3) to find X, y^ and z. To eliminate z from (he first two equations, multiply (1) by 2, and then add this to (2) , thus, 10a:— 8?/+42:= 96, by multiplying (1) by 2, 3a;+3j/— 40= 24, (2) 13x— 52/ =120, (5) by adding. To eliminate z from equations (1) and (3), multiply (1) by 3, and (3) by 2, and then subtract; thus, 15a;— 12?/+62r=144, by multiplying (1) by 3, 4a;— 10?/-|-62r^ 38, by multiplying (3) by 2, 11a;— 1y =106, (6) by subtracting. To eliminate y from equations (5) and (6), multiply (5) by 2, and (6) by 5, and then subtract; thus, 55a;— 10r/=530, by multiplying (6) by 5, 26a; -10^=2 40, by multiplying (5) by 2, 29a; =290; by subtracting. a;=10. SIMPLE EQUATIONS. 109 110— 2?/— 106, by substituting 10 for x in (6); whence, 2/=2. 50— 8+22=48, by substituting for x and y in (1); whence, Z—^. It is evident that the same method may be applied when the num- ber of equations is four or more. Hence, General Rule for Elimination by Addition and Sub- traction — 1st. Combine any one of the equations with each of the others^ so as to eliminate the same unknown quantity ; the number of equations and of unknown quantities will be one less. 2d. Combine any one of these new equations with each of the others., as before; the 7iumber of equations and of unknown quantities will be two less. 3d. Continue this series of operations until a single equation is obtained^ with one unknown quantity., and fnd its value. 4th. Substitute this value in the derived equations, for the values of the other unknoicn quantities. Remark — In some particular instances, solutions maybe ob- tained more easily and elegantly by other means. As specimens, we present the following: 2. Given — x-^y-\-z=za, (1) x-y^z^b, (2) x-\-y — 2;=c, (3) to find a:, y., and z. By adding the three equations together, and calling a-\^b-\-C=:8, We find x^y-\-z=8. (4) Then, by subtracting (1), (2), and (3), respectively from (4), and dividing by 2, We find . . . x=\{s—a), =\{b^c). y=l{s-b), =l{a-\-c). z=.^{s^c), =l{a+b). In a similar manner, solve the following examples: 3. x+y+z=22, y-^z^u^2\, iC-|-2-j-«=19, Ans. x= 5, U=. 4. no RAY S ALGEBRA, SECOND BOOK. 4. ^x-y-z=\% (1)^ 3i/-x-.=16, (2)[ 52— a:— y=24. (3) 3 Put a;4-2/+0=s, and add (1), (2), and (3) successively to the last equation. This gives 3a;=s-fl2 (a) 4^=5+16 (6) 60=s+24 (c) Multiplying these by 4, 3, and 2, we have 12a;=4s+ 48 12i/^3s+ 48 120=2s+ 48 Or, 12(a:-f2/+2;)=x9s-f 144, by addition. . . 12s^9s-fl44; 3s==144; 8= 48. Substituting this value of S in (a), (6), and (c), we find iC— 20, 1^=16, and 0=12. Solve the following by either method of elimination 5. a:-|-3/-f2=6, "^ Ans. a:=l, 4x-f33/-2=Y, 6. 3x+4^— 5^=32, 4x— 5^^+32=18, ^x—'^y—\z= 2. 7.^_9^+32-10«=21, 2x+73/-z-«=683, 3.^+y4_5z-|-2wrrrl95, 4a;— 6y— 22— 9«=516. 8. OJ+^yr^lO— J2, i(a;-2):^2y-7. y=2, 2==3. Ans. a;=;10, 3^= 8, z= 6. Ans. crr=100, 2=:-13, . Ans. a;^r=7, ;5:-3. SIMPLE EQUATIONS. Ill 9. 9x—2z-^u^41, lz—bu=ll. Ans. xr=.b^ 3=3, Examples to be solved by special methods : 10. 1^1 X X z y ^ =h. Ans. X- _2_ a-f-6— c' 2 2 b-{-c — a Suggestion.— Subtract (3) from (2), then combine the resulting equation with (1), to find X and y; z may be found similarly. 11. ~x-Yij^z^v=a, x—y-\rZ-\-v=^h, x-\-y Z-\-V:=C^ x-\-y-\-z—v=d, Ans. a:=J(s — a), y=l{s-h), wlierc s=:\(^a-\-h-{-c-\-d'). PROBLEMS PRODUCING SIMPLE EQUATIONS CONTAINING THREE. OR MORE UNKNOWN QUANTITIES. 1G1« For the method of forming the equations, see Arts. 154 and 159. 1. The stock of three traders amounts to $760 ; the shares of the first and second exceed that of the third by $240 ; and the sum of the second and third exceeds the first by $360; what is the share of each ? Ans. $200, $300, and $260. 2. What three numbers are there, each greater than the preceding, whose sum is 20, and such that the sum of the first and second is to the sum of the second and third, 112 RAY'S ALGEBRA, SECOND BOOK. as 4 is to 5 ; and the difference of the first and second, is to the difference of the first and third, as 2 to 3? Ans. 5, 7, and 8. 3. Find four numbers, such that the sum of the first, second, and third shall be 13; the sum of the first, second, and fourth, 15 ; the sum of the first, third, and fourth, 18 ; and lastly, the sum of the second, third, and fourth, 20. Ans. 2, 4, 7, 9. 4. The sum of three digits composing a certain number is 16 ; the sum of the left and middle digits, is to the sum of the middle and right ones as 3 to 3g ; and if 198 be added to the number, the order of the digits will be in- verted. Required the number. Ans. 547. 5. It is required to find tliree numbers such that i the first, ^ the second, and j the third, shall together make 46 ; J the first, | the second, and 4 the third, shall to- gether make 35 ; and | the first, i the second, and i the third, shall together make 28 1. Ans. 12, 60, and 80. 6. The sum of three numbers, taken two and two, are a, ?>, and c. What are the numbers ? Ans. -i(a+6— c), K^+c—h), and ^(h-^c—a). 7. A person has four casks, the second of which being filled from the first, leaves the first ^ full. The third being filled from the second, leaves it J full ; and when the third is emptied into the fourth, it is found to fill only -f^ of it. But the first will fill the third and fourth and have fifteen quarts remaining. How many quarts does each hold ? Ans. 140, 60, 45, and 80, respectively. 8. In the crew of a ship consisting of sailors and sol- diers, there were 22 sailors to every 3 guns, and 10 sailors over ; also the whole number of hands was 5 times the number of soldiers and guns together ; but after an engage- ment, in which the slain were one fourth of the survivors, there wanted 5 men to be 13 men to every 2 guns. Re- quired the number of guns, soldiers, and sailors. Ans. 90 guns, 55 soldiers, 670 sailors. SIMPLE EQUATIONS. 113 V. SUPPLEMENT TO SIMPLE EQUATIONS. Remark. — The principles employed in algebraic equations ma; be variously applied. We may, for example, by their aid demon- strate several of the theorems in fractions. Thus, to prove that —r = -r, Art. 118); put q=j. ♦ Then, bq=a, ^ndmbq=ma] .. q=r — a , ma ma a Hence, since 0'— 7-, and g=- ,, .-. — j- = -?-• ' ^6 mo mb b To prove that ^X^= 5^, (Art. 131); put p=-^ and g-^. Then, bp—a, and dq^C. Multiplying the last two equations, member by member, , ^ <^C , . , , , We have bdpq—ac\ .-. pq=j-j, which proves the rule. In a similar manner, the rules for Addition, Subtraction, and Division of fractions may be demonstrated. Other methods of application are given in Arts, following. I. GENERALIZATION. 16!S. Literal Equations are those in -which the known quantities are represented, either entirely or partly, by letters. Quantities represented by letters are termed general values, because the solution of one problem furnishes a general solution. A Formula is the answer to a problem, when the known quantities are represented by letters. A Rule is a formula expressed in ordinary language. 2d Bk. 10 114 RAYS ALGEBRA, SECOND BOOK, By the application of Algebra to the solution of general questions, many useful and interesting truths and rules may be established. Take the following as an example : 163. Divide a given number a into three parts, having to each other the same ratio as the numbers m, «, and p. Let mx, nx, and px^ represent the required parts. Then, mx^nx-^-px^a, a And .... iC=: ; from which we obtain, ma na . pa mx= , nx= , and px— — — m-\-n^p m-i^ri-^-p ^ 'm-\-n-]-p This formula, expressed in words, gives the following Rule for Dividing a Given Number into Parts having to each other a Given Ratio. — Multiply the given nnmher hy each term of the ratios respectively^ and divide the prod- nets hy the sum of the numbers expressing the ratios. Solve the following examples by this rule, and test its accuracy by verifying the results : 2. Divide 69 into three parts, having to each other the game ratio as the numbers 5, 7, and 11. Ans. 15, 21, and 33. 3. Divide 38] into four parts, having to each other the same ratio as the fractional numbers ], i, ], and 1. Ans. 15, 10, 1l, and 6. Solve the following general examples, express the formula in ordinary language, so as to form a general rule, and apply the rule or the formula, to the solution of the numerical problems. 4. The sum of two numbers is a, and their difference h. Required the numbers. . _, ah. ah A. Greater, ^-hg ; less, ^ — ^. SIMPLE EQUATIONS. 115 5. The joint capital of A and B in a firm, is $16000; but A's investment is $2000 more than B's. Required the capital of each. Ans. A's, $9000 ; B's, $7000. 6. The sum, of two angles is 120° 44' 52'', and their difi"erence is 26° 32' 18". Required the angles. Ans. Greater, 73° 38' 35"; less, 47° 6' 17". 7. The difference of two numbers is a, and the greater is to the less as m to n. Find the numbers, Ans. - 8. The difference in capacity of two cisterns is 678 gal., and the greater is to the less as 7 to 5. How much does each hold? Ans. Greater, 2373 gal. ; less, 1695. 9. The sum of two numbers is a, and their sum is to their difference as m to n. Required the numbers. Ans. Greater, :=:- ~T ; less, ^^-^ — - . 10. An estate, valued at $8745, was divided between a son and daughter in such a manner that the sum of their shares was to the difference as 5 to 3. What was the share of each? Ans. Son's, $6996 ; daughter's, $1749. 11. Divide the number a into three such parts, that the second shall exceed the first by 6, and the third exceed the second by c. a — 2h — c a-\-h — c a-\-h-\-2c Ans. g , g— , g . 12. At a certain election the whole number of votes cast was 602. B received 84 more votes than A, and C 56 more than B. How many did each receive ? Ans. A 126, B 210, C 266. 13. Divide the number a into four such parts, that the first increased by m, the second diminished by m, the third multiplied by w, and the fourth divided by m, shall be all equal to each other. . ma ma . a m^a (wi+l)2 ! (m-{-iy^ ' (m-j-1)^' (m+l)2' liG RAYS ALGElillA, SECOND DOOK. Let the four parts be represented by X — m, x-\-m^ — , and mx. 14. Divide the number 245 into four parts, such that the first increased by 6, the second diminished by 6, the third multiplied by 6, and the fourth divided by 6, shall be all equal to each other. Ans. 24, 36, 5, and 180. 15. A person has just a hours at his disposal; how far may he ride in a coach which travels h miles an hour, so as to return home in time, walking back at the rate of c miles an hour? . ahc .. Ans. , , miles. 6-f c 16. A person finds that he can row a skiflf 6 miles an hour with the current, and 3 miles an hour against it ; how far can he pass down the stream, and yet return to the point from which he set out, in 8 hours? Ans. 16 miles. 17. Given the sum of two numbers =«, and the quotient of the greater divided by the less =6. Required the num- bers. . ^ a ah This gives the following simple rule: To find the less nurnber, divide the sum of the numbers by (heir quotient increased by unity. 18. The sum of two numbers is 256, and the quotient of the greater by the less is 15. Required the numbers. Ans. 240 and 16. 19. A person distributed a cents among n beggars, giv- ing h cents to some, and c to the rest. How many were there of each ? . a — nc , , nh — a Ans. -,- at h cts., and —, at c cts. 6 — c b — c 20. A father divided $8500 ^mong 7 children, giving to each son $1750, and to each daughter $500. How many of his children were sons and how many daughters ? Ans. 4 sons, 8 daughters. SIMPLE EQUATIONS. 117 21. Divide the number n into two such parts, that the quotient of the greater divided by the less shall be q, with a remainder r. nq-\~r n — ?' Ans. ^^ ^i . 1+2' 1+? 22. Divide 1903 into two such parts that the quotient of the greater divided by the less shall be 12, with a re- mainder 5. Ans. 1757 and 146. 23. If A and B together can perform a piece of work in a days, A and C in ^ days, and B and C in c days : find the time in which each can perform it separately. . . . 2ahc _, . 2ahc „ . 2ahc Ans. A in — — , B in -j-^ , C in -— da. ac-\-bc — ab ab-\-uc — ac ab-\-ac — be 24. A tank is supplied with water from three pumps. The first and second will fill it in 30 hours, the first and third in 40 hours, and the second and third in 50 hours. in what time can each fill it separately ? Ans. First in 52rf*3, second in 70j?, third in l7l| hrs. 25. A, B, and C hold a pasture in common, for which they pay P $ a year. A puts in a oxen for m months ; B, h oxen for n months ; and C, c oxen for p months. Re- quired each one's share of the rent. Ans. A's, ^-^^— P $ ; B's, -^- — P%; and CX -^-- P%. ma-\-nb-{-pc From these formulas is derived the rule of Compound Fellowship. 26. A, B, and C engage in business together. A put into the firm $600 for 30 weeks, B $500 for 40 weeks, and C $800 for 28 weeks. They then divided a profit of $1812 between them. What was each man's share ? Ans. A's, $540; B's, $600; C's, $672. 27. A mixture is made of a lb. of tea at m shillings per 118 RAY'S ALGEBRA, SECOND BOOK. lb., b lb. at n shillings, and c lb. at p shillings : what will be its cost per lb. ? ^ ??m-f w/>4-«c Ans. , . , . From this formula is derived the rule termed Alligation Medial. 28. A drover bought 10 cattle at $30 apiece, 12 at $40, and 8 at $90. What was the average price per head ? Ans. $50. 29. A waterman rows a given distance a and back again in h hours, and finds that he can row c miles with the current for d miles against it : required the times of rowing down and up the stream, also the rate of the current and the rate of rowing. Ans. Time down, — j — - ; time up, — -— ; rate of current, — ^^vi — r-^: rate of rowinp', \, / . 2hcd ' ^' 2bcd 30. A vessel sailed with the wind and tide 60 miles, and returned ivith the wind and against the tide. She reached the same point in 12 hours, and the rate of sailing out and in was as 5 to 3. Required the time each way, and the strength of the wind and tide. Ans. Time out, 4\ hours ; time in, 7^ hours ; wind, 10^ miles per hour ; tide, 2| miles per hour. II. NEGATIVE SOLUTIONS. 164. It sometimes happens, in the solution of a prob- lem, that the result has the minus sign. This is termed a negative solution. We shall now examine a question of this kind. 1. What number must be subtracted from 3 that the re- mainder may be 7 ? Let x= the number. Then, 3—X— 7; whence, —x=7—3; or, z =—4. SIMPLE EQUATIONS. 119 Now, — 4 subtracted from 3, gives a remainder?; and the an- swer, — 4, is said to satisfy the question in an algebraic sense. The problem is evidently impossible in an arithmetical sense, and this impossibility is shown by the negative answer. But, since sub- tracting —4 is the same as adding -|-4 (Art. 48), the result is the answer to the following: What number must be added to 3, that the sum may be equal to 7 ? Let the question now be generalized, thus : What number must be subtracted from a, that the re- aaainder may be 6? Let X:= the number. Then, a—Xz=b\ whence, x=a—b. While b is less than a, the value of x will be positive; and the question will be consistent in an arithmetical sense. But if b becomes greater than a, the value of x will be negative; and the question will be consistent in its algebraic, but not in its arithmetical sense. When b becomes greater than a, the question, to be consistent in an arithmetical sense, should read : What number must be added to a that the sum may be equal to 6? From this we derive the following important general principles r 1st. A negative solidwn indicates smne arithmetical incorv- sistency or absurdity in tJw qv£stionfrom which the equation ima derived. 2d. When a negative solution is obtained, Hie question, to which it is the answer, may be so modified as to be consistent with arithmetixnl notions. After solving the following questions, let them be so modified that the results may be true in an arithmetical sense. 2. What number must be added to the number 30, that the sum may be 19? (x= — 11). 3. The sum of two numbers is 9, and their difference 25; required the numbers. Ans. 1*7 and — 8. 120 RAYS ALGEBRA, SECOND BOOK. 4. What number is that whose half subtracted from its third leaves a remainder 15? (x= — 90). 5. A father's age is 40 years; his son's age is 13 years; in how many years will the age of the father be 4 times that of the son? (cc=: — 4). III. DISCUSSION OF PROBLEMS. lOS. After a question has been generalized and solved, we may inquire what values the results will have, when particular suppositions are made with regard to the known quantities. The determination of these values, and the examination of the various results, to which different suppositions give rise, constitute the discussion of the problem. The various forms which the value of the unknown quantity may assume, are shown in the discussion of the following : 1. After subtracting h from a, what number, multiplied by the remainder, will give a product equal to c ? Let x= the number. Then, {a—b)x=c^ and x= a—b' This result may have five different forms, depending on the different values that may be given to «, fe, and c. To express these forms, let A denote, indefinitely, some quantity. I. When h is less than a. In this case, since a — h will be positive, the value of x will be of the form -|-A. II. When h is greater than a. In this case, a — h will be negative, and the value of ic, of the form — A. III. When h is equal to a. In this case, tho value of X is of the form -^, or, (Art. 136), x=^sxi , SIMPLE EQUATIONS. 121 IV. When c is 0, and b either greater or less than a. In this case, the value of x is of the form — , or, (Art, A ^ 136), x=0. V. When b is equal to a, and c is equal to 0. In this case, the value of x is of the form ^, which (Art. 13^) is the symbol of indetermination. The discussion of the following problem, originally proposed by Clairaut, will serve to illustrate further the preceding principles, and show that the results of every correct solution correspond to the circumstances of the problem. PROBLEM OF THE COURIERS. 1G6. Two couriers depart at the same time, from two places, A and B, distant a miles from each other ; the former travels m miles an hour, and the latter n miles : where will they meet? There are two cases of this problem, according as the couriers travel toward each other, or in the same direc- tion. I. When the couriers travel toward each other. Let P be the point where they meet, A Im^immmm^mmmhkI B and a=AB, the distance between the P two places. Let a;=AP, the distance which the first travels. Then, a — iC=BP, the distance which the second travels. But the distance each travels, divided by the number of miles trav- eled per hour, will give the number of hours he was traveling. X Therefore, — = the number of hours the first travels. m And = " " " " second travels. n 2d Bk. 11* 122 RAY'S ALGEBRA, SECOND BOOK. But they both travel the same number of hours ; therefore, X a-x m~' n ' nx^7na—7nx\ ma , na and a — Xz 1st. Suppose m—ri] then, 0:=^^ ^^o' and a— x=-^; that is, if they travel at the same rate, each travels half the distance. 2d, Suppose 71=0; then, x=z — =^a\ that is, if the second cou- rier remains at rest, the first travels the whole distance from A to B. In like manner, if m=0, a—X—a. 3d. Suppose rn^n, then the value of x will be greater than that of a— a:, since ma is greater than na; that is, the point P will be farther from A than B if m<^n, then the value of x will be less than that of a — X, or P will be nearer A. All of these results are evidently true, and correspond to the cir- cumstances of the problem. II. When the couriers travel in the same direction. As in the first case, let P be the point of meeting, each traveling from A toward -^ ■" P, and let a=AB, the distance between the places; a:=-AP, the dhtance the first travels ; x—a=BF, the distance the second travels. Then, reasoning as in the first case, we have X _^— «, m" n ' nx = mx — ma ; m^a , na X = ; and x—a-. m—n' m—n 1st. If we suppose m greater than w, the values of X and of x—a will both be positive ; that is, the couriers will meet on the right of both A and B. This evidently corresponds to the circumstances of the problem. 2d. If we suppose n greater than w, the value of x, and also that of x—a, will be negative. SIMPLE EQUATIONS. 123 Now, since the positive values of x and X — a implied that the couriers met at a point P, on the riyht of A and B, the negative values must indicate (Art. 47) that the place of meeting is at P^, on I 1 1 i V A B p the left of A and B. Indeed, where m is less than n, or the advance courier is traveling faster than the other, it is evident that they can not meet in the future. We may, however, suppose that they have met before. We may, therefore, on the principles explained in Art. 164, modify the question in one of two ways. 1st. We may inquire, Where have the couriers met? or, 2d. We may suppose the direction changed^ and call A the advance courier; that is, that they travel toward P''. We shall then have AB=:a, AP^=a:, and BP^— a-f a;. Forming and solving the equa- tion as before, we should obtain positive values of x and a-\-x. 3d. If we suppose m equal to n; then, ma , na x=-^= cc, and a:— a=— = oo. This evidently corresponds to the circumstances of the problem; for, if the couriers travel at the same rate, the one can never over- take the other. This is sometimes expressed, by saying they only meet at an infinite distance from the point of starting. 4th. If we suppose a=0; then, x= =0, and x—a= =0. m—n m — n This corresponds to the circumstances of the problem ; for, if the couriers are no distance apart, they will have to travel no (0) dis- tance to be together. 5th. If we suppose m=^n^ and a=:^0; then, a;=j{, and x—a=^. But this is the symbol of indeterminatxon^ and indicates (Art. 137) that the unknown quantity may have any finite value whatever. This, also, evidently corresponds to the circumstances of the prob- lem; for, if the couriers are no distance apart, and travel at the same rate, they will be always together ; that is, at any distance what- ever from the point of starting. 6th. If we suppose w=0; then, x= — =a; that is, the first cou- rier travels from A to B, overtaking the second at B. So, if m==Q, x—a^=—a. 124 RAY'S ALGEBRA, SECOND BOOK. 7th. If we suppose their rate of travel has a given ratio, as n=-^ ; then, X= =^za; that is, the first travels twice the dis- tance from A to B before overtaking the second. The results in the last two cases evidently correspond to the circumstances of the problem. IV. CASES OF INDETERMINATION IN SIMPLE EQUATIONS AND IMPOSSIBLE PROBLEMS. 167. An Independent Equation is one in which the relation of the quantities which it contains can not be ob- tained directly from others with which it is compared. Thus, a;4- 3^=19, are equations which are independent of each other, since the one can not be obtained from the other in a direct manner. a:+3i/^19, 2a:-f62/=:38, are not independent of each other, the second being derived directly from the first, by multiplying both sides by 2. 168. An Indeterminate Equation is one that can be verified by different values of the same unknown quantity. Thus, if we have, . . . X— 2/=3, By transposition, . . . Xz=z^^-\-y. If we make y=\\ then, a;=4. If we make 2/=^2; then, rc=5, and so on; from which it is evident that an unlimited number of Talues may be given to x and y^ that will verify the equation. If we have two equations containing three unknown quantities, we may eliminate one of them ; this will leave one equation containing two unknown quantities, which, as in the preceding example, will be indeterminate. Thus, in rr+Sy— 5^=20, X- 2/+30=16, If we eliminate a:, we have, after reducing, y—2z=l ; whence, y=l-\-2z. SIMPLE EQUATIONS. 125 If we make 0=1 ; then, 2/=3, and a:=20-f 52r— 3?/=16. If we make s=2; then, ^=5, and x=:15. So, any number of values of the three unknown quantities may- be found that will verify both equations. These examples are suf- ficient to establish the following General Principle. — When the number of unknown quan- titles exceeds the number of independent equations^ the prob- lem is indeterminate. A question that involves only one unknown quantity is sometimes indeterminate ; the equation deduced from the conditions being identical. (Art. 145.) The following is an example : What number is that, whose \ increased by the J is equal to the i^ diminished by the f-^'i Let X:= the number. _, XX 11a: 2x Then, ^ + 6=20r-l5' Clearing of fractions, 15a:-f 10a:=33a;— 8a;; or, 2oa;=25a;; which will be verified by any value whatever of x. 169. The reverse of the preceding case requires to be considered ; that is, when the number of equations is greater than the number of unknown quantities. Thus, we may have 2a:-f 3i/=23 (1.) ^x-2y= 2 (2.) 5a:+42/=40 (3.) Each of these equations being independent of the other two, one of them is unnecessary, since the values of x and ;y, may be found from either two of them. When a problem contains more conditions than are necessary for determining the values of the unknown quantities, those that are unnecessary are termed re- dundant. The number of equations may exceed the number of 126 RAY'S ALGEBRA, SECOND BOOK. unknown quantities, so that the values of the unknown quantities shall be incompatible with each other. Thus, if we have x-\- y=l2, (1.) 2a:+ y^ll (2.) 3a:-f22/=30 (3.) The values of X and ?/, found from equations (1) and (2), are a:=5, y^^^l-^ from (1) and (3), a;^G, and 2/=6; and from (2) and (3), a:i=4, and y='^. It is manifest that only two of these equa- tions can be true at the same time. EXAMPLES TO ILLUSTRATE THE PRECEDING PRINCIPLES. 1. What number is that, which being divided succes- sively by m and n^ and the first quotient subtracted from the second, the remainder shall be o ? . mnq Ans. x=. -. m — n What supposition will give a negative solution ? Will any sup- position give an infinite solution? An indeterminate solution? Illustrate by numbers. 2. Two boats, A and B, set out at the same time, one from C to L, and the other from L to C ; the boat A runs m miles, and the boat B, n miles per hour. Where will they meet, supposing it to be a miles from C to L ? ma . na . „ ^ Ans. — — - mi. irom C, or mi. from L. m-\-n m-\-n Under what circumstances will the boats meet half way between C and L? Under what will they meet at C ? At L? Above C? Below L? Under what circumstances will they never meet? Under what will they sail together ? Illustrate by numbers. 3. Given 2x—y=2, 5x— 3^=3, ^x^2y=Vl, 4rr4-3y =24 ; to find x and y, and show how many equations are redundant. (Art. 169.) Ans. a;=3, y=4. 4. Given a;+2y=ll, 2.t— 7/=Y, 3a;— ^^=17, a;+3^=19; to show that the equations are incompatible. (Art. 169 ) SIMPLE EQUATIONS. 127 V. A SIMPLE EQUATION HAS BUT ONE ROOT. ITO. Any simple equation involving only one unknown quantity, (x), may be reduced to the form mx=n; for all the terms containing x may be reduced to one term, and 11 all the known quantities to one term: whence, a;=-. ^ m Now, since n divided by m can give but one quotient, we infer that a simple equation has hut one root; that is, there is but one value that will verify the equation. VI. EXAMPLES INVOLVING THE SECOND POWER OF THE UNKNOWN QUANTITY. ITl. It sometimes happens in the solution of an equa- tion, that the second or some higher power of the unknown quantity occurs, but in such a manner that it is easily removed. The following equations and problems belong to this class : 1. {^^x)(x—b)=(x—2y. Performing the operations indicated, we have a;2_a;_20=a:-— 4a;^4. Omitting X^ on each side, and transposing, we have 3a:^24, or xS. ' ^fe?+4=^+i ^- -'■ 6x— 43' 4. .r Vo (3:r-19)^2a;4-19 Ans. a;. - aW^x:') , ax . ^ 5. -^- ^=ac+ -^ Ans. x=^-. hx ^ h G 128 RAY'S ALGEBRA, SECOND BOOK ^ ex"* dx"^ . ad — ce O. —-7-= ~r-j- Ans. x== ,. ■ . . a-\-ox €-\-/x cf — ud 7. It is required to find a number whicli being divided into 2 and into 3 equal parts, 4 times the product of the 2 equal parts shall be equal to the continued product of the 3 equal parts. Ans. 27. 8. A rectangular floor is of a certain size. If it were 5 feet broader and 4 feet longer, it would contain 116 feet more ; but if it were 4 feet broader and 5 feet longer, it would contain 113 feet more. Required its length and breadth. Ans. Length, 12 feet j breadth, 9 feet. YI. OF POWERS, ROOTS, RADICALS, AND IJ^EQUALITIES. I. INVOLUTION, OR FORMATION OF POWERS. 1T!S. The Power of a number is the product obtained by multiplying it a certain number of times by itself. Any number is the first power of itself. When the number is taken twice as a factor, the product is called the second power or square of the number. When the number is taken three times as a factor, the product is called the third power or cube of the number. In like manner, the fourth^ fifih, etc., powers of a num- ber are the products arising from taking the number, as a factor, four times, five times, etc. The Index or Exponent of the power is the number which denotes the power. It is written to the right of the number, and a little above it. FORMATION OF POWERS. 129 Thus, 3=31- . 3, is the let power of 3. 3X3=32= = 9, - 2d u " 3. 3X3X3=33= = 27, » 3d u » 3. 3X3X3X3=3^= . 81, " 4th u " 3. ixfxfx|=(|)^= = &'" 4th u "|. aX«X«X«=(«)^- -a* " 4th « " a. ac :2xac2xac2^(«c-)'''= =a-V^ " 3d u ac^. Prom the above, we have the following General Rule for Raising any Quantity to any Required Power. — Multiply the given quantity hy itsclj] until it is taken as a factor as many times as there are units in the exponent of the required power. As the application of this general rule frequently involves a tedious operation, it is best to reduce the labor attending it. It will, therefore, be most convenient to divide the subject into distinct cases. Case I. — To raise a Monomial Quantity to any Power. By inspecting the illustration above given, it will be seen that a coefficient is involved by repeated multiplica- tions, as in arithmetic, and the literal factors by repeated additions of the exponents. Thus, the 3d power of 3 is 3x3x3=27, but the 3d power of a2 is a2>4-^'"' aH6a'^^+15a^6^+20a363+15a264+6a65+66 .... (a+6)^ FORMATION OF POWERS. 133 If we involve a — 6, the result will be the same, except that the signs of the terms will he alternately plus and minus. The above results exhibit certain uniform laws of de- velopment, following which we may raise a binomial to any required power without the tedious process of multiplica- tion. These laws are as follows : 1st. Number of Terms. — The mimher of terms in any power of a binomial is one more than the exponent of the power. Thus, the 2d power has 3 terms, the 3d power 4 terms, etc. 2d. Signs of Terms. — If both terms of the binomial are positive, all the terms will be positive. If the second term is negative, the Ist, 3c?, etc., or the ODD terms, will be positive, and the even terms negative. 3d. Exponents. — TTie exponent of the leading letter is the same, in the first term, as the power to which the quan- tity is to be raised, and diminishes by unify, in the succeeding terms, disappearing in the last. The FOLLOWING letter is not found in the first term, but enters the second with an exponent of one, which exponent increases, by unity, in the succeeding terms, until it equals, in the last term, the exponent of the power. Thus, {a^bf=a^-\-oJ'b^a^b--\-a''b^-\-a-b^-{-ab^-\'b^, omitting coefficients. 4th. The Coefficients. — The coefficient of the first and last terms is always unity ; that of the second term is the same as the exponent of the LEADING LETTER in the first term. The coefficient of any other term may be found by the following rule : Multiply the coefficient of any term by the exponent of its leading letter, and divide the product by the number, express- ing the place of that term in the series for the coefficient of the succeeding term. 134 RAYS ALGEBRA, SECOND BOOK. The coefficients of all terms eqiialhj distant from the frst and last are equal. In the application of this theorem, we may first write the literal factors alone, and afterward supply the coefficients, according to the rules above given, or, we may carry for- ward both operations at the same time. Thus, Let it be required to raise x-\-y to the Tth power, or to expand (x-\~yy. Literal factors, oc^ , x^y, x^y-^ x^y^^ x^y^, x'^y\ xif^ y'^. The cotificient of the 1st term is uuity; .-. 1st term is iC^. " " " 2d " " 7 " 2d " " Ix^'y. u u u 3j u u 7X6 a gj u u 21x^y\ u a .t 4th « « 21X-^ u 4th a u 35a:4^3 Continuing thus, we have for the complete expansion, rr7-f72;*^2/+21a;'''2/2+35a:V4-35xV-f- 213:2^/^-1- 7a:?/*' +2/^. As a second example by the other method, Let it be required to expand (« — hy. The first term will be a'"'; the second, 6a^'b. For the third, multi- ply 6 by 5, and divide the product by 2, for the coefficient, and annex the literal factors. This gives 15a^b^. Multiplying 15 by 4 and dividing by 3, we have for the next term 20a^b^. Continuing this process, we find the next term to be 15a^b^, the next 6a6'*, and the last ^^ Giving the proper signs, we have af>_6a'^6-[-15a^62_20a363+15a26»_6a6^-(-66. The following additional facts may be noted, and may serve to render the application of the above principles stiH more simple : 1st. The sum of the exponents in every term is the same, and is always equal to the power of the binomial. Thus, in the first of the above examples, the sum of the exponents in every term is 7; in the second their sum is 6. 2d. If the power of the binomial be even, the number of terms will be odd ; but if the power be odd, the number FORMATION OF TOWERS. 135 of terms will be even. In the former case, there will be one middle term, and in the latter two, to the left and right of which the coefficients are the same. Thus, in the above examples, the coefficients are — For the 6th power, 1, 6, 15, 20, 15, 6, 1. For the 7th power, 1, 7, 21, 35, 35, 21, 7, 1. 3d. The smn of the coefficients, in every case, is equal to 2 raised to the required power of the binomial. Thus, in the above examples, 1-f 6+15H-20-}-15+6+l=64^2«, and l4-7+21+35-f35+21+7+l:^128.=27. Expand (a-{-hy. . . . Ans. a'-\-4:a^h-{-Qa'h^^4ah^-^hK Expand (x-\-i/y. Ans. a:«+6a:^-fl5a:y-f20ccy-fl5a;y-|-6.T/-f/. Expand («— a-)^ A. a^— 5a*x-f lOa^x'— 10aV+5aa;*— a:\ Expand (a-f-cc)^. Ans. a«4-8a^T-f28aV+56aV-f70aV+56aV+ 28aV-f8ax^H-x8. Expand (a—iy. Ans. a^—9a%-\-B6a-'h'—S4a^h'-^126a'h'—12Qa'¥ If one or both of the terms of the binomial have a coeffi- cient or exponent greater than unity, or more than one literal factor, the expansion may be made in the same way, after which the operations indicated must be completed. Thus, {2x^-{-5a^y={2x^y-^A{2x^)^{5a^)-\-6{2x^y{5a^Y-\~4{2x^) (5a2)3+(5a2)43^16a:i2_f_i60a:^a2-f600a:''a*+1000a:3a6+625a«. Or, put m=2x^ and n=5a^. Then, (2a;3-|-6a2)4=(m-f ti)^. Then, (w-f n)'<=m'* +4m3n-(- 67n2n2-f 4mn3-f n*. Returning to the values of m and n, we have m*=(2x^y=16x^\ 4m^n =4x(2rr3)3x 5a2 ^AxSx^X ^^^= 160a;%2. 6m2n2z=6X(2a:3)2x;5a2)2^6x4a;6x 25a4= 600x^aK 4m n^=4x 2x^ X^5a2)3^4y2a:3xl25a6z=1000a;3a6. n4=(5a2)4=625as. Hence, {2x^-{ba^y^lQx^-^ lQ0x^a^-\-600x^a*-j-1000x^a^-]-625a». 136 RAY S ALGEBRA, SECOND BOOK. In a similar manner, a quantity consisting of three or four terms may be involved, by first reducing it to the form of a binomial, as explained in Cases II and III. 1. Raise a;2_^3y to the fifth power, or expand (ic^-j-Sy^. Ans. a:'»-fl5a^y+90a;y-j-270a:y-f405xy-f243y''. 2. Expand (2a''-^axy. Ans. Sa^-{-12a^x-}-6a*x^-\-a^a^, S. Expand (2a~\-Sxy. Ans. 16a*-\-96a'x-\-216aW-{-216ax'-\-Slx^. 4:. Expand (^a—Sh)\ Ans. Jga*— |«36-f2^7a'^62_54a63-f 81Z/*. 5. Cube a+2Z>— c. Ans. a^-\-6a'h—Sa'c-{- Sh'-\- 1 2a¥—l 2h'c—c'J^ Sac" -\-6hc'—12ahc. 6. Expand (a-^-b-^c—dy. Ans. a*-\-4a^h-\- (ja'h^ + 4ah^ + h*-}- 4a'c + 12a'6c -\-12ah'c-{-4:b'c — 4a^d—12a'hd—12ah'd—Wd -\-6a'c'-\-12abc'-]-6b'c'~12a'cd—24:abcd—12b'cd -^6a'd'-\-12abd'-{-6b-'d'-\-4ac^ — 12acM-\-12acd^ —4ad^-\-Uc^ — 12bc''d-\-12bcd'—Ud'-{-c*—4:c^d -j-echP—4:Cd^-^d\ In many cases, as in some of the examples above given, it will sometimes be found most convenient to involve, by repeated multi- plicatioas, under the general rule. For further exercise, take the following : 1. If x-\-~=p, show that x^-\ — ^=p^ — 3/?. 2. If two numbers difi"er by unity, prove that the dif- ference of their squares is equal to the sum of the num- bers. 3. Show that the sum of the cubes of any three con- secutive integral numbers is divisible by the sum of those numbers. Note. — For a more general discussion of the Binomial Theorem, Bee Art. 310. EXTRACTION OF THE SQUARE ROOT. 137 II. EXTRACTION OF THE SQUARE ROOT. EXTRACTION OF THE SQUARE ROOT OF NUMBERS, 173. The Root of a number is a factor which multi- plied by itself a certain number of times will produce the given number. The Second or Square Root of a number, is that num- ber which multiplied by itself; that is, taken twice as a factor, will produce the given number. The Extraction of the Square Root is the process of finding the second root of a given number. 174. To show the relation that exists between the num- ber of figures in any given number, and the number of figures in its square root, take the first ten numbers and their squares : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The numbers in the first line are also the square roots of the numbers in the second. We see from this, that the square root of 1 is 1, and the square root of any number less than 100 is either one figure, or one figure and a fraction. Hence, When the number of places of figures in a number is not more than TWO, the number of places of figures in the square root will be ONE. The square root of 100 is 10 ; and of any number greater than 100 and less than 10000, the square root will be less than 100 ; that is, it will consist of two places of figures. Hence, When the number of places of figures is more than TWO, and not more than FOUR, the number of places of figures in the square root will be TWO. 2d Bk. 12 138 RAY'S ALGEBRA, SECOND BOOK. In the same manner it may be shown, that when tlie number of places of figures is more than /owr, and not more than six, the number of places in the square root will be three, and so on. 1T5. Every number may be regarded as being com- posed of tens and units. Thus, 76 consists of 7 tens and 6 units; and 576 consists of 57 tens and 6 units. Therefore, if we represent the tens by t, and the units by u, any number will be represented by t-{-u, and its square by the square of t-\-u, or {t-\'U)^. {t-\-uf=t^-^2tu-\-u^=.t2-\- {2t-]-u)u. Hence, 7%e square of any number is composed of two quantities, one of which is the square of the tens, and the other twice the tens plus the units multiplied hy the units. Thus, the square of 25, which is equal to 2 tens and 5 units, is 2 tens squared ^(20)2 = 400 (4 tens + 5 units) multiplied by 5^(40+5)5=225 ~625 1. Required to extract the square root of 625. Since the number consists of three places 625|25 of figures, its root will consist of two places, 400| according to the principle established in 20x2=40 225 Art. 174, we, therefore, separate it into two 5 225 periods, as in the margin. 45 Since the square of 2 tens is 400, and of 3 tens is 900, it is evident that the greatest square contained in 600 is the square of 2 tens (20) ; the square of 2 tens (20) is 400. Subtracting this from 625, the remainder is 225. The remainder, 225, consists of twice the tens plus the units, multiplied by the units; that is, by the formula, it is {2t-\-u)u^ of which t is already found to be 2, and it remains to find u. Now, the product of the tens by the units can not give a product less than tens; therefore, the unit's figure (5) forms no part of the double product of the tens by the units. Hence, if we divide the remaining figures (22) by the double of the tens (4), the quotient will be the unit's figure, or a figure greater than it. EXTK ACTION OF THE SQUARE ROOT. 139 Dividing 22 by 4 (2^) gives 5 (u) for a quotient. This unit's figure (5) is to be added to the double of the tens (40), and the sum multiplied by the unit's figure. Multiplying 40+5==45(2^-f w), by 5 (u), the product is 225, which is double the tens plus the units, multiplied by the units. As there is no remainder, we conclude that 25 is the exact square root of 625. In squaring and doubling the tens, it is customary 625|25 to omit the ciphers, and to add the unit's figure to 400 the double of the tens, by merely writing it in the 451225 unit's place. The actual operation is usually per- |225 formed as in the margin. 2. Required to extract the square root of 59049. Since this number consists of five places of 59049|243 figures, its square root will consist of three 4 places. (Art. 174.) We, therefore, separate it 441190 into three periods. |l76 In performing this operation, we find the 488 1 1449 square root of the number 590, on the same 11449 principle as in the preceding example. We next consider 24 as so many tens, and proceed to find the unit's figure (3) as in the preceding example. From these illustrations, we derive the following Rule for the Extraction of the Square Root of Num- bers. — 1st. Separate the given number into periods of two places each, beginning at the unit's place. (The left period will often contain but one figure.) 2d. Fi7id the greatest square in the left period, and place its root on the right, after the manner of a quotient in divi- sion. Subtract the square of the root from the left period, and to the remainder bring down the next period for a divi- dend. 3d. Double the root already found, and place it on the left for a divisor. Find how many times the divisor is con- tained in the dividend, exclusive of the right hand figure, and place the figure in the root and also on the right of the divisor. 140 RAYS ALGEBRA, SECOND BOOK. 4th. Multiply the divisor thus increased hy the last figure of the root; subtract the product from the dividend^ and to the remainder bring down the next period for a new divi- dend. 5th. Double the whole root already found for a new divi- sor^ and continue the operation as before, until all the periods are brought down. Note — If, in any case, the division can not be eifected, place a cipher in the root and divisor, and bring down the next period. ITG. In extracting the square root of numbers, the re- mainder may sometimes be greater than the divisor, while the last figure of the root can not be increased. To ex- plain this, Let a and a-|-l, be two consecutive numbers. Then, {a-f^lY^=a'^-\-2aA~l, the square of the greater. And (a)2=a2^ " " " less. Their difference is 2a-f 1. Hence, TJie difference ef the squares of two consecutive numbers is equal to twice the less number, increased by unify. Therefore, when atiy remainder is less than twice the root already found, plus one, the last figure can not be in- creased. Required the square root of 1. 2601. . . . Ans. 51. 15. 43046Y21. Ans. 6561. 2.7225. . . . Ans. 85. 16. 49042009. Ans. 7008. 1061326084. A. 32578. 3. 47089. . . Ans. 217. 4. 390625. . Ans. 625. 8. 948042681. Ans. 30709. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 177. Since |X|=ifV> t^e square root of rj% is |; that i/4 is, 1/25= Toe "-=|- Hence, we have the following EXTRACTION OF THE SQUARE ROOT. 141 Rule for Extracting the Square Root of a Fraction. — SJxtract the square root of both terms. When the terms of a fraction are not perfect squares, they may sometimes be made so by reducing. Thus, Find the square root of ff. Here, J-^ 4X5 9X.^ By canceling the common factor 5, the fraction becomes |, the square root of which is |. When both turms are perfect squares, and contain a common fac- tor, the reduction may be made either before or after the square root is extracted. Thus, v/if-|=i; or, M=|, and i/4_f. Required the square root of fi4 T2T- •2 2 5 400- Ans. |. Q 9 74 7 Ans '^^ ^- T0092- • • '^°^- oS- A 5 6169 Ans -37 ^' TOOOOOO- ^^^' 1000' ITS. A Perfect Square is a number whose square root can be exactly ascertained ; as, 4, 9, 16, etc. An Imperfect Square is a number whose square root can not be exactly ascertained ; as, 2, 3, 5, 6, etc. Since the difference of two consecutive square num- bers, a^ and a^-|-2a-f 1, is 2a-|-l ; therefore, there are always 2a imperfect squares between them. Thus, between the square of 5 (25) and the square of 6 (36), there are 10 {2a=i2y^5) imperfect squares. A quantity, affected by a radical sign, whose root can not be exactly found, is called a radical., or surd, or irra- tional root; as, \/2, i^5, etc. The root of such a quantity, expressed with more or less accuracy in decimals, is called the approximate value, or approximate root. Thus, 1.414-|- is the approximate value of |/2. lTO« It might be supposed, that when the square root of a whole number can not be expressed by a whole number, it might be exactly equal to some fraction. That it can not, will now be shown. 142 RAY'S ALGEBRA, SECOND BOOK. Let c be an imperfect square, as 2, and, if possible, let its square root be a fraction, -j, in its lowest terms. Then, |/c=-t; and C=^, by squaring both sides (Art. 148). Now, by supposition, a and b have no common factor; therefore, their squares, a^ and 6^, can have no common factor, since to square a number, we merely repeat its factors. Consequently, a2 ^„ must be in its lowest terms, and can not be equal to a whole ^- a^ number. Hence, the equation C=^, is not true, and the supposi- (X Hon on which it is founded, that is, that -^0^=^-, is false; there- fore, the square root of an imperfect square can not be a fraction. APPROXIMATE SQUARE ROOTS. ISO. To explain the method of finding the approxi- mate square root of an imperfect square, let it be required to find the square root of 5 to within J. If we reduce 5 to a fraction whose denominator is 9 (the square of 3, the denominator of the fraction 1), we have 5='*^^. Now, the square root of ^^ is greater than |, and less than |; hence, |, or 2, is the square root of 5 to within 1. To generalize this explanation, let it be required to ex- tract the square root of a to within a fraction -. Write a (Art. 127) under the form — ^, and denote the entire part of the square root of an^ by r. Then, an^ will be comprised between r^ and (r-|-l)2, and the square root of — y ^^^i he comprised r , r+l ^ between — and . n n V 7*4-1 1 T But the difference between- and — i— is — : therefore, — rep- n n n^ ^ n ^ resents the square root of a to within -. Hence, EXTRACTION OF THE SQUARE ROOT. 143 Rule for Extracting the Square Root of a Whole Num- ber to within a Given Fraction. — 1. Multiply the given number hy the square of the denominator of the fraction, which determines the degree of approximation. 2. Extract the square root of this product to the nearest unit J and divide the residt hy the denominator of the fraction. 1. Find the square root of 3 to within J. . Ans. 1|. 2. Of 10 to within -J Ans. 3. 3. Of 19 to within J Ans. 4J. 4. Of 30 to within j\ Ans. 5.4. 6. Of 75 to within -^Jg Ans. 8.66. Since the square of 10 is 100, the square of 100, 10000, and so on, the number of ciphers in the denominator of a decimal fraction is doubled by squaring it. Therefore, WJien the fraction which determines the degree of approxi- mation is a decimal, add two ciphers for each decimal place required; and, after extracting the square root, point off from the right one place of decimals for each two ciphers added. 6. Find the square root of 3 to five places of decimals. Ans. 1.73205. 7. Find the square root of 7 to five places of decimals. Ans. 2.64575. 8. Find the square root of 500. Ans. 22.360679+. 181. To find the approximate square root of a fraction. 1. Required to find the square root of ^ to within 4 — 4\/I— 2f The square root 3| is greater than ^ and less than ^; therefore, ^ is the square root of 4 to within less than ^. Hence, to find the square root of a fraction to within one of its equal parts, Rule. — Multiply the numerator by its denominator, extract the square root of the product to the nearest unit, and divide the result by the denominator. 144 RAY'S ALGEBRA, SECOND BOOK. 2. Find the square root of {'^ to within j\. Ans. 3. Find the square root of A J ^o within -^Jq. Ans. -f^. It is obvious that any decimal, or whole number and decimal, may be written in the form of a common fraction, and having its denominator a perfect square, by adding ciphers to both terms. Thus, .S=j%^j%%; .156=Vo'o%; 1.2=.lgg, and so on. Therefore, to extract the square root, as in the method for the approximate square root of a whole number (Art. 180), Rule. — 1. Annex ciphers to the decimal, until the number of decimal places shall be equal to double the number required in the root. 2. After extracting the root, point off from the right the required number of decimal places. 4. Find the square root of .4 to six places. Ans. .632455+. 5. Find the square root of 7.532 to five places. Ans. 2.74444+. When the denominator of a fraction is a perfect square, extract the square root of the numerator to as many places of decimals as are required, and divide the result by the square root of the denominator. Or, reduce the fraction to a decimal, and then extract its square root. When the denominator of the fraction is not a perfect square, the latter method should be used. 6. Find the square root of ^^g to five places. V^5=:2.23606+, i/16=4, i/fg^ ^^' ~ ^ ^ ^^-^ =.55901+ Or, -5g=.3125, and ;/ .3 125 =.55901+. 7. Find the square root of |. . . Ans. .774596+. 8. Find the square root of If . . Ans. 1.11803+. 9. Find the square root of 3^. . . Ans. 1.903943+. 10. Find the sqtiare root of ll'^. . . Ans. 3.349958+. EXTRACTION OF THE SQUARE ROOT. 145 EXTRACTION OF THE SQUARE ROOT OF ALGE- BRAIC QUANTITIES. EXTRACTION OF THE SQUARE ROOT OF MONOMIALS. 18!3. To square a monomial, (xlrt. 172), we square its coefficient, and multiply each exponent by 2. Thus, (3mn2)2^9m2n4. Therefore, ^9m^n^=Smn^. Hence, we have the following Rule for Extracting the Square Root of a Monomial. — Extract the square root of the coefficient as a niwiber, and divide the exponent of each letter hy '2i. Since +aX-f «=-|-«^, — aX— «=+«^; Therefore, ■^a^=-\-a, or —a. Hence, The square root of any positive quantity is either plus or minus. This is expressed by writing the double sign before the root. Thus, y'4a^=r=h2a; read, plus or minus 2a. If a monomial is negative, the extraction of the square root is impossible, since the square of any quantity, either positive or negative, is necessarily positive. Thus, i/ — 4, -|/ — b, are algebraic symbols, which indicate impossible operations. Such expressions are termed imaginary quantities. In an equation of the second degree, they often indicate some absurdity, or impossibility in the equation or problem from' which it was derived. 1. IQxhj^. Ans. d=4.T?/^ I 3. m^x^y^z^. Ans. -±zmxhf7}. 2. 25m2n2. Ans. drSmn. | 4. 1024a26V«.Ans.=b32aZ/V. / a \2 a a a^ la^ ^a^ a Since I -X |=tXt-=7-t,; therefore, -x-j-^z^-^-^-z^ziz^. Hence, \ b! ly^b b-' ' V6-' |/62 ^ ' To find the square root of a monomial fraction^ extract the square I'oot of both terms. 2d Bk. 13* 146 RAY'S ALGEBRA, SECOND BOOK. 5. Find the square root of — -. . . . Ans. ± — ■. 6. Find the square root of T7>~Tr.- • • Ans. rir=— ^. EXTRACTION OF THE SQUARE ROOT OF POLYNOMIALS. 183. In order to deduce a rule for . extracting the square root of polynomials, let us first find the relation that exists between the several terms of any quantity and its square. (a-(-6+c)2= a2-j-2a6-|-62-|-2ac+26c-f e2 =ra2^(2a-(-6)6+(2a -f26+c)c. {a+6+c'+d)2 =: rt2-f 2a6-f 62_^2ac+26c-f c2-f-2a(^+26cZ+2crf -j-d2^«2^(2a+6)6+(2a+26+c)c+(2a+26+2c+d)o?. Or, by calling the successive terms of a polynomial, r, r^, r^\ r'^\ and so on, we shall have (r+r^-|-r^''-[-r^^^)2=r2-j-(2r-]-7^)7^-f (2r -f 2r^-f r^^)r'^+(2r-f 2r^ + 2r'^+r''07^^^, where the law is mani- fest. In this formula, r, 7^, r^'^ r'^^^ may represent any algebraic quan- tities whatever, either integral or fractional, positive or negative. This formula gives the following law .-, The square of any polynomial is equal to the square of the first term — plus twice the first term^ plus the second, mul- tiplied hy the second — plus twice the first and second terms, plus the third, multiplied by the third — plus twice the first, second, and third terms, plus tjie fourth, multiplied by the fburth, and so on. Hence, by reversing tlie operation, we have the following Rule for Extracting the Square Root of a Polyno- mial. — 1st. Arrange the polynomial with reference to a cer- tain letter. 2d. Extract the square root of the first term, place the result on the right, and subtract its square from the given quantity. EXTRACTION OF THE SQUARE ROOT. I47 Sd. Divide the first term of the remainder hy double the part of the root already found ^ and annex the result to both the root and the divisor. Multiply the divisor thus increased by the second term of the root, and subtract the product from the remainder. 4th. Double the terms of the root already found for a partial divisor, divide the first term of the remainder by the first term of the divisor, and proceed in a similar manrier to find the other terms. 1. Find the square root of 4:x'^y'^-\-12x^y-\-9x'^ — SOxy^ Arranging the polynomials with reference to y, we have ROOT, 2by^~20xy^-{-'^x^?/^—S0xy^-\-V2x^y-\-9x^ \5y^^2xy-3x 2V 10?/2- 2xy\-20xy^-^Ax^y^ I - 2()a:?/H4a;2y2 10y^—ixy-3xl -S0xy^^l2x^y-^9x^ \-S0xy2-]- 12x^y^9x2 If the preceding example be arranged according to the powers of X, the root found will be 'Sx-{-2xy—5y^. This is correct also, as may be shown generally, thus : ^ {a^-\-2ax-^x^)^.db{a-\-x)=a-\-x, or —a — x. 2. x:'-^6ax-\-9a^ Ads. cc-f 3a. 3. 16x'^— 40a-y-f 25/ Ans. 4x—by. 4. 4x''z'—12xyz-i^9y' Ans. 2xz—Sy. 5. 49a*'"-«— 42a«'''-2-|-9a«'"+2. . . Ans. Va'^'"-'— 3a*'"+^ 6. l_|.2.T-[-7aj'-f 6a;^+9.x* Ans. l-\-x-{-Sx\ 7. 9a*—12a'b-\-S4:d'b'—20al/-{-2^b'. Ans. Sa'—2ab-i-bb\ 8. x^-\-4:x'-\~10x*-Jr20x'-^2bx'-^24x-^16. Ans. a;3+2x2+3a;+4. 9. 9x' — 6xy-{-S0xz-\-6xt-^y' — 10yz — 2yt-j-2bz'-^10zt -\-f. Ans. 3x—y-{-bz-\-t. 148 RAYS ALGEBRA, SECOND BOOK. 10. x'—2x'^i^-^~'-.-{-J^ Ans. :i;2— a;+i -^ 2^a'b^ bahc^ c* . hah & 11. -^ ^ + ^ ^^«-2--8- 12. -25 y--^-+94-49a:^ Airs. 7a;^-^H-a Id. 77—2+- Ans. . b^ a? b a 14. Reduce the following expression to its simplest form, and extract the square root : Qi—iy—2{a^^b''){a~by-\-2{a'^b'). . . Ans. d'-\-b\ 15. Find the square root of 1 — x^ to five terms. . ^ x^ x^ x^ hx^ Ans. 1- 2- g-jg-j28-' ''"■ 16. Find the first five terms of the square root oix^-^-a^. . , a^ a* , a^ 5a^ , 184. The following remarks will be found useful : 1st. Ab binomial can be a perfect square; for the square of a monomial is a monomial, and the square of a bino- mial is a trinomial. Thus, a2_j_^2 jg jjQt a perfect square, but if we add to it, or sub- tract from it, 2a6, it becomes the square of a-j-6 or of a—b. 2d. In order that a trinomial may be a perfect square, the two extreme terms must be perfect squares, and the middle term double the product of the square roots of the extreme terms. Hence, to find the square root of a trinomial when it is a perfect square, Extract the square roots of the extreme terms, and unite them by the sign of the second term. Thus, a^-\-4iax-\-Ax^ is a perfect square, and its square root is, a-\-2ac; 4x--{^8xy ]-9y^ is not a perfect square. For other illustra- tions, see Exs. 2, 3, 4, 11, and 13, Art. 183. EXTRACTION OF THE CUBE ROOT. 149 III. EXTRACTION OF THE CUBE ROOT. EXTRACTION OF THE CUBE ROOT OF NUMBERS. 185. The Cube, or third power of a number, is the product arising from taking it three times as a factor. (Art. 172.) The Cube Hoot, or third root of a number, is one of three equal factors into which it may be resolved. To extract the cube root of a number, is to find a num- ber which, taken three times as a factor, will produce the given number. 186. To show the relation that exists between the number of figures in any given number, and the number of figures in its cube root. The first ten numbers and their cubes are. Roots, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; Cubes, 1, 8, 27, G4, 125, 216, 343, 512, 729, 1000. We see from this that the cube of a number consisting of one place of figures, does not exceed three places. Again, comparing the numbers 10 and 100, we have, Numbers, .... 10, 100; Cubes, iOOO, 1*006006. Since the cube of 10 is 1000, and the cube of 99, which is less than 100, is less than 1000000; therefore, the cube of a number consisting of two places of figures, has more than three places, and not more than six places of figures. Again, since the cube of 100 is IOO6OO6, and the cube of 1000 is 1006000006; the cube of a number consisting of three places of figures has more than six places, and not more than nine places of figures. If, therefore, we begin at the unit's place of a number, and sepa- rate it into periods of three places each, the number of periods will show the number of places of figures in the root. The left period will often contain only one or two figures. 150 RAY'S ALGEBRA, SECOND BOOK. X8T- To investigate a rule for the extraction of the cube root. The first step is to show the relation that exists between any number composed of units and tens, and its cube. Let . . ^= the tens, and w= the units of a given number. Then, t-\-U= the number. And [t-\-u)^= the cube of the number. But (^+w)3.^^3^3^2^^3^i^2^^^3^j;3^^3^2^3^^^_|_^2)^, HencB, 77ie cube of any number consisting of tens and units, is equal to the cube of the tens, — plus three times the square of the tens, plus three times the product of the tens and mi its, jilus the square of the units, all three multiplied by the units. 1. Required to extract the cube root of 13824. Separating the number into periods by points, we find there will be two figures in the root. The greatest cube in 13 (thousand) is 8 (thousand); the cube root of which is 2 [t) , and its cube, 8 (thousand), corresponds to t^ in the formula. We then subtract this from the given number, and find a remainder 5824, which corresponds to [St^-^Stu -^u'^)u in the formula. The first term, St^, of this formula, is sometimes termed the trial divisor, as it is used to find the units figure u. If the remaining terms were only St^u, we could readily find u by dividing by St^; but if we divide by 3^2^ we may obtain a figure too large, on account of omitting the terms S(u-{^u'^, of which u is as yet unknown. But if we first obtain a figure too large, at a sec- ond trial we must take one that is less. Since the square of tens is hundreds, in using three times the square of the ten's figure as a trial divisor, we omit the figures (24) in the unit's and ten's places of the dividend. In this case, 12 is contained in 58 four times. This gives 4 (u) for the required unit's figure, and we now find the complete divisor, 3<2^3 i in •? i 1099 remaiudrr and subtrahend are each written ' -T"^ "^ 1-tt X' ~\-l^CC-X In two lines. | -j-3a^X^—Ga*X-\^Cl*'\ We first extract the cuhe root of x^, which gives x^ for the first term of the required root. Then, 3 times the square of this, =ioX*, constitutes the trial divisor for finding the remaining terms. Dividing — Gx^ by Sz"*, gives — 2x, the second term of the root. We then form the complete divisor by adding together S[x-)^-]-S ^x2x— 2a:) + (— 2a:)2^3a;4— 6a;3-f-4a;2. Multiplying this by —2x, EXTRACTION OF THE CUBE HOOT. 157 and subtracting, the first term of the second remainder is -fSa^a;'*, which divided by the trial divisor, gives -j-a^, for the third term of the root, and so on. ~ SECOND METHOD. The following rule, applicable both to numerical and algebraic quantities, may be found more convenient in some cases. The principle upon which it is founded will be obvious upon a careful inspection of the full expansion of the forms (a-\-by, (c?-f Z>-|-f)^, etc. 1. Arrange the poli/nomial, as in the previoiis rule. 2. Extract the cube root of the first term, etc., as before. 3. Eind the trial divisor and 2d term of the root, as before. 4. Cube the root already found, and subtract the result from the given polynomial. 5. Divide the first term of the remainder by the same trial divisor for the third term of the root. Cube the root already found, and subtract the result from the given polynomial. Continue this process until a quantity is found in the root which will be equal, when cubed, to the given polynomial. To illustrate this rule, take the example given above. icfi — 6a:'''-f 1 2a:4 -f-Sa^x^— 8a;-— 1 2a-a:H 1 2a2a;2-l- 3a%2_ ea^x-f- a« a;"*' |2;2_2a:+a2 ^x^\—{ix-'-\-Vlx^^ etc., 1st remainder. a;fi —Qx^-i^\2x^-^x\ cube of x^—2x. Si^l Za^x^—Vla^x^, etc., 2d remainder. a;'-._Ga:H12a;-»+3a2a:4_8a;-— 12a2.r-4-12a2a;2-f3a4a:2_6a^a:-|-a6 We first extract the cube root of X^, and find it x"^. Cubing this, subtracting, and dividing the first term of the remainder by SiC*, we obtain — 2x for the second term of tlie root. Cubing x"^ — 2a:, writ- ing it below, and subtracting, we have the second remainder. Dividing the first term of this remainder again by SrC*. we obtain a^ for the third term of the root. The cube of a;2— 2a:-f-a2 being equal to the given polynomial, the work is fin'shed. 158 RAYS ALGEBRA, SECOND BOOK. Remarks. — 1. A second method for extracting the square root, similar to the above, might be given, but it is less simple than the common rule. 2. The process of cubing the root may be conducted by Newton's Theorem, as explained in Art. 172. Find the cube root 2. Of a^-f 24a^6-j-192«Z>2-}-5126'. Ans. a-l 86. 3. Of Sa^—S4a'x-\-294ax'—S4Sx\ Ans. 2a— 1x. 4. Of ««— 6c/5+15a*— 20a3+15a^— 6a-|-l. Ans. a'—2a-\-l. 5. Of x^—9a^-\-S9x'—d9x'-]-lb6x'—lUx-\-64:, Ans. x'^ — 3.x-]-4. 6. Of (a-]-ly"x' — 6caP(a-\-iy'^x'-\-12chi'P(aJr'^y'' a;— 8c'a3^. Ans. (a-^iy"x—2caP. 7. Find the first three terms of the cube root of 1 — x. . ^ X x^ Ans. 1 — ^ — „ — 5 ^^^• IV. EXTRACTION OF THE FOURTH ROOT, SIXTH ROOT, N'^" ROOT, Etc. ]9!3. The fourth root of a number is one of four equnl factors, into which the number may be resolved ; and, in general, the u''' root of a number is one of the n equal fac- tors into which the number may be resolved. AVhen the degree of the root to be extracted is a mul- tiple of two or more numbers, as 4, 6, etc., the root can he obtained hy extracting the roots of more simple degrees. To explain this, we remark that {a^y=a^^^'—a^^, and in gen- eral («*")"=. a'"X«^a"i«. Hence, The TOL^^ iwwer of the m'^ iioioer of a number is equal to the mn'^ iiower of the number. Reciprocally, the mn'^ root of a number^ is equal to the n'* root of the m'^' root of that number; that is, y a—y,/ yd' EXTRACTION OF THE CUBE ROOT. 159 From this, it follows that ■^a=z:'\ y^a; and f/a='\ ;/«, or \f^Ct; in like manner y'a=\-|/ ^/a, and so on. 1. Find the 4th root of 65536, Ans. 16. 2. Find the 4th root of ISlOv .9601. . . Ans. 10.7. 3. Find the 6th root of 2985984. . . . Ans. 12. 4. Find the 8th root of 214358881. . . Ans. 11. 5. Find the 4th root of Sla*x^ Ans. Sax\ 6. Find the 4th root of a* -j- 4a'hx -f- Qa'b'x' -\- 4:ah'x^ -\-h*x*. Ans. a-\-hx. 1. Find the 4th root of x^— 4x^-{-10x' — 16x'-{-19 x'^ x'^ x'^x^ ^x' 8. Findthe6thro«tofa«+-— 6/a*+- 1 + 15/ a^-f- I -20. , 1 Ans. a . a 193. It ha? been shown already (Arts. 182, 183) that the square root of a monomial, or a polynomial, may he preceded either by the sign -|- or — . We shall now ex- plain the law in regard to the roots generally. If we take the successive powers of -\-a and ~a, we have -fa, -fa2^ -^a% -fa''. —a, 4-a2_ _a3, .^a\ . . . -j a-'", ~a^"+K From this we see that every even power is positive, and that an odd power has the same sign as the root. Conversely, it is evident, 1st. That every odd root of a monomial miist have the same sign as the monomial itself. Thus, f -f8a»=-f2a, f-8a^=—2a, ■^— 32a»o=— 2a2. 2d. That an even root of a jmsitive monomial may he either jyosifive or negative. Thus, ^'8la*b^^±:Sab\ f G4ai2^dr2a^ 16i7 RAYS ALGEBRA, SECOND BOOK. 3d. That every even root of a negative moiioTnial is im- possible; since no quantity raised to a power of an even degree can give a negative result. Thus, y/ — a^, ^ — 6, {/ — c, are symbols of operations which can not be performed. They are imaginary expressions, like l/^ ^116, (Art. 182.) TO EXTRACT THE N'^" ROOT OF ANY QUANTITY. 104. In raising any monomial to the ri''' power, (Art. 172,) we raise the numeral coefficient to the n^^ power, and multiply each exponent by ?i, thus, (2a^^*)^=8a^Z>^^. ■ Hence, conversely, to find the ?i'* root of a monomial, Extract the n<^ root of the coefficient^ and divide the expo- nent of each letter hy n. Rules for the extraction of any root of a numerical quantity, or algebraic polynomial, may be formed on the same principle as is that of the cube root, (Art. 191.) Thus, since (a+6)4z=:a4+4a''^^+6a262_|.4a63_^64^o^4^(4«3^.6a26+4a62 -f63)6. (a_|_5)r.^a5_|_(5cj4_^10a36+10a252^5ci63_^^fN5^ etc. The trial divisor for the fourth root would be of the form 4a-^, or four times the third power of the first term of the root, and the complete divisor of the form, 4a-'-f 6a26-^4a62_j_53_ For the fifth root, the trial and complete divisors would be of the forms, 5a^ and l^a^-\\^o?'b-\-\^0?b-'\-^ab'^A^b''^ and so for any higher root. A more simple method, however, would be like that which is called the Second Method for extracting the cube root, (Art. 191.) The trial divisors would be of the form 4a^, for the 4th root, 5a* for the 5th root, na"-^ for the nth root, or, in general, n times the (n— 1)'^ power of the first term of the root. Remark.— In the following examples, find the root of the numeral coefficient by inspection. It is Tinnecessary to give rules for extracting the 5th, 7th, etc., roots of numbers, as in the present state of science these operations are readily performed by Loga- rithms. RADICAL QUANTITIES 1. Find the 5th root of — 32( The 6th root of 7296V«. The 7th root of 128xy^ The 8th root of Q'o6la'L'\ The 9th root of — 512a;V^ The 10th root of 1024//°^= 2. 3. 4. 5. 6. 7. Extract 'b\ 8. Extract the 5th root of 32x5- lOx— 1. -80, 161 Ads. — 2ax-. Ans. ±3Z>c^ . Ans. 2x1/'^. Ans. ±3aR Ans. — 2xz'^. Ans. ±26;.-\ . Ans. a'^bc\ ;c*-f80x3— 40x'^ Ans. 2x — 1. V. RADICAL QUANTITIES. Note. — These quantities are generally called surds by English writers; while the French more properly term them radicals, from the Latin word radix, a root. 195. A Rational Quantity is either not aiFected by the radical sign, or the root indicated can be exactly as- certained ; thus, 2, a, |/4, and |^8 are rational quan- tities. A Radical Quantity is one affected by a radical sign, but whose indicated root can not be exactly expressed in numbers; thus, |/5=2. 23606797 nearly. 106. From Art. 193 it is evident that when a mono; mial is a perfect power of the ?i"^ degree, its numeral coeffi- cient is a perfect power of that degree, and the exponent of each letter is divisible by n. Thus, 4a~ is a perfect square, while Ga-^ is not; and Sa*"' is a per- fect cube, while 6a-\ 8a^, la^\ etc., are not. In extracting any root, when the exact division of the exponent can not be performed, it may be indicated by a fraction. Ihus,, 3 __ 4 ■j/a*^ may be written a^, and fa^ may be written a^ ; and, in gen- m eral, the n fci'X^a'XfTi—a, etc. This is evident from the definition of a root, (Art. 173.) REDUCTION OF RADICALS. Case I.' — To beduce Radicals to their Simplest Form. lOO. Reduction of radicals consists in changing the form of the quantities without altering their value. It is founded on the following principle : The square root of the j)i'oduct of two or more factors is equal to the product of the square roots of those factors. That is, |/a6=|/aXi/^; which is thus proved; Squaring both members of this equation, we have, (Art. 198,) ab=ayh, or ab—ab. Now, since the equation is true after both sides are squared, it was true before, (Art. 148, Ax. 6,) or -^ab^yay^yb. By this principle, ^/36=y'4x9=2x3 ; v/144=ry'9><16=3x4; ^8=|/4x2==-/4Xv' 2=2/2. Hence, we have the following Eule for the Reduction of a Radical of the Second Degree to its Simplest Form. — 1st, Separate the quantity to be reduced into two parts, one of which shall contain all the factors that are perfect squares^ and the other the remain- ing factors. 164 RAYS ALGEBRA, SECOND BOOK. 2d. Extract the square 7'oot of the perfect square^ and pre- fix it as a coefficient to the other part placed under the radical sign. To determine whether any numeral contains a fiictor fltat is a perfect square, divide it by either of the squares 4, 9, 16, etc. Reduce to their simplest forms the radicals in each of th« following examples : 1. ^12, ^18, ^45, /32, v^50^, y72a2b\ Ans. 2/3, 3/2, 3/5, 4/2, 5a/2«, 6a6/25. 2. /245, /i48, /8T0, /507Pc2, /ISOSo^P. Ans. 7/5, 8/7, 9/10, 136c/36, 19a26/5. In a similar mfinner, polj'nomials may sometimes "be simplified. Thus, /(3a3-6a2c+3ac2)=:/3a(a2— 2ac+c2)=r(a_c)/;ia. 3. /(as-a26), ^ ax^—Gax-^9a, ^ {x^--y2)i^r^y), Ans. a/(a— 6), (a:— 3) /a, {x-^y)-^ {x—y). To reduce a fractional radical to its simplest form, 1st. Render the denominator of the fraction a perfect square hy midtiplying or dividing both terms hy the same quantity. 2d. Separate into two factors^ one of which is a perfect square. 3d, Extract the square root of this factor^ and write it as a coefficient to the other factor placed under the radical sign. 4. Reduce -|/|, and ^/ , to their simplest forms. Vr-=»/|xl=>/l=/5V<5=v'i\X»/5--|/5. Ans. ly/2, J/c; 1/^ v/S, 3/30, |/10. A6' \56' \4cV \ 9«sW RADICALS. 165 ^^^•Z^V^*^' 66^^^"^' 2cj,>^«2/, 14^2(6^^) • 200. To reduce radicals of any degree to the most simple form. The principle of Art. 199 is, evidently, applicable to radicals of any degree. Thus, 1. Reduce -^54 to its most simple form. f54^f27x^=#27xf2=3^2. Similarly, f f-^iX|X|=f i|=f sVXlS-^f 18- Reduce each of the following to its simplest form : 2. f40, f81c\ f 128afic5, fie2m^n% ^144. Ans. 2f 5, 3cf3c, 4a2cf 2c2, Smnf6mn\ 2^9. 3- fh fl fh fi V f' vi vh ^ _ Ans. Ifi, lf6, ^f.36, if 15, 1 f 5i, | V^, Jf^32. 4. f 162, f 3888, ^Z'la^b^, {/729a\ Ans. 3f'2, 6^3, 2a6jj'2a6'i, 3a (/3a. SOI. The mji^^ root of any quantity may be simplified when it is a complete power of the m^^ or ji^^ degree, as shown, (Art. 192.) Thus, l/9a^--^-\ |/9a2=/3a. Also, l/a^-~2ab^b2=:\ ^a^~2ab-\~b^=fcr^. Reduce each of the following to its simplest form 1. ^36a2c2, ^.Slin^n^ ^4a2, f/16a2c4, ^1256^. Ans. |/6ac, Sriy/lm,', f2d, f 4ac2, ^65. 106 RAY'S ALGEBRA, SECOND BOOK. Case II. — To reduce a Rational Quantity to the FORM of a Radical. 30S. If we square «, and then extract the square root of the square, the result is evidently a. 2 3 That is, a==|/a2-__o^5 jn ^^^ manner, a=f^a^z=a^^ and gen- m, erallj, a='^ar=ar\ Hence, Rule for reducing a Rational Quantity to the form of a Radical. — Raise the quantity to a jpower corresponding to the given root, and write it under the radical sign. 1. Reduce 6 to the form of the square root. Ans. -j/BG. 2. — 2 to the form of the cube root. Ans. #^ — 8. 3. Sax to the form of the square root. Ans. y^9aV^ 4. m — n to the form of the square root. Ans. -j/m"^ — Imn-^n^, Similarly, a coefficient may be passed under the radical sign. Thus 2v/3=^4Xi/3=v/i^. Generally, a'^b=^'l/O^X'V~b=V^^^- 5. Express 5]/T, and a^|///, entirely under the radical sign. Ans. |/iY5, and |/a*6. 6. Pass the coefficient of the quantity 2|^5, under the radical sign. Ans. ]^40. Case III. — To reduce Radicals having Different Indices to Equivalent Radicals having a Common Index. 203. This is done by multiplying both terms of the fractional exponent by the same number, which, evidently, does not change its value. (Art. 118.) RADICALS. 167 1 Let, it be required to reduce ^2a, and ]^36, or (2a) ^ and (3^)'^ to quantities of equal value, having the same index. f 36= (36)4 =(36) 1^2:^12/ (36]3='^2763. Hence, Hule. — Reduce the fractional exponents to a common de- nominator; then the numerator of each fraction will repre- sent the power to which the corresponding quantity is to le raised, and the common denominator the index of the root to he extracted. i 1. Keducc |/3 and ^2, or 3^ and 2^ to a common index- Ans. ^27 and ;^ 4, or 27" and 4". 2. -^5 and |/4 Ans. |"/25 and y 64. J 3. a^ and h" Ans. |/a* and ^b. 4. r if a, v^56, and ^6^. Ans. 1*/^% \/62bh', and |>^2T6?. 5. r y/^, ^3/^, and if^. Ans. \^^«, p^< and '^a\ 6. rteducc 3^, 2^, and 5- to a common index. Ans. 3'^S 2^'^ 5'", or \7B56i, \^/512, ^^15625. ADDITION AND SUBTRACTION OF RADICALS. !S04. Required to find the sum of Sf/a and bf^a. It is evident that 3 times and 5 times any quantity, must make 8 times that quantity; therefore, 3f/a^5fa=8fa. But, if it were required to find the sum of S^/a and 5f/fi, since y a and f^a are different quantities, we can only indicate their addi- iion; thus, Sy/a^ bf/a. 168 RAYS ALGEBRA, SECOND BOOK. Similarly, Sy2-^Ty2~4y2^Gy^. But 3/5 and 4j/a=3j/5-|-4|/3. So also 3^/5 and 4^5^3/54-4^0. Radicals that are not similar, may often be made so; thus, |/12 and |/27 are equal to 2^/S and 3|/3, and their sum is 5|/3. The same principles apply to the subtraction of radicals. From the above we derive the following Rule for the Addition of Eadieals.— 1st. Reduce the. radicals to their simplest forms, and, if necessary, to a com- man index. 2d. If the radicals are similar, find the sum of their coejicients, and prefix it to the common radical; hut if they are not similar, connect them hy their proper signs. Rule for the Subtraction of Radicals. — Change the sign of the subtrahend, and proceed as in addition of radicals. 1. Find the sum of |/448 and ^TVl. -/448=/64x7= 8/7 |/T12=/T6X7=^V7 By addition, 12/7, Ans. 2. Find the sum of f/^ and f 81. . Ans. 5^3. 3. Of ^48 and f/102 Ans. 5^6. 4. Of i/fS^i/^ and yW^K Ans. (3a-Z>-f-5aZ>)i/2aZ). 5. Subtract i/TSO from |/405. . . . Ans. 3]/5. 6. Subtract f/40 from f/T3'5 Ans. ^5". Perform the operations indicated in each of tlie follow- ing : 7. /243-f/27+^48. Ans. 16/3. 8. y^24-|-/54— /90. Ans. /C 9. |/128— 2/50f/72-/18 Ans. y'l RADICALS. 169 10. ^4Sab2-\-by''7Ea^i/Sa{a—9bj^ Ans. a^Sa. 11. 2„ f+^|/60+^T5+/|: Ans. Vv/l^- 12. ^128— f686— f 16+4|f250 Ans. 15,^2: 13. 2^14-81^2: Ans. 3f 2. 14. 6f/4a2_^2|f2a+f/8a^. Ans. 9f 2ol 15. 2/3— ^v/12+4y'27— 2^-3g Ans. ^^^S. ^U. ^re+fST— 1^^=512+ f 192—7 f9 Ans. 10. 17. -x/^ + i-/(«^^-4«^^^+4a63) Ans. ^^ab. MULTIPLICATION AND DIVISION OF RADICALS. t20«S. The rule for the multiplication of radicals is founded on the principle (Art. 200) that The product of the n*^ root of two or more quantities is equal to the n"^ root of their product. That is, VaxVb=Vcib. (See Art. 198.) Hence, (Art. 53,) a'i/bXov'd^aXcXv'^X\^d=acl/bd. The rule for division is founded on the principle that The quotient of the n'^ roots of two quantities is equal to the n'^ root of their quotient. That is, J^r= = -\/'r; which is thus proved: '\/b >^ Raising both sides to the nth power, we have j-^^-ti which shows that the previous equation is true. Hence, we have the following Rules for the Multiplication and Division of Radi- cals. — If the radicals have dijf'erent indices, reduce them to the same index. Then., I. To l/Lulti-ply. —Multipli/ the coefficients together for the coefficient of the product, and also the parts under the radical for the radical part of the product. 2d Bk. 15* 170 RAY S ALGEBRA, SECOND BOOK. II. To Divide. — Divide the coefficient of the dividend hy the coefficient of the divisor for the coefficient of the quotient^ and the radical part of the dividend hy the radical part of the divisor for the radical part of the quotient. 1. Multiply 2i/«Z by Za^abc. 2 ^ab Sa^abc 2. Divide 4ay'^ab by 2|/c/c. 4a£a6^ l?^«?=2a J^=2avf =- ,/55. 2i/ac' 2 ^ ac \c \c^ c ^ 3. Multiply 2|f 3 by Sy^. Multiplying, . . . 6{/72, Ans. 4. Divide 6^/2 by 3if 2. 6/2=:6j/23^6^8. (1.) 3f2=:3^ 22=3^4. (2.) Dividing (1) by (2), wc have 2f 2. 5. Multiply 3|/ 12 by 5^1^. . . . Ans. 90^^. 6. Multiply 4^/12 by 3|r4. .... Ans. 24^/6. T. Multiply together 5|/3, 7|/|^ and |/2. Ans 140 8. Multiply Sfb by 4 fa. . . . Ans. 12^/'^*^ 9. Multiply together y% fS, and f 5. A. ^^648000. 10. Multiply together yx, y^x\ and i^'x^. Ans. yx', 11. Divide ^40 by i/2. Ans. 2fZ. 12. Divide 6^/54 by 3y^ Ans. 6y 3. RADICALS. 171 . Ans. 5^4 . Ans. f/S. . Ans. 2f/ 3. . Ans. |/2. b Ans ■v.- 13. Divide 70f 9 by 7^l8 14. Divide f/72 by y2. . 15. Divide 4f9 by 2]/ 3. 16. Divide f 72 by 1^3.. 17. Divide ^[ by ^^. . Polynomials containing radicals may also be multiplied; thus, 18. Multiply 3+V/5 ^y 2—1/5. 2— y'S 6+2/5" 6— ^5— 5=1— 1/5, Ans. 19. Multiply y/2+1 by |/2— 1 Ans. 1. 20. ll|/2— 4^15 by y^ 6-(-|/ 5. Ans. 2|/3— v^lO. 21. Raise |/2-i-y'3 to the 4th power. Ans. 49-|-20y 6. 22. Multiply r/l2+|/19 by l/l2— 1/19. Ans. 5. 23. Multiply a!'— .^1/2+1 by x'-{-xy/2-{-l. Ans. a->-f 1. 24. (x^-f l)(x2— ic^ 3+l)(x^-fa:|/34-l). Ans. x«-f 1. SOO. To reduce a fraction whose denominator contains radicals, to an equivalent fraction having a rational denom- inator. When the denominator is a monomial, as , it will become ra- tional if we multiply both terms by y/O. Again, if the denominator is fa, if we multiply both terms by f a'^, the denominator will become f a'Xf/a'^^i^a^^za. Thus, ^_ 172 RAY'S ALGEBRA. SECOND BOOK. In like manner, if the denominator is *\/ci^j it will become ra- tional by multiplying it by "y^a'"-". Therefore, When the denominator of the fraction is a mo7iomial, 7nultiply both terms hy such a factor as will render the ex- jwnent of the quantity under the radical equal to the index of the radical. Since the sum of two quantities, multiplied by their difference, is equal to the difference of their squares (Art. 80); if the fraction is of the form , , and we multiply both terms by b — i/c, the de- nominator will be rational. Thus, _ a{b—^c) ^ah~a^G 6+i/c~(6+|/c) (&-- i/c^"" ly'—G If the denominator is 6 — |/C, the multiplier will be b-\-y'c. If the denominator is ^6-f-i/c, the multiplier will be y^6 — |/c; and if it is |/6 — |/c, the multiplier will be y^6+|/c. If the denominator is of the form ^a-^-y/b-^-y'c, it may be rendered rational by two successive multiplications. The first will result in a quantity of the form m — ^/n, which may be made ra- tional as before. Reduce the following fractions to equivalent ones hav- ing rational denominators : i.-L 1/3- fS' Ans. ^-=Ji/3. 2. ^. Ans. yl^=W2: /6 ^ Ans. I] , 6 __ 4. r-. Ans. 11^16. 5. tV^. Ans. i-^y2. 3-2/2 6. V^±V?. Ans. r)-U2./6. V/3-/2 ^ 3/5-2/2 ' 2/5^/18' 8 3-hy3" ^ • /6-f-/2-/5' . Ans. 9+1/10. Ans. -/G-f /2-f-/5. RADICALS. 173 9. ^=:z 4- - Ans. 2a:. x^^x^—\ ' x—^x^—\ 10. '^!!±'_+'^_"_!z^ + i^ "!±i-'^^ An. 2.^. ^x^^\—^x^-\ yx^-j^l-{-^x^—l Remark. — By the preceding transformations, the process of finding the numerical value of a fractional radical is very much 2 abridged. Thus, to find the value of ~~^, we may divide 2 by the square root of 5, which is 2.2360679+. But ~= = J^, the true yo 5 value of which is found by multiplying 2.2360679 by 2, and divid- ing the result by 5. Reduce each of the following fractions to its simplest form, and find the numerical value of the result : 11. -^, and — ^ Ans. .894427+, and .707106+. ,„ v/25+,/12 POWERS OF RADICALS. SOT. Let it be required to raise i^Sa to the 3d power. Taking p'Sa as a factor three times, we have So, "^aX^VctX^cT ... to n factors, =7a\ Hence, Hule for raising a Radical Quantity to any Power.— Raise the qiiantitij under the radical to the given power ^ and affect the result icith the primitive radical sign. If the quantity have a coefficient, it must also be raised to the given power. Thus, the 4th power of 2f/M^Js_16f81a>^. This, by reduction, becomes 16f 27a6x3a2^48a2f 3a2. 174 RAY'S ALGEBRA, SECOND BOOK. If the index of the radical is a multiple of the exponent of the power, the operation may be simplified. Thus, {p2ay^={J y2aj^=^2:a, (Art. 192.) In general, Cv'TTf ^( ^^a ) ^'^'^^a. Hence, If the index of the radical is divisible hy the exponent of the power, we may perform this division, and leave the quan- tity under the radical sign unchanged. Thus, to raise ^ 3a to the 4th power, we have ^ 81a^^ \ y r=:y^3a, or, dividing 8 by 4, we obtain at once y'Sa. 81 a4 1. Kaise f^2a to the 4th power. 2. S^2ab^ to the 4th power. ■^ac^ to the 2d power. l/ac'' to the 4th power. . ■y/Sc'^ to the 3d power. . yx — y to the 3d power. . . Ans. 2af/2a. Ans. 162ahY^aI/. . . . Ans. cy'a. . . . Ans. aV. . . . Ans. cy'S. Ans. (x — y)\/^ — y^ ROOTS OF RADICALS. 208. Since !;!/ l/a^y'a (Art. 192), therefore, to tract the roots of radicals, we have the following ex- Rule. — Multiply the index of the radical hy the index of the root to he extracted, and leave the quantity under the radi- cal sign unchanged. Thus, the square root of f^2d is ^ ^2a=^2a. If the radical has a coefficient, its root must also be extracted. If the quantity under the radical is a perfect power of the same degree as the root to be extracted, the process may be simplified. Thus, ^ ^8^3 is equal (Art. 192) to -^ f'8a3-=^2a. RADICALS. 175 1. Extract the cube root of y^d^b. . . . Ans. \/d^h. 2. The 4th root of 16a«lf 2^. . . Ans. 2^^^ ^ 2^ 3. The square root of fWa}. . . . Ans. f^Ta. 4. The cube root of 64fSlF. . . . Ans. 4^^ 2^. 5. The cube root of (m-\-n')y'm-\-n. Ans. y^m-\-n. IMAGINARY, OR IMPOSSIBLE QUANTITIES. SCO. An imaginary quantity (Arts. 182, 193) is an even root of a negative quantity. Thus, |/ — a, and y^— 6'', are imaginary quantities. The rules for the multiplication and division of radicals (Art. 205) require some modification when imaginary quantities are to be mul- tiplied or divided. Thus, by the rule (Art. 205), |/"^aX i/— «=>/— «X— «— ya^—ztia. But, since the square root of any quantity multiplied by the square root itself, must give the original quantity, (Art. 198,) therefore, y/^^XV~^=~^- SIO. Evert/ imaginary quantity may he resolved into two factors, one a real quantity, and the other the imaginary ex- pression, |/ — 1, or an exj)ression containing it. This is evident, if we consider that every negative quantity may be regarded as the product of two factors, one of which is — 1. Thus, — a=raX — 1, -^^r^ft^^-l, and so on. Hence, ^— a2:=p/a^X— l=i/«^Xi/— l = =t=a/ — 1. Since the square root of any quantity, multiplied by the square root itself, must give the original quantity; Therefore, (^:rTy2^ ^ZTXi/^^=— 1. Also, (^3I]3_(^^2x^^=:=-Vlir=— i/^TT. (>/-i)M/-i)' (v/-i)M-i)(-i)=+i- Attention to this principle will render all the algebraic opera- tions, with imaginary quantities, easily performed. Thus^ ^11^ X /=& ---/ax V—^ X i/6 X i/^ = >/«6x 176 RAY'S ALGEBRA, SECOND BOOK. OPERATION. If it J)e^ required to find the product of a+6^/— 1 a-j-by/ — 1 by a — 6/^, the operation is a— h ^ — 1 performed as in the margin. a'^-\-ab /~~1 a^-^b\ Since a^-\-b'^-=z[a^b^'—i){a — 6^/— 1), any binomial whose terms are positive may be resolved into two factors, one of whicli is the sum and the other the difference of a real and an imaginary quantity. Thus, m-Yn={ym-\-^n'^—l)[^m^^lfi^^^)^ 1. Multiply -j/ — a^ by -p,/ — h'^ Ans. — ah. 2. Find the 3d and 4tli powers of ay/^l. Ans. — a^\/- — 1, and a*. 8. Multiply 2,/^ by 2.^'-^. . . Ans. — 6|/ 6. 4. Divide 6^/^=^ by 2|/=4. .... Ans. ly'E. 6. Simplify the fraction z "r^' • -^"^- )/ — 1- 6. Find the continued product of x-\-a, x-\-ay — 1, X — «, and X — a-j/ — 1. Ans. cc* — a*. 7. Of what number are 24+ Y|/^, and 24—7]/^, the imaginary factors? Ans. 625. VI. THEORY OF FRACTIONAL EXPONENTS. 211. The rules for integral exponents in multiplica- tion, division, involution, and evolution, (Arts. 56, *70, 172, and 194,) are equally applicable when the exponents are fractional. Fractional exponents have their origin (Art. 19G) in the FRACTIONAL EXPONENTS. 177 extraction of roots, when the exponent of the power is not divisible by the index of the root. Thus, the cube root of a- is a^. So the n^'^ root of a"' is a". 2 4 '" The forms a^, a^, and a '*, may be read a to the power of f, 771 a to the power of |, and a to the power of minus — ; or, a expo- nent |, a exponent |, a exponent MULTIPLICATION AND DIVISION OF QUANTITIES WITH FRACTIONAL EXPONENTS. 2X2» It has been shown (Art. 56) that the exponent of any letter in the product is equal to the sum of its expo- nents in the two factors. It will now be shown that the same rule applies when the exponents are fractional. 2 4 1. Let it be required to multiply a^ by a**. a^^f^^='^/~a'\ a^^l/a*=.'(^^, (Art. 205.) 2 4 33 But this result is the same as that obtained by adding the expo- nents together. 2 4 2.4 111+13 22 Thus, a3>— 1, and — 3>— 5. Two inequalities are said to subsist in the same sense, when the greater quantity stands on the right in both, or on the left in both ; as, 5>3 and 7>>4. Two inequalities are said to subsist in a contrary sense, when the greater stands on the right in one and on the left in the other ; as, 5>1 and 4<8. SIO. Proposition I. — If the same quantity^ or equal quantities^ he added to or subtracted from both members of an inequality, the resulting inequality will continue in the same sense. Thus, 7>5. Adding 4 to each member, . . . 11>9. Subtracting 4 from each member, . 3^1. Also, — 5<^ — 3; and by adding and subtracting 4, — l<-fl, and — 9<--7. Similarly, if a>6, then a-f c>6-f-c, or a— c>6 — c. Hence, Any quantity may be transposed from one side of an in- equality to the other, if at the same time its sign be changed. S20. Proposition II. — If two inequalities exist in the same sense, the corresponding members may be added together^ and the resulting inequality will exist in the same sewse. Thus, if 7>6, and 5>4; then, 74-5>6+4, or 12>10. When two inequalities exist in the same sense, if we subtract the corresponding members, the resulting in- INEQUALITIES. 183 equality will exist, sometimes in the same, and sometimes in a contrary sense. First, 7>3 By subtracting, we find the resulting inequality 4>-l exists in the same sense. 3>2 Second, 10>9 In this case, after subtracting, we find the 8>3 resulting inequality exists in a contrary ~2<6 ^^''^^• In general, if a>6 and c>d, then, according to the particular values of a, 6, c, and d, we may have a—c^b—d, a — c<6 — d, or a~c=b — d. S31. Proposition III. — If the two members of an in- equality be midtij)lied or divided by a positive number, the residting inequality icill exist in the same sense. Thus, 8>4 and 8X3>4x3, or 24>12. Also, 8--2>4--2, or 4>2. This principle enables us to clear an inequality of frac- tions. If the multiplier be a negative number, the resulting inequality will exist in a contrary sense. Thus, — 3<— 1, but _3X— 2>— lX-2, or 6>2. From this principle we derive 23S. Proposition IV. — TJie signs of all the terms of both members of an inequality may be changed, if at the same time toe establish the residting inequality in a contrary sense. For this is the same as multiplying both members by — 1. SI33. Proposition V. — Both members of a positive in- equality may be raised to the same power, or have the same root e^Jracted, and the resulting inequality icill exist in the same sense. Thus, 2<3 and 22<32, 2'^<<^^ or 4<'0, 8<27; and goon. Also, 25>16, and /25>i/16, or 5>4; and so on. 184 RAY'S ALGEBRA, SECOND BOOK. But if the signs of both members- of an inequality are not positive, the resulting inequality may exist in the same, or in a contrary sense. Thus, 3>— 2, and 32>(— 2)2, or 9>4. But, — 3<-2, and (_3)2>(— 2)2, or 9>4. EXAMPLES INVOLVING THE PRINCIPLES OF INEQUALITIES. 1. Five times a certain whole number increased by 4, is greater than twice the number increased by 19 ; and 5 times the number diminished by 4, is less than 4 times the num- ber increased by 4. Required the number. Let a:= the number. Then, 5a:+4>2a;+19, (1) 5a:— 4<4a:+4. (2) 5a:— 2a:>19— 4, from eq. (1) by transposing, 3a:>15, by reducing, 3:^5, by dividing both members by 3. 5a:— 4a:<4+4, from eq. (2) by transposing, ^<8, by reducing. Hence, the number is greater than 5 and less than 8, consequently eithei- 6 or 7 will fulfill the conditions. '2. If 4x— 7<2.2:-f 3, and 3a:+l>13— rr, find x. Ans. a-=4. 3. Find the limit of x in 7a:— 3>32. Ans. a:>5. 4. Of'a: in the inequality 5-f-|a;<84-|a; Ans. a:<36. (Z—\—C —\—6 5. Show that J~ ~|~ > the least, and < the greatest d C € of the fractions, j, ^, -., each letter representing a posi- tive quantity. Suppose y- to be the greatest, and -^ the least, of the fractions, ace ace c e c ,a a c ^a e ^a I' d J ^"'="' -b>a a=rf />rf' ""* 6=6' dt' "^tI'' ">!• (^'•'•221-) ab ad af a = -^, c<-^, e<-^. (Art. 221.) a + c + e>(6fc?4-/)^. (Arts. 219, 220.) a_{_c + €<(64-d+/)^. (Arts. 219, 220.) a_(_c + e c a + c-fe a Hence, r—, — ^rr-j^^ ^i J and ^ *— r-:c< t"- ' b-\-d^S d' b-\-d-{-f^b 6. It is required to prove that the sum of the squares of any two unequal magnitudes is always greater than twice their product. Since the square of every quantity, whether positive or negative, is positive, it follows that (a_6)2, or a2— 2a6+62>0. Adding, -\-2ab to each side (Art. 219), a2_|_62^2a6, which was required to be proved. Most of the inequalities usually met with, are made to depend ultimately upon this principle. 7. Which is greater, y'S+i/Ti or yS-^S^/T? Ans. the former. 8. Given J(a;+2)-|-Jrc-^(a:+l)-|-», to find X. Ans. x=:S. 9. The double of a certain number increased by 7, is not greater than 19, and its triple diminished by 5, is not less than 13. Required the number. Ans. 6. 10. Show that every fraction -f- the fraction inverted, is o:reater than 2 ; that is, that - 4-->>2. ^ h ^ a 11. Show that a'^-{-h'^-\-c'^^ah-\-ac-\-hc, unless a=h=c. 12. If x'^=a'^-^l'^, and i/'^z=c^-{-d'^, which is greater, xi/ or ac-{-hd? Ans. xi/. 13. Show that ahc'^(a-]-h — c)(a-[-c — h)(h-\-c — a), un- less a=:h=c. 2d Bk. 16 186 RAYS ALGEBRA, SECOND BOOK. YII. QUADRATIC EQUATIOIS^S. 234. A Quadratic Equation, or an equation of the second degree, is one in which the greatest exponent of the unknown quantity is 2 ; as, x^-{-x=a. An equation containing two or more unknown quantities, in which the greatest sum of the exponents of the un- known quantities in one term is 2, is also a Quadratic Equation ; as, x2/=a, xij — x — y=-c. 225m Quadratic equations, containing only one unknown quantity, are divided into two classes, p«/-e and affected. A Pure Quadratic Equation is one that contains only the second power of the unknown quantity, and known terms; as, x'^2=¥l—Ax\ and ax''^h=cx'—d. A pure quadratic equation is also called an incomplete equation of the second degree. An Affected Quadratic Equation is one that contains both the first and second power of the unknown quantity, and known terms ; as, hx'^-\-^tx=iZ^, and ax"^ — hx'^-\-cx — dx=e — f. An affected quadratic equation is also called a complete equation of the second degree. !336« The general form of a pure equation is ax^=h. The general form of an affected equation is ax'^-\-hx=c. Every quadratic equation containing only one unknown quantity may be reduced to one of these' forms. For, in a pure equation, all the terms containing x^ may be collected into one term of the form, ax^ j and all the known quantities into another, as h. QUADRATIC EQUATIONS. 187 So, in an affected equation, all the terms containing x^ may be reduced to one term, as ax^ ; and those contain- ing X to one, as hx ; and the known terms to one, as c. PURE QUADRATIC EQUATIONS. 22T. — 1. Let it be required to find the value of x in the equation, }^x^ — 3-|-p\ic'^^12| — x^. Clearing of fractions, 4x2— 36-f5a:2=rl 53—1 2a:2; Transposing and reducing, 21a:2— 189; Dividing, a:2=9; Extracting the square root of both members, X—zizZ] that is, a:=r-|-3, or a:— —3. Verification. 1(^-3)2- 3+y'5^(-|-3)2=rl2|-(H 3)2. 3— 3+3|=12|-9; or33^3|. Since the square of —3 is the same as the square of -f 3, the value x=z — 3, will give the same result as x^=-f^. 2. Given ax'^-\-h=id-\-cx?^ to find the value of x. Transposing, .... ax'^—cx'^—d—b\ Factoring, {a — C)x'^ — d — 6; r.' '.' 2 ^-* Dividing, x^= ; >a— c From the preceding examples, we derive the following Rule for the Solution of a Pure Equation. — Reduce the equation to the form ax^=:b. Divide hy the coefficient of x^, and extract the square root of both members. 228. If we solve the equation ax^=h, we have, X^=z±:\l-: that is, X—-{-\-, and X— — \l~. 188 RAYS ALGEBRA, SECOND BOOK. The equation may be verified by substituting either of these values of X. Hence, we infer, 1st. That in every "pure equation tlte unJawwn quantity lias two values^ or roots, and only two. 2d. Thxit these roots are equal in value, hut have contrary 1. \\x^—U=h:c'^10 Ans. a;=±3. 2. \(x^—Vl)=\x'—l Ans. .-c^rtG. 3. (a;-f 2)^=4x+5 Ans. x=^l 4- i--V + T-T9-==25 Ans. ^-±.3. 1 — Zx l-\-lx _ X-\-l X — 7 7 . , o xi' — ^x x'-\-lx x' — 73 6. -^+-^z=c. . . . Ans. x=^^x/h'c:'—2ahc. h-\-x b — X c ^ 1. xi/6-\-x'=l-\-x'' Ans. x^±^, 2a.' a - 8. x-^./a'-irx'=--- ,—-— ,. . . . Ans. a:=±oi/3. 2 2 a — i/a'^ — x'^ , . _,2ai/b 10 ^ =^h. . .... Ans. a:=±-^-f^ . a-\-i/a'—x' ^+1 QUESTIONS PRODUCING PURE EQUATIONS. !339. For the statement of the equation, see Art. 154. 1, What two numbers have the ratio of 2 to 5, the sum of whose squares is 261 ? Let 2x and bx— the numbers. Then, 4x^-\-25x^=29x^=261 ; AVhence, x^r^9, and x=S. Hence, 2X—Q, and 5a:=:15 the required numbers. QUADRATIC EQUATIONS. 189 2. The square of a certain number diminished by 17, is equal to 130 diminished by twice the square of the num- ber. Required the number. Ans. 7. 3. Required a certain number, which being subtracted from 10 and the remainder multiplied by the number itself, gives the same product as 10 times the remainder after subtracting 6| from the number. Ans. 8. 4. What number is that, the J part of whose square being subtracted from 30, leaves a remainder equal to | of its square increased by 9 ? Ans. 6. 5. There are two numbers whose difference, is | of the greater, and the difference of their squares is 128 ; find them. Ans. 18 and 14. 6. Divide 21 into two such parts, that the square of the less shall be to that of the greater as 4 to 25. Let X and 21— a:= the parts. Then, rc2: (21— a:)2 : :4:25; Or, (Auith., Art. 200,) 25a:2z=4(21— a:)^; Extracting square root, 5a:=2{21 — x)\ Whence, aj=6, and 21— a;=15, 7. Divide 14 into two such parts, that the quotient of the greater divided by the less, shall be to the quotient of the less by the greater, as 16 to 9. Ans. 6 and 8. 8. What number is that which being added to 20 and subtracted from 20, the product of the sum and difference shall be 319? Ans. 9. 9. Find two numbers, whose product is 126, and the quotient of the greater by the less 3.^ . Ans. 6 and 21. 10. The product of two numbers is p, and their quo- tient q. Required the numbers. . — , /» Ans. |/p<2 and ^r-. 11. The sum of the squares of two numbers is 370, and the difference of their squares 20B- Required the num' bers. Ans. 9 and 17. 100 RAY'S ALGEBRA, SECOND BOOK. 12. The sum of the squares of two numbers is c, and the difference of their squares d. Required the numbers. Ans. l\/'2{c^d), and iy2{c—d). 13. A certain sum of money is lent at 5 % per annum. If we multiply the number of dollars in the principal by the number of dollars in the interest for 3 mon., the prod- uct is 720. What is the sum lent? Ans. $240. 14. It is required to find 3 numbers, such that the prod- uct of the 1st and 2d =a., the product of the 1st and 3d =6, and the sum of the squares of the 2d and 3d =^c. 15. The spaces through which a body falls in different periods of time, being to each other as the squares of those times, in how many sec. will a body fall through 400 ft., the space it falls through in one sec. being 16.1 ft.? Let X=i the required number of seconds. Then, 16.1 : 400 : : P : a;2 ; whence, a:=4.98-f- sec. In what time will it fall 1000 ft.? Ans. 7.88+ sec. 16. What two numbers are as 3 to 5, and the sum of whose cubes is 1216? Let 3a: and bx— the numbers; Then, 27x3+125^:3== 152ic3=1216; Whence, a:3=8, and a:=r^8=2. Hence, the numbers are 6 and 10. This is properly a pure equation of the third degree; but ques- tions producing such equations are generally arranged with those of the second degree. 17. A money safe contains a certain number of drawers. In each drawer there are as many divisions as there are drawers, and in each division there are four times as many dollars as there are di:awers. The whole sum in the safe is $5324 ; what is the number of drawers? Ans. 11. QUADRATIC EQUATIONS. 191 18. A and B set out to meet each other ; A leaving the town C at the same time that B left D. They traveled the direct road from C to D, and on meeting, it appeared that A had traveled 18 miles more than B; and that A could have gone B's journey in 15| days, but B would have been 28 days in performing A's journey. What is the dis- tance between C and D? Ans. 126 miles. 19. Tw^o men, A and B, engaged to work for a certain number of days at different rates. At the end of the time, A, who had played 4 days, received ^5 shillings ; but B, who had played 7 days, received only 48 shillings. Had B played only 4 days, and A 7 days, they would have received the same sum. For how many days were they engaged? Ans. 19. 20. A vintner draws a certain quantity of wine out of a full vessel that holds 256 gal. ; and then filling the vessel with water, draws off the same number of gal. as before, and so on for fpur draughts, when there were only 81 gal. of pure wine left. How much wine did he draw each time? Ans. 64, 48, 36, and 27 gal. AFFECTED QUADRATIC EQUATIONS. S30« — 1. Required to find the value of x in the equation, a.;_6a:+9=4. It is evident, from Art. 184, that the first member of this equa- tion is a perfect square. By extracting the square root of both, members, We find, .... a:— 3^dz2; Whence, .... a;=3 ±2=3 4-2=5, or 3-2=1. Verificaiion. (,5)2-6(5)-f 9=4; that is, 25— 30+9=4. (l)2_6(l)+9=4; thfitis, 1—6+9=4. Hence, x has two values, +5, and +1, either of which verifies the equation. 192 RAY'S ALGEBRA, SECOND BOOK. 2. Required to find the value of x in the equation, As the left member of this equation is not a perfect square, we can not find the value of X by extracting the square root, as in the preceding example. We may, however, render the first member a perfect square by adding 9 to it. This may be done provided the same number be added to the other member, to preserve the equality. The equation then becomes, a:2_6a:4 9=36. Extracting the square root, a;— 3:=.=t:6. Whence, :r=:3dt6^=-( 9, or — 3, either of which values of x will verify the equation. !S31. We will now proceed to explain the method of completing the square. Since every afi'ected equation (Art. 226) may be reduced to the form, ax^-^hx^iC^ he Dividing both sides by a, x^^—Xzzz—. b c For the sake of simplicity, let — =2r), and ~—-Q. As each of these fractions may be either positive or negative, the equation must assume one of the four following forms : x^-\-2px=q. (1) x^—2px=q. (2) x'^-\-2px=~q. (3) x^—2px=—q. (4) Hence, Every affected equation may he reduced to the form x'^±2px=dzq. It will now be shown that the first member of this equation may always be made a perfect square. We may consider x'^-\-2px as the first two terms of the square of a binomial, the third term being unknown or lost. Extracting the root of .r-, we find that the first term of the bino- mial must be X. We next observe that 2px is twice the product of the first term by (he second; therefore, p, which is half the. coeffi- cient of Xj is the second term of the binomial, and its square, p^^ . QUADRATIC EQUATIONS. 193 added to x'^-\~2px, will render it a perfect square. But, to preserve the equality, we must add the same quantity to both sides. This gives, . . . x--^2px-{^p^=q-^p^j Extracting the square root, x-\'P=^zizy/q-'rp^', Transposing, X——pzh^q-^p'^. It is obvious that in each of the remaining three forms, the square may be completed on the same principle. Solving equations (2), (3), and (4), and collecting together the four diflferent forms, we have the following table: (1) x^^2px=q. x=—pdLi/q-\-p^. (2) x^—2px=q. a:=4-pzfc|/g+p2; (3) x^-\-2px=—q. x=—pdoi/—q-\-p^ (4) x^-2px=—q. x=-^pzti^ —q-\ p^. From the preceding we derive the following Rule for the Solution of an Affected Equation.— 1st. Reduce the equation, hy clearing of fractions and transjwsi- iion, to the form ax^-|-bx=c. 2d. If the coefficient of x^ is minus, change the signs of all the terms, or midtiply each term hy — 1. 3d. Divide each side of the equation hy the coefficient of x^. 4th. Add to each memher the square of half the coefficient of X. 5th. Extract the square root of hoth sides, transpose the known term, to the second 7nemher, and find the value of x. Remark. — Although from the equation rc^^m^, we have dba;=r=hm; that is, -\-x=^m{\), -\-x=—m{2), —x=^m{^), and — a;=— r?2(4), it is evident that equations (1) and (4) are the same equation, as also (2) and (3). Hence, -f rr— ±m, embraces all the values of x. For the same reason it is necessary to take only the plus sign of the square root of {x-\-pY. 1. Given 17a:— 2x^=32— 3.t, to find x. Transposing, — 22:2-4- 20a:= 32; Changing signs, and reducing, a:^— 10a:— — 16; Completing the square by adding (y))2=r25 to both sides, a:2_l0a:-(-25=— 16+25=9 ; 2d Bk. :7- 194 RAY S ALGEBRA, SECOND BOOK. Extracting the root, a;— 5= ±3; Whence, .... a:=5±3=8, or 2. Verification. 17(8)— 2(8)2=32-3(8), or + 8=+ 8. 17(2)-2(2)2=32— 3(2), or +26=+26. 2. Given 3x^—2x^65, to find x. Dividing by 3, . . . .x'^~%x=^^; Completing the square, x'^—%x^[^Y=^^^{lY=\^. Extracting the root, . a:~^:=rty. Whence, a:=^=h:L4^5, or —4^. Both of which values verify the equation. 3. Given 4.d'~2x' ^2ax=l^ah—lU\ to find x. Transposing, . . . — 2a;2-f 2aa:=— 4a2^18a6— 1862; Dividing by -2, . . x-—ax=^2a^—^ab^^b'^\ Completing the square, x~—ax-\—T-=^—7 9a6-|-962; Extracting root, . . a:— ^=±1 -^ — 36 I; AVhence, ^=? + ( T ~^^ )=2a— 36, or —a-f 36. 4. Given .'r-f-|/'(5cc-|-10)=8, to find x. By transposition, . . |/(5a:-f-10)=8— a;; By squaring, . . . 5a:+10~64— 16a:+a;2; Or, a;2-21a:=— 54; Completing the square, x'^-2\x-\-{'^^ )2=rl4 1 —54=226 ; Extracting the root, . a:— 2^1 =±1^5; Whence, a:=2^i _j_i^5^3^6^-18, ^r j=3. These two values of a: are the roots of the equation, x"^ — 21a:=r — 54 but they will not both verify the original equation. For, the proposed equation might have been a:dr]/(5x-fl0)=8; and the operations which have been employed would result in the same equation, x'^ — 21a;=: — 54, whether the sign of the radical part be -|- or -'• Hence, in the equation X-\--y/{hx-\'\0)^=%, the value of a; is 3 ; but Id the equation a:— j/(5a:4-10)=i8, the value is 18. QUADRATIC EQUATIONS. 195 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. x'—16x=—60. x'—6x=6x-^2S. 10 4-350— 12a:=^0 2x=^+-.. . . X 3a:^+10a:=5V. (x— 1)(.7:— 2)=^!. ix^— ix+2^9. ^ , 110 x-f22 3~" 4 Ox-6 16. 17. 18. l'7a:^4-19x=1848. . 19. 20. 21. 3 ■10'^«~^" Sx^\x:'=10. 22. ^2_3,^ + ^-.^4^ g^.- .T-f-4 1—X _ 4:X-\-7 9 23. a:+- X — 3 1 ^-^x ^^x 13 X 1 X X 26. •I + «-?=0. a £c a Ans. x^=6, or — 10. Ans. ic=10, or — 6. Ans. x= — 6, or — 10. . Ans. x=io, or 10. Ans. £c=:14, or — 2. Ans. x=:70, or 50. . Ans. x=Sy or — 1. Ans. x=S, or — 6J. Ans. a;=^(3zt:^5). Ans. a:=4, or — 3^. Ans. a:=ll, or — 10. . Ans. x^=2, or ^|. Ans. a:=3, or — 2|. Ans. a:=9[f, or — 11. . Ans. z=l, or — i. Ans. x=r-6dz2i/—l. Ans. ic=:^4, or — 3|. Ans. ic=21, or 5. Ans. a:=|/3, or l\/S. . . Ans. x=S, or — |. Ans. 5c=l=±:|/(l— a'). 196 RAY'S ALGEBRA, SECOND BOOK. ZO. ='zax — cx^ Ans. a;= . c c 27. x^ — (a-\-h~)x-\-ab=:^0. . . . Ans. x=a, or h. 28. (a—h)x'—(a-\-h)x-^2b=0. Ans. x=l, or -— . 29. inqx"^ — ninx-\-pqx — jip=0. Ans. x=-, or SO. -^^-(a^-b^)x== ^ m c^-\-b^ (ab') ^-\-(a'b) - Ans. x=:a, or — b. 31. adx — acx'^=bcx — bd. . . Ans. x=~, or . c a 12 32. |/ (0^4-5)= /no -x- • A°s- ^=4, or -21. «>«5. ;= = 7=^ Ans. a:r=4. 4+1/^ 1/^ 34. i/'s^—2i/X=X Ans. X=z4:. 35. y^x-\-a — i/x-\-b=y^2x. Ans. x=—'^zhli/2a'-\-2b\ Q^i /-r-i / ^-"^ A 4a 3a 00. ya-\-x-\-ya — x= r~ — ■ — . Ans. ic:=-^, or -=-. 233. Hindoo Method of Solving ftnadratics.— When an equation Is brought to the form ax'^-\-bx=rCj it may be reduced to a simple equation, without dividing by the coefficient of cc', thus avoiding fractions. If we multiply every term of the equation ax^-l-bx=:C, by four times the coefficient of the first term, and add to both sides the square of the coefficient of the second term, we shall have 4a2a:2-|-4a6a;+62^4ac-]-62 Now, the first member of this equation is a perfect square, and by extracting the square root of both sides, we have 2ax-\-b^zdc:y/4ac-\-b^, which is a simple equation. Hence, the QUADRATIC EQUATIONS. 197 Hindoo Eule for the Solution of Quadratic Equa- tions. — 1st. Reduce the equation to the form ax^-fbx=rc. 2d. Multiply both sides by four times the coefficient of x^. 3d. Add the square of the coefficient of x to each side, extract the square root, and finish the solution. 1. Given 2a:2— 5a:=3, ^o find x. Multiplying both sides by 8, four times the coefficient of a:^, "Wehave 16a:2— 40a;=24. Adding to each side 25, which is the square of the coefficient of X, We have .... 16a:2—40a:+ 25=49; Extracting the root, 4a:— 5= ±7; whence, a:=3, or — J. Find the value of the unknown quantity in each of the following examples by the Hindoo Rule : 2. 3x2+5a:=2 Ans. x=\, or —2. 3. rr2+a:=30 Ans. x=^b, or —6. 4. x^ — x=^2 Ans. x=^9, or — 8. 40 9*7 5. -^-f^=13 Ans. x=^9, or i|. X 5 X '13 By an inspection of the forms given in Art, 231, it will be seen that the value of the unknown quantity may be found without the formality of completing the square, by the following Rule. — Reduce the equation to the form x^-f-^px^q. The unknown quantity will then be equal to one half the coefficient of its first power taken with a contrary sign, plus or minus the square root of the square of the number last written to- gether with the known quantity in the second member of the equation taken icith its proper sign. Thus, let a:2-fl6a;=— 60. _ Then, a:=-8±:v/64-60=-8db/4=— 8dr2. a;=— 6, or —10. After some exercise in completing the square, it is best to employ this last method. 198 RAY'S ALGEBRA, SECOND BOOK. PROBLEMS PRODUCING AFFECTED EQUATIONS. 2SSm — 1. A person bought a certain number of sheep for $40, and if he had bought 2 more for the same sum they would have cost $1 apiece less. Required the num- ber of sheep, and the price of each. Let a;:= the number of sheep. 40 Then, — = the price of one. 40 And ^ = the price of one, on the second supposition. m, . 40 40 , , , Therefore, — —^zr^ — — 1, by the question. Solving this eq., a;^— lzt:9=8, or — 10, number of sheep. 40 40 40 And — =iP=$5, price of each. Also, — = — — = — 4. X ^ ^ X — 10 The negative value, — 10, to fulfill the conditions of the question in an arithmetical sense, must be modified, on the principles ex- plained in Art. 164, thus : A person tells a certain number of sheep for $40. If he had sold 2 fewer for the same sum he would have received $1 apiece more for them. Required the number sold. 2. Find a number such, that if 1*7 times the number be diminish^! by its square, the remainder shall be *70. Let a:= the number. Then, \'ix—x'^=lQ. Or, x'^~llx=—lQ. Whence, a;^7, or 10. In this case, both values of x satisfy the question in its arithmeti- cal sense. Thus, 17X 7— 72^.119- 49^70. Or, 17X10— 102:^170-100:.z:70. 3. Of a number of bees, after |, and the square root of \ of them, had flown away, there were two remaining ; what was the number at first? QUADRATIC EQUATIONS. 199 To avoid radicals, let 2a:2 represent the number of bees at first; 16x2 Then, — - _[-a;+2=2a;2. y Whence, X=:i6, or — 1^; but the latter value, being fractional, though satisfying the equation, is excluded by the nature of the question; the number of bees is 2x62=72. 4. Divide a into two parts, whose product shall be h'K Let x=: one part ; then, a — a;= the other. Therefore, x{a — x), or ax — x'^=Jfi. Whence, a;=^(art^a2— 462) . ^hat is. x=^(a±i/a^—4b^), and a-Xz:^^{az:fzy/a^—Ab^), are the parts required, and the two parts are the same, whether the upper or lower sign of the radical quantity be used. Thus, if the num- ber a is 20, and b 8, the parts are 16 and 4, or 4 and 16. The forms of these results enable us to determine the limits under which the problem is possible; for it is evident that if 462 i^q greater than a2^ j/ci2 — 462 becomes imaginary/. The extreme possible case will be, when i/a^ — 462:=^0, in which case x—^a, and a—x=^a\ also, 62=zia2^ ^^d b—^a. In the following examples, that value of the unknown quantity only is given, which satisfies the conditions of the question in an arithmetical sense: 5. What two numbers are those whose sum is 20 and product 36? Ans. 2 and 18. 6. Divide 15 into two such parts that their product shall be to the sum of their squares as 2 to 5. Ans. 5 and 10. 7. Find a number such, that if you subtract it from 10, and multiply the remainder by the number itself, the prod- uct shall be 21. Ans. 7 or 3. 8. Divide 24 into two such parts that their product shall be equal to 35 times their difference. Ans. 10 and 14. 9. Divide the number 346 into two such parts that the sum of their square roots shall be 26. Ans. 11^ and 15'^ SroGESTioN. — Let a;= the square root of one of the parts, and 26— a;, of the other. 200 * RAYS ALGEBRA, SECOND BOOK. 10. What number added to its square root gives 132? Aus. 121. 11. What number exceeds its square root by 48 J ? Ans, 56J 12. What two numbers are those, whose sum is 41, and the sum of whose squares is 901 ? Ans. 15 and 26. 13. What two numbers are those, whose difference is 8, and the sum of whose squares is 544? Ans. 12 and 20. 14. A merchant sold a piece of cloth for $24, and gained as much per cent, as the cloth cost him. Required the first cost. Ans. $20. 15. Two persons, A and B, had a distance of 39 miles to travel, and they started at the same time ; but A, by traveling | of a mile an hour more than B, arrived one hour before him ; find their rates of traveling. Ans. A 3i, B 3 mi. per hr. 16. A and B distribute $1200 each among a number of persons ; A gives to 40 persons more than B, and B gives $5 apiece to each person more than A ; find the number of persons. Ans. 120 and 80. 17. From two towns, distant from each other 320 miles, two persons, A and B, set out at the same instant to meet each other ; A traveled 8 miles a day more than B, and the number of days before they met was equal to half the number of miles B went in a day ; how many miles did each travel per day? Ans. A 24, B 16 mi. 18. A set out from C toward D, and traveled 7 miles a day. After he had gone 32 miles, B set out from D to- ward C, and went every day -j'^ of the whole journey; and after he had traveled as many days as he went miles in one day, he met A. Required the distance from C to D. Ans. VG, or 152 miles. 19. A grazier bought a certain number of oxen for $240, rbx=:C, reduces to bx—C, an equation of the first degree, which can have but one root. The supposition that it has two, gives one value infinite, which is equivalent to saying, the equation has but one finite root. If we had at the same time a=iO, 6^0, C— 0, the equation would be altogether indeterminate. This is the only case of indetermina- tion occurring in quadratic equations, ' S39. We shall now apply the principles above stated, in the discussion of* the followin"; 208 RAY'S ALGEBRA, SECOND BOOK. Problem of the Lights. — It is required to find, in a line BC, which joins two lights, B and C, of different in- tensities, a point which is illuminated equally by each. P^^ B P C P^ It is a principle in optics, that the intensity of the same light at different distances, is inversely as the square of the distance. Let a be the distance BC between the two lights. Let b be the intensity of the light B at the distance of 1 ft. from B. Let C be the intensity of the light C at the distance of 1 ft. from C. Let P be the point required. Let BP =X] then, CP =a—x. By the principle above stated, since the intensity of the light B at the distance of 1 foot, is 6, at 2, 3, 4, . . . . feet, it must be A- 7{- ir. • • • • ) hence, the intensity of B and of C, at the 4 9 16 be distance of x and of a — x feet, must be -tt and .,. ' x^ [a — x)^ But, by the conditions of the problem, these two intensities are equal. Hence, we have for the equation of the problem, which reduces to x'^ {a—xf x^ 6' Whence, =:: — — , or — ^_. ^ ^b |/6 This gives the following results : , ay/lb , awTi 1st. x=i--zi^ — -] whence, a — x=—:^- — :::., 2d. x= —^ — -; whence, a—x= ., '^ -. |/6— |/c y/b—yc "We shall now proceed to discuss these values. QUADRATIC EQUATIONS. 209 I. Let h^c. CLr/' b The first value of X, — =1, is positive, and less than a, for ^ - is a proper fraction. Hence, this value gives for the point illuminated equally, a point P situated between B and C. We per- ceive, also, that the point P is nearer to C than B ; for, sincv &>c, we have i/6+/6>y/6+|/c^ or '2i/byy^-\-y^e, and —z£z =:>2) ^'^d) consequently, ■ — ^ ="> — This is manifestly correct, for the required point must be nearer the light of less intensity. The corresponding value of a — a:, (Xi/C . Ob — :=^ — - IS positive, and evidently less than tt^. /6+^c ^ ' ^ 2 The second value of x, - . - — -, is positive, and greater than a ; yb—yc This value gives a point P^, situated on the prolongation of BC, and in the same direction from B as before. In fact, since the two lights emit rays in all directions, there will be a point P'', to the right of C, and nearer the light of less intensity, which is illumin- ated equally by the two lights. The second value of a—x, -^~— ^, is negative, as it ought to yb—x/c be, and represents the distance CP'', in the opposite direction from G, (Art. 47.) II. Let 5+vo>vbH^b; •••,75q:;7si/c— i/6 .-. — = x^l, y — j/ o' and — =^ !>«. This represents CP^^, and is the sum of the dis- |/c— 1/6 tances CB and BP^'', in the same direction from C as before. m. Let h=c. The first values of x and of a — x, reduce to ^, which shows that the point illuminated equally is at the middle of the line BC, a re- sult manifestly true, upon the supposition that the intensities of the two lights are equal. The other two values are reduced to ^ =00 . (Art. 136.) This result is manifestly true, for the intensities of the two lights being supposed equal, there is no point at ^ny finite distance, except the point P, which is equally illuminated by both. IV. Let l=c, and a=0. The first system of values of x and a — a:, become 0. This is evi- dently correct, for when the distance BC becomes 0, the distances BP and CP also become 0. The second system of values of x and a — X. become ^; this is the symbol of indetermination, (Art. 137.) QUADRATIC EQUATIONS. 211 This result is also correct, for if the two lights are equal, and placed at the same point, every point on either side of them will be illuminated equally by each. V. Let ci=0, h not being =c. All the values of a; and a — X reduce to 0; hence, there is no point equally illuminated by each. In other words, the solution of the problem fails in this case, as it evidently should. This might also have been inferred from the original equation ; for if we put a=Q, -^5=7 r, becomes -^ = -^, which can never ^ 'a:2 (re— a)2 x^ x^' be true except when 6=c, as in Case IV. SSO"*. Examples for discussion and illustration. 1. Required a number such, that twice its square, in- creased by 8 times the number itself, shall be 90. Ans. 5, or — 9. How may the question be changed, that the negative answer, taken positively, shall be correct in an arithmetical sense? 2. The diiference of two numbers is 4, and their prod- uct 21. Required the numbers. Ans. -f-3, +7, or —3 and —7. 3. A man bought a watch, which he afterward sold for $16. His loss per cent, on the first cost of the watch, was the same as the number of $'s which he paid for it. What did he pay for the watch? Ans. $20, or $80. 4. Required a number such, that the square of the num- ber increased by 6 times the number, and this sum, in- creased by 7, the result shall be 2. Ans. x= — 1, or — 5. What do the values of a; show? How may the question be changed to be possible in an arithmetical sense? 5. Divide the number 10 into two such parts, that the product shall be 24. Ans. 4 and 6, or 6 and 4. Is there more than one solution? Why ? 212 RAY'S ALGEBRA, SECOND BOOK. 6. Divide the number 10 into two such parts that the product shall be 26. Ans. S-j-y — 1, and 5 — -y/ — 1. What do these results show? 7. The mass of the earth is 80 times that of the moon, and their mean distance asunder 240000 miles. The at- traction of gravitation being directly as the quantity of matter, and inversely as the square of the distance from the center of attraction, it is required to find at what point on the line passing through the centers of these bodies, the forces of attraction are equal. Ans. 21o86o.5-[- miles from the earth, and 24134.5— '' " " moon. Or, 270210.44- " " " earth, and 30210.4 -f " beyond the moon from the earth. This question involves the same principles as the Problem of the Lights, and may be discussed in a similar manner. The required results, however, may be obtained directly from the values of Xj page 208, calling a=240000, 6=80, and c=l. TRINOMIAL EQUATIONS. S40. A Trinomial Equation is one consisting of three terms, the general form of which is ax"^-\-bx'*=:c. Every trinomial equation of the form that is, every equation of three terms containing only two powers of the unknown quantity, and in which one of the exponents is double the other, can be solved in the same manner as an affected equation. As an example, let it be required to find the value of x in the equation X* — 2})x'^=q. QUADRATIC EQUATIONS. 213 Completing the square, a:<— 2pa:2-^p2_g_|_p2^ x-— p=v'g-fp2. a;2=+p±/g+p2. 241. Binomial Surds. — Expressions of the form A±j/B, like the value of x^ just found, or of the form |/A±|/B, are called Binomial Surds. The first of these forms, viz., A=t]/B, frequently re- sults from the solution of trinomial equations of the fourth degree ; and as it is sometimes possible to reduce it to a more simple form by extracting the square root, it is neces- sary to consider the subject here. We shall first show that it is sometimes possible to ex- tract the square root of A±|/B, or to find the value of V A±|/B. Let us inquire how such binomial surds may arise from involution. If we square 2=t|/3, we have 4±4|/3-j-3, which, by reduction, becomes 7dt4|/3. Hence, ^7d=4|/3=2=ii:p/3. In the same way it may be shown that -«(5dt2|/6=|/2db|/3. It thus appears that the form Adzy/B may sometimes result from squaring a binomial of the form azfc|/6, or |/a=irj/6, and uniting the extreme terms, which are necessarily rational, into one. In such cases, A is the sum of the squares of the two terms of the root, and y^B is twice their product. To find the root, therefore, put x^-^y^—A and 2a:?/^|/B, and pro- ceed to find X and y, the terms of the root. Thus, Extract the square root of . . . 7-f4j, 3. Put x^^y^=7 (1), and 2xy^AyyS: Adding, we have a;2-f 2:ri/4-2/2=7+4/37 Subtracting, we have x^— 2x7/-|-?/2— 7— 4/37 214 RAY'S ALGEBRA, SECOND BOOK. Extracting the root, iC-j-^^^7-|-4|/3 (2). ^-2/=v 7_4,/3 (3). Multiplying (2) by (3), x^-y^=^^d—48=yl=l (4). By adding and subtracting (1) and (4), we have 2x^=^8 .-, x=2 and 2y^=z6 .-. y^zy'W. Hence, 2-|-^3 is the root to be found. 1. Extract the square root of 15-f-6|/6. Ans. 3-f-|/6. 2. Of 34~24i/2 Ans. 4-3|/2: 3. Of 14±4i/6. Ans. y2ziz2y'W. We shall now proceed to demonstrate more fully that the square root of A=t|/B may always be found in a simple form, when A^ — B is a perfect square. To do this it is necessary to prove the following theorems : Theorem I. — The value of a quadratic surd can not he partly rational and partly irrational. For, if possible, let |/'a:=a-f|/6; ,•. squaring both sides, _ X a"^ b x=a^-\'2ayb-{-b] .-. yb= ^ -; that is, an irrational quantity is equal to a rational quantity, which is impossible. Theorem II. — In any equation of the form xd:r|/y=:azt: -j/'^b, the rational quantifies on opposite sides are equal, and also the irrational quantities. For if X does not =a, let x=ra-|-m; Therefore, a-f m+y/^— a+^/fi; .-. m-{-j/y=y/W; that is, the value of a quadratic surd is partly rational and partly irrational, which has been shown by Th. T, to be impossible ; hence, x=a, and y/y=y''b. We shall now proceed to find a formula for extracting the square root of A-f |/B. QUADRATIC EQUATIONS. 215 Assume .... 'JA.-{-■^/'B=-^/x^■^/'^, A-]-'i/h'=^x-^y-\-2-i/xy, by squaring. By Th. II, a:+^=A(l); and 2^^=/B(2); Squaring equations (1) and (2), we have Axy — B; Subtracting, a:2_2a;y+2/^=A2— B; or, (a;— i/)2=:A2— B. Let A2— B be a perfect square =C2; then, C=i/ A2 — B. Therefore, . . (a;— ^)2— C^, or x—y~^\ But, .... iC-fy=A; Whence, . . . Xz= — ^ — ; and 2/= — ^r— . __ /A + C _ /A— C And . . i/a:=±'Y 2 I ^'^^i |/2/==t \~2~'' Therefore, i/aj+v^y, or ^A+i/B = rt^/A+2±^ a-4:j Similarly, V^^W. or ^/A-^B=d=^4L+^:^ ^^^. Or, . .^^W^=^[^^^^\ (K.) And .Va-./B=±(^/^_V^). (L.) By substituting particular values for the general ones in these formulas, examples may be easily solved. 1. Extract the square root of 31-|-10|,/6. Here, A=r31, /B==10/6; .-. A2— B=C2z=961— 600rr:361; and C=19. .-. A+C:=50, A— Cr=12. 216 RAYS ALGEBRA, SECOND BOOK. Taking the formula and substituting, we have Va^(V^%/-^). Proof .— (5+ i/6)2^25+10|/ 6+6=31 +10/6. 2. Reduce ^ np-\-2m^ — 2my'np-\-m\ to its simplest form. Here, A=np-}-2m^, and B^=i4'm^{np-\-m^). A2_B=in2p2^ and C=np, (formula L.) .-. A+C=2rip+2m2, A— C=2m2 .-. x—npJf-m^, y=m^. Formula (L) gives =h(|/^p+w2— m). 3. Find the square root of ll + 6|/2r Ans. 3-f |/2 4. Of 8±2|/2 Ans. y 2±1 5. Of l7+2i/60 Ans. 2|/3+|/5 6. Of cc— 2;,/^^ Ans. i/^Hl—l 1. Of 2a^/^l. (A.^.0.) . Ans. |/a(l-f ^=1) 8. Of x-\-j/-\-z-\-2i/xz-\-yz. . Ans. yx-{-y-{-y^z, 9. Find the value of ^28+10|/3+^67-16i/3. Ans. 13. When A2 — B is not a perfect square, or when the binomial surd is of the form |/Adt|/B, the root will be more complex than the original form. Remark. — By the above method the square root of any bino- mial or residual, as a + 6, a— 6, a'^-\^b^, etc., may also be found, in a complex form. S4!3. We shall now resume the subject of Trinomial Equations. The general form of trinomial equations is x^-\-2px'*=.q ; but there are several varieties of this form, QUADRATIC EQUATIONS. 217 of which the following are the principal : viz., x-\~-i/ x=^q, p. an x*-\-px^=q, x'^-\-px'^:=q, x^^-{-2)X^=q, x''"-}-p^""=9'5 (j^^ -\-px-\-qy-\-h(x'^-\-px-\-q')=r, and (x'^-\-2JX-\-qy~"-\-h(x'^-\~2^^ +qy=k. Some of these varieties, if developed, would produce very compli- cated expressions, yet they may all be solved by the general method given in Art. 240. 1. Given x^ — 6x^^16, to find the value of x. Assume, .... X^—y; then, X^=7/-, and Whence, . . . . 2/ ~8, or — 2. Therefore, . . . x3=8, or —2; and x^2, or —fZ Or, the example may readily be solved without introducing a new letter. Thus, Completing the square, x'' — 6a:'^-f 9=25. Extracting the root, x^ — 3= ±5 a;3=:8, or —2, and x=2, or —f2. It will be shown hereafter, (Art. 396,) that in such examples as the preceding, there are four values of X not deteimined. ^ 2. Given 5.c — 4|/cc=33, to find the value of x. Assume, .... ^'x=^y; then, x=z^y^, and 52/2-42/=33; Whence, .... y— 3, or — 'i; Consequently, . . X—9, or ^X^. 3. Given i/x-\-12-\-fx-\-12=G, to find the value of x. Assume, f/x-\-12=y; then, -i/x-^12=y^, and 2/2+2/^6; whence, y=2, or— 3; Therefore, ^a;-fl2=2, or —3. Whence, X-{-12^'[^ or 81 ; and X=.i, or 69. 2d Bk. 19* 218 RAY'S ALGEBRA, SECOND BOOK. Or, without introducing a new letter 2/, we may proceed to com- plete the square. Thus, l/a;+12+f/a:-i-12+i=6+i=:2_5; -2' ^a;+12=-^±|=+2, or -3. a:+12=:16, or 81. Whence, a:=4, or 69. 4. Given Sx'-\-i/Sx''-^l=bb, to find the value of x. Adding 1 to each member, the equation becomes 3a:2-^l+^3a:2+l=56. The equation may now be solved like the preceding. The values of x are +4, —4, +|/2l, and —^21. Find the values of x in each of the following examples : 5. x'—2bx'=—14:4:. . . . Ans. x==:±^S, or ±4. 6. bx*-^1x'^67S2. Ans. x=±:6, or ±-pL|/=3740. 7. 9x«— ll;r^.==488. . . Ans. x^2, or ^1^=183. 8. x3_aji^i5500. . . Ans x=:2b, or (—124)^. 9. x^4-x^'=i056. . . . Ans. x=64, or (—33)^ 10. x-{-b=i/x-{-b-\-6. . . . Ans. x=4:, or — 1. 11. 2|/x'-^— Sx-f-ll^x^— 3.X+8. Ans. x=.2, 1, or ^±A/^31. 12. x'^— 7x4-^/^^— Yx+18=-24. Ans. x=9, —2, or Klz^ylTS). 13. (x^— 9)2==3+ll(x2— 2). Ans. a;=±5, or ±2. Ans. a;=4, 2, or ^(— 7±|/lV.) 15. .T*(l-h^|-(3:.H^)-^0. Ans. x^S, — 3^, or J(— 1±^/_251.) QUADRATIC EQUATIONS. 219 16. x/( l-^) = ^-. Ans. x=±^a±iy2). Sometimes it may be necessary to substitute a new letter two or more times, or to complete the square, without substitution, three or more times. The following is an example: 17. x*-i-bx'-\-4i/x*-\-bx'=:60 Ans. x=.±2, ±3^/-!, and ±-^|(— 1±:T/177) S43. Equations sometimes occur in which the com- pound term is not at first apparent, but which may be reduced to the form of a trinomial equation by the follow- ing method : Extract the square root to two or three terms, and if we find a remainder (omitting known terms, if necessary,) which is any mul- iiple or any part of the root already found, the given equation may "be reduced to a trinomial, of which the co7npound term will be the root already found. If the greatest exponent of the unknown quantity be not even, it must be made so by multiplying both members of the equation by the unknown quantity. -to 4 1. Given x^ — 4:ax^ — 2a'^x-^12a^= , to find x. X Multiplying botlj sides by X, and transposing, we have Proceelling to extract the square root, we have the following OPERATION. x*—Aax^ 2a-x--\-12a^x—16a^x-—2ax. X* 2x^ 2ax \ — 4aa;3 2a^x^ - 4ax?'{-4a'^X" Remainder, . . —6a^x^ -^12a^x IGa^; Or, —6a^{x^—2ax) 16a*. Ilencc, the given equation may be written thus: [x^ 2axY—Ga^{x^—2ax)—16a*^0. Or, . . . {x-—2ax)^—6a'2{x^-2ax)^16a*. Proceeding with the solution, we find X— 4a, —2a, or adra^/"^. 220 RAY'S ALGEBRA, SECOND BOOK. 2. x*— 2a;3— 2a:2+3xr=108. Ans. a;:=4, —3, or -'(Irfc^/ZISS). 3. x'—2a^-\-x=S0. Ans. x=S, —2, or i (Idzy-HTO). 4. x'— 6a:^+lla:— 6=0. . . . Ads. x=1, 2, or 3. 5. x*—Qx^-{^bx'^12x=60. Ans. x=b, —2, or i(3±^^=T5). 6. x*—Sx'-j-10x'-\-24:x=—b. Ans. a;:=;5, — 1, or 2±|/5. 7. 4x*4-'^=4x3+33. Ans.x=2, — ^, or j(l±y^=43). 14 ta;^"^ 3x ~2x'^ ^' Ans. x=i, 3, or -^(7^^/69). SIMULTANEOUS QUADRATIC EQUATIONS CONTAINING TAVO OR MORE UNKNOWN QUANTITIES. S44. ftuadratic Equations, containing two or more unknown quantities, may be divided into two classes, pwre and affected. Pure Equations embrace those that may be solved with- out completing the square. Affected Equations embrace those in the solution of which it is necessary to complete the square. The same equations may sometimes be solved by both methods. PURE EQUATIONS. 24:5m Pure equations may in general be reduced to the solution of one of the following forms, or pairs of equa- tions. ao^tp}. (2-)^Tp}. c^-);!::::}. QUADRATIC EQUATIONS. 221 We shall explain the general method of solution in each of these cases. To solve x^y=^a (1), and xy^=h (2), we must find X — y. Squaring Eq. (1), . . x'^^'tjcy-Yy'^=a^\ Multiplying Eq. (2) by 4, \xy =r46; Subtracting, . . . x-—lxy^y'^—a?—^b, Or, (x-?/)2=:a2_45. AVhence, .... x-y=riz^a^—Ab; But, x^y=a\ Adding, and dividing by 2, X=^\a:±L\^ a^ — 46. Subtracting, and dividing by 2, y^=z\az^\^/ aP-^^h. The pair of equations (2) is solved in the same manner, except that in finding iC-|-?/, we must add 4 times the second equation to the square of the first. The pair of equations (3) is solved merely by adding and sub- tracting, then dividing by 2 and extracting the square root. 1. Given x^^y'^=l^, and rc-f^=7, to find x and y. Squaring the 2d Eq., a;2-f 2a:^-f-2/^=:49; But, a;2 +?/2=rr25 (1). Subtracting, ... 2a:?/ =^24, (2). Taking (2) from (1), x^~~1xy^-y'^= \ Whence, .... x—y^r±z\ (3). But, x^y=l (4). Adding and subtracting (3) and (4), and dividing by 2, a:=4, or 3; and y—^, or 4. 2. Given x^-\-xy-\-y''=^\{^'), and a:+|/^-f 7/==13(2), to find X and y. Divide Eq. (1) by (2), a:-|/^-f-2/= 7. (3). But, a:+i/^-f2/=13. (2). By subtracting, . . . 2|/'S^^6. Whence, ^xy—3, and xy—9. (4). 222 RAYS ALGEBRA, SECOND BOOK. By adding [2) and (3), . . . x-^y^ 10. (5). Squaring, (5), . . . a:2-f 2a2/+^2^100; Multiplying (4) by 4, . 4x1/ ^ 36; x2—2xi/^-y^-^ 64, .-. x-y=±:8. But, a:+7/=10; whence, X—9, or 1 ; and 2/=l, or 9. Equations of higher degrees than the second, that can be solved by simple methods, are usually classed with pure equations of the second degree. 1 1 3 3 3. Given x'^-\-y^=^6, and x^-\-y^=^126, to find x and y. In all cases of fractional exponents, the operations may be simpli- fied by making such substitutions as will render the exponents in- tegral. To do this, put the lowest power of each unknown quantity equal to the first power of a new letter. 11 3-3 In this example, let x^—V, and^^=Q; then, a;4— p3 and y^=Q\ The given equations then become, P-fQ= 6 (1), ps-f Q3^126 (2). Dividing Eq. (2) by (1), P2— PQ-f Q2z=21; Squaring Eq. (1), . . P2+2PQ+Q2.^36; Subtracting, .... 3PQ=15, .-. VQ=5. Having P-|-Q=:6, and PQ=5, by the method explained in form (1), we readily find P^=5, or 1; and Q=:l, or 5. Whence, a:=625, or 1 ; and y=^l, or 3125. 4. Given (a^y)(x'—f)=160 (1), lx-{-7/Xx''-\-f)=bS0 (2), to find X and y. ajS — x^—xy--]-y^=160 (1), by multiplying. a^-\-x^y-\-xy^-\-y^=580 (2), " « 2x^y ^2x1/^=420 (3), by subtracting. Add (3) to (2), rc-H 3a:22/4-3tri/2+y3^1000. Extract cube root, . . . x-\-y—lO. From (3), .... a:?/(a:4?/)-_210; .-. a:?/=21. From x-\-y—10, and X1/—21, we readily find a:—?, or 3; and 2/=3, or 7. QUADRATIC EQUATIONS. 223 Solve the following by the preceding or similar methods : 5. x—y=2, 6. a:^+/=13,| xy = ^.) 7. ^xJry=1. 4x'-^y'=2b. 8. x^-f=.16, 1 . x-y=2. 9. x+y= 11,) x'-\-y'=40l. 3 / 10. 1(a^-\-y')=9(x'—y') x^y — y'^x=lQ. 11. x'-{-xy=Si, ) . x'—y' =24. I . 12. x'-^f=lb2, ) sy x'-xyJ^f=19.\ 13. x'-\-y'-\-xy=20S, x -^y =16. 14. x^—if=7xy,) . ^_3^_2. I . 15. a:*+a;y+y=91, 16. x—y=^x-^y^y, 3 3 a:-— 7/^=37. 11.xi-^yi= 5, J-fy^=13. 18. 05^3/'= 5, a: -[-y =35. Ans. 33=15, or — 13; 3/=13, or — 15. Ans. a;=±3; y=±2. Ans. a;=:2, or | ; t/=S^ or 4. . . Ans. x=5; Ans. x=7, or 4; ^=4, or 7. . . Ans. £c=4; . . y=2. Ans. a:;==t7 ; y=±5. Ans. a;:=5, or 3; y=S, or 5. Ans. a;=:12, or 4; ^= 4, or 12. Ans. a;=4, or — 2; 3/=2, or —4. Ans. icz=:rh3, or ±1 ; 3/=dtl, or ±8. Ans. a:=16, or 9; y= 9, or 16. Ans. a;=16, or 81; y=2l, or 8. Ans. a:= 8, or 27; y=27, or 8. 224 RAYS ALGEBRA, SECOND BOOK. 19. x^-\-y-^=. 4, 20. x'^2f=Zhl, xy=^ 14. 21. x-^y= 4,) . 22. x{y^z)=a, ^ Ans. a;r=9, or 1 ; ?/=l, or 9. Ads. x=^, 9. 3/=2, or 7. Ans. x=^S, or 1 ; y=l, or 3. Ans ^— f-J (^+^-^)(^+^-^) Ans.a:_=i=^ 2(^>-j-c-a) ' / (a-f?>-c)(6+c-«) ^- \ 2(a-fc— ^.) ' ^_± /(64-c-«)(«+^-Z') 2(a-f;>— c) AFFECTED EQUATIONS. S46. The most general form of quadratic equations, ^ " containing two unknown quantities, is ax^ -\-hxy-\-cx-\- dy"^ -\- ey -\-f=- . By arranging the terms according to the powers of x^ and dividing by the coefficient of the first term, two quad- ratic equations containing two unknown quantities, may be reduced to the following forms : a:^+(ay+& )x-^^cf^dy-^^e r^O (1), x^J^{a:yJ^y)x^c'f^d'y^e'=.^ (2). To find the values of either of the unknown quantities, we must eliminate the other. We shall now show that this operation pro- duces an equation of the fourth degree. By subtracting the second equation from the first, and making a—a'—a^^, h—b'=b'\ etc., we have Whence, X— QUADRATIC EQUATIONS. 225 As this value of x contains ?/2, that of x^ will evidently con- tain y*, which value of a:^, substituted in the first equation, neces- sarily gives rise to an equation of the fourth degree. Hence, The solution of two quadratic equations, containing two unknoicn quantities^ depends upon the solution of an equation of the fourth degree^ containing one unknown quantity. As there are no direct methods of solving equations of any higher degree than the second, those of the class now under consideration can not be solved except in particu- lar cases, and then only by indirect methods, or special artifices. We now proceed to point out some of these special cases, in addition to those already referred to in Arts. 242, 243, and 245, with some of the more common artifices em- ployed. S47. There are two cases in quadratics which may always be solved as equations of the second degree, viz. : Case I. — When one of the equations rises only to the first degree. Given ax-\-hy=c (1), dx^-\-exy-\-fy'^-\-gx^hy=k (2), to find x and y. From eq. (1), we may obtain a value of X in terms of y. Sub- stituting this value, for x and X- in (2), the new equation will evi- dently contain only y ^^^ V'- Case II. — When both equations are homogeneous. (See Art. 30.) Given ax^-['h xy-]^cy'^=:^d (1), a'x^-\-h'xy-[-c'y^=^d' (2), to find x and y. Put y^=^tx, where ^ is a third unknown quantity, termed an auxiliary quantity. Substituting this value of y in the two equa- tions, we have a 0:24. h tx'^^c t^x'^=x'^{a +6 t^c t'^)^d (3), a'xi-Yb'tx2-^&t'^x^~-^--x'^{a'^b't^&f^)^d' (4). 226 RAY'S ALGEBRA, SECOND BOOK. From eq. (3), we find . . . x'^= j- -, (5). From eq. (4), we find ... x^=-^^,jj-^^ (6). ^, . d d' Therefore, .... j-: -r,— , ,,^ 77,. Or, .... d{a'+b't^&t'^)=d\a^bt-Ycr-), a quadratic equation, from which the value of t may he found, (Art. 281.) and thence x from (5) or (6), and y from the equa- tion yr=dx. 248. When two quadratic equations are symmetrical with respect to the t^wo unknown quantities ; that is, when the two unknown quantities are similarly involved, they may frequently be solved by substituting for the unknown quantities the sum and diflference of two others. 1. Given , x -\-y =za (1), x^-\-y^^=h (2), to find x and y. Let x=iS-\-z, and y-S—z\ then, s=^ (3), x^=s^^^s^z^\Os-'z'^-^\Os^z^-YhHZ^-\-z\ 7/' ^ .S'^' _5.s<2;+10.s'32;2_ lOs'^z^^^az^- z^; x'^-{-y->^2s^-\-20s^z^-\-10sz*=b. By substituting the value of s—-^, and reducing, we find a2 2_166— o-'i ^ "^2^ ~ 80a • Completing the square, wc find the value of Z] and from (3), that of X and y. 349. An artifice that is often used with advantage, consists in adding such a number to both members of an equation as will render it a trinomial equation that can be resolved by completing the square, (Art. 240). The following is an example: QUADRATIC EQUATIONS. 227 x'-\-f=20 (2), to find X and y. Since / -+^ \ =^-^2+^; add 2 (o each side of eq. (1), and then ^ to complete the square. Whence, - +'^=±3— 1:=.5 or — g. Let ^+M; then, ^i+-/-hl)=--lSxy (1), (x'-{-/Xxy-{-l)=z20Sxy (2), to find X and y. Let x-{^y=s, and xy=p; then, s(iHl)=18p, (1), and (s2— 2p)(p2_^l)=208p2 (2). From the square of (1), take (2), and after dividing bj 22>, we have «2_|p2_|.i^58p (3). But, .... 2s(p-fl)=36p, from(2), And .... 2p= 2p. Adding, . . . (s fp+l)2:^96p, QUADRATIC EQUATIONS. 235 But, ... . p + l=lf; ISp Therefore, . . . s=4|/6p — "^ , or s- — 4sy'6p=— 18p; From which, . . s=3|/t)p, or yUp. But, .... p+l=4|/6p— s=3/^, or yBp. Whence, . . . p^26zbv/675, or 2d=|/3, and s=±v/{6(26zfc|/675j}, or d=i/{6(2d= v/3)J. Having x-\-y and xy, the values of a: and ?/ are easily found., (Art. 246); two of the values are :c=7rh4,/3^ ?/=2rfr/3. 19. 2(x-^i/y+l={x^^i/)(xi/-^x'-{-f) (1), a;+3/=3 (2). Ans. x=2, y=l. on 1 , 3 .1 , N3 A l-^2adtzyl2a-S 20. l-fa:3=of(l+ic)'. Ans. x= — ■ — ts-^ ^ • ^ ^ Z(l — a) 31. 4_1 L-2a-"=l. a:^ it \ x Ans. a:=]ll±i/l— 8«±\'2±2(1— 8a)^4-8a}. 22. a:;4-3'+^i/(a;4-y)4-^'^y^==85, Ans. x=^6, or 1. ^3'+(^+y)'+a^i'(«H-y)=97. 3/=l, or 6. 2c+a^ c Ans. cc=:^ — — ?— ,, or \a-\-b)d' {a—h)d' 24. (.T'+l)(x2+l)(a:+l)=30x^ __ Ans.a:=A(3±]/5). 25. x'+/=35, Ans. x=^, 2, or l±i|/22; a;^-f/=13. y=2, 3, or lq=i|/22. 236 RAY'S ALGEBRA, SECOND BOOK. 26. -q^^a, Ans. x=J j 2aZ>c(a^-H>c-aj)_ | ^ a^-l-y \ ( (^ab-i-ac—bc){ab-^hc—ac) j ' ^cyz^ _ j( 2ahc{ab'\-hc~ac) | a)+2~ ' ^~ \ I {ac-^bc—ab)(ab-\-ac—bc) j ' X1/Z / f 2abc{ab-\-ac — be) ) y-{-z ' \ 1 {ac-\-bc — ab)(ab-^bc — ac) ) ' 27. (x^-^i)y=(f+r)c^, Ans. x= i{^r3+3+^>3-: y=i{r3.^/r3+3±^V9-l} VIII. RATIO, PROPORTION", AND PROGRESSIONS. SS4. Two quantities of the same kind may be com- pared in two ways. By considering, 1st. How much the one exceeds the other. 2d. How many times the one is contained in the other. The first method is termed comparison by difference; the second, comparison by quotient. The first is sometimes called Arithmetical ratio; the second. Geometrical ratio. If we compare 2 and 6, we find that 2 is four less than 0, or that 2 is contained in 6 three times. Also, the arith- metical ratio of a to b is b — a; the geometrical ratio of a to 6 IS -. a The term Ratio, unless it is otherwise stated, always signifies g-sometrical ratio. RATIO AND PROPORTION. 237 255m Ratio is the quotient which arises from dividing one quantity by another of the same kind. Thus, the ratio of 2 to 6 is 3, and the ratio of a to ma is m. S56. When two numbers, as 2 and 6, are compared, the first is called the antecedent, and the second the conse- quent. When spoken of as one, they are called a couplet. When spoken of as two, they are called the terms of the ratio. Thus, 2 and 6 together form a couplet, of which 2 is the first term, and 6 the secojid term. 257. Ratio is expressed in two ways : 1st. In the form of a fraction, of which the antecedent is the denominator, and the consequent the numerator. Thus, the ratio of 2 to 6 is expressed by £ ; the ratio of a to 6, by -. 2d. By placing two points between the terms. Thus, the ratio of 2 to 6, is written 2:6; the ratio of a to h, a\h, etc. 258. The ratio of two quantities may be either a whole number, a common fraction, or an interminate decimal. Thus, the ratio of 2 to 6 is fi, or 3, of 10 to 4, is |. The ratio of 2 to ]/5 is \-; or ^'^ , or 1.118-f . 2 2 2SO. Since the ratio of two numbers is expressed by a fraction, of which the antecedent is the denominator, and the consequent the numerator, whatever is true with regard to a fraction is true with regard to the terms of a ratio. Hence, 1st. To multiply the consequent, or divide the antecedent of a ratio hy any number, multiplies the ratio hy that number. 238 RAY'S ALGEBRA, SECOND BOOK. 2d. To divide the consequent or to multiply the antecedent of a ratio hy any number^ divides the ratio by that number. 3d. To multiply or divide both the antecedent and conse- quent of a ratio by any number, does not alter the ratio. !3GO. When the terms of a ratio are equal to each other, the ratio is said to be a ratio of equality; when the second term is greater than the first, a ratio of greater inequality ; when it is less, a ratio of less inequality. Thus, the ratio of 2 to 2 is a ratio of equality. The ratio of 2 to 3 is a ratio of greater inequality. The ratio of 3 to 2 is a ratio of less inequality. Hence, a ratio of equality may be expressed by 1; a ratio of greater inequality, by a number greater than 1 ; and a ratio of less inequality, by a number less than 1. S61. When the corresponding terms of two or more ratios are multiplied together, the ratios are said to be compounded, and the result is termed a compound ratio. Thus, the ratio of a to 6, compounded with the ratio of c to d, is 6 dbd a c~aG' A ratio compounded of two equal ratios is called a dupli- cate ratio; one compounded of three equal ratios, a tripli- cate ratio. Thus, the duplicate ratio of — is — X — = — oj the triplicate ratio G/ Ct CL CL . 63 IS - -A. The ratio of the square roots of two quantities is called a subduplicate ratio; that of the cube roots, a subtriplicate ratio. Thus, the subduplicate ratio of 4 to 9 is |; and that of a to 6 is 1-:^; the subtriplicate ratio of a to 6 is ^-^-. V« fa RATIO AND PROPORTION. 239 2^2* Ratios may be compared with each other by re- ducing the fractions which represent them to a common denominator. Thus, the ratio of 2 to 7 is greater than the ratio of 3 to 10, for the fractions | and LO, reduced to a common denominator, are y and 2J)^ and the first is greater than the second. PROPORTION. S63. Proportion is an equality of ratios ; that is, when two ratios are equal, their terms are said to be proportional. Thus, if the ratio of a to 6 is equal to the ratio of C to d ; that b cl is, if — = — ; then, a, 6, C, «, form a proportion. Proportion is written in two ways : 1st. By placing a double colon between the ratios ; Thus, a: b . : c : d. Read, a is to 6 as c is to d. 2d. By placing the sign of equality between the ratios ; Thus, a : 6=C : d. Read, the ratio of a to 6 equals the ratio of c to d. From the preceding definition it follows, that when four quantities are in proportion, the second divided by the first, must give the same quotient as the fourth divided by the third. This is the primary test of the proportionality of four quantities. Thus, if 3, 5, 6, 10, are the four terms of a proportion, so that 3 : 5 : : 6 : 10, we must have l^^^'zP. If these fractions are not equal to each other, the pro- portion is false. Thus, the proportion 3 : 8 ; : 2 : 5 is false, since |>|. 240 RAY S ALGEBRA, SECOND BOOK. Remark. — The words ratio and proportion should not be con- founded. Thus, two quantities are not in the proportion of 2 to 3, but in the ratio of 2 to 3. A ratio subsists between two quantities, a proportion between four. 264. Each of the four quantities in a proportion is called a term. The first and last terms are called the ex- tremes; the second and third terms, the means. S6«>. Of four quantities in proportion, the first and third are called the antecedents^ and the second and fourth, the consequents (Art. 257) ; and the last is said to be a fourth proportional to the other three taken in their order. !366. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third. The middle term is a mean proportional between the other two. Thus, if a: b : : b: c, then 6 is a mean proportional between a and c; and C is called a third proportional to a and b. When several quantities have the same ratio between each two that are consecutive, they are said to form a continued proportion. 207. Proposition I. — In every proportion^ the product of the means is equal to the product of the extremes. Let a \ b : \ c : d. Since this is a true proportion, we must have (Art. 263) a~ c' Clearing of fractions, bc^=.ad. Illustration by numbers. 2 : 6 : 5 : 15; and 6x5=2x15. ,. , , 6c ad , ad be „ Taking bc—ad, we tnd d= — , c=-^, o= — , «=^- Hence, RATIO AND PROPORTION. 24 1 If any three terms of a proportion he given, the remaining term may he found. 1. The first three terms of a proportion are cc-f ?/, a-} — 3/^, and X — y; what is the fourth? Ans. x^ — Ixy-^-y"^. 2. The 1st, 3d, and 4th terms of a proportion are (ni — ?i)^, m^ — ?i^, and m-\-n ; required the 2d. Ans. m — n. 3. The 1st, 2d, and 4th terms of a proportion are ^^±4, a^_6, and (^=VM^Z^ • required the 3d. a— 1/6 a-\-yb Ans. 1. This proposition furnishes a more convenient test of proportion- ality than the method given in Art. 263. Thus, 2 : 3 : : 5 : 8, is not a true proportion, since 3X5 is not equal to 2x8. !S68. Proposition II. — Conversely, If the jyroduct of two quantities is equal to the product of two others, tvfo of them may be made the means, and the other two the extremes of a proportion. Let bcz=ad. Dividing each of these equals by ac, we have be ad OG ac' b d Or, - = -. ' a c That is (Art. 263), . . . a : b : . c : d. By dividing each of the equals by ab, cd, bd, etc., we may have the proportion in other forms. Or, since one member of the equation must form the extremes and the other the means, we have the following Rule. — Take either factor on either side of the equation for the first term of the proportion, the two on the other side for the second and third, and the remaining factor for the fourth. 2d Bk. 21* 242 RAY'S ALGEBRA, SECOND BOOK. Thus, from each of the equations bc::=zCid, and 3x12=^4x9, we may have the eight following forms : a: b: :c:d. 3 4 • : 9: 12. a: C: :b:d. 3: 9 : 4: 12. d:b: : c: a. 12 4 : 9: 3. d:c: :b.a. 12 9 : 4: 3. b:a: : d: C. 4: 3 : 12: 9. b.d: : a: C. 4: 12 : 3: 9. C: a : :d:b. 9: 3 : 12: 4. c:d: : a: b. 9: 12 : 3: 4. SOO. Proposition III. — If three quantities are in pro- portion, the product of the extremes is equal to the square of the mean. If a: b:: b: C] Then, (Art. 267), . . . ac=^bb^b^. It follows from Art. 268, that the converse of this proposition is also true. Thus, if ac=62, a : b : : b : c. Hence, If the product of the first and third of three quantities is equal to the square of the second., the second is a mean pro- portional between the first and third. STO. Proposition IV. — If four quantities are in pro- portion, they will he in proj>ortion hij ALTERNATION; that is, the first will he to the third as the second to the fourth. Let Then, (Art. 263), . . Multiply both sides by c. Divide both sides by b, That is, (Art. 263), . . If 2:6:: d; . a : b : : C : d. b d a~ g' be a c _d a~b' a : C: : b : d. 10:30; then, 2 :10 : :6:30. RATIO AND PROPORTION. 243 S71. Proposition V. — If four quantities are in propor- tion, they will he in proportion hy Inversion ; that is, the second will he to the first as the fourth to the third. Let a . b : : C :d. Then, (Art. 263), . . a G ' T . , • a c Inverting the fractions, . . j^z=z-^. That i«, (Art. 263), . , b : a : : d : C. If 5 : 10 : : 6 : 12; then, 10 : 5 : : 12 : 6. 6 d It follows from this proposition, that the equation — = — may be converted into a proportion in either of two ways, thus : a : b : : c : d, or b • a : \ d : G. 1S7^* Proposition VI. — If two sets of proportions have an antecedent and consequent in the one, equal to an ante- cedent and consequent in the other, the remaining terms will he proportional. Let a . b . : G: d (1), And a.b: .e.f (2); Then will . . . . G : d : e f. From (1), - = -; from (2), ~^'-. Hence, ~=* \ Which gives . . . G . d : : e : f. If 4 : 8 : : 10 : 20 and 4 ; 8 : : 6 : 12; then, 10 : 20 : : 6 : 12. 273. Proposition VII. — If four quantities are in pro- -portion, they will he in proportion hy COMPOSITION ; that is, the sum of the first and second will he to the first or 'second, as the sum of the third and fourth is to the third or fourth. 244 RAY'S ALGEBRA, SECOND BOOK. Let a\b'.\c:d (1), Then will a+6 : 6 : : c+d : d. ^ ,-,. b d From (1) - = -. , . • (^ c Inverting the fractions, . t^ = ^- Adding unity to both members, r+l=;y+l- ^ , • . ,. • a+6 cA-d Reducing to improper tractions, — ^- — =: ' . Hence, (Art. 271), . . a^b : b : : c-^d : d. If 3 : 6 : : 9 : 18; then, 3+6 : 6 : : 9+18 : 18, or 9 : 6 : : 27 : 18. In a similar manner it may be shown that a-\-b : a : : c-{-d : c. S74:. Proposition VIII. — If four quantities are in pro- portion, they will he in proportion hy Division ; that is, the difference of the first and second will he to the first or sec- ond, as the difference of the third and fourth is to the third or fourth. Let a: b:\ C: d (1), Then will a—b : b : : (i—d : d. b d From (1), - = -. , . . « c Inverting the fractions, . r= j- Ct G Subtracting unity from both members, j- — 1=^ — 1. T. , . . . . <^—h G—d Reducing to improper fractions, . . — v— = — -r— • This gives (Art. 271), a~b : b : : G—d : d. If 18 . 6 . : 30 : 10; then, 18-6 :6 ; : 30-10 : 10, or 12 : 6 ; 20 : 10. In a similar manner it may be shown that a — 6 : a : : c — d : c. S7S« Proposition IX. — If four quantities are in pro- portion, the sum of the first and second will he to their difference as the sum of the third and fourth is to their dif- ference. RATIO AND PROPORTION. 245 Let a: d : : c: d (1), Then will .... a-f6 : a—b : : e-f d : c — d. From (1), by Composition and Division, (Arts. 273, 274,) a-f6 : 6 : : C-f C? : d; And a—b : b : : c — d : d. By alternation, . . a-f 6 : C-\-d : : b : d; And a — b : C—d : : b : d. From which, (Art. 272), a+6 : C-^d : : a—b : c—d. Or, by alternation, a+6 : a— 6 : : c-\-d : c— d. If 6 : 2 : : 12 : 3 ; then, 6+2 : 6 -2 : : 12+4 : 12—4, or 8 : 4 : : 16 : 8. !S76. Proposition X. — If four quantities are in propor- tion^ like i^owers or roots of those quantities will also he in proportion. Let a.b :: C\ d, Then will a" : 6" : : c" : d'*. From the 1st, -==-. Raising each of these equals 6" d" to the n<'' power, . . • ^. = -^»- That is, a" : S** : : c" : d" Where n may be either a whole number or a fraction. If 2 : 6 : : 10 : 30; then, 22 : 62 : : 102 : 302, or 4 : 36 : : 100 : 900. If 8:27: : 64: 216; then, fS": f27:: ^64:^216; or 2: 3:: 4:6. STT. Proposition XI. — If two sets of quantities are in proportion, the products of the corresponding terms will also he in proportion. Let a:b::C:d (1), And m-.n-.'.r-.s (2), Then will am : bn : : cr : ds. For from (1), | = | (3); and from (2), ^ = ^ (4). Multiplying (3) by (4) — = — ; this gives, am : bn : : cr : ds. (XTih C/ Let . . . . . Then, . . . Since a : h : : c Since a : h \ : m 246 RAY S ALGEBRA, SECOND BOOK. If 3 : 9 : : 2 : 6, and 5 : 15 : : 4 : 12; then, 15 : 135 : : 8 : 72. 278. Proposition XII. — Li any number of proportions having the same ratio ^ any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the con- sequents. . a : b : : C: d: : m: n, etc. . a : b : : a-\-c-^m : 6-f rf-fn. d, we have bc—ad (Art. 267). w, we have bm=^an, Also, ab=ab. The sum of these equal- ities gives .... ab-\-bc-\-bm=^ab'\-ad^an. Factoring, . . . b{a-\-c-\-7n)=:za[b-\-d^n). This gives (Art. 268), a : b : : a-^C-\-m : b-\-d-{-n. If 5 : 10 : : 2 : 4 : 3 : 6, etc. ; then, 5 : 10 : : 5+2+3 : 10+4+6, or 5 : 10 : : 10 : ZO. EXERCISES IN RATIO AND PROPORTION. 1. Which is the greater ratio, that of 3 to 4, or 3^ to 4'^? Ans. last. 2. Compound the duplicate ratio of 2 to 3 ; the triplicate ratio of 3 to 4 ; and the subduplicate ratio of 64 to 36. Ans. 1 to 4. 3. What quantity must be added to each of the terms of the ratio m : n, that it may become equal to p : q? , mg — np Ans. — -. p—q 4. If the ratio of a to h is 2|, what is the ratio of 2a to h, and of 3a to 4h? Ans. 1], and 3i}. 5. If the ratio of a to Z> is 1^, what is the ratio of a-\-h to b, and of b — a to a? Ans. |, and |. 6. If the ratio of m to n is :j, what is the ratio of m — n to 6m, and also to 5»? Ans. 14, and 6|. RATIO AND PROPORTION. 247 7. If the ratio of by — 8a; to Ix — by is 6, what is the ratio of a; to ^? Ans. 7 to 11. 8. What eight proportions are deducible from the equa- tion ah=^a^ — x^. Ans. a : a-j-rc : : a — x : b, a : a — x : : a-]-x : b, b : a-{-x : : a — x : a, etc. 9. If x^-{-y^=2aXj what is the ratio of x to y? Ans. X : y : : y : 2a — x. 10. Four given numbers are represented by a, 5, c, d ; what quantity added to each will make them proportionals ? . be — ad Ans. -j- a — b — c-\-d 11. If four numbers are proportionals, show th»t there is no number which being added to each, will leave the resulting four numbers proportionals. 12. Find X in terms of y from the proportions x:y \:a^ :b\ and a : b : : f^c-\-x : f^d-\-y. 13. Prove that equal multiples of two quantities are to each other as the quantities themselves, or that ma '. mb : : a : b. 14. Prove that like parts of two quantities are to each other as the quantities themselves, or that - : - : : a : b. n n 15. If a : 5 : : c : (7, prove that ma : mb : \ nc : nd, and also that ma : nb : : mc : nd, m and n being any multiples. 16. Prove that the quotients of the corresponding terms of two proportions are proportional. STd. The following examples are intended as exercises in application of the principles of proportion. 1. Resolve the number 24 into two factors, so that the sum of their cubes may be to the difference of their cubes as 85 to 19. 248 RAY'S ALGEBRA, SECOND BOOK. Let X and y denote the required factors; then, xy=2A^ and X^-[-y^ . x^ — y^ Therefore, (Art. 275), 2x^ : 2y^ Or, x^ : y^ Or, (Art. 276), .... x : y 19; IG: From which y=^X] then, substituting the value of y in the equa- tion xy=24:, we find x=zh6', hence, y=ziz4:. 2. Given l^±l+l^^2, to find x. if x+ 1— f/a;_l Resolving this equation into a proportion, we have fx-\-l~fx—l : fx-{-l-\-f'X—l : : 1 : 2; .-. (Art. 275), 2f ^+1 : 2fx—l : : 3 : 1 ; Or, f ^+1 : ^^Hl : : 3 : 1 ; Or, (Art. 276), . . . x-{-l : x—1 : : 27 : 1 ; (Art. 275), . . . 2a: : 2 : : 28 : 26 ; Whence, 52a:=56, or a;=lJ^. ir } : 8. x-\-y : x—y : : 3 : 1, x^ — ^3^=56. 4. x-\-y : a: : : 7 : 5, 1 . . xy-{-y'=126. j . . 5. (x-\-yy : (x~yy : : 64 : 1, a;y=63. 6. —j/a'—x'' a-^y'd'—x' . Ans. a:=:4, Ans. a;=±15, Ans. .7;==t9, 6+1 • .ns x 2ah 7. y/^j^-T/g-^^l ^^3 ^_ ^a-\-x-\-^a — x ^ 8. It is required to find two numbers whose product is 320, and the diflference of whose cubes is to the cube of their difference, as 61 is to 1. Ans. 20 and 16. RATIO AND PROPORTION. 249 280. Harmonical Proportion. — Three or four quan- tities are in Harmonical Proportion when the first has the same ratio to the last, that the difference between the first and second has to the difi"erence between the last and the last except one. Thus, a, 6, C, are in harmonical proportion when a : c : : a — b : b — c ; and a, 6, c, cZ, when a : d : : a—b : c — d. 1. Let it be required to find a third harmonical propor- tional X, to two given numbers a and b. We have, a : x : : a — 6:5 — x\ Therefore, (Art. 267), a{b—x)^x{a—b)\ ^^^^^^^^' ^^2£z6- 2. Find a third harmonical proportional to 3 and 5. Ans, 15. 3. Find a fourth harmonical proportional ic, to three c;iven numbers, a, h, and c. ^ ac ^ Ans. x= 2a— b' 381. Variation, or, as it is sometimes termed, Gen- eral Proportion, is merely an abridged form of common Proportion. Variable Quantities are such as admit of various values in the same computation. Constant, or Invariable Quantities have only one fixed value. One quantity is said to var?/ directly as another, when the two quantities depend upon each other in such a man- ner, that if one be changed the other is changed in the same ratio. Thus, the length of a shadow varies directly as the height of the object which casts it. Such a relation between A and B is expressed thus, 250 HAY'S ALGEBRA, SECOND BOOK. A oc B, the symbol oc being used instead of varies^ or varies as. 282. There are four different kinds of Variation, which are distinguished as follows : I. A X B. Here A is said to vary directly as B, or, simply A varies as B. Ex. — If a man works for a certain sum per day, the amount of his wages varies as the number of days in which he works. II. A oc ^. Here A is said to vary inversely as B, Ex. — The time in which a man may perform a journey will vary inversely as the rate of traveling. III. A X BC. Here A is said to vary as B and C jointly. Ex. — The wages to be received by a workman will vary jointly as the number of days he works, and the wages per day. T> IV. A X ^. Here A is said to vary directly as B, and inversely as C. Ex. — The time occupied in a journey varies directly as the distance, and inversely as the rate of travel. These four kinds of variation may be otherwise modi- fied ; thus, A may vary as the square or cube of B, in- versely as the square or cube, directly as the square and inversely as the cube, etc. Ex. — The intensity of the light shed by any luminous body upon an object will vary directly as the size of the luminous body, and inversely as the square of its distance from the object. (See Art. 238.) RATIO AND PROPORTION. 251 In the followiBg articles, A, B, C, represent corresponding values of any variable quantities, and a, 6, c, any other corresponding values of the same quantities. SS3. If one quantify vary as a second, and that second as a third, the first varies as the third. Let A oc B, and B oc C, then shall A oc C. For A : a : : B : ft, and B : ^ : : C : c; therefore, (Art. 272), A : a : : C : c ; that is, A oc C. In a similar manner it may be proved that if A oc B, and B oc ^, that A oc ^. 284. If each of two quantities vary as a third, their sum, or their difference, or the square root of their j^'i'odiict, will vary as the third. Let A oc C, and B oc C ; then, A±B oc C; also, |/AB oc C. By the supposition, . . . A : a : : C : e Therefore, A : a : : B : b Alternately, (Art. 270), . . A : B : : a : 6 By Composition or Division, AzfcB : B : : adzb : 6; Alternately, A±B : adzb : : B : 6 : : C : C; That is, AdbB x C. Again, A : a : ; C : C; And, B:6::C:e; Therefore, (Art. 277), . . AB : a6 : : C2 : c^; And, (Art. 276), .... ^/AB : ^ab~: : C : C; That is, "i/AJToc C. By a similar method of reasoning, the following propo- sitions may be proved: 285. If one qnantity vary as another, it will also vary as any multiple, or any part of the other. That is, if A oc B; then, A oc ?7iB, or oc — . 252 RAYS ALGEBRA, SECOND BOOK. SSO* If one quantity vary as another, any power or root of the former will vary as the same power or root of the /after. Let A cc B ; then, A" cc B", n being integral or frac- tional. 28T. If one quantity vary as another, and each of them be multiplied or divided by any quantify, variable or invari- able, the products or quotients will vary as each other. A B Let A oc B ; then, qA. oc qB, and — oc — . 288. If one quantity vary as two others jointly, either of the latter varies as the first directly, and tlie other in- versely. A A Let A Gc BC ; then, B oc — , and C oc y-- 389. If A vary «s B, A is equal to B multiplied by some constant quantity. Let A oc B ; then, A=mB. If we know any corresponding values of A and B, the constant quantity m may be found. SOO. In general, the simplest method of treating varia- tions, is to convert them into equations. 1. Given that y oc the sum of two quantities, one of which varies as x, and the other as x^, to find the corre- sponding equation. Because one part oc a:, let this =zmx^ and the other part oc x^, " " =z,nx-. Therefore, y—mx-\-nx'^, where m and n are two unknown invariable quantities which can only be found when we know two pairs of corresponding values of X and y. RATIO AND PROPORTION. 253 2. If i/=r-{-s, where r cc x and s a — , and if, when x=^l, X y=Q, and when x=.2^ y^9> what is the equation between X and yl n n Let r=7nx, and s=- .-. y—mxA — . ' X ^ ^x But if a;=l, ?/=6, .-. 6=im+n; And if a;=2, 2/=9, .-. 9=2m+^. 2 Hence, w=4, n— 2, and 2/=4a:4- -. 3. If y cc .T, and when a;=2, y=:^a ; find the equation between x and y. Ans. y=2ax. 4. If 3/ oc -, and when a;^^, 3/=8 ; find the equation X ^ between x and y. Ans. y=-. X 5. If 3/= the sum of two quantities, one of which varies as X. and the other varies inversely as x^ ; and when x=.\, y=6, and when x=^2, y=^ ', find the equation between X and y. ^4 Ans. y=2x-\--. 6. Given that y= the sum of three quantities, of which the 1st is invariable, the 2d varies as x, and the 3d varies as x"^. Also, when x=l, 2, 3, y=Q, 11, 18, respectively; find y in terms of ic. Ans. y=S~{-2x-\-x^. 7. Given that s gc f, when / is constant; and s oc/, when t is constant; also, 2s=/j when ^=1. Find the equation between /, s, and t. Ans. s=-^ft\ Remarks.— 1. The above examples may all be proved. Thus, if in Ex. 5, we put x=:^l in the answer, y will equal 6. If we put x=2, y=5. 2. The Principles of Variation are extensively applied in mechan- ical philosophy. 254 RAY'S ALGEBKA, SECOND BOOK. ARITHMETICAL PROGRESSION. 291. An Arithmetical Progression is a series of quan- tities whicli increase or decrease by a common difference. Thus, 1, 3, 5, 7, 9, etc., or 12, 9, 6, 3, etc., and a^ a-f-cZ, a-j-2c?, etc., a, a — (7, a — 2(Z, etc., are in Arithmeti- cal Progression. The series is said to be increasing or decreasing^ accord- ing as d is positive or negative. 293. To investigate a rule for finding any terjn of an arithmetical progression, take the following series, in which the first line denotes the number of each term, the second an increasing arithmetical series, and the third a decreas- ing arithmetical series. 12 3 4 5 a, a~^d, a-f2c?, a~^3d, a-|-4c?, etc., a, a~d, a— 2d, aSd, a~4d, etc. It is manifest that the coefficient of d in any term is less by unitT/ than the number of that term in the series ; therefore, the n^^ term r=a-f-(n — l)d. If we designate the n^^ term by /, we have l^=a-{-{n — l)d, when the series is increasing, and l=a — (n — l)d, when the series is decreasing. Hence, Rule for finding Any Term of an Arithmetical Series. — Multij)ly the common difference by the number of terms less one; when the series is increasing, add this product to the first term; when decreasing, subtract it from the first term. The equation l = a-\-(ii — l)c7, contains four variable quantities, any one of whicli may be found when the other three are known. ARITHMETICAL PROGRESSION. 255 S93. Having given the first term a, the common dif- ference d^ and the number of terms n, to find S, the sum of the series. If we take any arithmetical series, as the following, and write the same series under it in an inverted order, we hLve S= 1+3 + 5+ 7+ 9+11, S^ll+9 + 7+ 5+ 3+ 1. Adding, . . 28=12+12+12+12+12+12. 2S=12X the number of terms, =12x6=72. Whence, . . S=:2 of 72=^36, the sum of the series. To render this method general, let ^= the last term, and write the scries both in a direct and inverted order. Then, S=a+(a+d)+(a+2(^)+(a+3d). . . +^, And, S=; + (l—d) + {l—2d) + (l—Scl). . . +a. 2S=(H«)+(^+«)+(^+«)+(^+a). . . +(Ha), 2S=(^+a) taken as many times as there are terms (r?) in the series. Hence, .... 2S=(^+a)n; S=(Z+a)^=(^)n. Hence, Rule for finding the Sum of an Arithmetical Series,— Multiply half the sum of the two extremes hy the number of terms. It also appears that The sum of the extremes is equal to the siim of any other two terms equally distant from the extremes. 294. The equations l=a^{n—\)d, and Sr^(a-^?)^, furnish the means of solving this general problem : Knowing any three of the five quantities^ o, d, I, n, S, which enter into an arithmetical series^ to determine the other two. The following table contains the results of the solution of all the diflFerent cases. As, however, it is not possible to retain these in 256 RAY'S ALGEBRA, SECOND BOOK. the memory, it is best, in ordinary cases, to solve all examples in Arithmetical Progression by the above two formulae: No. Given. Required. Formulae. 1. 2. 3. 4. a, d, n a, cZ, S a, n, S d, n, S l=a-\-{n—l)d, l=-ld-^^[2d^J^{a-ldY], ^= a, n^ 2 5. 6. 7. 8. a, cZ, n a, rf, I a, n, I d, n, I s S=An{2a+(7i— l)d}, ^~ 2 ' 2d ' %=\n\ll—[n—\)d\. 9. 10. 11. 12. a, 71, I a, w, S a, ?, S n, I, S cZ ^ 2(S-a/i) n(n_l) ' '^-2S_^-a' 2(n^-S) n(n-l)- 13. 14. 15. IG a, d, I a, d, S a, I, S d, I S n -ty^ (2a— cZ)2-|-8dS— 2a+rf ""- 2d 2S 2^f-c^=fc/(2^+d)2-8c^s ^ 2rf 17. 18. 19. 20. C/, 71, I d, 71, S d, I, S w, I, S a a=l—[n—\)d, S (rt— l)c« a-- ^^ — o — J n 2 a _-??-.. n ARITHMETICAL PROGRESSION. 257 1. Find the 15"* term of tlie series 3, 7, 11, etc. Ans. 59. Here, a=3, n — 1=14, and d=zA. Substituting these values in formula (1), we have ^=3+14x4=3-f o6==59. 2. Find the 20"* term of the series 5, 1, —3, etc. Ans. —71. 3. Find the S''^ term of the series f , f^, -^, etc. Ans. -^^ 4. Find the 30"^ term of the series —27, —20, —13, etc. Ans. 176. 5. Find the n'^ term of 1 + 3+5-f 7. Ans. 2n— 1. Of 2-f 2H2^ + Ans. J(n-f 5). Of 13+122 + 12^ + . . . . Ans. K^O—Ji). 6. Find the sum of 1-f 2-|- 3-|-4, etc., to 50 terms. From formula (1), we find ?=:50. Substituting this in formula (2), we Lave S=(l -[-50)25=1275, Ans. Or, use formula 5. 7. Of 7+V+ 2^+, etc., to 16 terms. Ans. 142. 8. Of 12+8+4+, etc., to 20 terms. Ans. —520. 9. Of 2+2i+2j+,etc., t0 7iterms. Ans.^vC^+ll). 10. Of ^— 2_y_, etc., to n terms. A. 4s(13— 7vO- 11 f\Q '^~~^ . ^*~2 n — 3 , ^ ^ ^ 11. Ul 1 ^, etc., to n terms. n n n ^_^ Ans. —^. 12. If a falling body descends 16j':7 ^"set the 1st sec, 3 times this distance the next, 5 times the next, and so on, how far will it fall the 30th sec, and how far altogether in half a min. ? Ans. 948ii, and 14475 ft. 13. Two hundred stones being placed on the ground in a straight line, at the distance of 2 feet from each other; 2d Bk. 22 258 RAY'S ALGEBRA, SECOND BOOK. how far will a person travel who shall bring them sepa- rately to a basket, which is placed 20 yards from the first stone, if he starts from the spot where the basket stands ? Ans. 19 miles, 4 fur., 640 ft. 14. Insert 3 arithmetical means between 2 and 14. Here, a=:2, ^=14, and n^5. From formula (1), we obtain d=S. Hence, the three means will be 5, 8, and 11. To solve this problem generally, let it be reqiiired to insert m arithmetical means between a and I. Since there are m terms between a and I, we shall have n=m-\-2, and formula (1) becomes I ^:^ a -\- (7n4-l)d. Hence, d^=- -. ^ ^ I V I ; J 7/1+1 Therefore, 77ie common difference will he equal to the difference of the extremes divided hy the number of means plus one. 15. Insert 4 arithmetical means between 3 and 18. Ans. 6, 9, 12, 15. 16. Insert 9 arithmetical means between 1 and — 1. Ans. I, I, etc., to —4. 17. How many terms of the series 19, l7, 15, etc., amount to 91? Ans. 13, or 7. From (2) and (1), find n, or use formula 14. Explain this result. 18. How many terms of the series .034, .0344, .0348, etc., amount to 2.748? Ans. 60. 19. The sum of the first two terms of an arithmetical progression is 4, and the fifth term is 9 ; find the series. Ans. 1, 3, 5, 7, 9, etc. 20. The first two terms of an arithmetical progression being together =18, and the next three terms =12, how many terms must be taken to make 28? Ans. 4, or 7. 21. In the series 1, 3, 5, etc., the sum of 2r terms: the sum of r terms : : x : 1 ; determine the value of x. Ans. 4. GEOMETRICAL PROGRESSION. 259 22. A sets out for a certain place, and travels 1 mile the first day, 2 the second, and so on. Five days afterward B sets out, and travels 12 miles a day. How long and how far must B travel to overtake A? Ans. 3 days, or 10 days; and travel 36 miles, or 120 miles. Explain these results. GEOMETRICAL PROGRESSION. S95. A Geometrical Progression is a series of terms, each of which is derived from the preceding, by multiply- ing it by a constant quantity, termed the ratio. Thus, 1, 2, 4, 8, 16, etc., is an increasing geometrical progression, whose common ratio is 2. Also, 54, 18, 6, 2, etc., is a decreasing geometrical pro- gression, whose common ratio is \. In general, a, ar^ ar'^, ai^^ etc., is a geometrical progres- sion, whose common ratio is r, and which is an increasing series when r is greater than 1 ; but a decreasing series when r is less than 1. It is evident that In any given geometrical series, the common ratio will he found hy dividing any term by the term next preceding. I296. To find the last term of a geometrical progression. Let a denote the first term, r the common ratio, I the n'^ term, and S the sum of n terms ; then the respective terms of the series will be 1, 2, 3, 4, 5, . . . n— 3, n—2, n— 1, n, a, ar, ar^^ ar^^ ar* . . . ar"-'*, ar"-^^ ar"-"^^ ar"-^. That is, the exponent of r, in the second term, is 1, in the third term 2, in the fourth term 3, and so on. Hence, the n}^^ term of the series will be l=^ar''~^. Hence, 260 RAY'S ALGEBRA, SECOND BOOK. Rule for finding the Last Term of a Geometrical Series. — Multiply the first term hy the ratio raised to a power whose exponent is one less than the number of terms. Required to find the 6'^ term of the geometrical progres- sion whose first term is 7, and common ratio 2. 2^=32; and 7x32^=224, the 6"^ term. SOT* To find the sum of all the terms of a geometri- cal progression. If we take the series, 1, 3, 9, 27, 81, and represent its sum by S; then, 8=1-1-3 + 9 + 27 + 81 (a). Multiplying by the ratio 3, 38=3+94-27+81+243 (6). Subtracting (a) from (6), 3S— S=243— 1 ; whence, S=121. To generalize this method, let a, ar, ar^, ar^^ etc., be any geometrical series, and S its sum ; then, S=a+ar+ar2+ar3. . . . +ar"-2+ar'»-i. Multiplying this equation by r, we have rS=ar+a7'2+ar3. . . . +ar"-i+ar'". Subtracting, rS — S^ar** — a\ whence, S=— '^ =— . Since, Izz^ar"*-^, we have rl=^ar^\ n,, « o, ar'^—Gi rl — a „ Therefore, S= ^ = =-. Hence, ' r— 1 r— 1 ' Eule for finding the Sum of a Geometrical Series. — Multiply the last term hy the ratio^ from the product subtract the first term, and divide the remainder by the ratio less one. Find the sum of 6 terms of the progression 3, 12, 48, etc. i=3x4^=3072. ... S=^ = ?5!|^3^4095, An. S9S. If the ratio r is less than 1, the progression is decreasing, and the last term /, or ar''~^, is less than a. In GEOMETRICAL PROGRESSION. 261 order that both terms oi the iraction =■, or =- may r — 1' r — 1 -^ be positive, change the signs of the terms, (Art. 124), and S:=3 , or =z-^ -. Therefore, for finding the sum of 1 — r 1 — r the series, when the progression is decreasing, Kule. — Multiply the last term by the ratio, subtract the product from the first term, and divide the remainder by one minus the ratio. SOO. When the series is decreasing, and the number of terms infinite, I is infinitely small, or 0. Therefore, r/=::0, and S== becomes 8=^^ . Hence, 1 — r 1 — r Bale for finding the Sum of an Infinite Decreasing Series. — Divide the first term by one minus the ratio. Find the sum of the infinite series l+^+l+gH-, etc. Here, a=l, r=^, and S=:.j = :^ 1^^^^' ^°s. 1 r i J That the sum of an infinite decreasing series may bo finite, will easily appear from the following illustration : Take a straight line, AK, and bisect it in B; bisect BK in C; CK in D, and so on continually; then will AK=zAB+BC+CD+, etc., in infinitum, =AB+JAB-f JAB, etc., in infinitum, =:2AB, which agrees with the example. 300. The equations, l=zar"-~^, and S=:- ^-, furnish this general problem : Knowing any three of the five quantities a, r, n, I, and S, of a geometrical progression, to determine the other two. J^32 RAY'S ALGEBRA, SECOND BOOK. The following table contains all the values of each unknown quantity, or the equations from which it may be derived : No. 10. 11. 12 13. 14. 15. 16. Given. 17. 18. 19. 20. a, r, n a, r, S a, n, S r, n, S a, r, 71 a, r, I a, n, I r, n, I r, n, S r, I, S n, I, S a, n, I a, w, S a, I, S n, I, S a, r, I a, r, S a, ^, S r, ^, S Rcauired. Formulfe. ^^ a+(r-l)S ^(S— ^)"-i_a(S_a)n-i:^0, (r— l)Sr«-i r'*— 1 S= ■ r-l rl—a Va" ~y.n y.n-1 a= (r-l)S r»— 1 ' a=r^— (r— 1)S, a(S— a)'^-!— ^(S— ^"-^=0. /-=' „ S , S— a _ a ^ a ' S— g 'S=T» log. ia-^(r—l)S']—log. a log. r log. I— log. a .4-1, log. (S—a)—lng. (S -7) ^ Jog. l~log. [^r-(r-l)S ] ^ GEOMETRICAL PROGRESSION. 263 By observing, in any particular example, what are given and re- quired, tlie proper formulae may be selected from the above table. Nos. 3, 12, 14, and 16 may require the solution of an equation higher than the second degree. Kos. 17, 18, 19, and 20 are obtained by solving an exponential equation, (Art. 382) but are introduced here to render the table complete. The two formulae l=.ar^-^ (1), and S J^£^, or, (Art. 298,) ^ (2\ are, however, sufficient for the solution of all examples ia Geometri- cal Progression, and may easily be retained in the memory. 1. Find the 8'^ term of the series 5, 10, 20, etc. Aus. 640. 2. The "1^ term of the series 54, 27, 13i, etc. ^ Ans. ||. 3. The 6'^ term of the series 3|, 2j, 1^, etc. Ans. ^. 4. The 7^'* term of the series —21, 14, — 9^, etc. Ans. — 4|f. 5. The 7i'* term of the series J, ^, |, etc. Ans. o^^i* 6. Find the sum of 1-f 3-|-9-[-, etc., to 9 terms. From (1), ^=1X3*^=6561. From (2), S=?^^?^i^=9841, Ans. 7. Of 1+4+16+, etc, to 8 terms. Ans. 21845. 8. Of 8 + 20-[-50 + , etc., to 7 terms. Ans. 3249|. 9. Of 1 + 3+9+, etc., to n terms. Ans. ^(3»— 1). 10. Of 1— 2+4— 8+, etc., to n terms. Ans. J(1=f2"). 11. Of X — y-\-- ^+, etc., to n terms. A„s,4-{i-(-?y{. 12. The first term is 4, the last term 12500, and the ^/umber of terms 6. Required the ratio and the sum of all the terms. Ans. Ratio =^^5 ; sum =:15624. 264 RAY'S ALGEBRA, SECOND BOOK. Find the sum of an infinite number of terms of each of the following series : 13. Of 5+^ + 1+, etc Ans. |. 14. Of 9H-6 + 4+, etc Ans. 27. 15. Of|-i + ^- etc Ans. |. 16. Of a+Z/+-+-2+) etc An a a — h 17. The sum of an infinite geometric series is 3, and the sum of its first two terms is 2| ; find the series. Ans. 2+1+1+ • . . or 4-|+|-. . . . 18. Find a ojeometric mean between 4 and 16. Ans. 8. Here, a=4, ^=rl6, and n=:3; or, (Art. 269) the mean =--y^4xl6. 19. The first term of a geometric series is 3, the last term 96, and the number of terms 6 ; find the ratio, and the intermediate terms. Ans. r=2. Int. terms, 6, 12, 24, 48. If it be required to insert m geometrical means between two numbers, a and I, we have n=im-|-2; hence, n — l^m-|-l, and r^=::'"+*'y'— . Or, we may employ formula (1). 20. Insert two geometric means between i^ and 2. ^Ans. I, -^ 21. Insert 7 geometric means between 2 and 13122. Ans. 6, 18, 54, 162, 486, 1458, 4374. 301* To find the value of Circulating Decimals; that is, decimals in which one or more figures are continually repeated. In such decimals the ratio is JL, y^, jq^^^. etc., according as one, two, or more figures recur. Thus, / ^1 ^1 ^1 \ .253131 .... =_^»,+ ( _ +jg, + j^+, etc. ) HARMONICAL PROGRESSION. 265 The part within the parenthesis is an infinite series, having «=T^o^Oo ^^d "-=1^ Hence, (Art. 299,) S=^^3 ,_. Therefore, .253131 .... =j%%+^UTi=Ur6=mi- This operation may be performed more simply, as follows: Let S=.25313131 .... Multiplying by 10000, 100008=2531.3131 . . . Dividing by 100, . . 100S= 25.3131 . . . Subtracting, .... 9900S=-2506 .-. S^fsge. 1. Find the value of .636363 Ans. j\. 2. Find the value of .54123123. . . Ans. Jffg^. 302. Harmonical Progression. — Three or more quan- tities are said to be in Harmonical Progression, when their reciprocals are in arithmetical progression. Thus, 1, |, ^, ^, etc.; and 1, 2 i, j^ etc., are in harmonical progression, because their reciprocals 1, 3, 5, 7, etc.; and 4, 3|, 3, 2^, etc., are in arithmetical progression. 303. Proposition. — If three quantities are in harmoni- cal progression, the first term is to the third as the differ- ence of the first and second is to the difference of the second and third. For if a, b, c, are in harmonical progression, — , -, — , are in arithmetical progression ; therefore, = = J. Hence, multiplying by abc, ac — bc=r.ab — ac; or c(a — b)=a(b — c). This gives (Art. 268), a : c : : a — b : b — c; therefore, A Harmonical Progression is a series of quantities in harmonical proportion (Art. 280) ; or such that if any three consecutive terms be taken, the first is to the third as the difference of the first and second is to the differ- ence of the second and third. 2d Bk. 23* 266 RAY'S ALGEBRA, SECOND BOOK. Hence, all problems with respect to numbers in harmon- ical progression, may be solved by inverting them, and considering the reciprocals as quantities in arithmetical progression. We give, however, below, two formulae of frequent use r 1. Given the first two terms of a harmonical progres- sion, a and i, to find the n^^ term. Here, a, 6, and ?, the first two and inf^ terms become (Art, 302), -, , and in formula (1) (Art. 294). Also, C?=t = — v-' Therefore, _==-4-(n— 1) — ^ = ^ -^ -\ ' I a^^ ' ab ab ' Whence, l=i ' {n—\)a—{n -2)6' By means of this formula, when any two successive terms of a harmonical progression are given, any other term may be found. 2. Insert m harmonic means between a and I. Here, since m^^n — 2, and ni-{-l=^n — 1, we have, as above, 1 1 , . 1XJ ^ ^ «— ^ «— ^ T = — \-i^ — l)»j aiid <^=7 i — 7 = 7 r-is- ,1 I a^^ ' ' {n—l)al {m,~\-l)aV whence, the arithmetical progression is found; and by inverting its terms, the harmonicals are also found. 3. Insert two harmonic means between 3 and 12. Ans. 4 and 6. 4 Insert two harmonic means between 2 and 1. Ans. ^ and §. 5. The first term of a harmonic series is ^, and the 6'* is j'^; find the intermediate terms. 6. a, J, c, are in arithmetical progression, and 6, c, )(lx2 . . 2)(1X2 . . r)x, etc. * 308. The Combinations of quantities are the different collections that can be formed out of them, without refer- ence to the order in which they are placed. Thus, a6, ac, he, are the combinations of the letters a, 5, c, taken two together; ah and ha, ac and ca, he and c6, though diflFerent permutations, forming the same combina- tion. Proposition. — To find the number of comhinatiom that can he formed out of n letters, taken singly, taken two to- gether, three together, and r together. Let C„ Cj, . . . Or denote the number of combinations of n things taken singly, taken two together, .... and taken r together. The number of combinations of n letters taken singly, is evi- dently n; that is, C,=n. The number of permutations of n letters taken two together, is rt{n — 1); but each combination, as a6, admits of (1X2) permuta- C2=- PERMUTATIONS AND COMBINATIONS. 273 tions, a6, ha] therefore, there are (IX^) times as many permuta- tions as combinations. Hence, n(n— 1) " Tx2"- Again, in n letters taken three together, the number of permuta- tions is n{n — l)(n— 2); but each combination of three letters, as abc, admits of 1x2x3 permutations; therefore, _ n{n-l){n-2) ^3- 1^2X3 • So, for n letters, each of which contains r combinations, n(?8~l)(n-2) . . . [n-(r-l)] '~ 1X2X3 r 300. Intimately connected with the subject of the pre- ceding articles, is that of the Doctrine of Chances, or the Calculus op Probabilities. This, however, being too abstruse for an elementary treatise, is omitted in this work. 1. How many permutations of 2 letters each can be formed out of the letters a, 6, c, t/, e? How many of 3? Of 4? Ans. (1) 20. (2) 60. (3) 120. 2. How many combinations of 2 letters each can be formed out of the letters a, 6, c, d, e? How many of 3? Of 4? Of 5? Ans. (1) 10. (2) 10. (3) 5. (4) 1. 3. In how many ways, taken all together, may the letters in the word NOT be written ? In the word HOME ? Ans 6, and 24. 4. How often can 6 persons change their places at din- ner, so as not to sit twice in the same order? Ans. 720. 5. In how many different ways, taken all together, can the 7 prismatic colors be arranged? Ans. 5040. 6. In how many different ways can 6 letters be arranged when taken singly, 2 by 2, 3 by 3, and so on, till they are all taken? Ans. 1956. Suggestion . — Take the sum of the different permutations. 274 RAY'S ALGEBRA, SECOND BOOK. 7. How many different products can be formed with any two of the figures 3, 4, 5, G ? Ans. 6. 8. The number of permutations of n things taken 4 to- gether = 6 times the number taken 3 together ; find n. Ads. n=0. 9. How many different sums of money can be formed with a cent, a three cent piece, a half dime, and a dime ? Ans. 15. Suggestion. — Take the sum of the different combinations of 4 things taken singly, 2 together, 3 together, and 4 together. 10. With the addition of a twenty-five cent piece, and a half dollar, to the coins in the last example, how many different sums of money may be formed? Ans. 63. 11. At an election, where every voter may vote for any number of candidates not greater than the number to be elected, there are 4 candidates and only 3 persons to be chosen ; in how many ways may a man vote? Ans. 14. 12. On how many nights may a different guard of 4 men be posted out of 16? and on how many of these will anj particular man be on guard? Ans. 1820, and 455. 13. How many changes may be rung with 5 bells out of 8, and how many with the whole peal? Ans. 6720, and 40320. 14. Out of l7 consonants and 5 vowels, how many wordil can be formed, having two consonants and one vowel in each? Ans. 4080 BINOMIAL THEOREM, WHEN THE EXPONENT IS A POSITIVE INTEGER. 310. We have already explained (Art. 172) the method of finding any power of a binomial, by repeated multiplica- tion, and by Newton's Theorem, as proved experimentally. BINOMIAL THEOREM. 275 We shall now proceed from the theory of Combinations (Art. 308), to demonstrate the Binomial Theorem in its most general form. The Binomial Theorem teaches the method of develop- ing into a series any binomial whose index is either in- tegral or fractional, positive or negative ; as, (a+x)", (a+a.)-^ {a-\-xy\ (a-^x)-\ where a or x may be either plus or minus. The following investigation applies only to the case where the exponent is positive and integral; the other cases will be considered hereafter. (See Art. 319.) By actual muhiplication, it appears that {x-{-a){x-\-b)—x^-\^a i x^ah. + 61 In like manner, {x-\-a){x-^b){x-\-G) +6 x'^-\-ab x-\-abc. -f «c -f6c Also, {x^a){x-{-b)(x^c){x-\-d) =zx*-\-a x^-\-ab x'^+abc 4-6 -^ac -\-abd -fc -fad -\-acd J,d -f-6c -{-bed -\-bd -fccZj x-\-abcd. An examination of either of these products, shows that it is com- posed of a series of descending powers of X, and of certain coeffi- cients, formed according to the following law: 1st. The exponent of the highest power of x is found in the first term, and is the same as the mtmher of binomial faetors, and the other exponents of x decrease by 1 in each succeeding term, 2d. The coefficient of the first term is 1 ; of the second, the sum of the quantities a, 6, c, etc. ; of the third, the 276 RAY'S ALGEBRA, SECOND BOOK. sum of the products of every two of the quantities a, &, c, etc. ; of the fourth, the sum of the products of every three, and so on ; and of the last, the product of all the n quan- tities a, b, c, etc. Suppose, then, this law to hold for the product of n binomial fac- tors a;+ a, x-\-b, ^+c, x-[-k- so that {x-{-a){x-^b){x-^c) (a;-f ^)=^-a;^^4-Aa;''-i4-Ba;"-2+Ca;"-3+ -f K, Where . . , A=a-f-6+c+ .... -f A;. B=ab-\^ac-\-ad-{- C=a6c-f a6cZ-f- Etc. = etc Y.=^abcd Jc. If we multiply both sides of this equation by a new factor x-\-lj we have {x-^a){x-\-b){x-j-c) {xi-'k){x-\-l) j^l I _^a; I -I-B; 1 .... +KZ. Here, . . A+^ =a+6+c+ » . . . +7^+?; B-f A^ =ab+ac-{-ad . . . -^al-\-bl . . . -j-Jcl. Etc. = etc Kl =abcd JcL It is evident that the same law, as above stated, still holds. Hence, if the law holds when n binomial factors are multiplied together, it will hold when n-|-l factors are taken; but it has been shown, by actual multiplication, to hold up to 4 factors; therefore, it is true for 4+1, that is, 5; and if for 6, then for 5-[-l, that is, 6; and so on generally, for any number whatever. Now let . . . b, c, d, etc., each —a; Then, A=a-f a+a-f-a+j etc., to n terms =na. B=a^-\-a'^-{-, etc., =a2 taken as many times as ^ is equal to the No. of combinations of n things taken j- = . — . two together, which is (Art. 308), ^ C=a3-|-a3_[_^ etc., =a^ taken as many times as is equal to the No. of combinati of the things taken three together, which (Art. 308), Etc. = et6. K=zaaa .... to n factors ^a" aany n Lions ( _ n(n—l)(n—2)a ^ ;hisj 1-2-3 ~* BINOMIAL THEOREM. 277 Also, (a:-fa)(a;+6)(a;+c) {x^l)={x-\-aY. + +«". By changing a: to ^a^xK Corollary 4. — If n be a positive integer, and r=n-\-2 ; then, {n — r-\-2) becomes 0, and the (n-|-2) term vanishes; therefore, the series consists of (n-\-V) terms altogether; that is, • The number of terms is one greater than the exporient of the power to ivhich the binomial is to be raised. Corollary 5. — When the index of the binomial is a posi- tive integer, the coefficients of the terms taken in an in- verse order from the end of the series, are equal to the coefficients of the corresponding terms taken in a direct order from the beginning. If we compare the expansion of (a-^x)^, and (a:-f a)"", we have {xJrar=x--^nx--^a-\-'^-^^=^ Since the binomials are the same, the series resulting trom their expansion must be the same, except that the order of the terms will be inverted. It is clearly seen that the coefficients of the corre- sponding terms are equal. Hence, in expanding such a binomial, the latter half of the ex- pansion may be taken from the first half. Example. — Expand (a — Z>)^ Here the number of terms (n-fl) is 6; therefore, it is only neces- sary to find the coefficients of the first three, thus: (a— 6)5=a5_5«45^^(^362_i0a2^.|.5«64_5.'>. BINOMIAL THEOREM. 279 Corollary 6. — The sum of the coefl5cients, where both terms are positive, is always equal to 2'*. For if we makea;=a=l; then, . . ra;+a)»=r(l+l)»»=2". 311. From an inspection of the general expansion of (a-f x)", it is evident that If the coefficient of any term he multiplied hy the expo- nent of the first letter of the binomial in that term, and the product he divided hy the number of the term, the quotient will be the coefficient of the next term. For examples, see Newton s Theorem, Art. 172. 31!3. To expand a binomial aifected with coefficients or exponents, as (2a'^ — 86^)*, see Newton s Theorem, Art. 172. 313. By means of the Binomial Theorem, we can raise any polynomial to any power. Thus, let it be required to raise a—h-\-c to the third power. Let a — b=m, etc., as already explained, Art. 172. 1. Expand (a+5)«, (a—hy, and (5— 4a-)*. (1) Ans. a'-\-Sa'b-\-2Sa'b'Jr^Qc^'f>'-\-l0a'b'-^b6a'h^ Jr2Sa'h^~\-SaP-\-h\ (2) Ans. a''~7a'b-^21a'b'—S^a'L'-\-Sba^b*~21a'h' j^^ah^—h\ (3) Ans. 625— 2000a- -h 2400x^—1 280a:3-f 256a;*. 2. Kequired the coefficient of x^ in the expansion of (^-j_y)io. Ans. 210. 3. Find the 5'^^ term of the expansion of (c^ — d^y^. Ans. 495c^«2c+3ac2+6fec='— c'. 9. Prove that the sum of the coefficients of the odd terms of (a-f ^)"» is equal to the sum of the coefficients of the even terms. X. INDETERMINATE COEFFICIENTS: BINOMIAL THEOREM, GENERAL DEMONSTRATION : SUMMATION AND INTERPOLA- TION OF SERIES. 314. Indeterminate Coefficients.— The method of de- veloping algebraic expressions, by assuming a series with unknown coefficients, and finding the values of the assumed coefficients, is termed the method of Indeterminate Coejfi- cients. It depends on the following THEOREM. If A+Br^-fCa^^+Dor'-f, etc , =rA'+B'ic-f CV4 DV+, etc., for every possible value of x (A, B, A', B', etc., not containing x, and x being a variable quantity) we shall have A=A', B=B', C=C', etc.; that is, The coefficients of the terms involving the same powers of x in the two series, are respectively/ equal. For, by transposing all the terms into the first member, we have A—A'-\-{B—B')x-\-{C—C')x^-\-iJ)—T)')x^-\-, etc., =.0. If A — A^ is not equal to 0, let it be equal to some quantity p; then, we have {B—B')x^{C—C')x^^{D—Ty)x^-j-^ etc., =—p. INDETERMINATE COEFFICIENTS. 281 Now, since A and Af are constant quantities, their difference, p^ must be constant; but — p=(B— B^)a:-|-(C— C^)a:2+, ^xe., a quan- tity which may evidently have various values, since it depends upon X] therefore, the same quantity [p) is both fixed and variable, which is impossible. Hence, there is no possible quantiiy (p) which can express the difference A— A^; or, in other words, A— A^=0 .-. Arr:A^ Hence, {B—B')x-\-{C—C')x^^{I)—jy)x^-j-, etc., =0. By dividing each side by x, we have B-B'+(C— C0a:+(D-iy)a:2-f, etc., =0. Reasoning as before, we may show that B^iB'; and so on, for the remaining coefficients of the like powers of X. Corollary. — If we have an equation of the form A^Bx-\-Cx'-^Dx'-\-Ex'-\-, etc., ==0, which is true for avi/ value whatever of x; then, Arrr^O, B=0, C±=0, etc.; that is, each coefficient is separately equal to zero. For the right hand member may evidently be put under the form 0-f0a:-(-0a:2-^0a:3-[-, etc.; then, comparing the coefficients of the like powers of x, we have A=0, Bz=lO, C=0, etc. 31S. Let it be required to develope — —7— into a series a-\-ox without a resort to division. The series will consist of the powers of X multiplied by certain undetermined coefficients, depending on either a or 6, or both of them, and x will not enter into the first term; therefore, assume ^^=A+Ba:+Ca:2+Da:3+, etc. Multiply both sides by the denominator a-\-hx, and arrange the terms according to the powers of x\ we thus obtain a=ka-\-Ba I x-\-Qa I x'^-\-Da I x^-\-^ etc. +A6| +B6| +C6| 2d Bk. 24 282 RAY'S ALGEBRA, SECOND BOOK. But by the preceding theorem and corollary, a=Aa ; hence, A^l ; Ba+A6=0: " B= ; a' Ca+B6=0; « C=:+^,; 53 Substituting these values in the assumed series, we find a b b^ b^ b^ 1 X-}- —x^ :.x^-] — -x^, etc., the sa'me as would a^bx a a'^ a^ a* be obtained by actual division. 316. A series with indeterminate coefficients is gener- ally assumed to proceed according to the ascending in- tegral and positive powers of x, beginning with x° ; but in many series this is not the case. The error in the assump- tion will then be shown, either by an impossible result, or by finding the coefficients of the terms which do not exist in the actual series, equal to zero. Thus, if it be required to develope ^ 5' ^^^^ ^^ assume the series to be A+Bic-f CiC^-j-DrcS-f EiC*-]-, etc., we have, after clearing of fractions, l=3Aa;-h(3B— A)a:2-f (3C— B)a:34., etc.; from which, by equating the coefficients of the same powers of x, 1=0, 3A=0, etc. The first equation, 1==0, being absurd, we infer that the expres- sion can not be developed under the assumed form. But, clearing of fractions, and equating the coefficients of the like powers of X, we find A=|, B=^, C^^i^, D=g'p etc. Hence, Sx 1 1/1 X x\ x^\ \ x-\x\ x x\ ^ INDETERMINATE COEFFICIENTS. 283 Or, since the division of 1 by the first term of the denominator gives — , or Sx~^, we ought to have assumed oX Ax-^-\-B-\-Cx-\-'Dx-^, etc. Sx — x'^ 1 x2 Again, if we assume ^ ^=A-fBa;+Ca:2-f Da:3-f , etc.; we shall find the true series to be l—2x--\3x*—5x^-\-, etc., the coefficients B, D, F, etc., becoming zero. 317* Evolution by indeterminate coefficients. Example. — Extract the square root of a^-\-x^. Assume y {a'^^x'^)=k^J^x-\-Qx--\-J)x^^^x^^^ etc. Squaring both sides, we have, a2_|_a;2^A2-f 2ABa;+2AC i a;2-]-2AD 1 a:34-2AE x^-\-, etc. 4-B2 I -f 2BC I +2BD + 02 .-. A2=:a2, 2AB=.0, 2AC+B2==1, 2AD+2BC=0, etc. And, A=a, B=0, C^^-' ^=0, Er=— — ^ — , etc. Therefore, v/(«2-t-a;2)=a-{-|-_^34-, etc SIS. Decomposition of Rational Fractions. — Frac- tions whose denominators can be separated into certain factors, may often be decomposed into other fractions whose denominators shall consist of one or more of these factors. To illustrate by an example. ^x 14 Decompose .^ ' ^ — — ^ into two other fractions whose denominators shall be the factors of x^ — 6;r-f8. Since x'^—^x-\-^={x^2){x-^\ (Art. 234), assume 5a;_14 _ A B 284 RAY'S ALGEBRA, SECOND BOOK. Beducing the fractions to a common denominator, We have 5:r-14 _ A(a:-4)+B (:r.-2). Or, ox—U=A{x—A)-{-B{x—2)={A-\-B)x—4A—2B. Now, since this equation is true for any value whatever of X, we may equate the coefficients (Art. 314); this gives A-|-B^5; _4A— 2B=— 14; whence, A=2, and B^^. And . . . ^2_Qx-{-8~~x~2'^x^' By the method of Indeterminate Coefficients, show that 2. l+2x ^ ^i^3^^4^2_^7^8^11^4_^18^5^^ etc. J. X X 3. n_l"^)3 "^l'+^'^H-3V-{-4V-f5V+, etc. cc x^ Sa^ S'bx* 4. i/l— ^=1-2~2~^""2^1^~2~T^8~ ^*^- 5. ^/(l+^+^^):=l+|+|!_|^^+, etc. « 1+^ 1 , 2 X X" X a;-'— 7x+12~~x— 4 x— 3* 8. ,-,_,^;_,,= ^^--i + 1 (x^— 1X^5—2) ~ 3(0;— 2) 2(a;— 1) "^ 6(a;H-l)' BINOMIAL THEOREM. 285 BINOMIAL THEOREM, WHEN THE EXPONENT IS FRACTIONAL OR NEGATIVE. SIO. We shall now proceed to prove the truth of the Binomial Theorem generally ; that is, to show that ^\'hethe^ n be integral or fractional, positive or negative. Since a-\-b=a{ \-\ — ); Therefore, (a-f 6)"=a'' / l-j— ) ''^a'\\-\-x)\ if x=~. Hence, if we can find the law of the expansion of {\-\-x)^ we may obtain that of (a-J-fe)", by writing - for a:, and multiplying by a". a AVe shall therefore prove that, in all cases, /i I X,. 1 I . ^(^—1) 9 , n(n— l)(n— 2) , , (l-fa:)"=l-i-na:-|-A___^a:2-f -A-^-^^-^ — 'x^J^, etc. The proof may be divided into two parts : 1st. To show that (l-fa:)"=l-f wic-f, etc. 2d. To find the general law of the coefiicients. First. — To prove that the coefficient of the second term of the expansion of {\-\-xY is n, whether n be integral or fractional, positive or negative. Let the index be positive and integral ; then, since by multiplica- lion we know that (1+^)2=1+2^+-, etc., (\J^xf=l-\-^x-{-, etc.; Let us assume that {\^xY-'^=\-\-{n — \)x-\-, etc. Multiply both sides of this equality by 1-f-^; then, {l^x)-^l^x)=\l^{n^\)x^, etc. }(l-|-a:); Or, {\-\-xY=z\-\-nx-\-, etc., by multiplication. 286 RAYS ALGEBRA, SECOND BOOK. Hence, if the proposition is true for any one index n — 1, it will be true for the next higher index n. Now, by maltiplication, it is true for the index 3, it is therefore true for the index 3-f 1—4; for the index 4-(-l=5, and so on. Hence, by continued induction, it is always true for n when it is integral and positive. P Next, let n be a fraction =— . p Also, let {l-\-x)q—l-\-ax^, etc., =1-|-Aa;, where Ax is put to represent all the terms after the first. p Since .... (l-|-a:)?=l+Aa;, .-. by raising both sides to the 2' power, . . . {\-^x)P={l-\-kxf\ .-. l-fpaj+j etc., =l+gAa;+, etc., =l^q{ax-\-, etc.,)-]-, etc., =l-[-qax^^ etc. By equating the coefficients of the like powers of x (Art. 314), p^qa .-. aJ^, and {l^x)^=\^^x+, etc. Lagtly, let n be negative; then, (Art. 81), l\-\-x)-''z=—- — — = .= i =l—nx-\-, etc., by division. ^ ' ' (l-|-a;) l-\-nx-\-, etc. ^' ' '' Therefore, {l-{-xy—\-{-nx^, etc., whatever be the value of n. Therefore, (a-|-6)''=a" I 1-j— V=a«(l_|_72_-|-, etc.), =a'»-{-na"-i6-}-, etc., and the first two terms of the series are determined. Second. — To find the general law of the coeflBcients. Let (l+a-)'*=l-(-r?a;-f Bx^-f Ca;3-f Dx*+, etc., where B, C, D, etc., depend upon n. For Xy put x~\-Zy and consider {x-\-Z) as one term; then, |l_f_(a;4-0) }»=^l4-n(a:+2;)4-B(a:+2;)2+C(a;-}-2:)3-|-, etc. But . . (a-|-6)"=:a"-fna'*-i6-j-, etc.; .-. {x-^zY=x'^-{-2xz-\-,eic.\ {x^z)^=^x^-[-^x'^z-{-, etc.; {x^zy=x^-]-Ax^z-\-, etc.; .-. {l+(a:-f0)}''==rl-fna:-[-Ba:2-f Ca:3-1-Dic^+, etc., -f (n-f-2Ba:-l-3Ca:244Da;3-f , etc.)2;-|-, etc., =z(l+a;)'»-|-(n-f 2Ba;-f3Cx2-|-4Da:3-f , etc.)0-f-, etc., (A). BINOMIAL THEOREM. 287 But, considering (1+a;) as one term, {\-\-x^zY=[{l-\-x)-\-z]'^\ and {(l-fa;)-f^]»=iz(l+a:)"+n(l+a:)^-J2;+, etc. (B). Equating the coefficients of z in (A) and (B), n-f 2Ba;+3Ca;2-f4Dx3+, etc., =/i(l+a:)"-J. Multiplying both sides by 1+iC, we have n-f 2Ba:+3Cx2+4Da;3-}-, etc. -) :=n(l+a:)'» -f na;4-2Bx2-f3Cic3_(-, etc. J =n(l+na;+Ba;2+Ca;8-|-, etc.) By equating the coefficients of the same powers of ic, we have 2B4-7i=n2 .-. 2B=n2— n=w(n— 1). ^ n(yi_l) ^-12' 3C+2B^Bn .-. 3C=B(n— 2); B(n-2) _ n(n— l)(n— 2) ^- 3 ~ 1 • 2 • 3 ' Also, 4D+3C=nC .-. 4D=.C(n-3); * _ C(7i— 3) n(n— l)(n— 2)(n— 3) , . „ „ ^ D^- -^— ^ = -5^ , ^''^ ^ ^, ; ; and so on for E, F, G, etc. 4 l-2'3-4 ' »'' ... (l+.)-^l+n.+"-(»-^l)x3+ "("-^f3-^> :^+. etc., and .-. putting - for X, (a+6)"=a"( 1+- J, h n(n~-l)b^ n(n—'i){n—2)b^ =a-+«a--6+r^a'-^6H "'T^yr^' «-^frH, etc. If — 6 be put for 6, then since the odd powers of — b are negative ^Art. 193) and the even powers positive, («-6)'"=a»-«a"-.6+!i^a-»6^-"i^^ —r'-- \ r /a a-\-b From the nature of the case, r is necessarily integral ; if [n-\-Yib (n-f-l)& is fractional, take r = the first integer > — —j- , and the r** term will be the greatest. If ■ ]" • is an integer, and we take n will 1 r=%- ~\ , \ then the r"* term = the (r4-l)'^ and each of these is greater than any other term ; for this can only occur when r 'a II. If n is a positive fraction : If — > 1 there is no greatest term, for the series will evidently diverge. But if - < 1 the series will have its greatest term (or terms) whose position may be ascertained as in I. III. If n be negative, either an integer or a fraction > 1 : The multiplier that changes the r'* term into the (r-f-1)'^ viz., — ^ .- may be written — ( )-i and as the numerically r a "^ \ r /a/ greatest term is sought, disregard the sign of the multiplier; then. as in I, the r'^ term will be the greatest when .- is first < 1 r a „ , h{n—Ti) or r first >-^ — j-\ a — o As in I, if J • be a whole number there are two equal terms ' a—b ^ each greater than any other; and, as in II, if — be > 1, there is no greatest term. BINOMIAL THEOREM. 289 IV- If n be negative and < 1, and - < 1, the first term is the greatest; for in this case the multiplier is < 1 for all values of r, that is, each term is less than the preceding. Note. — If b is negative, since it is the numerical value of the term that is to be considered, wfi mav disregard the sigu of b and apply the appropriate one of the preceding rules. [Cf. Todbunter's Algebra, Art. 520.] Examples. — Find the greatest term in each of the following expansions : (71+1)6 35 1. (2+f)6. Here 6.5.4 53 20000 a+6 11 1.2.3'^ 3«"~ 2. (1+|)V. 81 3. (f+lj^ 4. (l+i)-i2. Uere^.^^=^4 12-13 1 78 1-2 •52~25' 5. (1+f)-^ a — b Here ^^^^^^=5 a—b than any other term. 6. (l-,V)-i r=4 gives the greatest term= Ans. 2*^. Ans. 5"' and 6'\ -3 gives the greatest term: 53=0. Hence, the formula na-\ — \^ — ^I^iH — —^i — jr^q — ^2 gives n^—Sn^-\-2n n(n+l)(n+2) n+n2_n+ g— — =-i_X.^A_!U. (A) 330* To find the number of halls in a square pile. A square pile, as V — EFH, is formed V of successive square horizontal courses, such that the number of balls in the sides of these courses decreases con- tinually by unity, from the bottom to the single ball at the top. If we commence at the top, the number of balls in the respective courses will be as follows: 1»«. 2<^. 3'^. 4(>;-^ )D3+, etc. The intervals between the given numbers is always to be consid- ered as unity, and p is to be reckoned in parts of this interval; hence, p will be fractional. Sufficient accui\acy is generally obtained by making use of Dj and D2 only, in the above formula. Ill practice, however, the following is generally adopted: Take the two functions of the series which precede, and the two which follow the term required, and find from them the three first differences, and the two second differences. Call the second of the three first differences d, the mean of the two second differences d\ the fractional part of the interval p', and second term 6. We then have from the above formula, t^bJ,p\d^-^d'), INFINITE SERIES. 305 Applying this formula to the above example, we have Nos. Logarithms. 1st Diff. 2d Diff. Mean of 2d DilT. 102 103 104 105 2.0086002 2.0128372 2.0170333 2.0211893 42370 41961 41560 -409 -401 —405 Here, iy-=.55, rf=41961, d^=— 405, and 6=2.0128372. ^=2.0128372 +.55(41961+^X405). <=2.0128372+.0023129=^2.0151501, Ans. 1. Find the 2*^ term of the series of which the 4'^* dif- ferences vanish, the 1*', 3'^, 4'^ and b^^ terms being 3, 15, 30, 55; and find the 6^ 7^ and 8"^ terms. Ans. 7; and 93, 147, and 220. 2. Find the 5'^ term of the series of which the 6"^ dif- ferences vanish, and the 1*', 2^^, 3*^, 4'^, 6'^ and 7'* terms are 11, 18, 30, 50, 132, 209. Ans. 82. 3. Given the logarithms of 101, 102, 104, and 105; viz., 2.0043214, 2.0086002, 2.0170333, and 2.0211893, to find the logarithm of 103. Ans. 2.0128372. 4. Given the cube roots of 60, 62, 64, and 66; viz., 3.91487, 3.95789, 4, and 4.04124, to find the cube root of 63. Ans. 3.97905. 5. Having given the squares of any two consecutive whole numbers, show how the squares of the succeeding whole numbers may be obtained by addition. INFINITE SERIES. 336. An Infinite Series is a series consisting of an unlimited number of terms. The Sum of an infinite series is the limit to which we approach by adding together more terms, but which can 2d Bk. 26 306 RAY S ALGEBRA, SECOND BOOK. not be exceeded by adding together any number of terms whatever. A Convergent Series is one which has a sum or limit. Thus, l+i+i + .+Jj+3'3+gL+, etc., is a convergent series, whose limit is 2, since the sum of any number of its terms can not exceed 2. A Divergent Series is one which has no sum or limit; as, l_^2-f4+8+16+32+, etc. An Ascending Series is one in which the powers of the leading quantity continually increase ; as, a-\-hx-\-cx^-\-ddi?-\-. A Descending Series is one in which the powers of the leading quantity continually diminish ; as, ' ' ' ' ' ^ X x^ x^ ^ S37. There are four general methods of converting an algebraic expression into an infinite series of equivalent value, each of which has been already exemplified ; viz., 1st. By Division^ Art. 134; 2d. By Extraction of Roots, Art. 183; 3d. By Indeterminate Coefficients, Arts. 315-7; and, 4th, By the Binomial Theorem, Art. 321. 338. The Summation of a Series is the finding a finite expression equivalent to the series. The General Term of a Series is an expression from which the several terms of the series may be derived ac- cording to some determinate law. Thus, in the series :p-)- 5- + K--|-;r+ the general a, 1 J o 4 term is — because by making X—\, 2, 3, etc., each term of the X series is found. Again, in the series 2 • 2-f 2 • 3-f2 • 4+2 -5+ the general term is 2(a;-|-l). INFINITE SERIES. 307 As different series are in general governed by different laws, the methods of finding the sum, which are applicable to one class, will not apply universally. We present two methods of most general application. First Method. — In a regular decreasing geometrical series, whose first term is a, and ratio r, the sum is = (Art. 299). Second Method. — By subtraction. Ex. 1. — Find the sum of the infinite series ^^ — r>-|-o — i I ^ 2 • 8 ' 3 • 4 + 4^5 + 5^-^'^'^- Then, | + l+i+ J-f, etc., = S-J. Subtracting 273 + 3T4 + 4-75 + 5T6+' ^^c., = ^, Ans. Ex. 2. — Find the sum of the infinite series J I 1 3 ' a- 5 + 6^ + 7^+''''- . • Then, i+i+i+^+ etc., :r^ S -1. Subtracting j"^ + 3^ + 5^+, etc, = 1, and ^-L + ^^ In such series, the first factor in the successive denominators is variable, while the second factor exceeds the first by a constant quantity. The general term is therefore , , where n is vari- ^ -> ^ n{n-\-py able and p constant. Since 1^^=-P?^^ ... '^ =l{g__^|. From which we derive the foUowinGr 308 RAY'S ALGEBRA, SECOND BOOK. Rule. — Having found the values of q, n, and p, in the given series, express the series whose general formulas are - and — ^ — ; subtract the latter from the former, and divide n n-\-p' J J ^ the result hy p for the sum of the series. 1. Required the sum of the series ■= — ^-f" 6 — s: + ''= — tt+j etc., ad infinitum, that is, to infinity. Here, . . . q=l, p=2, and n=A, 3, 5, 7, etc. Then, . . . ^=1+1+1+4+, etc, ad inf. Subtracting, ^— = -~-^ — r=l ; n{n+py :^z^ sum of given series. The sum of n terms of the same series is found in a manner nearly similar. Thus, l=^+m+^ d-i ^^=i+5+h 2^i^ + 2n+i :^Cp^Bq\-Ar; E:-Dp+Cg+Br; F=Ep-\-Dq+Cr', etc., And, by combining these equations, the values of p, q, and r are readily found, (Art. 158.) In a similar manner the scale may be determined in series of the higher orders. In finding the scale of a series, we must first ascertain by inspection whether the series is in G. P. ; if not, then 312 RAY'S ALGEBRA, SECOND BOOK. make trial of a scale containing two terms, then one of three, four, and so on, until a correct result is obtained. We must be careful not to assume too many terms ; for in that case every term of the scale will take the form ^. 343. To find the sum of an infinite recurring series whose scale of relation is known. Let A-I-Bx+Cx^+Da^'-J-Ex*, etc., be a recurring series of the second order, p and q being the terms of the scale. Then, . . . A=A; Ca:2=Bpa:2-f-Aga;2; T)0(?=Cpx-'-[-Bqx^; etc., ad infinitum. Represent by S the required sum, and add together the corre- sponding members of the preceding equations, observing that Ba;-|- Cx'^-\-Y)x^-\-f etc., =S — A ; then, we have S=A+Brc+ (S— A) pa^-f Sga;2 ; . • . S— Spa;— Sgx2= A+Ba:— Apa; ; Or, ... . S=^±J?^=^. ' 1 — px—qx^ If we make q^=^, (remembering that B=Ap), the formula be- comes 8=:; , which is, as it ought to be, identical with the l—px' ' * ' formula of Art 299. Remark. — Every recurring series may be supposed to arise from the development of a rational fraction, and the summation of such a series may be regarded as a return to the generating fraction. There are several methods of accomplishing this return : of these the preceding is regarded as the most suitable for an elementary work. 1. Find the sum of l + Sx+Sa^^+Tx-^+Px*, etc. Here, A=l, B=3, C=5, ^=7, etc. And, hence, (Art. 341.) p=2, (/=— 1. A \-Bx~kpx _ \ -f-3a;— 2t; _ 1 -\-x men, fe- ■^_^^,_^^^2 -i_2.T+a;2-(l-a;)2* REVERSION OF SERIES. 313 In each of the following series, find the scale of rela- tion, and the sum (S) of an infinite number of terms : 2. l-j-6x-\rl2x'Jr^8x'-i-120x'-\-, etc. 3. l-]-2x-^Sx'-\-^x^-^bx'-\-ex'-\-, etc. Ans. p=2, q=-l; S==-J_. 4. ^H -, — 1-, etc. Ans. The series is iu G. P. p= ; S= — '■ — ^ , c ' c-j-bx 5. x-\-x^-\-3^-\-, etc. Alls. The series is in G. P. p=l ; S^ ^ 1 — X 6. X — x^-\~x^ — x*-\-, etc. Ans. The series is in G. P. p=—l ; S=— ^. 7. l + 3a:+5a;2+7x3+9a;*+, etc. Ans. p=2, q=—l, S: ■^^'^ 8. l^+2^a;+3V+4V+5V+6V4-, etc. Ans. p=d, q=~S, r=l ; S.r= J-t^ ■(1-x) REVERSION OF SERIES. 344. To Revert a Series is to express the value of the unknown quantity in it by means of another series in- volving the powers of some other quantity. Let X and y represent two undetermined quantities, and express the value of y by a series involving the powers of x ; thus, 7/=ax-\-bx'^-\-coi^-\-dx*-{-, etc., (1), in which a, b, c, d, etc., are known quantities; then, to revert this series is to express the value of x in a series 2d Bk. 27* 314 RAY'S ALGEBRA, SECOND BOOK. containing the known quantities a, 6, c, d, etc., and the powers of y. To revert this series, assume x=Ay-]-By^-{-Cy^-]-T)y^, etc. (2), in which the coefi&cients A, B, C . . . aie undetermined. Eind the values of y^, y^, y* . . . from (1), thus, y-=a^x^-\-2abx^-\-{b^-{-2ac)x*-{-. .... 2/3= a^x^-{-Sa-bx^-{- .... y*= d^x^-^ .... etc. Substituting these values in (2), and arranging, we have- a;4-f , etc. 0=Aa rc+A6 a;24- Ac ic3+ kd -1 Ba2 -f2Ba6 + B62 + Ca3 4- 2Bac + 3Ca26 + Da4 Since this is true, whatever be the value of x^ and the coefficients of iC, a:2, x^^ etc., will each =0, (Art. 314, Cor.), we have Aa-1 =0, .-. A=-, a' A6+Ba2 =0, .-. B=-A Ac+2Ba6+Ca3 ==0, .-. d 262_ac Ad-fBd2_|_2Bac+3Ca26+Da<=0, .-. d=_'L'^'~^^^+^^ a7 1 6 „ 262_ac „ a2d-5a6c-}~563 . Hence, a:=- ?/- -gg/H ^5 ^ ^7 ^ +, etc. (3) 345. If the given series has a constant term prefixed, thus, y=^a!-\-ax-{-hj?-\-c7?-\-dx^-{- assume y — a!=.z.^ and we have z=zax-\-hxP--\-c^-\-dx^-\-^ etc. But this is the same as (1) in the preceding article, except that z stands in the place of y\ hence, if 2 be substituted for y in REVERSION OF SERIES. 315 (3,) [Art. 344], the result will be the required development of X] and then, y — a' being substituted for Z, the result is 2 f) 21)2 (2c ^=-(2/-«0-^(2/-«0'+ —^^{y-ct'r- etc. 346. When the given series contains the odd powers of X, assume for x another series containing the odd powers of y. Thus, if y=zcex-{-hx^-\-cx''-\-dx''-{- to develope x in terms of ^, assume x=A^+Bf-\-C7/-{-J)i/'+ Then, by substituting the values of y, 2/^, etc., derived from the former equation, in the latter, and equating the coefficients to zero, we find 1 6 Sb^—ac , a^d—8abc+12b^ ^ If both sides of the equation be expressed in a series, as ^2/H-^2/'+C2/^+j etc., =a^x-\-b^x^^&x^-{-, etc., and it be required to find y in terms of X, we must assume, as before, y=zAx^Bx^-\~Cx^-^Dx*-{-, etc., and substitute the values of y, y^, y^, etc., derived from this last equation, in the proposed equation; we shall then, by equating the coefficients of the like powers of x, determine the values of A, B, C, etc., as before. The following exercises may be solved either by substi- tuting the values of a, 5, r, etc., in the equations obtained in the preceding articles, or by proceeding according to the methods by which those equations were obtained. 1. Given the series i/=x — x'^-\-x^ — x^-\~ .... to find the value of X in terms of y. Ans. x=i/-\-y'^-^y^-\-y^-\-, etc. Find the value of a*, in an infinite series in terms of y: 2. AVhen i/=x-\-x'^-{-oc^-\-, etc. Ans. x=y — y'^~\-y^ — y*'~\~'!^ — ) ®^^' 316 3. When 7/=2x-\-Sx'^4:a^-\-5x-'-]-, etc. Ans. ^-c^^y— /gy+i'/g/— , etc. 4. When 3/=:l— 2a:+3x^ Ans. a:=-J.(y-l) + 3(3,_l)^__9_(^__l)3+, etc. 5. When y=a:+^.x2+Ja;3_|__^i_^4_|_^ el-^, Ans. cc^:^— ^/+J/— 1/4-, etc. 6. When y+^y+^y+^y • • • =gx-\-hx'^-\-kx^-\-lx*' . . . Ans x=^ I (^^ -^y I W-^9-^K^f~h)y XI. CONTINUED FRACTIONS : LOGARITHMS : EXPONENTIAL EQUATIONS: INTEREST, AND ANNUITIES. CONTINUED FRACTIONS. 34T. A Continued Fraction is one whose denomina- tor is continued by being itself a mixed nuinher, and the denominator of the fractional part again continued as be- fore, and so on ; thus, 1 1 1 -r "+1 in which a, 5, c, c?, etc., are positive whole numbers. Continued fractions are useful in approximating to the values of ratios expressed by large numbers, in resolving exponential equations, indeterminate equations of the first degree, etc. CONTINUED FRACTIONS. 317 34:8« To express a rational fraction in the form of a continued fraction. 80 Let it be required to reduce y^m ^^ ^ continued fraction. If we divide both terms of the fraction by the numerator, we ^+30 7 1 If we omit ^n, the denominator will be too small and =, the value 6\j 5 of the fraction, will be too large. 7 Again, if we divide both terms of the fraction ^ by the numer- ator, we find r-,^=^ = . ' 157 ^ 1 2 1 4 If we omit =, the value will be expressed by :, =oT» which is less than the true value of the fraction. Hence, generally, By stopping at an odd reduction^ and neglecting the frac- tional part, the result is too great ; hut hy stopping at an even reduction, and neglecting the fractional part, the result is too small. Since -= =^, we find so 1 =-c^=- T 1»< reduction, too great; ' 5-1 2d " too small; 4-\-- 3^ . « too great; 3-1- 4^^ " true value. By this process we find 13 1 49 1 30 = ^^1 204-^^1 3+5 6+g 318 RAY'S ALGEBRA, SECOND ]500K. 349« The different quantities 111 a+- a-\ ^^ etc., are called converging fractions, because each one in succes- sion gives a nearer value of the given expression. -* The fractions — , ^, — , etc., are called integral fractions. a b c^ ^ ^ 330. To explain the manner in which the converging fractions are found from the integral fractions. 1. - :=— 1*' conv. fraction. a a 2. , 1 =— — — f 2'' conv. fraction. a+^ ab-\-l 1 a-\- zi , , , = ^ ■■ , ^, , — S'^ conv. fraction. 1 c(aZ;+l)-j-a c By examining the third converging fraction, we find it is formed from the 1'', and 2<^, and from the 3^ integral fraction, as follows: Num. =3^^ quot.Xnum. of 2<^ conv. fract.-|-num. of 1*' conv. fract. Denom.=::3<^ quot.Xden. of 2'^ conv. fract.-|-den. of l"' conv. fract. P Q R To prove the general law of formation, let p^ ^T/j p/ he the three converging fractions corresponding to the three integral fractious — , V, and -, and, as has already been shown, R _Q c+P CONTINUED FRACTIONS. 319 1 S Let us now take the next integral fraction -r, and let -^ express R the 4'A converging fraction. Then, it is obvious that z^ will become S 1 ^ by substituting c-f--^, instead of C; hence, S^_ Q [ <^+a / "^ ^ _ (Q c+P ) d+Q _ Rc^-fQ Q^(o4)+P From this we see that the same rule applies to the 4'^ converging fraction, and so on. Hence, for the n'^ converging fraction, Multiple/ the denominator of the n*^ integral fraction hy the numerator of the (n — 1)'* converging fraction^ and add to the product the numerator of the (n — 2)"^ converging fraction. This will give the numerator of the d'* converging fraction. Multiply the denominator of the n^^ integral fraction hy the denominator of the (a — 1)"* converging fraction^ and add to the product the denominator of the (n — 2)'^* converg- ing fraction. This will give the denominator of the n''* con- verging fraction. Ex. — To find a series of converging fractions for ^^^ly. The integral fractions are ^, |, i, |, |, i, ^. The converging fractions are 1, -J, |, /y, ^ j%\, ^^^. 351. If the 2^ converging fraction (Art. 350) be sub- tracted from the 1*', the remainder will be found to be a fraction having for its numerator +1, and for its denom- inator the product of the two denominators ; and if the 3*^ be subtracted from the 2'^, the resulting fraction will have — 1 for its numerator, and the product of the denom- inntors for its denominator. 320 RAY'S ALGEBRA, SECOND BOOK. By a process of reasoning similar to that employed in Art. 850, it may be shown, in a general manner, that T^e difference between any two consecutive converging frac- tions is always a fraction having -^-1, or — 1, for its numer- ator, and the product of the two denominators for its denom- inator, according as the fraction subtracted is in an even or odd place. S52m To show that every converging fraction is in its lowest terms; and to find the approximate value of the frac- a tion Y' o A C If — and — be any two consecutive converging fractions, by Art. 351, |_5:3.+2^, or -^; that is, AD-BC=±1. Now, if A and B have a common divisor greater than 1, it will divide their multiples AD and BC, and their difference rirl, (Art, 100); or, a quantity greater than 1 is a divisor of 1, which is im- possible; hence, — is in its lowest terms. Since the denominators of the convergents continually increase, and their values continually diminish, and since the true value of J lies between any two consecutive convergents, it is evident that by continuing the series, any degree of approximation to the true value may be obtained. 3»>3. To express |/N, t^Ae?i N=a'+1, in the form of a continued fraction. 1 ^a^4.1_a+^/a2^1_a^a+-;==^ (Art. 206), 2a-fj^ — |-, etc. Ex. -/l7=|/4^-f-l=4 CONTINUED FRACTIONS. 1 321 8-j-T^-|-, etc., the converg- . ^ . 1. 11 . A 1 8 65 ing tractions to be added to 4, are q, ^, ^oo5 ^^^• 3S4. To convert y^N, where Nrnia^-j-^j i^^o a con- tinued fraction. Ex. Convert |/19, or |/16-|-3, into a continued frac- tion. 1 Hence, 1/19=4+-. 1 _ ,/T9+4 1 «-^/19_4 3—''+ 5- Hence, _^_ _3(/19-h2)_ ^19+2 _^ , 1 ~^19— 2~ 15 ~ 5 ~^G ~^19-3~ 10 ~ 2 "*"d- • • v/19^4+^ 1+^+, etc. Proceeding in the same manner, the successive values of a, 6, c, cZ, e, and / will be found 2, 1, 3, 1, 2, 8. The value of g is the same as that of a ; consequently, the succeeding values will recur in the same order as before. The converging fractions are 4, S, L3, IS^ 61^ etc. 33S. To find the value of a continued fraction^ when the denominators q, r, s, etc., of the integral fractions recur ad infinitum in a certain order. 322 RAY'S ALGEBRA, SECOND BOOK. 1 Ex. 1.— Let .+'- , 1 r ^_j_z, etc., ad infinitum. Then, .... ^ =x, or — ~ — r^=^- From this equation, the value of x is easily found. 3S6. To find in the form of a continued fraction the value of X, which satisfies the eqiiation a^=b. Ex. — Required the value of a; in the equation 10-^^2. By substituting and 1 for a:, it appears that a:>0 and <\, 1 1 Let . . x=--, then, 10'/=2, or 2^=10. y Since 23=8, and 2''=16, one of which is less and the otlier greater than 10 ; therefore, 2/>3, and <4; let 2/=3-| — ; z Then, 2H^=10; 1 \ Or, 2^.2^=10, or 2^ = lyO^l. 25; Therefore, (1.25)''=2. Again, it appears that 2;>3, and <4; let 0=3-^ — ; then, ( 1.25 )3+« =(1. 25)3(1. 2.5)« =2 .-. (1.25)«z= ^-^^1.024; Therefore, (1.024)"=:1.25, and by trial «>9 and <10. 1_ 3+: Hence, a;_ 3+9+, etc. This gives a:=J— y3g+, || -=^.30107 nearly, etc. CONTINUED FRACTIONS. 323 Reduce each of the following to a continued fraction, and find the successive integral and converging fractions : 1. 130 Ans. Integral fractions 1 '3' h hh- 421 Converging " h 1% - y. ^1?- 2. 130 Ans, Integral fractions h h hk- 291 Converging " h i, ih m- 3. 157 Ans. Integral fractions h h h i, h 972 Converging " 1 6' A 2 1 6 8 1 o 7 5 1305 43iT' 975 4. The height of Mt. Etna is 10963 feet, of Vesuvius 3900 feet; required the approximate ratio of the height of the former to that of the latter. Ans ' J -S- ifi 37 90 127 3900 -^"o- 3' 3' 14' 45' T04' 253» ~367» 10963* 5. The height of Mt. Perdu, the highest of the Pyrenees, is 11283 feet; that of Mt. Hecla is 4900 feet; required the approximate ratio of the height of the former to that of the latter. Ans. i, |, ig^ ||, -^^3, etc. 6. When the diameter of a circle is 1, the circumfer- ence is found to be greater than 3.1415926, and less than 3.1415927 ; required the series of fractions converg- ing to the ratio of the circumference to the diameter. Ans. ^, 3^, ^o«, andilf. Show that this last ratio, J if, is true to within less than three ten-millionths of the circumference. Suggestion.— In examples of this kind the integral fractions, corresponding to both fractions, should be found, and then the con- verging fractions calculated from those integral fractions that are the same in both series. 7. Express approximately the ratio of 24 hr. to 5 hr., 48 min., 49 sec, the excess of the solar yr. above 365 da. An« 1 "7 8 31 39 655 694 1349 20 92 9 Hence, after every 4 years, we must have had 1 intercalary day, as in leap year; after every 29 years, we ought to have had 7 inter- calary days; after every 33 years we ought to have had 8 inter- 324 RAY'S ALGEBRA, SECOND BOOK. calary days. This last was the correction used by the Persian astronomers, who had 7 regular leap years, and then deferred the eightli until the fifth year, instead of having it on the fourth. 8. Find the least fraction with only two figures in each term, approximating to l^H- Ans. |A. 9. The lunar month, calculated on an average of 100 years, is 27.321661 days. Find a series of common frac- tions approximating nearer and nearer to this quantity. Ans 27 82 765 3907 pfp ^ns. -J-, -3-, -2g-, -y4 3 , eic. 10. Find a series effractions converging to y2. A n « 1 3 7 17 4 1 p+p 11. Show that y5 is > |g|, and < ff|9. 12. If 8-==32, find a: ^ ^ Ans. s. 13. If 3-=15, find a; Ans. 2.465. LOGARITHMS 3S7* This method of computation was invented by Jjonl Napier, but subsequently much improved by Mr. Henri/ Briggs, whose system is now universally adopted in numerical computations. The advantage, secured in the use of logarithms, arises from the application of the law of exponents, by which multiplication, division, involution, and evolution are per- formed by addition, subtraction, multiplication, and divi- sion. cii Thus, a^X «'=«', -3=«*, {ay=a^\ fd'^=.a^. If some number, arbitrarily assumed, be taken as a hase, then The Logarithm of any mimher is the exponent of that power of the base, which is equal to that number. Thus, if a is the base of a system of logarithms, N, N'', N'''', etc., any numbers, and LOGARITHMS. 325 a2=^N, a^-^^W, a^=N^^, then, 2, 3, and X are called the logarithms of N, N'', and N^^, in the system whose base is a. The base of " Briggs' Logarithms," or the common system, is the number 10, Assuming this, we shall have (10)0=1 (10)1=10 (10)2=100 (10)3=1000 (10)4=10000 hence, is the log. of 1 ; " 1 " » log. of 10; " 2 " " log. of 100; " 3 " " log. of 1000 ; " 4 " " log. of 10000; Etc., Etc. The logarithm of every number between 1 and 10 is, evidently, some number between and 1 ; that is, a proper fraction. The logarithm of every number between 10 and 100 is some num- ber between 1 and 2: that is, 1 plus a fraction. The logarithm of every number between 100 and 1000 is 2 plus a fraction ; and so on. SSS. The integral part of a logarithm is called the in- dex or characteristic of the logarithm. Since the logarithm of 1 is 0, of 10 is 1, of 100 is 2, » of 1000 is 3, and so on ; therefore, for any number greater than unity, The Characteristic of the logarithm is one less than the number of integral figures in the given number. Thus, the logarithm of 123 is 2 plus a fraction ; the logarithm of 1234 is 3 plus a fraction, and so on. 339. The computation of logarithms, in the common system, consists in finding the values of x in the equation 10"^=N, when N is successively 1, 2, 3, etc. One method of finding an approximate value of x has been explained in Art. 356, but other methods more ex- peditious will be given hereafter. 326 RAYS ALGEBRA, SECOND BOOK. The following table contains the logarithms of numbers from 1 to 100 in the common system : N. Log. N. Log. N. Lug. N. 76 Log. 1 0.000000 26 1.414973 51 1.707570 1.880814 2 0.301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 63 1.724276 78 1.892096 4 0.602060 29 1.462398 54 1.732394 79 1.897627 5 0.698970 30 31 1.477121 55 1.740363 80 1.903090 6 0.778151 1.491362 56 1.748188 81 1.908486 7 0.845098 32 1.505150 57 1.766876 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 69 1.770852 84 1.924279 10 1.000000 1 35 36 1.544068 60 1.778151 85 1.929419 n 1.041393 1.556303 61 1.785330 86 1.934498 12 1.079181 37 1.568202 62 1.792392 87 1.939619 13 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 1 39 1.591065 64 1.806180 89 1.949390 15 1.176091 40 1.60206.) 65 1.812913 90 91 1.964243 16 1.204120 41 1.612784 66 1.819544 1.969041 17 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832609 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 20 1.301030 45 1.653213 70 1.845098 96 1.977724 21 1.322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.867333 97 1.986772 23 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.380211 49 1.690196 ' 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061 100 2.000000 In works on Trigonometry, Surveying, etc., where a set of loga- rithmic tables is given, the characteristic is usually omitted, and must be supplied by the rule given in Art. 358. 360. General Properties of Logarithms. — Let N and N' be any two numbers, x and x' their respective loga- rithms, and a the base of the system ; or, take any two numbers in the common system. Then, (Art. 357), lO^r-100000, 10^^100, (1), LOOAKITHMS. 327 Multiplying equations (1) and (2) together, we find 10' =10000000, r=a^+*'=NN^. But, by the definition of logarithms, 7 and x^x^ are the loga- rithms of 10000000 and of NN^ respectively. Hence, TliG sum of the logarithms of tico numhers is equal to the logarithm of their product. Similarly, the sum of the logarithms of three or more factors, is equal to the logarithm of their product. Hence, to multiply two or more numbers by means of logarithms. Rule. — Add together the logarithms of the numhers for the logarithm of the product. 361. Taking the same equations, (Art. 360), we have 10^^:100000, a-=N .... (1), 10-^^=100, a^=W .... (2). Dividing equation (1) by equation (2), we find 103=1000, .... a^^=f-. But, by the definition of logarithms, 3 and X—x^ are the loga- N rithms of 1000 and of ^. Hence, to divide by means of logarithms, Kule. — From the logarithm of the dividend suhfract the logarithm of the divisor for the logarithm of the quotient. 1. Find the product of 9 and 6 by means of logarithms. ♦ By the table (page 326), the log. of 9 is . . . . 0.954243 " " the log. of 6 is . . . . 0.778151 The sum of these logarithms is 1.732394 and the number corresponding in the table is 54, 2. Find the quotient of 63 divided by 9, by means of logarithms. 328 RAY'S ALGEBRA, SECOND BOOK. The log, of 63 is 1.799341 '• log. of 9 is 0.954243 The difference is 0.845098 and the number corresponding to this log. is 7. By means of logarithms 8. Find the product of 7 and 8. 4. Find the continued product of 2, 3, and 7. 5. Find the quotient of 85 divided by 17. 6. Find the quotient of 91 divided by 13. 36!S. Resuming equation (1), (Art. 360), we have 10'=:100, a"=N. Raising both sides to the 3d and to the rrV-^ power, we find lGfi=1000000, a'^=N^ But, (Art. 357), 6 and mx are the logarithms of 1000000 and of N"* respectively Hence, to raise a number to any power by nicans of logarithms, Hule. — Multiply the logarithm of the given number by the expoyient of the required power for the logarithm of the power of the number. 363. Take the same equation 10^=1000000, a'=N. Extracting the 3d and n<^ root of both sides, we have « 102:^100, a":=N". X - But, (Art. 357), 2 and — are the logarithms of 100 and of N" respectively. Hence, to extract any root of a number. Rule. — Divide the logarithm of the given member by the index of the required root for the logarithm of the root of the number. LOGARITHMS, 329 1. Find the third power of 4 by means of logarithms. The logarithm of 4 is 0.602060 Multiply by the exponent 3 3 The product is 1.806180 which is the logarithm of 64. 2. Extract the fifth root of 32 by means of logarithms. The logarithm of 32 is 1.505150 Dividing by the index 5, the quotient is ... . 0.301030 which is the logarithm of 2, the required root. ^ Solve the following examples by means of logarithms : 3. Find the square of 7. 4. Find the fourth power of 3. 5. Extract the cube root of 27. 6. Extract the sixth root of 64. Other examples may be taken from arithmetic. It is, however, the province of algebra to explain the principles of logarithms, and the methods of computing the tables, rather than their use in actual calculations. 364. By means of negative exponents, we can also ex- press the logarithms of fractions less than 1. Thus, in the common system, since (10)-^=-', =.1 , therefore, —1 is the log. of .1 (10)-2=_ij(j =.01 , " —2 " log. of .01 (10)-3=_^i^_ =.001 , " —3 " log. of .001 (^^)-'=l^hm=-^^^ " -4 " log- of -0001: Etc., ' Etc. The logarithm of any fraction between one and one-tenth, between one-tenth and one-hundredth, etc., may be expressed thus, Log. (f^)== log. (JoX7)=: log. J^+ log. 7—1+ log. 7. I-og- (t!u)= l^g- (fkX3)- log. -fio+ l«g- 3=-2+ log. 3. 2d Bk. 28 330 RAY'S ALGEBRA, SECOND BOOK. It is customary not to perform the subtraction indicated, but to unite the logarithm of the numerator to the negative characteristic. Thus, Log. 0.7 =—1+ log. 7=— 1.845098, or 1.845098. Log. 0.03 =—2+ log. 3=— 2.477121, or 2.477121. Log. 0.004=— 3+ log. 4=— 3.602060, or 3.602060. Hence, the characteristic of the logarithm of a decimal fraction is a negative number^ and is numerically equal to the distance of the first significant figure from the decimal point. 36S. On the piinciple above explained, we may deduce the following General Rule for finding from the Tables the Loga- rithms of any Decimal Fraction.—!. Find the logarithm of the figures composing the decimal as if the fraction were a whole number. 2. Prefix the negative characteristic according to the rule given in Art. 364. 360. The following examples, illustrative of the prin- ciples already explained, will afford a useful exercise : 1. Log. (a.b .c.d. . )= log. a-\- log. b-\- log. c-j- log. d. 2. Log. I -y- 1= log. a-f- log. b-\- log. c — log. d — log. e. 3. Log. (cC^ .h^ . c^ . )=m log. a-f 71 log. b-\-2^ log. c. — ^ — I z=m log. a-]-n log. b — p log. c 5. Log. (a^ — x'^)= log. [(a-}-^)(« — ^)]= log- («-f^) + log. (a—x). 6. Log. |/a''' — x^=}i log. (a-{-x)-\-^ log. (a — a-). LOGARITHMS. 331 7. Log. (a^Xif^^Sf log. a. 8- Log. ^^^^Wllog. (a-a.)-3 log. (a-fc.) j. 367. Let us resume the equation a^=N. 1st. If we make x=l, we have a^=:N=a ; hence, log. .=1 ; that is, TTAa^ever he the base of the system, its logarithm in that system is 1. 2d. If we make x=0, in the equation a*==N, we have a^=l^—-.l ; hence, log. 1=0 ; that is, In any system the logarithm of 1 is 0. 368. In the equation a=^=N, consider a>l, as in the common and the Naperian systems, and x negative ; we then have a-^=i=N, and ^=a^ =0, or log. 0=— oo . I* a Hence, the logarithm of 0, in a system whose hase is greater than 1, is an infinite number and negative. In a similar manner, it may be shown that in a system whose base is less than 1, the logarithm of is infinite and positive. 369. As the positive and negative characteristics are taken to designate whole numbers and fractions, there re- mains no method of designating negative quantities by logarithms ; or, as N, in each of the equations a*=N and a~*=N, is positive, Negative numbers have no real logarithms. 332 COMPUTATION OF LOGARITHMS. 370. Before proceeding to explain the methods of computing logarithms, we may observe that it is only neces- sary to compute the logarithms of the prime- numbers. For, the logarithm of every composite number is equal to the sum of the logarithms of its factors. Hence, the logarithms of 1, 2, 8, 6. 7, etc., being known, we can find those of 4, 6, 8, etc. Thus, 4=22 hence log. 4=2 log. 2, (Art. 362); 6=2X3 (( log. 6= log. 2-j- log. 3; 8=23 . (( log. 8=3 log. 2; 9=32 (( log. 9=2 log. 3; 10=2X5 (( log. 10= log. 2+ log. 5. 1. Suppose the logarithms of the numbers 2, 3, 5, and 7 to be known ; show how the logarithms of the compo- site numbers from 12 to 30 may be found. 2. Of what numbers between 30 and 100, may the loga- rithms be found from those of 2, 3, 5, and Y; and why? Ans. Of 23 diflferent numbers, from 32 to 98. 371* In the common system, the equation a^=N (Art. 357) becomes 10"=N. If we multiply both sides by 10, we have 10'XlO=10^+^=10N; Also, . . . 10^X100=10^X10^=10'^^=100N. Hence, in the common system, the logarithm of any number will become the logarithm of 10 times, 100 times, etc., that number, by increasing the characteristic by 1, 2 etc. From this results the advantage of Briggi system. Thus, the log. of 3 is 0.477121, « « 30 " 1.477121, « " 300 « 2.477121. LOGARITHMS. 333 Also, the log. of .2583 is —1.412124, 2.583 " 0.412124, « 25.83 " 1.412124. 37S* If we compare the different powers of 10 with their logarithms in the common system, we have Numbers 1 , 10 , 100, 1000, 10000, Logarithms 0,1, 2,3, 4 , and so on. Hence, while the Jiumbers are in geometrical progression^ their logarithms are in arithmetical progression. Therefore, if we take a geometrical mean between two numbers, and an arithmetical mean between their loga- rithms, the latter number will be the logarithm of the former. Thus, tlft geometrical mean between 10 and 1000 is |/I0xl000 =100, and the arithmetical mean between their logarithms, 1 and 3, is (l-|-3)--2=2. In general, if a*=N, and a^':z=W] then, Log. of y'Niris^i^. By means of this principle, the common, or Briggean, system of logarithms was originally calculated. Ex. — Let it be required to calculate the logarithm of 5. First. — The proposed number lies between 1 and 10; hence, its logarithm will lie between and 1. The geometrical mean is i/(lXl0)=3.162277; the arithmetical mean is (0-f l)-=-2=0.5. Hence, the log. of 3.162277 is 0.5. Secondly. — Take the numbers 3.162277 and 10, and their loga- rithms .5 and 1, we find The log of 5.623413 is 0.75. Thirdly.— T&kQ the numbers 3.162277 and 5.623413, and their logarithms 0.5 and 0.75, we find The log. of 4.216964 is 0.625. ^34 RAY'S ALGEBRA, SECOND BOOK. Fourthly.— Take the numbers 4.216964 and 5.623413, and their logarithms 0.625 and 0.75, we find The log. of 4.869674 is 0.6875. By continuing this process, always taking the two numbers near- est to 5, one of which is less and the other greater, after twenty-two operations, we obtain the number 5.000000-(-, and its corresponding logarithm 0.698970+. Having the log. of 5 we readily find that of 2, or '^ (Art. 361). To find the log. of 3, take the numbers 2 and 3.162277, and their logarithms, and proceed as in finding the log. of 5. 37S. Logarithmic Series.— The most convenient method of computing logarithms is by means of Series, which we shall now proceed to explain. Let ic be a number whose logarithm is to be expressed in a series, and let us apply the method of Indoierminato Coefficients (Art. 314). If we assume log. x—A-\-Bx^Cx'^-{-Dx^-j-, etc., and make x=0, we have, log. 0=A=qo (Art. 368). Hence, 00 =:A, which is absurd. If we assume log. Xz=A.x-{-Bx^~\-Cx^-\-, etc., and make a;=rO, we have log. 0=:^0; that is, (Art. 368), qo =^0, which is also absurd. Hence, it is impossible to dcvelope the logarithm of a number in powers of that number. But if we assume Log. {l-\-x)=Ax^Bx-~\-Cx^'-{-T)x*-}-, etc. . . (1) and make X:^0, we have log. 1=:0, which is correct (Art. 367). In like manner, also assume Log. {l-\-z)=Az-\-Bz^-{-Cz^--{-'Dz*-}-, etc. ... (2) Subtracting equation (2) from (1) we get Log. (l-frc)— log. {l-\-z)=A(x—z)-^B{x^—z^) -f C(a;^— 2;3)+, etc. . . . (3). LOGARITHMS. 335 The second member of this equation is divisible by x — z Art. 83); let us reduce the first member to a form in which it shall also be divisible by the same factor. By Art. 361, Log. (1-f :^)- log. (1+^)= log. ( ^ ) - log. ( 1+^ ). as a sinsrle aua +2; X — z Now, regarding as a single quantity, we may assume 1 -\-Z / ^ X—Z \ . X—Z , ^/ X—Z \2 , ^ / X—Z \3 Substituting this for log. (1-|-^) — log- (l+2^)j in equation (3), and dividing both sides by x — z, we obtain =zA-{-B{x-\-z)-{-C(x2-\-xz-]-z^)-^, etc. Since this equation is true for all values of X and z, it must be true when x=zZ. Making this supposition, we have A.-— - =A+2Ba:4-3Crr2-|-4Da:3-f5Ea;4+, etc.; or, performing the division of 1 by l+^j we have A(l— a:+x2_ic3-frt— . . . . ).z-A+2Ba;-f 3Ca:2+4Da:^+. Equating the coefficients of the like powers of x (Art. 314), A=A, B=_^, c4 D=-^ The law of this series is obvious, the coefficient of the nth term being ±— , according as n is odd or eve7i. A. A A Hence, log. (l-fa:)— Aa;— ^-a:2-f-a:3— -x^-f .... a;2 ic^ x\ X' x^' , . ^.^ =A(=^-2+T-4 + 5-6-+ • • • ) W There still remains one quantity, A, undetermined. This is as it should be, for the logarithm of a given number is indeterminate unless the base of the system be given. 336 RAY'S ALGEBRA, SECOND BOOK. The value of A depends on the base of the system, so that when A is given, the base may be determined ; or, when the base is known, A may be determined. If we denote the series in the parenthesis in equation (4) by rc^, we may write Log, {l-{-x)=Ax^. Hence, The logarithm of a number consists of two factors^ one of rchich depmids on the number itself^ and the other on the base of the system in which the logarithm is taken. That factor which dej>ends on the base is called the Modulus of the system of logarithms. Lord Napier, the inventor of logarithms, assumed the modulus equal to unity, and the system resulting from such a modulus, is called the Noperian^ or Hyperbolic system. For all values of x above x=\. the series (5) diverges, and is, there fore, inapplicable. Designating the logarithms in this system by log^., we have Log^ (l+^)=i-f + ^-T+' '''' (^^ Thus, if a:^0, we find log^. 1=^0, as in Art. 367. If we make x=l^ we have Log^ 2^1-1+1-1 + 1-, etc. 374. The preceding series converges so slowly that it would be necessary to take a great number of terms to obtain a near approximation. But we may obtain a more converging series in the following manner : Resuming equation (5), Log'. (1+^)=^-^+^-^ + ^-, etc. . . . (5). Substituting — x for a:, in this equation, we obtain Log'. (l_^)=_f-f -f-^_f -,cto. . (6). LOGARITHMS. S37 Subtracting equation (6) from (5), and observing that Log^ (l+iC) — log^. {l—x)=z log^l ■—- \, we have , 1+x I X x^ x'^ x^ x\ \ lA-x ^ , 2x , 14-x ^ , 1 1 Since _n^l4---, let -1^=1+-, .-. x. l^x ' 1—x' 1—x ' 0' ' ■ 20+1' and log^. ^^ log^ {^^\)= ^<- ( ^ ) = log^. {Z^l)— log^ 2;. By substitution, the preceding series becomes Log-. (.+l)-log',.=2{2^ + 3^3 + g^,+ .. }, Log'. (.+1)= log'. z^2{^^ + p^,3 + 5(2iW+ • • ^'^• 375. By means of this series, the Naperian logarithm of any number may be computed, when the logarithm of the preceding number is known. But the log', of 1 is 0, (Art. 36Y) ; therefore, making z=l, 2, 4, 6, etc., we ob- tain the following NAPERIAN, OR HYPERBOLIC LOGARITHMS. Log^ 2=.log^. 1+2 {1 + ^, + ^, + ^'^^+ . . 1^0.693147 Log^ 3=.log^ 2+2(^ + 3^^+ J^, + J^,-|- . .}=1.098612 Log^ 4=2. log. 2 =1.386294 Log^ 5=log^ 4+2{^ + 3ip + ^. + J^,+ . .}=1.609438 Log^. 6=log^ 2+ log^. 3 =1.791759 Log^7=log^. 6+2 {^^ + 3-433+5^13^+ . . }=1.945910 2d Bk. 29* 338 RAYS ALGEBRA, SECOND BOOK. Log^. 8.^3 log^. 2, or log^ 2+ log^ 4 . . . . =.-2.079442 log^. 9=2 log'. 3 =2.197225 Log'. 10^ log^ 2+ log^. 5 =2.302585 In this manner the Naperian logarithms of all numbers may be computed. When the numbers are large, their logarithms are computed more easily than in the case of small numbers. Thus, in calculating the logarithm of 101, the first term of the series gives the result true to seven places of decimals. 3T6. To explain the method of computing common loga- rithms from Naperian logarithms. We have already found (Art. 373, Equation 4), ., , , , I X X^ X^ X^ X-' x^ \ Denoting the Naperian logarithm by an accent, -yve have X x^ x^ X* x'" x^ + 5-T+- ■ • Log^. (l+a:)=A^^j-- + -- Since the series in the second members are the same, we have Log. (1+a:) : log^ (1+^^) : : A : A'. Therefore, Tlie logarithms of the same number, in two different si/s- ie77is, are to each other as the moduli of those systems. But in Napier's system the modulus A'--l. Therefore, Log. (l-l-:r)=A log'. (1+a;). Hence, To find the common logarithm of any numher, multiply the Naperian logarithm of the number by the modulus of the common system. It now remains to find the modulus of the common system. From the equation, log. (l-\-x)=A. log', (l+ic). Wo find A=!2?i41+£) „e„„, log'- (1 f «) LOGARITHMS. 339 The modulus of the common system is equal to the common log. of any number divided hy the Naperian log. of the same Humher. But the common logarithm of 10 is 1, and we have calculated the Naperian logarithm of 10, (Art. 375); therefore, log. 1 _ 1 _ 4342944 ^"l.) 5D 5X.00071489 nnnn...-. ,^x TiWTT? =^XS^ =.000056.4; (E.) 7E _ 7X.00005674 -00000400. (F ^ pF+Tp 9xP -.00000490, (F.) 9F 9X.00000490 fv^nf^nf,iK /n ^ iT(2P+I7 =^11X3^ =.00000045; (G.) IIG 11X-0QQ00045 00000004. rn^ Therefore, common logarithm of 2 . . =^.30102999. Exercise. — In a similar manner let the pupil calculate the common logarithms of 3, 5, V, and 11. For the results to 6 places of decimals, see the Table, page 326. 3TS* To Jind the base of the Naperian system of loga- rithms. If we designate the base by e, we have, (Art. 376), Log. e : log', e : : A : A'. But A=r=.4342944, A'=l, and log'. c=\, (Art. 367); hence, Log. c : 1 : : .4342944 : 1 ; whence, log. ^^=.4342944. Taking the number of which the logarithm is .4342944, from the table of common logarithms, we find e=2.71828182. TVe thus see that in both the common and the Naperian systems of logarithms, the base is greater than unity. Napier's logarithms are used in the Calculus, but not in the common operations of multiplication, division, etc. POSITION. 341 3TO. The student may prove tlie following theorems : 1. No system of logarithms can have a negative base, or have unity for its base. 2. The logarithms of the same numbers in two different systems have the same ratio to each other. 3. The difference of the logarithms of two consecutive numbers is less as the numbers themselves are greater. SINGLE AND DOUBLE POSITION. Note. — This subject is introduced in connection with that of logarithms, because the rule for Double Position is applied to the solution of exponential equations. 380. Single Position.— The Rule of Single Position is applied to the solution of questions which give rise to an equation of the form ax=m (1). If we assume x/ to be the value of x, and denote by m' the result of the substitution of x' for x, we have ax'=m' (2). Comparing equations (1) and (2), we have m' : m : : ax' : ax : x' : x ; that is, As the result of the supposition is to the result in the ques- tion, so is the supposed number to the number required. Example. — What the number, whose third, fourth, and sixth part being added, the sum will be 45 ? Ans. 60. 381. Double Position. — In Double Position, the result, although it is dependent on the unknown quantity, does not increase or diminish in the same ratio with it. 342 RAY'S ALGEBRA, SECOND BOOK. The class of questions to which it is particularly appli- cable, gives rise to an equation of the form ax-\-h=m (1). If we suppose x' and x" to be near values of X, and e' and e" to be the errors, or the differences between the true result and the re- sults obtained by substituting x' and a;'' for x, we have ax' -^-b^m^e' (2), ax"-\-b=7n-\-e" (3). If we subtract equation (1) from (2), and (3) from (2), we have a{x'—x )=e' (4), a{x'—x")=e'—e'' (5). From these equations, we easily obtain (6). e'—e" e' By subtracting equation (1) from (3), we also find a{x"—x)z=e'\ and thence, x'—x" x"-~x Hence, (Art. 263), Tlie difference of the errors is to the difference of the two assumed numbers, as the error of either result is to the difference between the true result and the cor- responding assumed number. When the question gives rise to an equation of the form ax-\-b^=z7n, this rule gives a result absolutely correct; but when the equation is of a less simple form, as in exponential equations (Art. 383), the result obtained is only approximately true. Corollary. —The common arithmetical rule is deduced from the following value of x, found either from equa- tion (6) or (7) : _e'x"—e"x' EXPONENTIAL EQUATIONS. 343 EXPONENTIAL EQUATIONS. 382. An Exponential Equation is an equation in which the unknown quantity appears in the form of an exponent or index ; as, a-^=6, x*=:a, a^^=^c^ etc. Such equations are most easily solved by means of loga- rithms. Thus, in the equation . . . a^=6, We have (Art. 362), . . x log. a= log. 6; ^ log. b Or, 2;=-^ — . log. a Ex. 1. — What is the value of x in the equation 2'=64? Here, x log. 2= log. 64 ; log. 64 1.806180 ^ ^ Whence, . . . x^^^ = -^^^^=Q, Ans. 383. If the equation is of the form a;^=a, the value of X may be found by Double Position, as follows : Find by trial two numbers nearly equal to the value of X ; substitute them for x in the given equation, and note the results. Then, from (7), we have (Art. 263) the proportion, As the difference of the errors, is'^ the difference of the two assumed numbers, so is the error of either result, to the correc- tion to he apjylied to the corresponding assumed number. Ex. 1.— Given x'^^lOO, to find the value of a;. The value of x is evidently between 3 and 4, since 3^=27, and 4'=256; hence, taking the logarithms of both sides, X log. x= log. 100:=2. 344 RAY'S ALGEBRA, SECOND BOOK. By trial, we readily find that x is greater than 3.5, and less than 3.6: then, let us assume 3.5 and 3.6 for the two numbers. First Supposition. x=3.3- log. a:=.544068 multiply by 3.5, we find X log. X =1.904238 true no. =2.000000 =—.095762 Second Supposition. x=S.6; log. a;=.556303 multiply by 3.6, we find X. log. X =2.002690 true no. =2.000000 error -f .002690 DiflF. results : DifF. assumed nos. : : Error 2d result : Its cor. .098452 : 0.1 : : .002690 : .00273 Hence, 3.6 -.00273=3.59727 nearly. By trial we find that 3.5972 is less, and 3.5973 greater than the true value; and by repeating the operation with these numbers, we would find a;=3.5972849 nearly. 2. Given 20"=:100, to find x. Ans. a:r:=1.53724. 3. Given x^=b, to find x. Ans. a:=2.129372. 4. How many places of figures will there be in the num- ber expressing the 64''' power of 2 ? Ans. 20. 5. Given a^-'+'^=c, to 6nd x. Ans. ^_ ^g- ^~ . • ^g- ^ 6. log. a 6. Given a'^^Z/"'=c, to find x. Ans. x= — "j r. m. log. a-j- *i- log. o 7. Given x-\-y=a, and in^y=n, to find x and y. Ans. x=^(a-\- log. ?ih- log. m), y=^l{)2fx-i)^(^_j>)2x^ to find x. Ans. .^1+l^M. ^log. (a-f 6) 14. Given (a*— 2a262-f£*)'"'=(«— ^)'^(«+^)~'5 *« ^^"^ ^^ Ans ^^ ^Qg- (^-^) log. (a4-6)* 15. Given xy^=y^^ and x^=^yi^ to find a? and y. Ans...= (|jA, j=(|| P-9. IG. Given 3^^' ^"^+^^=1200, to find x. Ans. a::r-4.33, or —0.33. INTEREST AND ANNUITIES. 3^4. The solution of all questions in Interest and Annuities may be simplified, and also generalized, by means of algebraical formulae ; but certain problems in Compound Interest and Annuities may be very much abridged by the use of logarithms. 346 RAYS ALGEBRA, SECOND BOOK. Let P= the principal, or sum at interest in dollars. r= the interest of 1 ^ for one year. t= the time in years that P draws interest. A= the amount, at the end of t years. 385. Simple Interest. — Since tr represents the inter- st of 1 $ for t years, and P^r, the interest of P $ for I years ; therefore, A==P-j-P/r=P(l + ^r) (1). From this equation, any three of the quantities P, r, <, A, being given, the fourth may be found. Thus, _A_ ._A— P _A— P Examples may be given from any treatise of arithmetic. 386. Compound Interest. — Let R=l-f ?-, the amount of 1 $ for one year ; then, R will be the principal for the second year ; and since the amount, in each case, is pro- portional to the principal for the same time ; therefore, 1 : E : : R : the amount of 1 J in 2 years:=R2. 1 : R : : R^ : R^, the amount of 1 f in 3 years. And, in like manner, R' is the amount of 1 $ in ^ years. The amount of Pf will be P times the amount of 1 $. Hence, A=P.R'=P(l-f-r)'; whence, Log. A= log. P+^. log. (1-f r) (1). Log. P= log. A-^. log. (1+r) (2). log. A— log . P o '- log.(l+0~ ^^^' _. ,-, , X log. A — log P ..^ Log. (l+r)=-^^ ^ ^1- (4). Corollary L— The interest =A— P=PR'— P^P(R'- 1). INTEREST AND ANNUITIES. 347 Corollary 2. — If t^e interest is paid half-yearly^ tz=2t. and '■=^. Hence, A=p/ 1+^ Y{h). li^2i\di quarterly, A=:Vi 1-j-^ ^(6). Corollary 3.— From the equation A=rP.R<, we can readily find the time in which any sum, at compound in- terest, will amount to hvice^ thrice^ or m times itself. Thus, if A.=2P; then, 2P=PR< .-. R<=2, ^tc, is divisible by X — a, the given equation is divisible by x — a. GENERAL THEORY OF EQUATIONS. 353 Corollary. — Conversely, if the equation ^u_|_^^H-i_|_B^n-2_j_^ . . . -^T:r-f V=:=0, (n) is divisible by X — a, then a is a root of the equation. For if the equation (n) is divisible by X — a, if we call the quo- tient Q, we have (x — a)Q=0 (n), v/hich may be satisfied by mak- ing X — a=0, whence x=:a. D'Alembert's Proof of Prop. I, — If said division leave a re- mainder, call it R, and the quotient Q; then, equation [n) becomes (a:— a)Q4-R=0. But X — Cf=0, .-. R=zO; that is, there is no remainder on dividing equation (n) by x — a. Illustration. — In the equation x^-j-x"^ — 14x — 24=:0, the roots are — 2, — 3, and 4 ; and the equation is divis- ible by x-\-2, x-\-S, and x — 4. 396. Proposition II.-— ^71 equation of the n'^' degree has n roofs. Let a be a root of the equation cc'^-f Aa;'»-i4-Ba:"-2-f-Ca:"-'5+. . . . ^-Tic-f V=0 (n). By Art. 395 this equation is divisible by X — a. If we perform the division, and denote by Aj, Bj, etc., the coefficients of the powers of X in the quotient, equation (n) becomes (a:— (2)(a;"-i+Aia:«-24-Bia;"-34-, . . . -|_Tia;+Vi)=^0. Hence, x"-^-\-A^x''~^-^B^x''-^^ ^Tia:-f Vi=0. Now, this equation must also have a root, which may be denoted by 6, and is (Art. 395) divisible by x — b. Hence, (a:— 6}(u;""2^A2a;*»-3+B2a;"-4+. . . . +t2^+V2)=0. Placing the second factor of this equation equal to zero, taking c, a third root, and dividing by X — C, we shall have an equation of a degree still lower by a unit. It is evident that if this operation be continued, the exponent n will be exhausted, and the last quotient will be unity; hence, call- ing the last root I, we shall have 2d Bk. 30 354 RAY'S ALGEBRA, SECOND BOOK. {x—a){x—b){x—c){x—d), . . . {x—l)=0, which is satisfied by making a:=a, b, c, d, . . . . or Z; that is, the equation has n roots, a, 6, c, d, etc. Corollary I. — If we know one root of an equation, by dividing (Art. 395) we may find the equation containing the remaining roots. Thus, one root of the equation a:^— 12a;24-47a:— 60=0, is 5, and by dividing it by x—6, the quotient is a:^— 7a;-f 12=0, the roots of which may be found, viz., -)-3 and -|-4. Corollary II. — When any equation, whose right hand member is zero, can be separated into factors, the roots of the equation may be found by placing each of the fac- tors equal to zero. Thus, if a:2-f4a:=0, we have a:(a:-|-4)=0, whence a:=0, and x=—4. (See Art. 253.) 1. One root of the equation x^ — llrr'^-l-^Sic-j-SS^^O is — 1 ; find the equation containing the remaining roots. Ans. 3^2— 12.T-f 35:^.0. 2. One root of the equation x^ — 9x'^-\-26x — 24=::0 is 3 ; find the remaining roots. Ans. 2 and 4. 3. Two roots of the equation x'-j-2x^—41x''—42x-\-SQ0 =-0, are 3 and — 4 ; required the remaining roots. Ans. 5 and — 6. Remark. — Two or more of the n roots may be equal to each other. Thus, the equation x^ — 6x^-\-l2x — 8=0, is the same as (a;_2)(a:— 2)(a;— 2)=0, or {x — 2)3=0. Hence, the three roots are x=2, x=2, and<^/ — 1, or ±]/^, enter an equation by pairs. Corollary II. — Since irrational and imaginary roots al- ways occur in pairs where the coefficients are real, it fol- GENERAL THEORY OF EQUATIONS. 359 lows that every equation of an odd degree must have at least one real root. Corollary III. — Corresponding to any pair of imaginary roots a±^^ — 1^ we have in the eq. the quadratic factor, [x—^a^hy'—l) } {x— (a— 6i/^=a) ]=(x—ay f 6^ ; Hence, every eq. of an even order, with real coefficients, is composed of real factors of the second degree. 1. One root of the equation x^ — 26;t-|-60=^0 is — 6; required the other roots. Ans. 3±|/ — 1. 2. One root of a:'— Yo^^-flSx— 3=0, is 2— 1/3 ; find the other roots. Ans. 2-[-i/3 and 3. 3. One root of o;^— 3x^—42.-^—40= is — ^(S-f-y^^M) ; find the other roots. Ans. — A(3— |/— 31), 4, and — 1. 4. Two roots of a^— 10a:*H- 29^3— 10a:2—62x-f 60=0 are 3 and ^2 ; find the other roots. A. — 1/2, 2, and 5. 402. Proposition VIII— Descartes' Rule of the Signs. — No equation can have a greater number of POSITIVE roots than there are VARIATIONS of sign; nor a greater num- ber 0/ NEGATIVE roots than there are PERMANENCES of sign. In the equation x — a:^0, where the value of x is -fa, there is one variation, and one positive root. In the equation x-{-a=^^, where the value of x is — a, there is one permanence, and one negative root. In x^ — (a-\-b~)x-\-ah=.{), where the values of x are -{-a and -\-b, there are two variations and two positive roots. In x'^-]r(a-\-b)x-^ab=z{), where the values of x are —a, and — 6, there are two permanences, and two negative roots. In x^ — X — 12=0, where .^=-1-4, and — 3, there is one variation, and one positive root, one permanence^ and one negative root. 360 ' RAY'S ALGEBRA, SECOND BOOK. If we form an equation of the third degree, (Art. 397), whose roots are -f 2, -f 3, -|-4, we shall have x^—dx^ -\-26x — 24=0, where there are three variations, and three positive roots. But if we form an equation whose roots are — 2, — 3, -f-4, we shall have a?-\-x^ — l^x — 24=0, where there is one variation, and one positive root, and two permanences, and two negative roots. To prove the proposition generally, let the signs of the terms in their order, in any complete equation be -j--j- — + — + + -|-5 ^^<^1 Ist a new factor X — a=0, corresponding to a new positive root be introduced, the signs in the partial and final products will be + +- + - + + + + - + + - + - + + + + d=- + - +±=i= Now, in this product, it is obvious, that each permanence is changed into an ambiguity ; hence, the permanences, take the ambiguous sign as you will, are not increased in the final product; but the number of signs is increased by one, and therefore the number of variations miftt be increased by one. Hence, the introduction of any positive root introduces at least one additional variation of sign. Let us now begin with the equation x — a=^0, which contains one positive root, and has one variation of sign. Then, since every additional positive root introduces at least one additional variation of sign, the number of positive roots can never exceed the number of vari- ations of sign. Again, if we change the signs of the alternate terms, the roots will be changed from positive to negative, and, conversely, (Art. 400), the permanences and variations, in the proposed equation, will be interchanged. But since the changed equation can not have a greater number of positive roots than there are variations of sign, the proposed equation can not have a greater number of negative roots than there are permanences of sign. GENERAL THEORY OF EQUATIONS. 361 Corollary I. — In an equation of the m'^* degree, since the sum of* the variations and permanences is equal to m, the number of real roots in any equation can not be greater than its degree. Corollary II. — If the number of real roots be less than the degree of the equation, the remaining roots are im- aginar?/. Take, for example, the equation x'-{-16=0, or a)2±0:c+16=i0. Taking the upper sign, there are no variations ; hence, there is no positive root : taking the lower sign, there are no permanences ; hence, there is no negative root. But the equation has two roots (Art. 396); they must, therefore, both be imaginary. Take, again, the cubic equation a:3_|_Bx+C=0, or x'dtzOx'-\-Bx-^C=0. Reasoning as before, we find that there can be but one real root, which is negative. Therefore, the other two roots must be im- aginary. 403. Proposition IX. — If two nnmbers. when suhsiituted for the unknown quantity in an equation^ give results affected with different signs, one root, at least, of this equation lies between these numbers. Let the equation, for example, be x^ — x^-{-x — 8=i0. If we substitute 2 for x in this equation, the result is — 2; and if we substitute 3 for x, the result is -|-13. It is required to show that there must be one real root, at least, between 2 and 3. The equation may evidently be written thus, (^x3-\-x)—{x^-]-8)=:0. Now, in substituting 2 for ic, x^-^x=10, and x^-\-8=12; Therefore, X^-\-X<:^X^-\-8; Also, in substituting 3 for X, x^-\ x—SO, and 2:24-8— 17; Therefore, x^-\-xyx^-^8. 2d Bk. 31* 362 RAY'S ALGEBRA, SECOND BOOK. Now, both members of the inequality increase while x increases, but the first increases more rapidly than the second, since when a:=:2, it is le8& than the second, but when a::=3, it is greater. Con- sequently, for some value of x between 2 and 3, we must have x'^-\-x^zz.x'^-\-% and this value of x is, therefore, a real root. In general, suppose X=zO to be a polynomial equation involv- ing ic, and that 'p and Q', when substituted for a:, give results with contrary signs. Let P be the sum of the positive, and N the sum of the negative terms. When a;=p, let P — N be negative, or P<^N; and when x=g, let P — N be positive, or P>N. Now, there must be some value of x between p and g, which renders P=zN, or satisfies the equation X=0. This value of X is, therefore, a real root of the equation. Corollary. — If the difference between p and q is equal to iinity^ it is evident that we have found the intngral part of one of the roots. 1. Find the integral part of one value of x in the equation X*— 42^3 4-3a;2-|-x— 5=0. If x=7:3, the value of the expression is — 2; but if a;=r:4, the value is 47. Hence, 8 is the first figure of one root. 2. Required the first figure of one of the roots of the equation x^ — hx^ — x-|-l=0. Ans. 5. TRANSFORMATION OF EQUATIONS. 404. The Transformation of an Equation is the changing of it into another of the same degree, whose roots shall have a specified relation to the roots of the given equation. Thus, in a«-j-Aa;«-^+Bx"-2. ^ ^ ^ _^Tx+V=0 ; (1) if — y be substituted for x^ the equation will be trans- formed into another whose roots are the same as those in (1), but with contrary signs, for ?/= — x. If - be substituted for X\ then, y=-, and the roots of the new equation in y will be the reciprocals, of those of equation (1). TRANSFORMATION OF EQUATIONS. 363 4:OS. Proposition I. — To transform an equation into one whose roots are the roots of the given equation multiplied or divided hy any given quantity. Let a, Z>, c, etc., be the roots of the equation a;«+Aa:"-i+Bx«-2. . . . -fTa:+yz=0. (1). Assume yz=hx^ or a:=j. Substituting this value for a:, in (1), yn yu-i yn-2 ^y |+A|^+Bf^ +|+^=«' Hence, 2/'*+AA-2/^i+B%"-2. . . . 4-TA-"-i|/-f-/i:"Vz^0. Since y^zlcx^ the roots of this equation are ka^ "kb, 7tC, etc. It is evident that this equation may be derived from (Ij; or that the transfox-mation of (1) is effected, by multiplying the successive terms by 1, A", A"-, Z'^, etc., and changing X into y. In the case of division, assume 2/— y, ov x^^Tcy^ aud substitute. Corollary. — By this transformation an equation may be cleared of fractions, or the cotifl&cient of the first term may be made unity. 1. Let it be required to transform the equation into one which is clear of fractions, and which has unity for the coefficient of the term containing the highest power of X. Multiplying by 6, Qx^-\-^px'^+2qx-{-Qr=0. Putting y^Qx, or x=yy, 6^+3p^V2g^-f 6r=0; Multiplying by 62, ?/3-f 3j92/2+12g?/+216r=.0. 2. Find the equation whose roots are each 3 times those of the equation a;*-]- 7x^—40^+ 3=0. Ans. y+63/— 108y+243==0., 364 RAY'S ALGEBRA, SECOND BOOK. 8. Find the equation whose roots are each 5 times those of the equation x*-\-2x^ — ^x — 1=0. Ans. y-flO/_8753/— 625r=0. 4. What is the equation whose roots are each 4 of those of re'— 3x2+4x+10=0 ? Ans. 4/— 6^^+4^+5=0. 5. Transform eq. x^ — 2a;^+Ja; — 10=0, into one having integral coefficients. Ans. y — %^+3^ — 270=0. 4:06. Proposition II. — To transform an equation into one whose roots are greater or less hy any given quantity than, the corresponding roots of the proposed equation. Let £c"+Ax"-^+Ba;"-2 _|_Ta;+V=0, be an equa- tion whose roots are a, 6, c, etc. The relation between X and y will be expressed by the equation yz=xdtir. As the principle is the same in both cases, let y^^X — r, or x=:zy-\-r. Substituting y-\-r for x, we have (2/+r)»+A(2/+r)«-i+B(2/+r)«-2 _|_T(^+r)+V=.0. Developing the different powers of y-\^r by the Binomial Theorem, and arranging the terms, we have y'^j^nr + A n-l I ^(^-1) y.2 1-2 +(n— l)Ar y. +7^ +Ar«-i +Br'^2 -fTr Now, since y=x — r, the values of y in this equation are a — r, b — r, c — r, etc. 40T. Corollary. — By means of the preceding transfor- mation, we may remove any intermediate term of an equar tion. Thus, to transform an equation into one which shall want the second term, r must be assumed so that nr+ A=0. TRANSFORMATION OF EQUATIONS. 365 To take away the third term, put ^n^n — l)r^-\-(n — l)Ar-|- 1. Transform the equation a:^ — 'Jx-\-^=^0 into another whose roots shall be less by one than the corresponding roots of this equation. Ans. y-j-3y — 4i/-\-lz=^0. 2. Find the equation whose roots -are less by 3 than those of the equation x*~-dx^— lbx^-\- 4^9 x— 12=^0. Ans. y-f-9/-f 12y— 14y=0. 3. Transform eq. x^ — 6x'^-\-Sx — 2=0 into another whose second term shall be absent. Ans. y — 4y — 2=0. 408. There is an easier and more elegant method of transformation, which we will now proceed to explain. Let the proposed equation be Aa;*-fBa;3-fCa;2-[-Da;+E=0, (1) and let it be required to transform it into another, whose roots shall be less by r; then, i/=x — r and x=i/-\-r. By substituting y-\-r, instead of x, we have A(2/+^)'+B(2/+r)-^+C(2/-i-r)2+D(2/-f?*)+E-0. By developing the powers of 2/+^? and arranging, as in Art. 406, the transformed equation will take the form A2/<+Bi?/-'5+C,2/2+I>i2/+Ei=0, (2) where A is evidently the same as in (1), while Bj, Ci, Di, and Ei, are unknown quantities to be determined. For y, substitute its value x — r, and equation (2) becomes A(a:— r)4-[-Bi(a:— r)3-fCi(rc— r)2+Di(a:— r)+Ei=:0. (3) Now, since the values of X are the same in (1) and (3), these equations are identical. Hence, any operation may be performed on (1) or (3) with the same result. Now, as the object is to obtain the values of Bj, Cj, etc., let (3) or (1) be divided by X—r, and the quotient will be A(a;— r)3+Bi (x—r)^-\-C^ (a:— r)+Di, with the remainder E^ ; hence, E^ is determined. 366 Divide this quotient by X — r, and the next quotient will be A(:r_r)2+Bi(a:-r)+Ci, with a remainder D^; hence, Di is determined. Continuing the division by X — r, we obtain Cj and Bj, and thus find all the coefficients of equation (2). To illustrate, let us now solve Ex. 1, Art. 407, by this method. Transform the equation ic' — 7cc-|-7=0 into another, whose roots shall be less by 1 than the corresponding roots of this equation. Here, 2/=^ — 1, and we proceed to divide the proposed equation and the successive quotients by X — 1. The successive remainders will be the coefficients of 2/ in the transformed equation, except that of the highest power, which will have the same coefficient us the highest power of x in the proposed equation. x—\ )x^—ix^ 7 (a;2+a:— 6 X^—X^ Ist quot. +a:2— 7a; X^— X rc— l)a;2-fa:— 6(a:-f-2 X^—X 2d quot. -f2a;— 6 2a:-2 — 6a:+7- — 6a:+6 iBtrem. =+1 2d rem. = — 4 a:— l)a:-f2(l, ndquot. x-\ 3d rem. r-^-j-o Since the successive remainders are -f-3, — 4, and -fl, we have A=l, Bi3=-}-3, Ci= — 4, and D]=-)-l. Hence, the transformed equation is y'^^Zy^—\y-\-\=:S). This method of transforming an equation may be greatly shortened by Horner's Synthetic Method of Division, which we shall now proceed to explain. 409. Synthetic Division.— This may be considered as an abridgment of the method of division by Detached Coefficients (Art. 77). To explain the process, we shall first divide 5x* — \2x^-\-Zx^-\-4iX — 5 by x — 2, by detached coefficients. TRANSFORMATION OF EQUATIONS. 367 By changing the sign of Divisor. Quotient, the second term of the divi- 1—2)5—12+3+4—5(5—2—1+2, 8or, and adding each partial 5—10 or bx^—2x'^—X+2. product, except the first term, —2+3 which, being always the same — 2+4 as the first term of each divi- -^ i ^ dend, may be omitted^ the op- l_|-2 eration may be represented ■ '~t~ as in the margin below: ^^ . Let it be observed that the 1+2)5—12+3+4—5(5—2—1+2 figures over the stars are the •*+10 coefficients of the several 9 , q terms of the quotient; also, .:;. * that it is unnecessary to bring down the several terms of the dividend. Hence, the last operation ^^ ^ may be represented as fol- ' "~^'* lows : —1 1+4 *_2 +2)5—12+3+4—5 +10-4—2+4 _ 2-1+2—1 In this operation, 5 is the first term of the quotient, +10 is the product of +2, the divisor, by 5; the sum of +10 and — 12 gives — 2, the second term of the quotient; +2X — 2= — 4, and —4 and +3 gives — 1, the third term of the quotient, and so on. The last term, — 1, is the remainder. Supplying the powers of re, the quotient is bx^ — 2x'^ — ^+2, with a remainder — 1. A similar method may be used when the divisor contains three terms, but the process is more complicated. If the coefficient of the first term of the divisor is not unity, it may be made unity by dividing both dividend and divisor by tiie coefficient of the first term of the divisor. If any term is wanting, its place must be supplied with a zero. 410. In application of these principles, 368 RAY'S ALGEBRA, SECOND BOOK. 1, Let it be required to find the equation whose roots are less by 1 than those of the equation x^ — ta:-f 7. Since the second term is wanting, its place must be supplied with 0. The divisor is X — 1; hence, we divide by -j-l- OPERATION BY SYNTHETIC DIVISION. -fl) 1 ±0 —7 +7 +1 +1 -6 +1 +2 +2 —4 .-. —4= 2d R. -f 3 .-. -f 3= 3^ R. Hence, the required coefficients are 1, -(-3, — 4, and -\-l. .'. y^-\-^y'^ — 4y-|-l=0 is the transformed equation required. 2. Transform the equation 5a-*-j-28a;3-f 51a;2-f-32x— 1^=0, into another having its roots greater by 2 than those of the given equation. Here, 7/i=a;-f-2; hence, we divide by —2, thus, —2) 5 +28 +51 +32 —1 _10 —36 —30 —4 +18 +15 +2 — 5 .-. — 5= 1*' R. _10 —16 + 2 + 8 — 1 + 4 .-. +4= 2'i R. -10 + 4 _ 2 + 3 .-. +3= S'^ R. —10 —12 .-. -12= 4'A R. Hence, A=5, Bi=— 12, Ci=+3, D,=+4, and E,=— 5 .-. the transformed equation is 5?/'' — l'2.y^-{-^y'^-\-Ay—b=:^0. 3. Find the equation whose roots are less by 1.7 than those of the equation u? — 2x'^-\-Zx — 4=0. TRANSFORMATION OF EQUATIONS. 369 If we transform this equation into another whose roots are less by 1, the resulting equation is 2/^+2/^+2^/ — 2=0. We may then transform this into another whose roots are less by .7, or the whole operation may be performed at once, as follows : +1.7) 1 —2 +1.7 +3 .51 -4 +4.233 — .3 +1.7 +1.4 +1.7 +2.49 +2.38 +4.87 .• + .233 .-. +.233= 1st R '. +4.87= 2d R. +3.1 .-. 3.1= Sd R. Hence, the equation is 2/H3.1^2_^4.872/+.233=0. 4. Find the equation whose roots are each less by 3 than the roots of a^—2lx—S6=0. Ans. ^3_|_9y_90^0. 5. Required the equation whose roots are less by 5 than those of the equation x*—lSx^—S2x''-\-11x-\-9=0. Ans. y+2/-152/—1158y— 2331=^0. 6. Required the equation whose roots are less by 1.2 than those of the equation x^—6x*-\-lAa^-^1. 92x^—11. SI 2x —.79232=0. Ans. f—1f-\-27/—S=0. Transform the following equations into others wanting the 2d term. (See Art. 407.) 7. x'—6x'-{-7x—2=0, Ans. y— 5y— 4=0. 8. x'—6x'-\-12x-{-19=^0. Ans. y»+27=0. Transform the following equations into others wanting the 3d term : 9. a^—Qx'-j-9x—20=0. Ans. y+3y— 20=0, or /— 3/— 16=0. 10. x^— 4x2+ 5a:— 2=0. Ans. y^ — ^y^=0, or i/^4-y^ — 27=^* 370 RAY'S ALGEBRA, SECOND BOOK. 411. Proposition III. — To determine the law of Derived Polynomials. Let X represent the general equation of the n^^ degree ; that is, X=:a;«+Aa:«-i+Bx"-2. . . . -fTx+Vr^O. If we substitute x-\-h for a:, and put Xi to represent the new value of X, we have Xi=(a:+/t)"+A(a;+/i)''-i+B(a;+7i)«-2-f, etc., and if we expand the different powers of X-\-h by the binomial theorem, we have Xi= +Aa:»-J +B:r"-2 -]-, etc. + (/i— l)Aa:«-2 +(7i— 2)Ba;'*-3 -f-, etc. I /l2 w(n— l)a:"-2 ^-7^+, etc. +(w— l)(n— 2)Aa;"-3 +(n— 2)(n— 3)Ba;'*-4 etc. But the first vertical column is the same as the original equation, and if we put X', X'', X"\ etc., to represent the succeeding col- umns, we have X =ra:'*+Aa:»-i+Ba:"-2-f-, etc., X' =na;"-i + {n—1 ) Aa:"-2+ (n— 2)Ba;"-34-, etc., X"=n{n—\ )x"-2-^ (n—1 ) (n— 2) Ax"-3-|-, etc.. Etc., etc. By substituting these in the development of Xj, we have Xi=X+X'Hj^/i2-f ^-^713+, etc. The expressions X', X", X'", etc., are called derived polynomials of X, or derived functions of X. X' is called the first derived polynomial of X, or first derived function of X ; X" is called the second^ X!" the iliird, and so on. It is easily seen that X' may be derived from X, X" from X', etc., hy mnltiplylng each term hy the exponent of x ii^that term^ and diminishing the exponent hy unity. TRANSFORMATION OF EQUATIONS. 371 412. Corollary.— If we transpose X, we have Xj— X X" =X.'h-{-z; — 7^^^+j 6tc. Now, it is evident that h may be X'' taken so small that the sign of the sum X'A-f- — ^^^-f, etc., will be the same as the sign of the first term X'A. For, since X'h+^X"h^-]-, etc., =A(X'+^X"/i+, etc.), if h be taken so small, that ^X''h-{- ^X'^h'^-^, etc., becomes less than X' (their magnitudes alone being considered), the sign of the sum of these two expressions must be the same as the sign of the greater X^ 413. By comparing tlie transformed equation in Art. 406, with the development of Xj in Art. 411, it is easily seen that Xi may be considered the transformed equation, 1/ corresponding to x, and r to h. Hence, the tranformed equation may be obtained by sub- stituting the values of X, X', etc., in the development of Xj. As an example. Let it be required to find the equation whose roots are less by 1 than those of the equation x^ — ^x~{-7=^0. Here, ... X =a:3.-7a:+7, X"'=6, X' =3a:2— 7, Xiv=0. X"=^6x, Observing that h=zl, and substituting these values in the equa- v// v/// tion Xi=^X-fX^^+:p-^/t2-f h^^, etc., we have X^={x^—7x 1 n -f-7)-K3a;2 — 7)l+(6a;).,— ^-f =x^j^Zx:^—ix^l, in which ihe value of a; is equal to that of x in the given equation diminished ^yi. By this method, solve the examples in Art. 410, 372 RAY'S ALGEBRA, SECOND BOOK. EQUAL ROOTS. 414. To determine the equal roots of an equation. We have already seen (Art. 396, Rem.) that an equa- tion may have two or more of its roots equal to each other. "We now propose to determine when an equation has equal roots, and how to find them. If we take the equation (x — 2)^=0 (1), its first derived polynomial is Z(x — 2)^=0. Hence, we see that if any equation contains the same factor taken three times, its first derived polynomial will contain the same factor taken twice; this last factor is, therefore, a common divisor of the given equation, and its first derived polynomial. In general, if we have an equation X=0, containing the factors {x — ay^{x — 6)", its first derived polynomial will contain the fac- tors m[x — a)'^~'^n{X — 6)"-l ; that is, the greatest common divisor of the given equation, and its first derived polynomial, will be [x — a)'"~i(a;— 6)"-\ and the given equation will have m roots, each equal to a, and n roots, each equal to b Therefore, to determine whether an equation has equal roots, Find the greatest commori divisor between the equation and its first derived 'polynomial. If there is no common divisor^ the equation has no equal roots. If the G.C.D. contains a factor of the form x — a, then it has two roots equal to a; if it contains a factor of the form (x — ay it has three roots equal to a, and so on. If it has a factor of the form {x — a){x — 6) it has two Voots equal to a, and two equal to 6, and so on. 1. Given the equation x^ — x^ — 8.x-|-12=0, to determine whether it has equal roots, and if so, to find them. We have for the first derived polynomial (Art. 411), Sa:^— 2a:— 8. The G.C.D. of this and the given equation (Art. 108) is x—2. Hence, iC— 2=0, and a:=-|-2. Therefore, the equation has two roots equal to 2. LIMITS OF THE ROOTS OF EQUATIONS. 373 Now, since the equation has two roots equal to 2, it must be divis- ible by {x—2){x—2), or (x-2)2. (Art. 395). Whence, a;3_x2_8a:-fl2==(a:— 2)2(a:+3)=0, and a:+3=.0, or x=— 3. Hence, when an equation contains other roots besides the equal roots, the degree of the equation may be depressed by division, and the unequal roots found by other methods. The following equations have equal roots ; find all the roots. 2. a:»— 2x2— 15a;4- 36=0. . , _ Ans. 3, 3, —4. 8. x'—^x'^^x-\-\2=0. . . . Ans. 2, 2, —1, —3. 4. a;*— 6x3+12x2— 10x+3=0. Ans. 1, 1, 1, 3. 5. a;*_7x3-f-9x2+27x— 54=0. Ans. x=3, 3, 3, —2. 6. x*+2x3— 3x2— 4x+4=0. Ans. —2,— 2^-1-1, +1. Y. X*— 12x^4-50x2— 84x+49=0. A. 3±i/2, 3±|/2. 8. x^— 2x*+3x3— 7x2+8x— 3=0. Ans. 1, 1, 1, _i±iy^— 11. 9. x''-}-3x5— 6x*— 6x3-f 9x2-f 3x— 4=0. Ans. 1, 1, 1, —1, —1, —4. Suggestion. — In the solution of equations of high degree, the principles above explained may be extended. Thus, in the last example, the G.C.D. is X^ — X^ — x-f-1. Proceeding, we may, 1st, find the common measure of this and its first derived polynomial, and thus resolve into factors; or, 2d, find the G.C.D. of the first and second derived polynomials. If it is of the form X — a, one of the factors of the original equation will evidently be (X — a)^, etc. By the 1st method, we find x^— x2— x-fl=(x— l)(x2— l)=(x— 1)2 (x+1); by the 2d, (x— 1)3 is a factor of the original equation; hence, (x — 1)2 is a factor of x3_x2— x-(-l. LIMITS OF THE ROOTS OF EQUATIONS. 415. Limits to a Root of an Equation are any two numbers between which that root lies. A Superior Limit to the positive roots is a number numerically greater than the greatest positive root. 374 RAY'S ALGEBRA, SECOND BOOK. Its characteristic is, that when it, or any number greater than it, is substituted for x in the equation, the result is positive. An Inferior Limit to the negative roots, is a number numerically greater than the greatest negative root. The substitution of it, or any number greater than it, for x, produces a negative result. The object of ascertaining the limits of the roots is to diminish the labor necessary in finding them. 416. Proposition I. — The greatest negative coefficient^ increased hy unity ^ is greater than the greatest root of the equation. Take the general equation ^n_|_Ax«-i-|-Bx»-2 j^Tx-\-Y=0, and suppose A to be the greatest negative coefiicient. The reasoning will not be affected if we suppose all the ooefficients to be negative, and each equal to A. It is required to find what number substituted for x will make a;»>A(x"~i-f-a;"-2_|_^»-3^ ^ ^ ^ _|_^_|_1). x^ 1 By Art. 297, the sum in parenthesis is *- — y ; hence, we must X — J. havea;»>A(|^), or a:">. A a:'* 1 x—V But if x''=- — -, we find a:=A4-l; .-. A+1 substituted for x will X — J. „ Aa:" , ^, „^ Ax" A render x'= — , and, consequently, a;"_> ^ ^. By considering all the coefficients after the first negative, we have taken the most unfavorable case ; if any of them, as B, were posi- tive, the quantity in parenthesis would be less. 41T. Proposition II. — If we take the greatest negatice coefficient^ extract a root of it whose index is equal to the number of terms preceding the frst negative term, and in- crease it hy unify, the result will he greater than the greatest positive root of the equation. LIMITS OF THE ROOTS OF EQUATIONS. 375 Let Cx^~^ be the first negative term, C being the great- est negative coefficient ; then, any value of x which makes cc">C(x»-'--f-a;»-'-i +^^+1) (1) will render the first of the proposed equation >>0, or positive ; because this supposes all the coefficients after C negative, and each equal to the greatest, which is evi- dently the most unfavorable case. By Art. 297, the series in parenthesis = ^ — . Hence, ^ \ x—1 /' ^ x—1 x—1 ' this inequality will be true if a:'*^ r — , or > ^r—; or, by multiplying both members by iC— 1, and dividing by 0:"-''+', when (a;-l)a:'--i=C, or >C (2). But x—1 is C; OriC — 1={/C, or >v C; Orar^l + v'C; or >l + v/C: Find superior limits of the roots of the following equa- tions : 1. x'—bx'-\-^1x'—Sx+S9=0. Here, C=5, and r=l .-. 1+^C=1+5tz=6, Ans. 2. x'-^1x'—12a^—49x'Jro2x—lS=0. Here, l+;:/C^l+ 1^49^1 + 7=8, Ans. 3. o^-'+llx^— 25x— 67=0. By supposing the second term -fOa;', we have r=3 ; hence, the limit is l-[-^67, or 6. 4. Sa^—2x'—Ux-^4=^0. Dividing by 3, a:^— |ic2_y^_|_4^0. Here, the limit is l + V' ^^ ^-v 370 RAY'S ALGEBRA, SECO^^D EOOK. 418. To determine the inferior limit to the negative TOots, change the signs of the alternate terms; this will change the signs of the roots (Art. 400); then, The superior limit of the roots of this equation, by changing its sign, will be the inferior limit of the roots of the proposed equation. 419. Proposition III. — If the real roots of an equation^ taken in the order of their magnitudes^ he a, b, c, d, etc.^ a heing greater than b, b greater than c, and so on; then, if a series of numbers, a', b', c', d', etc., in which a' is greater than a, b' a number between a and b, c' a number between b and c, and so on, be substituted for x in the proposed equa- tion, the results will be alternately positive and negative. The first member of the proposed equation is equivalent to (x — a)(x — b)(x — c')(x — d) =0. Substituting for x the proposed series of numbers a^, b\ &, etc., we obtain the following results: {cf —a){a^ —b){a^ — c){a^ — d), etc =-f product, since all the factors are -|-. (J)^ — a){b^ — b){b^—c){b^—d), etc = — product, since only- one factor is — . {& — a){& — b){G^ — c){&—d), etc =+ product, since two factors are — , and the rest -\-. {d^—a){d^~b){d^ — c){d^—d), etc =— product, since an odd number of factors is — , and so on. Corollary 1. — If two numbers be successively substituted for X, in any equation, and give results with contrary signs, there must be one, three, five, or some odd number of roots between these numbers. Corollary 2. — If two numbers, substituted for x, give results with the same sign, then between these numbers there must be two, four, or some even number of real roots, or no roots at all. THEOREM OF STURM. 377 Corollary 3. — If a quantity 2', and every quantity greater than g', render the results continually positive, q is greater than the greatest root of the equation. Corollary 4. — Hence, if the signs of the alternate terms be changed, and if p, and every quantity greater than p^ renders the result positive, then — p is less than the least root of the equation. Illustration. — If we form the equation whose roots are 5, 2, and — 3, the result is x^ — 4iX^ — lla;+30=0. Now, if we substitute any number whatever for x^ greater than 5, the result is positive. If we put a;=:5, the result is zero, as it should be. If we substitute for rc, any number less than 5, and greater than 2, the result is negative. Putting a:— 2, the result is zero. Substituting for rc, any number less than 2, and greater than — 3, the result is positive. Substituting — 3, it is zero. Substituting a number less than — 3, the result is negative. From Cors. 3 and 4, it is easy to find when we have passed all the real roots, either in the ascending or descending scale. STURM'S THEOREM. 4L20» To find the number of real and imaginary roots of an equation. In 1584, M. Sturm gained the mathematical prize of the French Academy of Sciences, by the discovery of a beautiful theorem, by means of which the number and sit- iiaiion of all the real roots of an equation can, with cer- tainty, be determined. This theorem we shall now proceed to explain. Let X=rc«-|-Aa;«-i-j-Ba:'»-2 _|_Tx+V=rO, be any equation of the ?i'^ degree, containing no equal roots ; for if the given equation contains equal roots, these may be found (Art. 414), and its degree diminished by di- vision. 2d Bk. 32 378 RAY'S ALGEBRA, SECOND BOOK. Let the first derived function of X (Art. 411) be denoted by Xj. Divide X by Xj until the remainder Xi)X (Qi is of a lower degree with respect to x XjQj than the divisor, and call this remain- Z^ ^ r\ -^^ der — X^; that is, let the remainder, with its sign changed, be denoted by Xo. ^2)^1 (Q2 Divide Xj by X2 in the same man- X0Q2 ner, and so on, as in the margin, de- x, X Qo=— X noting the successive remainders, with their signs changed^ by X3, X4, etc., ^Z)^2 (vs until we arrive at a remainder which ^sQ.i does not contain rr, which must always X9 XoQozzr X4 happen, since the equation having no equal roots, there can be no factor containing X. common to the equation and its first derived function. Let this i-emainder, having its sign changed, be called X^^j. In these divisions, we may, to avoid fractions, either multiply or divide the dividends and divisors by tiny positive number, as this will not affect the signs of the functions X, Xj, Xo, etc. By this operation, we obtain the series of quantities X, Xi, X2, Xo. . . . Xh-1 (1). Each member of "this series is of a lower degree with respect to X than the preceding, and the last does not contain X. Call X the primitive function, and Xi, X2, etc., auxiliary functions. 431. Lemma I. — Two consecutive functions^ Xj, X2, /or example^ can not both vanish for the same value of x. From the process by which Xi, X2, etc., are obtained, we have the following equations : X =X,Qi-X2 (1) Xi =X2Q2-X3 (2) X2 =X3Q3-X, (3) X,_i=X,Q.-X,+i yr). If possible, let Xi— 0, and X2=0; then, by eq. (2) we have Xo=0; hence, by eq. (3) we have X4=0; and proceeding in the same way, we shall find Xj^O, Xg^O, and finally X^i=0. But this is impossible, since X^-f 1 does not contain iC, and therefore can not vanish for any value of x. THEOREM OF STURM. 379 4t22* Lemma II. — If one of the auxiliary functions van- ishes for any particular value of x, the tico adjacent functions must have contrary signs for the same value of x. Let us suppose that X3=0, when a:=a; then, because X9^X3Q3 — X4, and X3=i0; therefore, X2^=— X4; that is, X2 and X4 have contrary signs. 4S3. Lemma III. — If any of the auxiliary functions vanishes when x=a, and h be taken so small that no root of any of the other functions in series (1) lies between a — h and a-j-h, then will the number of variations and perma- nences^ when a — h and a-j-h are substituted for x in this series^ be precisely the same. Suppose, for example, the substitution of a for x causes the function X3 to vanish ; then, by Art. 421, neither of the functions X2 or X4 can vanish for the same value of X] and since when X3 vanishes, Xg and X4 have contrary signs, (Art. 422); therefore, the substitution of a for x in X2, X3, X4, must give ^2 > ^3 , X4 , or Xg, X3, X4. + - , - + And since h is taken so small that no root either of X2=0, or X^=iO, lies between a — h and CL-\-h^ the signs of these functions will continue the same whether we substitute a — h or a-\-h for x (Art. 419). Hence, whether we suppose X3 to be -j- or — by the substitution of a — h and a-\-h for x^ there will be one variation and one permanence. Thus, we shall have either X2 , X3 , X4 , or Xj , X3 , X4. + ± - - ± + So that no alteration in the number of variations and perma- nences can be made in passing from a — h to a-\-h. 4!S4. Lemma IV. — If a is a root of the equation X=^0, then the series of functions X, X^, Xo, etc.^ will lose one variation of signs in passing from a — h to a-f-h ; h being taken so small that no root of the function Xj=^0 lies between a — h and a-i-h. 380 RAY'S ALGEBRA, SECOND BOOK. For X substitute a^li in the equation X=0, and denote the result by H. Also, put A, A^, A^^ for the values of X and its derived func- tions when a-\-}i is substituted for x\ then (Art. 411), H=A-f A^/i+JA^^/i2-|-, etc. But, since a is a root of the eq. X=0, we shall have A=:0, while A^ can not be 0, since the eq. X=iO has no equal roots. Hence, YL=h'}i-\-\k.''}i^^, etc., =7^(A^+^A^^/i+, etc)- Now, li may be taken so small that the quantity within the paren- thesis shall have the same sign as its first term A'', (since A^ ex- presses the first derived function of X, corresponding to X^, in Art. 412); therefore, the sign of X, when a:=a4-/», will be the mme as the sign of X^. If we substitute a — }i for X in the equation X:=0, and denote the result by IF, we then have, by changing li into — \ in the expres- sion for H, \V=~h{iV—\K.''li-^, etc). Now, it is evident that for very small values of ^■, the sign of IF will depend upon the first term — A^/t, and, consequently, will be contrary to that of A''. Hence, when X^=a — ^, there is a variation of signs in the first two terms of the series X, Xj ; and when x=^a-\-h^ there is a continuation of the same sign. Therefore, one variation is lost in passing from X=z^a—1l to a-\-h. If any of the auxiliary functions should vanish at the same time by making x=:a^ the number of variations will not be affected on this account (Art. 423), and therefore, one variation of signs will still be lost in passing from a — h to a-{-h. 425. Sturm's Theorem. — If any hco numbers^ p and q, (p being less than q) he substituted for x in the series of functions X, Xj, X2, etc., the substitution of p for x giving k variations, and that of q for x, giving k' variations; then, k — k' will be the exact number of real roots of the equation X=0, which lie between p and q. Let us suppose that — 00 is substituted for x, and sup- pose that X continually increases and passes through all degrees of magnitude till it becomes 0, and finally reaches -|- oc. THEOREM OF STURM. 381 Now, it is evident, that so long as iC, -with its minus sign, is less than any of the roots of Xr^O, Xi=0, etc., no alteration will take place in the signs of any of these functions (Art. 419); but when X becomes equal to the least root (with its sign) of any of the auxiliary functions, although a change may occur in the sign of this function, yet we have seen (Art. 423) that it is the order only, and not the number of variations which is affected. But when X be- comes equal to any of the roots of the primitive function, then ono variation of signs is always lost. Since, then, a variation is always lost whenever the value of x passes through a root of the primitive function X=0, and since a variation can not be lost in any other way, nor can one be ever in- troduced, it follows that the excess of the number of variations given by x=p, above that given by Xz=q {p<^q), is exactly equal to the number of real roots of X=0, which lie between p and q. Corollary. — If the equation is of the n'^ degree, and m represents the number of real roots, then (Art. 396), the number of imaginary roots will be n — m. 4:!S6« To determine the number of real roots. Substitute — oo and -j- go for a; in the several functions, since the roots must all be comprised between these limits. In this case, the sign of each function will be that of its first term. If we substitute for cc, the number of variations lost from — oo to 0, will be the number of negative roots ; and from to -|- 00, the number of positive roots. 4:2T« To determine the situation of each real root; that is, the figures between which it lies. Substitute 0, — 1, — 2, — 3, etc., for x, in series (1), till we find a number which produces as many variations as ic= — 00 produced. This will be the limit of the nega- tive roots. Substitute 1, 2, 3, etc., till we find a positive number which gives the same number of variations that x=-\- cx> does. This Avill be the superior limit of the positive roots. 381 382 RAY'S ALGEBRA, SECOND BOOK. By observing where variations are lost, we find the situation of the roots. If two or more variations are lost between two of the substitutions, take smaller numbers, until only one is lost. This is termed the separation of the roots. 1. Find the number and situation of the real roots of the equation 4x^— 12^2 -fllx— 3=0. Here, we have X = 4a;3— 12a:2-f Ha:— 3, and (Art. 411) Xi=12a:^—24x -fll. Multiplying X by 3, to render the first term divisible by the first term of Xj, and proceeding as in the method of finding the G.C.D., (Art. 108), we have for a remainder — 2x-\-2. Canceling the factor -|-2, and changing the signs (Art. 420), we have X2=x — 1. Divid- ing Xj by X2, we have for a remainder — 1 ; therefore, X3=-}-l. Hence, X = 4a:3— 12a:2+lla;— 3. Xi=12a:2— 24a;+ll. X2-= x—1. Put — Qo and -f oo for X, and we have (Art. 426), for a:=: — QO, — -|- — -|- three variations, .-. k =3. X=-{- cc, -\- -\- -{- -\- no variation, .-. k'=iO. .♦. k — k'=3 — 0=3, the number of real roots. For x=Oj — -f- — -|-) three variations .♦. k =3. Hence, there is (Art. 426) no real root between and — oo. This we might also have learned from Art. 402, since there is no perma- nence in the proposed equation. It is best to substitute integral numbers first, and afterward frac- tional, if two or more roots are found to lie between two whole numbers. Or, substitute fractions at once, thus : X For x= — GO the signs are — x= — ^=+1 - ^=4-^ ^=+1 + ic^-j-l — X, X2 X3 + — + giving 3 var. + — + ({ 3 " + — + (( 3 " + — + — — + « 2 " THEOREM OF STURM. 383 Fora:=-fli^ ..... — — + + giving 1 var. x^+n + + + x=-^H + + + + " '« a:=+oo + + + + " " The roots are ^, 1, and 1 j. If these numbers had not been sub- stituted, the loss of one yariation in passing from | to f ; one in passing from | to 1|^; and one in passing from 1^ to If, would have given the situation of the roots. A careful study of this example will serve to illustrate the the- orem. Thus, we see that there are three changes of sign of the primitive functioUj two of the first auxiliary function, and one of the second. Again, while no variation is lost by the change of sign of either of the auxiliary functions, each change of sign of the primitive function occasions a loss of one variation. 2. How many real roots has the eq. x' — Sx'^-]-x — 3=0 ? Here, X = x^—3x^-^x-3 Xi=3a;2— 6a;H-l X2= x^2 X3=— 25. For x= — 00 the signs are — + ,2 variations, .-. k=2. a;=-|- 00 the signs are + + H » 1 variation, .-. k'=l. .-. k — k'^2 — 1=^1, the number of real roots. One variation is lost in passing from 2 to 4 and X— when a:=^3 i therefore, the root is -{-3. Find the number and situation of the real roots in each of the following equations : 3. x'—2x''—x-{-2=^0. Ans. Three. —1, +1, +2. 4. Sx' — S6x'^46x — lb=^0. Ans. Three. One be- tween and 1, one between 1 and 2, one between 2 and 3. 5. rc^— 3a;2— 4x+ll=0. Ans. Three. One between —2 and — 1, one between 1 and 2, one between 3 and 4. 6. z^^-2x — 5=0. Ans. One between 2 and 3. 384 RAY'S ALGEBRA, SECOND BOOK. Y. o.-^— 15x— 22rr=0. Ans. Three. One root is —2, one between — 2| and — 2^, one between 4 and 5. 8. x'—4:x^—8x-i-2S=0. Ans. Two. One between 2 and 3, and one between 3 and 4. 9. a:*— 2a;3— 7a)2-f-10x+10=0. Ans. Four. The limits are (-3, -2); (0, -1); (2, 3); (2,3). 10. a^—10x^-^6x-\-l=0. Ans. Five. The limits are (-4, -3); (-1, 0); (-1, 0);.(0, 1); (3, 4). XIII. RESOLUTION OF JS^UMERICAL EQUATIONS. 4;28« In the preceding articles we have demonstrated the most important propositions in the theory of equations, and in some cases have shown how to find their roots. The general solution of an equation higher than the fourth degree, has never yet been effected. In the prac- tical application of Algebra, however, numerical equations most frequently occur ; and when the roots of these are real, they can always be found, either exactly or approxi- mately. The way for doing this has been prepared in the preceding articles, by finding the limits of the roots, and separating them from each other. RATIONAL ROOTS. 4SO. Proposition I. — To determine the integral roots of an equation. If a be an integral root of the equation Aa-*-| B^'-h^*^' -fDa;-fE=0, we shall have Aa*+Ba'4-Ca'^-f-Da--|-E=0i E therefore, — = — Aa' — Ba'* — Qa — D. RESOLUTION OF NUMERICAL EQUATIONS 385 Now, since the second member of the last equation is evidently a E whole number, E is divisible by a. Put — =E^; transpose D to the CL first member, and divide by a\ this gives E'+D • = — Aa"— Ba— 0; .-. a is also a divisor of E'-f D. E'+D Put — L_z=D, transpose C, and divide by a; this gives D'-l-C ' — — Aa— B: .-. a is a divisor of D'-fC. a D'-4-C C'-l-B Again, put =C', transpose B, divide by a, and — ~—= — A. C'4-B Lastly, making =^'i and transposing A, we have B'-f-A=0. If, then, all these conditions are satisfied, a is a root of the pro- posed equation; but if any one of them fails, a is not a root. Hence, we have the following Rule for finding the Integral Roots of an Equation. — Divide the last term of the equation hy any of its divisors a, and add to the quotient the coefficient of the term containing x. Divide this sum hy a, and add to the quotient the coefficient ofx\ Proceed in this manner unto the first term, and if a be a root, all these quotients will be whole numbers, and the result will be 0. Corollary 1. — It will be easier to ascertain whether -\-l and — 1 are roots, by trial. Also, by ascertaining the* limits to the positive and negative roots (Art. 417), we may reduce the number of divisors. Corollary 2. — If the first coefficient be not unity, the equation may have a fractional root. To determine if this be the case, transform the equation into one having its first coefficient unity (Art. 405, Cor. 1), and its roots integers (Art. 399). Corollary 3. — "When all the roots except two are in- tegral, divide the equation and find the others (Art. 396, Cor. 1). 2(1 Bk. 33* 386 RAY S ALGEBRA, SECOND BOOK. 1. Find the rational roots of the equation x'-\-Sx'—4x—12=0. Here, by Art. 417, no positive root can exceed 1-|-^12, or 4, and the limit of the negative roots is l-[-3=4. It is also found, by trial, that -fl and — 1 are not roots. We then proceed to arrange the divisors of — 12, among which it is possible to find the roots, and proceed as follows: Last term — 12 Divisors -f 2 , -f 3 , -f 4 , — 2 , — 3 , - 4 Quotients _ 6 , — 4 , — 3 , -f 6 , -f4 , -f3 Add —4 —10 , —8 , — 7 , +2 , — , -1 Quotients — 5 , * , * , — 1 , * Add 4-3 — 2 , -f 2 , +3 , Quotients — 1 — 1 , — 1 , Add -f 1 , 0,0, Since —8, — 7, and — 1, are not divisible by -|-3, 4-4, and — 4, we proceed no further with these divisors, as it is evident that they are not roots of the equation. The loots are -\-2, — 2, and — 3. Find the roots of the following equations : 2. a;'— Y-c^-f 36=.0 Ans. 3, 6, and —2. When any term is wanting, as the 3d term in this example, its place must be supplied with 0. When there are equal roots, they may be found (Art. 414), or having found one, reduce the degree c,f the equation by division, and proceed as before. 3. a:'— 6.r;-f llo;— 6r=0. . . . 4. ar'+x2— 4a;— 4=0. , , , 5. a^—Sx'—46x—72=0. . . 6. a^s—Sx'— 18.t4-Y2=.0. . . V. a;*— 10x»+35x2— 50a;-j-24:=.0 . Ans. 1, 2, 8. Ans. 2, -1, —2. Ans. 9, —2, —4. . Ans. 3, 6, — 4. Ans. 1, 2, 3, 4. HORNER'S METHOD OF APPROXIMATION. 387 8. x'-^4x'—x'—lQx—12=0. Ans. 2, —1, —2, —3 9. cc*— 4x3— 19x^+46x-hl20=0. Ads. 4, 5, —2, —3 10. x'—2lx'-^Ux-i-120=Q. Ans. 3, 4, —2, —5 11. x*+a;3— 29a:'^— 9x+180=:0. - Ans. 3, 4, -3, -5 12. a^—2x'—4x-\-S=.0. Ans. 2, 2, —2 13. x'Jr^x'—Sx-^lO^r^O. Ans. —5, l^]/"^ 14. x*-9x'Jr'^7x'-\-21x—60^0. Ams. 4, 5, =±=v 3 15. 2a:'— 3x2-f-2a;— 3:^.0. Ans. |, ±]/=I 16. 3a:'— 2:i;2— 6^+4=0. Ans. f, ±|/2 17. 8x3— 26x^+1 la:-f-10=--0. Ans. |, |(3±|/4l) 18. 6x*— 25.x'+26a;2+4x— 8=0. Ans. 2, 2, f, —J 19. X*— 9.x'4-yx^+ya:— V=0. Ans. |, |, 3±3|/2 IRRATIONAL ROOTS— METHODS OF APPROXIMATION. Having found all the integral roots, we must have re- course to methods of approximation, the best of which is Horner's. 4SO. Horner's Method of Approximation. — The prin- ciple of this method de-pends on the successive transforma- tions of the given equation, by Synthetic Division (Art. 410), so as to diminish its roots at each step of the operation. Let the equation, one of whose roots is to be found, be Px«-f Qx»-i ^T.x+V=0. Suppose a to be the integral part of the root required, and r, S, ^ ... the decimal digits taken in order, so that x=za-^r-{-8-\-t. . . Find a by trial, or by Sturm's theorem, and transform the equation into one whose roots shall be diminished by a (Art. 410). Let P^'^^Q'^"-! ^Ty-\^Y'=^0 he the transformed equa- tion; then, the value of ^ is the decimal r-\-S'\-t ; and since 388 RAY S ALGEBRA, SECOND BOOK. this root is contained between and 1, we may easily find its first digit r. Again, let the loots of this equation be diminished by r, and let the transformed equation be P2:"+Q'^2;"-i _|_T"2;4-V'^=0. Now, the value of z in this equation is 8-\-t. . . . , and the value of s lies between .00 and .1 ; that is, it is either .00, .01, .02, . . . „ or .09. But since the figure 8 is in the second place of decimals, z^^ z^ . . . . will be small, and we may generally find 8 from the equation T^^2;+V^^=0; or, Sr=— Vh-T, nearly. Having found S, diminish the roots of the last equation by S, and then from the last two terms, T^^^2;^-(-V^^^, of the resulting equation, •find t the next decimal figure, and so on. 431. The absolute number, or last term, is sometimes called the dividend, and the coefficient of the first power of the unknown quantity, (as, T" or T"',) the incomplete or trial divisor. The correctness of the values of s, t, etc., obtained by means of the trial divisor, will always be verified in the next operation. If too great or too small, the quotient figure must be increased or diminished. The accuracy with which each succeeding decimal figure may be found, increases as the value of the figure decreases. In general, after finding three or four decimal figures, the rest may be obtained with suflQcient accuracy by dividing \^^ by T^^. 43!S. By changing the signs of the alternate terms (Art. 400), and finding the positive roots of the resulting equation, we may obtain the negative roots of the proposed equation. Remark. — It is generally easier to find the first decimal figure of the root by trial than by Sturm's theorem. 433. To illustrate this method, let it be required to find the positive root of the equation o--^ — 4x — 10.*768649=0. We readily find that X must be greater than 5, and less than 6j HORNER'S METHOD OF APPROXIMATION. 389 therefore, a=^5. We then proceed to transform this equation into another whose roots shall be less by 5. (See Art. 410.) 5) 1-4 —10.768649 +5 + 5 +1 — 5.768649 +5 +6 1st Trans, eq. . . 2/2+6^ — 5.768649=0. Here we may find the value of y nearly, by dividing 5.7 by 6, which gives .9; but this is too great, because we neglected y^. If we assume 2/=-8, and deduct 2/^=.64 from 5.7, and then divide by 6, we see that y must be .8, Let us now transform the equation into another whose roots shall be less by .8. .8) 1 +6 .8 —5.768649 +5.44 +6.8 .8 7.6 — .328649 Trans, eq. . . z^.-\-7.6z~ -.328649=0. The approximate value of z in this equation is the second decimal figure of the root. This is readily found by dividing the absolute term by the coefficient of z, the first term, z^, being now so small that it may be neglected. Thus, .328-^7.6=.04=s. We next diminish the roots of the last equation by .04. s .04) 1 +7.6 .04 +7.64 .04 - 328649 .3056 .023049 3d Trans, eq. . +7.68 .... 2:^2 1 7.68.:;' ~.023049z 390 RAY'S ALGEBRA, SECOND BOOK. Here ^ is nearly .023^7.68=:.003=r<. By diminishing the roots of the last equation by .003, we have t .003) 1 +7.08 —.023049 .003 .023049 4-7.683 .0 The remainder being zero, shows that we have obtained the exact root, which is 5.843. By changing the sign of the second term of the proposed equation, we have a:^+4a: — 10.768649=0. The root of this equation may be found in a similar manner; it is 1.843, Hence, the two roots are +5.843 and —1.843. Ex. 2. — To illustrate this method further, let us form the equation whose roots are 3, -f-|/2, — 1/2, which gives ic' — Zt^ — 2a;-(~^=^- ^^^ ^^ "^^ ^^ required to find, by Horner's method, the root which lies between 1 and 2 ; that is, |/27 One root lies between 1 and 2; hence, «=1, and the first step is to transform the equation so as to diminish its roots by 1. a 1) 1 -3 -2 +6 +1 _2 _4 ~Il2 "^4* "+2 -1 -5 T' 5 Hence, ?/''rt0?/2_5y+2=0, is the first transformed equation. By dividing the absolute term 2 by 5, the trial divisor or coefficient of ?/, we find 7*=:.4, and proceed to transform the equation so as to diminish its roots by .4. HORNER'S METHOD OF APPROXIMATION. 391 .4) 1 zbO .4 —5 .16 -4.84 + .32 —4.52 +2 —1.936 .4 .4 + .064 V'' .064 .8 .4 ^ T^'-~4.52 =m 1.2 This gives z^-}-1.2z^—4.52z-\-Mi^0, for the 2d transformed equation; and for s, the next figure of the root, .01. Transform this equation so as to diminish its roots by .01. .01) +1.2 .01 .01 1^22 .01 T23 -4.52 .0121 -4.5079 .0122 -4.4957 -f.064 —.045079 +.018921 ^0189 4:495' .004 This gives ^'S+l. 232^— 4.49572;'+.01 8921:^0, for the 3d trans- formed equation ; and for the next figure of the root ^=.004. Transform this equation so as to diminish its roots by .004. t .004) +1.23 .004 1.234 .004 1.238 .004 1.242 —4.4957 + .004936 -4.490764 + .004952 —4.485812 +.018921 —.017963056 .000957944 We may obtain several of the succeeding figures accurately by division; thus, .000957944--4.485812==.0002135, which is true to the last decimal place, as will be found by extracting the square root of 2. Hence, a;:^l. 41 42135. 392 RAY'S ALGEBRA, SECOND BOOK. In practice it is customary to make some abridgments. Thus, mark with a * the coefficients of the unknown quantity in each transformed equation instead of rewriting it. Also, when the root is required only to five or six places of decimals, use about this number in the operation. 3. Given a:*—Sx^-{-14x^-\~4:X—S=0, to find a value of x. 5.236068; —8 5 OPERA +14 -15 - 1 10 9 35 *44 2.44 TION. -f4 —5 —1 45 *44 9.288 —8 —5 —3 5 *— 13 10.6576 2 5 *— 2.3424 1.93880241 7 5 53.288 9.784 *— .40359759 .39905490 *12 .2 46.44 2.48 -63.072 1.554747 «_ .00454269 .00400954 12.2 .2 48.92 2.52 64.626747 1.566321 ?i:_ .00053315 12.4 .2 *51.44 .3849 -66.193068 .31608 12.6 .2 51.8249 .3858 66.50915 .31656 *12.8 .03 52.2107 .3867 «66.82571 12.83 .03 *52.5974 .08 12.86 .03 52.68 .08 12.89 .03 52.76 *12.92 IIORNERS METHOD OF APPROXIMATION. 3, substituted for cc, give results, the one too small, and the other too great, so that one root lies between a and h. (Art. 403.) Let A and B be the results arising from the substitu- tion of a and b for x, in the equation X=;0. Let x=a-{-h, and b=a-^Jc] then, if we substitute a-}-h and a-{-7c for x, in the equation X=:0, we shall have X=.A-f A'A-f-JA^'A^-f , etc. B==A+A7^+iA"A;'^+, etc. Here, A^, A^^, etc., are the derived functions of A (Art. 411). Now, if h and k be so small that their second and higher powers may be neglected without much error, we shall have X-A=:--A^/i nearly; B— A=A^/^ " . Whence, B-A X— A:: A'Jc: A'h : 7c: %; Or, . . B— A h : : X— A : /i, (Art. 270) Or, . . B-A b—a : : X— A : h, since k= a. Hence, we have the following Knle. — Find, by trial, two numbers which, substituted for X, give one a result too small, and the other too great. Then say, As the difference of the results is to the difference of the suppositions, so is the difference between the true and the first result, to the correction to be added to the first sup>- position. Substitute this approximate value for the unknown quan- tity, and find whether it is too small or too great; then, take two less numbers, such that the true root may lie between them, and proceed as before, and so on. It is generally best to begin witli two integers which differ from each other by unity, and to carry the first approximation only to one place of decimals. In the next operation make the ditference of the suppositions 0.1, and carry the 2d quotient to two places, and so on. NEWTON S METHOD OF APPROXIMATION. 397 1. Given x^-{-x''-\~x=100, to find x. Here, X lies between 4 and 5. Substitute these two numbers for X in the given equation, and the result is as follows : 4 . . . . . X . . . . . . 5 64 . . . . . . x^ . . . . . . 125 16 . . . . . . a;2 . . . . . . 25 4 . . . . . . X . . . . . . 5 84 . . . . . results . . . . . 155 55 . . . . . . 5 . . . . . . 100 84 4 84 71 : "1 : : ~Tq : 0.22, therefore, a:=4.2, the first approximation. Substituting 4.2 and 4.3 for x, and proceeding as before, we get for a second approximation x=4.264:. Assuming a:=4.264 and 4.265, and continuing, we obtain a:=4.2644299, nearly. Find one root of each of the following equations : 2. x'-\-S0x=.420 Ans. x=6.1'J010S. 3. 144x3— 973a^=:319. 4. x'-^10x-'-\-bx=2600. 5. 2x'-ir^x'—4x=10. . 6. x'—x''-\-2x'-\-x=4. . . . Ans. .T=2.75. Ans. a-=ll. 00679. . Ans. a:=l. 62482. . Ans. a:=rl.l4699. 7. f1x'-^4x'JrVl0x{2x—l)=-2S. A. a;^4.51066. 437. Newton's Method of Approximation. — Thio method, now but little used, is briefly as follows : Find, by trial, a quantity a within less than 0.1 of the value of the root. Substitute a-j-i/ for x in the given equa- tion, and it will be of this form A+AV+iAV+iA"y+, etc., :=0 (Art. 411), where A, A', A", etc., are what the proposed equation, the first derived polynomial, etc., become w^hen x=a. 398 RAY'S ALGEBRA, SECOND BOOK. From this equation, by transposing and dividing, A A^'' K^^^ We find ?/=:— ^/_i — 2/2— ^— 2/"*— , etc.; and since y is <0.1, ?/2 will be <0.01, 2/'''<0.001, and so on. Therefore, if the sum of the terms containing 2/^, 2/^ etc., be less than .01, we shall, in neglecting them, obtain a value of y within .01 of the truth. Putting y=—~^ we havea:=a— ^. This will diflFer from the true value of x by less than .01. Now, put b for this approximate value of x, and let X::^b-\-Z', we have then as before B-f B^2;+^B^^2:2_^^B^^^03_[_^ etc., =^0; and as z is supposed to be less than .01, z'^ will be <.0001. If^ then, we neglect the terms containing 2;2, z^, etc., we shall obtain a probable value of Z within .0001 ; and so on. Applying the succes- sive corrections, we obtain the value of x. Newton gave but a single example, viz. Required to find the value of x in the equation x^ — 2.t 5=0. Ans. 0^=2.09455149. CARDAN'S RULE FOR SOLVING CUBIC EQUATIONS. 43S. In its most general form, a cubic equation may- be represented by x?-\-2yx'^-\-qx-\-r=^0 ; but as we can always take away the second term, (Art. 407,) we will suppose, to avoid fractions, that it is reduced to tiie form a:»_|_3^a;-f2r=0. Assume x=zy-{z^ and the equation becomes , y3_^2;3_|_3y2;(2/-f2:)+3(7(2/-f2)4-2r=0. Now, since we have two unknown quantities in this equation, and have made only one supposition respecting them, we are at liberty CARDAN S SOLUTION OF CUBIC EQUATIONS. 399 lo make another. Let^ therefore, yz:=^ — g. Substituting, we have ^•'5_|_2:3_[_2r=0; but since yz=—q, z^'z y. q Hence, 2/3— ^-|-2r^0; Whenee, 2/'3= -r-fy'r^ip^. And similarly, , . z^^^ — r — \'"f'^'\-(j^', the radical being positive in one, and negative in the other, by reason of the relation yz-= — q. And since a;=2/-|-2;, we have This formula will give one of the roots. The others may be found by reducing the equation (Art. 396, Cor. 1). -f- 439. If r^-fgs be negative, that is, if r'+$''<0, the values of x become apparently imaginary when they are actually real, and we shall now show that Cardan's Method of Solution does not extend to tJiose cases in which the equation has three real and unequal roots. Suppose the one real root (Art. 401, Cor. 3), to be a; and the other two arising from the solution of a quadratic to be 6-|-y/3c, und b—^/Sc, in which, if 3c be positive, the roots are real, and if 3c be negative, they are imaginary ; and because the second term of the equation is 0, we have (Art. 0=a+(6+ |/3c) + (6— i/3c)=ra+26 ; 3g=aX26+^^— 3c=— 362— 3c ; 2r=—a{b^—3c):=2b^~Qbc. Hence, we have r^-\-q^={b^~3bcy—{b^-^e)^=—9b^e-\-eb-e^—c^ =._c(362_.c)2 ... ^/r2+g3^(362_c)^irc. Now, this expression is real when C is negative, and imaginary when c is positive, or when the equation has three real roots. 400 RAY'S ALGEBRA, SECOND BOOK. If c=0, the roots are a, 6, and 6; hence, Cardan's Rule is appli- cable to equations containing two equal roots. 440. In illustration of the apparent paradox that when the roots of the quadratic equation, Art. 438, are imagin- ary, the roots of the cubic equation are all real, take the following Example. — To find the three roots of the equation By substituting y-\-z for x, we have And, since . . . 3i/z=15, y^-{-z^—4=0. From the solution of these equations, we obtain y^=2-\-lly'^—i. By actual multiplication, we find that p=2-{-y^ — 1 ; likewise, s^>rz=2— 11/^, and z=2—^^^. Hence, x=y-^z={2i-y^-[)-^{2—^^^)=4. By dividing the given equation by X — 4, we find the other two roots are x=—2-\-y'3, and —2 — ^3. As no means have yet been discovered for reducing the imaginary forms to real values. Cardan's rule fails when all the roots are real. This is the Irreducible Case of cubic equations. 441. The following examples, containing one real and two imaginary roots, may be solved b'y Cardan's rule. When the equation contains the second term, remove it (Art. 40 Y), and reduce the equation to the form x'-^Sqx-\-2r=0. Then, x^f{—r-\-^r^^^)-{-f{—r—^/'f2-j-q^), will be the real root of the proposed equation. Having the real root, the imaginary roots may be found by reduc- ing the equation to a quadratic (Art. 396, Cor. 1). RECIPROCAL OR RECURRING EQUATIONS. 401 1. Solve the equation v^-j-Sv''^9v—lS=0. Substituting x — 1 for v (Art. 407), we have x^-\-6x — 20=0. Comparing this with the equation x^-\-Sqx-]-2r^:zzO, we find q=2, 7*:^— 10 ; hence, a;=:f (1 0+ y'lOS) 4- f (10— i/108j=2.732— .732=2. Whence, ^;=a>— 1=2— 1=1. The other two roots are easily found to be — IdtSy' — 1. 2. 0)3— 9a:+28..:=0. . . . Ans. x=—4, 2±^=8. 3. x'-{-6x—2=0. . . Ans. a:=/4— /2=.32748. 4. x'—Gx'Jr'^Sx—10=0. Ans. x=2, 2±y/3i. 5. a^—9x'^6x—2=.0 Ans. x=SM66l4. Remark . — Cardan's Rule, together with those of Ferrari, Euler, Descartes, and others, are regarded, since the discovery of Horner's method, and Sturm's theorem, as little more than analytical curi- osities. RECIPROCAL OR RECURRING EQUATIONS. 442. A Recurring or Reciprocal Equation is one such that if a be one of its roots, the reciprocal of a will be another. Proposition I. — In a recurring equation the coefficients^ when taken in a direct and in an inverse order ^ are the same. Let a;^+Aa;"-i+BiC"-2 ^Sa:2-f Ta:+V=0, ^e a recurring equation; that is, one that is satisfied by the substitution of the re- ciprocal of X for X. This gives and multiplying by X'*, 1+Aa:+Ca;2 -f-Sa;''-2+Ta;«-i+Va:''=0, which proves the proposition. 2d Bk. 84 402 RAY'S ALGEBRA, SECOND BOOK. Such equations are called Recurring Equations, from the forms of iheir coe^cienls ; and Reciprocal Equations, from the forms of their roots. Proposition II. — A recurring equation of an odd degree^ has one of its roots equal to -}-l, when the signs of the like coefficients are different^ hnt equal to — 1, when their signs are alike. Since every power of -fl is positive; when the signs of the like coefficients are different, if we substitute -f 1 for x the corresponding terms will destroy each other. AVhen the signs of the like coefficients are the same, since one will belong to an odd, and the other to an even power, if we substitute — 1 for X, the corresponding terms will destroy each other. Such equations may, therefore, be redmced one degree lower by dividing by x — 1, or x-]-!. Proposition III. — A recurring equation of an even degree^ ivhose like coefficients have opposite signs, is divisible by x^ — 1, and tJierefore two of its roots are -j-l, and — 1. Leta:2'*-fAa;2"-i-|-Ba:2"-2 _Ba;2— Aa;— 1=0, be an equa- tion of the kind specified. It may evidently be arranged thus, and be divisible by x^~l (Art. 83). (a:2n_|)_^Aa;(a:2»-2_l)_>.Ba;2(a:2"-4_l)-|-, etc. . . .=0. Corollary. — An equation of this form may therefore be reduced two degrees lower by either common or synthetic division. Proposition IV. — Every recurring equation of an even degree above the second^ may he reduced to an equation of half that degree, when the signs of Hie correspoTuiing tenns are alike. For, a;2n_Aa;2»-i_^Bz2»-2_, ete 4-Bx2— Az-f 1=0, by RECURRING EQUATIONS. 403 dividing by a:", and collecting the pairs of terms equi-distant from the extremes, becomes of the form ( ^"+^' ) -^ ( ^"- +^ ) +" ( ^""+x-=-^ ) -> ^"=- =°- Let a:-j — —2; then, x^-\ — 2=^^ — 2, by squaring; also, and generally ( x-'+i ^ = ( a;-i + ^ )z-( ^"-'+^ )• Hence, each of the binomials may be expressed in terms of z, and the resulting equation will be of the n''* degree. 1. Given x* — bx^-\-6x'^ — bx-\-l=:^0, to find x. Here, .... a:2-5rc-f 6— - + ^ = 0, Let X-] — —z; then, z^ — 50+4=iO, and z=4, or 1 ; Putting x-\ — equal to each of these values of 2, we obtain for the four values of X, 2=bi/3~and ^(1±t/— 3). The second of tliese values is the reciprocal of the first, and the fourth of the third, as may be shown, thus : 1 1 2-T./3 2-/3 2+v^'"2+v/"3^2-v/3~ 4-3 =^2-i/3. EXAMPLES IN RECURRING EQUATIONS. 1. a^*—10.7:3-^26.T2— 10^:4-1=0. Ans. x=S^2y'% 2±|/3. 2. x*—^x^-^2x'—.^x-j-l=0. Ans. x=2, 1, ±1/^. 404 RAY S ALGEBRA, SECOND BOOK. 3. x'~Sx'-^Sx—l=0. Ans. x=^l, l(3±|/5). 4. x'-~llx'-\-11x'-{-11x''—llx-\-l=0. Ans x~ 1 ^±1^ ^-V^^' ^+l/^ 3-T/^ 5. 4ic«— 24a:^+57a^*— 73a:34-57a;2-.24x-h4=r0. Ans x-^^ ^ 2 ^ 1±1^? l-l/^ BINOMIAL EQUATIONS. 443. Binomial equations are those of the form 3/"±A:=0. Let *[/A=a; thatis, A=a"; Then, 2/"±a"=0. Let . . , y=ax', then, a"a;"dra"=:0. Or, a:"=tl=0, which is a recurring equation. 444. — I. The roots of the equation x"=tl=:0, are all unequal ; for the first derived polynomial nx"~^^ evidently has no divisor in common with ic"±l, and therefore there are no equal roots (Art. 414). II. — If n be even, the equation a;" — 1=0, or x"=l, has two real roots, -|-1 and — 1, and no more, because no other real number can, by its involution, produce 1. By dividing x"— 1=0 by (x-\-l)(x—l)—x'^—l, we have a recurring equation, having n — 2 imaginary roots. BINOMIAL EQUATIONS. 405 For example, the equation x^=l, or x^ — 1=0 divided by ic* — 1 gives x*-\-x^-^l:=0; whence, , /(-i±j/=ri "=*Vl — 2^ I- This gives for the six roots of 1 +1, -1, III. — If n be odd, the equation x^ — 1=0 has only one real root, viz.: -fl ; for -|-1 is the only real number of which the odd powers are -\-l. Dividing x^ — 1=0 by x — 1, we have a recurring equation, having n — 1 imaginary roots. For example, the equation x^=l, or x^ — 1=0, divided by X — 1, gives a:"^-j-a:-f-l=0; whence, a;= ^ . Hence, the three third roots of 1 are 1 -1+1/-3 -1-^-3 ^' 2 ' 2 • IV. — If n be even, the equation .t"-}-1=0, or ic"= — 1, has no real root, since ^ — 1 is then impossible. Hence, all the roots of this equation are imaginary. 406 RAY'S ALGEBRA, SECOND BOOK. For example, the four roots of the recurring equation ic*-f 1=0 (Art. 442), are -^14-1/^=1 —l-y^l l+v/= i l~y—l 1/2 ' ^/2 ' -/2 ' 1/2" ' V. — If n be odd, the equation x"'-{-l=0, or x^= — 1, has one real root, viz.: — 1, and no more, because this is the only real number of which an odd power is — 1. For example, a:^-j-^l=0, divided by x-^1, gives x^ — x -j-l=0 ; whence, x= ^ . Therefore, the three third roots of — 1, are — 1, o 5 ana ^ . Binomial equations have other properties, but some of them can not be discussed without a knowledge of Ana- lytical Trigonometry. 1. Find the four fourth roots of unity. Ans. +1, -1, +i/— 1, — i/— 1. 2. Find the five fifth roots of unity. 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