>AY'y j^ ' -#^? #^2-^^i»*^* ~- :;'-')JMi*W-i*»y^MiiAK-^;si^, FMATTCAL *^*i*rf -*^ W ,*^ ^W '^^ »«^W -f*^fc* ^*+p' -r^ fl^W ^ff^T^^ I ■•■■f* cs f ^ f ' ».-^« 'iil^„ "••(S' :h ! > ' ■^1 ^^> Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031285806 Cornell University Library ^V19314 Ray's new higher algebra 3 1924 031 285 806 olin,anx ECLECTIC EDUCATIONAL SERIES. RAY'S NEW HIGHER ALGEBRA. ELEMEI^TS 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, of Mathematics, Hamden Sidney "T OOLIiEOE. VAN ANTWERP, BRAGG & CO., 137 WALNUT STREET, 28 BOND STREET, CINCINNATI. NEW YORK. ® Ray's Mathematical Series. ARITHMETIC. Bay's New Primary Arithmetic Bay's New Intellectual Arithme Bay's New Practical ArithmetJ*. g-\ ^ Bay's New Higher Arithmetic. ^ * TWO-BOOK SERIES. S : \ ^-: Bay's New Elementary Arithmette:^ — Bay's New Practical Arithmetic ALGEBRA. Bay's New Elementary Algebra. Bay's New Higher Algebra. HIGHER MATHEMATICS. Bay's Plane and Solid Geometry. Bay's Geometry and Trigonometry. Bay's Analytic Geometry. Bay's Elements of Astronomy. Bay's Surveying and Navigation. Ray's Differential and Integral Calculus. Bnterpd according to Act of Congress, in the year IS52, by W. B. Smith, in the Clerk's Office of the District Court of tiie United St.ites, for the District of Ohio. Entered according to Act of Congress, in the year Iftfifi, bv Saboebt, Wtlson & HiNKLE, in the Clerk'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 acquaintatice 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 Ilulee that maj' 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. I^o eflfort 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, 80 that the learner may not be required to rely on a prin-' 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 Kadicals, 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, wliich 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. WooDWAKD College, Jhnj, 1S,J2. Publishers' Notice. — This work, originally published as Ray's Algebra, Part II., was revised, in lS(i7, by Dr. L. D. Potter. Portions of the work were revised in 1875, by Prof. Del. Kemper. CONTENTS. I.— FUNDAMENTAL RULES. Definitions and Notation Exercises on the Definitions and Notation Examples to be written in Algebraic SjTnbols . Addition — General Rule. — Subtbaction — Rule Bracket, or Vinculum Observations on Addition and Subtraction . . Multiplication — Preliminary principle . . , Rule of Coefficients — of Exponents Rule of the Signs — General Rule . Multiplication by Detached Coefficients Remarks on Algebraic Multiplication Division — Rule of Signs — Coefficients- — Exponents Division of a Monomial by a Monomial Division of Polynomials by Monomials Division of one Polynomial by another Division by Detached Coefficients . . articles. 1— 36 36 36 37— 45 46 47— 51 52— 53 55— 56 Go- 61 62 es- 65 67— 70 71 74 75— 76 77 II.— THEOREMS, FACTORING, Etc. Algebeaic Theorems — Square of the sum of two quantities . 78 Square of the difference of two quantities ... 79 Product of Sura and Difference 80 Transfer of factors in a fraction . 81 Any quantity whose exponent is is :=1 ... 82 (i"» — 6"» is divisible by is divisible by a-)-5 85 a2m+i_j_j2»H-i is divisible by a+6 86 Factoring — Of Numbers — Of Algebraic quantities .... 87 — 95 Greatest Common Divisor 96 — 108 IiEAST Common Multiple 109 — 113 (V) CONTENTS. III.— ALGEBRAIC FRACTIONS. 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 — L'^0 Multiplication and Division of Fractions 131 — 132 Reduction of Complex Fractions I33 Resolution of Fractions into Series I34 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 — Rule 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- ^-'ties . . . 160 Problems producing Simple Equations containing three or more unknown quantities 161 v.— SUPPLEMENT TO SIJIPLE EQUATIONS. Generalization — Formation of Rules. — Examples .... 162 163 Negative Solutions — Discussion of Problems — Couriers . . 164--166 Cases of Indeterminatinn and Impossible Problems . . . 167—169 A Simple Equation hos but One Root 170 CONTENTS. vii VI.— FORMATION OF POWEES— EXTRACTION OF ROOTS- RADICALS— INEQUALITIES. autici.es. Involution or Formation of Powers — Newton's Method . . 172 Squakk Eoot 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 Monomiols—Of Polynomials 190—191 Fourth Root — Sixth Root— Nth Eoot, etc 192 Signs of the Roots — Nth Root of Monomials 193—194 Radical Quantities — Definitions — Reduction of Eadicals . 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 Eoots of Radicals— Imaginary Quantities . . . 207 — 210 Theory of Fractional Exponents 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 Y — Examples 217 — 224 VII.— QUADRATIC EQUATIONS. Definitions — Pure Quadratic Equations — Problems .... 224 — 229 Affected Quadratic Equations .... 230 Completing the Square — General Eule — Hindoo Method . . 2:il— 232 Problems producing Affected Equations 233 Discussion of General Equation — Problem of Lights . . . 234 — 239 Trinomial Equations — Definitions 240 Binomial Surds — Theorems — Square Eoot of A dry- ii . . . 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 252 Special Artifices and Examples 253 viii CONTENTS. VIII— RATIO— PROPORTION-— PROGEESSIONS. ARTICLES. Ratio — Kinds — Antecedent and Consequent . . . . 2o4 — 256 Ratio — MuHii)liealion and Division of ... . ... 259 Ratio of Equality — Of greater and less Inequality . . 260 Ratio — Compound — Duplicate — Triplicate ....... 261 Ratios — Corajiarison of . ... 262 Proportion — Definitions . . . . 203 — 2fi6 Product of Means equal to Product of Extremes 267 Proportion from two Equal Products 268 Product of the Extremes equal to the Sqiiare of the Mean . 269 Proportion by Alternation — By Inveision . . 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 Proportioniils are in Proportion . . 276 Products of Proportionals are in Proportion .... 277 Continued Pro]tortion — Exerciteb — Problems .... . 27S — 279 IIarmonical Proportion 280 "\'ariatiox — Prnpositions — Exercises ... . . 281 — 290 Aritiimktical Progression — Increasing and Decreasing . . 291 last Term — Rule — Sum of Series — Rule — Table . 292—29-1 To insert m Arithmetical Means between two numbers, Ex. 16 29-1 Geometrical Progressiox — Increasing and Decreasing . . 295 last Term— How to find it— Sum of Series— Rule . . . 296—297 Sum of Decreasing Infinite Geometrical Series . . . 299 Table of General Formulte . ... . . 300 To find a Geometric Mean between two numbers, Ex. 18 . 300 To insert m Geometrical Means between two numbers, E.x. 19 300 Circulating Decimal? — To find the value of 301 Harmomcal PROGRrssioN — Proposition . . . . 302 — 303 Problems in Arithmetical and Geometrical Progression . 304 IX.— PEE.MUTATIONS- COMBIXATIOXS— BINOMIAl THEOREM. Permutations . .... 305 — 307 Combinations 308 — 309 CONTENTS. ix AIlTirLES. Binomial Theokem when the Exponent is a Positive Integer 310 Binomial Theorem ajiplied to Polynomials- 311 X.— INDETERMINATE COEFFICIENTS— BINOMIAL THEOEEM —GENERAL DEMONSTRATION— SUMJIATION AND INTERPOLATION OF SERIES. Indeterminate Coefpicients — Theorem — Evolution . . . 314 317 * Decomposition of Rational Fractions . 318 Binomial Theorem for any Exponent — Application of . . . 319 — 321 E.^traction of Roots by the Binomial Theorem 322 Limit of Error in a Converging Series 323 Differential Method of 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 3.33 — 33i Summation of Infinite Series 336 — 338 Recurring Series 339 — 3-13 Reversion of Series 344 — 346 XL— CONTINUED FRACTIONS— LOGARITHMS-EXPONENTIAL EQUATIONS— INTEREST AND ANNUITIES. Continued Feacttotis . 347 — 356 Logarithms — Definitions — Characteristic Table . . . . 367 — 359 Properties of Logarithms — Multiplication — Division . . . 360 — 361 Formation of Powers — Extraction of Boots 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^Simple Interest 381 — 385 Compound Interest — Increase of Population 386 — 387 Compound Discount — Formula: — Annuities 388 — 391 CONTENTS. XII.— GENERAL THEORY OF EQUATIONS. Definitions — General Form of Equations . . An Equation whose Root is a is divisible by x — a . An Equation of the nth Degree has n and only n roots Relations of the Roots and Coeificients of an Equation What Equations have no Fractional Roots . . To change the Signs of the Roots of an Equation Number of Imaginary Roots of an Equation must be Descartes' Rule of the Signs Limits of a Root — A method of finding .... Transformation op Equations . . . Synthetic Division — Transformation of Equations Derived Polynomials — Law of — Transformation by Equal Roots . . Limits of the Roots op Equations . . . Limit of the greatest Root— Of the Negative Roots Sturm's TuEORiiii .... by ARTICLES. 393— 39i 395 396—397 398 399 400 401 402 403 404—403 409—410 411—413 414 415 416^18 420—427 XIII.— KESOLUTION OF XUMEEICAL EQUATIONS. Eational Roots — Rule for finding .... 429 Horner's Method op Appeoxim ition 430 434 Approx]mation' hy Double Position 436 Newton's Method of Approximation 437 Cardan's Rule for Solving Cubic Equal ioijs 438 441 Ekciprooal or Recurring Equations 442 Binomial Equations . , 443 444 HIGHER ALGEBRA. I. DEFINITIONS. Article 1. Mathematics is the science of the exant relations of Quantity as to Magnitude or Form. 2. Quantity, as the subject of mathematical investicja- 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 are 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 "quantity" whenever used in algebra, is synonymous with "number." G. The algebraic symbols are of two kinds: 1. Symbols of numbers ; 2. Symbols of relation. Numbers are usually represented by letters ; as, a, 5, 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 -^ V^- 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, x=b, denotes that the quantity represented by X equals 5. 9. The Sign of Addition, +, is read plus. It denotes that 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 minus. It denotes that the quantity to which it is prefixed is to be subtracted. Thus, a — b denotes that h is to be subtracted from a. 11. The signs -(- and — are called the signs. The former is called the posi/ivf, the latter the negatloe 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, -|- 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 Jihe signs ; those having different signs, unlilce signs. Thus, -)-a and -\-h, or — a and — I), have like signs.; while -j-c and — d have unlike signs. 14. The Sign of Multiplication, X, is read into, or raultipUed hy. 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, h, and c, is denoted by aX^'X^j Of a.L.c, or ahc. DEFINITIONS. 13 15. Factors are quantities that are to be multiplied together. Thus, in the product ah, there are two fac- tors, a and 6; in the product 3x5x7, there are three factors, 3, 5, and ^. 16. The Sign of Division, -^, 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-=-6, or -, signifies that a is to be divided by 6. 17. The Sign of Inequality, >, denotes that one of the two quantities between which it is placed is greater than the other. The ojjcning of the sign is toward the greater quantity. Thus, a^h, denotes that a is greater than 6. It is read a greater than I. Also, c<^d, denotes that c is less than d, and is read, c less than d. 18. A CoefiB.cient is a number or letter prefixed to a quantity, to show how many times it is taken. Thus, if a is to be taken 4 times, instead of writing a-\-a-\-a-{-a, write 4a ; also, az-\-az-\-az^Saz. The coefficient is called nvmeral or literal, according as it is a number or a letter. Thus, in the quantities Bx and mx, 5 is a numeral and m a literal coefficient. In Saz, 3 may be considered as the coefficient of az, or 8a as the coefficient of z. When no numeral coefficient is expressed, 1 is understood. Thus, a is the same as la, and ax the same as lax. 19. A Power of a quantity is the product arising from multiplying the quantity by itself one or more times. When the quantity is taken twice as a factor, the product is called the second power ; when three times, the third power ; and so on. 14 RAY'S ALGEBRA, SECOND BOOK. Thus 2X2^ 4, is the second power of 2. 2x2/;-^— ^1 '^ ^^^ third power of 12. 2x2x2x2=16, is the fourth jower of 2. Also aX«= f"^i i^ ''"* scco'id I'ower of a. a"^a'X.a=aaa, is the third power of a; and so on. The second power is often termed the square, and the third power, the cube. An Exponent is a small figure or letter placed on the right, and a little above a quantity, to express its power. Thus, aa=:a', aaa^a', etc. a" indicates that a is taken as a factor as many times as there are units in m. When no exponent is expressed, 1 is understood. Thus, a is the same as a', each signii'ying the first power of a. 20. A Boot of a quantity is a factor, which, multiplied by itself a certain number of times, will produce the given quantity. The root is called the square or seiviid root, the mibe or third root, the fourth root, etc., according to the number of times it must be taken as a factor to produce the given quantity. Thus, 2 is the second or square root of 4, since 2x2=4 ; a is the fourth root of a*, since a> uX"X«=oi*- 21. The Radical Sign, ]' or ■^ , when prefixed to a quantity, denotes that its root is to be extracted. An Index is a figure or letter placed over a radical sign to denote what root is to be taken. Thus, ■j-'Q, or f i>, denotes the square root of 9, which is 3. 1^8, or f 8, denotes the cube root of 8, \vhich is 2. I^ a, or i^'u, denotes the fourth root of a When the radical sign has no index over it, 2 is under- stood ; thus, J (( and ^-''« signify the same thing. 32. An Algebraic Quantity, or an Algebraic Exqjrcs- DEFINITIONS. 15 sion, is any quantity written in algebraic language, that is, by means of symbols. Thus, 5a, is the algebraic expression of 5 times the number a; 36-)-4c, is the algebraic expression for 3 times the number 6 in- creased by 4 times tlie number r; 3a2 — Tab, for 3 times the square of a, diminished by 7 times the product of the number a by the number b. 23. A Monomial is a quantity not united to any other by the sign of addition or subtraction ; as, 4a, d'hc, — 4a;!/, etc. A monomial is often called a simple quantity, or term. 24. A Polynomial, or Compound Quantity, is an alge- braic expression composed of two or more terms ; as, a-\-h, c — x-\-y, etc. 25m A Sinomial is a quantity having two terms ; as, a-j-t, x'-j-y, etc. A Residual Quantity is a binomial, the second term of which is negative ; as, a — h. 26. A Trinomial is a quantity consisting of three terms; as, a-\-h — c. 2T. The Numerical Value of an algebraic expression is the number obtained by giving a particular value to each letter, and then performing the operations indicated. Thus, in the algebraic expression 4a — 3f, if a=5 and c=6, the numerical value is 4x5—3x6=20—18=2. 25. The value of a polynomial is not affected by chang- ing the order of the terms, provided each term retains its sign. Thus, P — 2a6-|-c is evidently the same as h'-\-c — 2ah. 29. The Degree of any term is equal to the number of literal factors which it contains. 16 RAY'S ALGEBRA, SECOND BOOK. Thus, ba is of &« first degree; it contains one literal factor. ax is of the second degree; it contains two literal factors. Za^lj-c='iaaabbc. 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^' — ~xy- is homogeneous; each term being of the third degree. X- — ixy'^ is not homogeneous. 31. An algebraic quantity is said to be arranged ac- cording to the dimensions of any letter it contains, when llic exponents of that letter occur in the order of their magnitudes, either incnd^ing or decreasing. Thus, ax--{-a-x^a'':c", is arranged according to the ascending powers of a; and 6.c^' — 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 — (« — 6) means that a — b is to be subtracted from 10. 5{a-if-b — c) means that a^b — c is to be multiplied by 5. LKi^(b—c) means that b — c is to be added to 5a. When the parenthesis is used, the sign of multiplication is generally omitted. Thus, (a — h)y^(a-\-h), is written i"-l)Ca+h). A Vinculum, , is sometimes used instead of a parenthesis. Thus, a-\-byc5 means the same as b(a-\-h). Sometimes the vineulum is placed vertically ; it is then called a hnr. Thus, a X-, is the same as (a — h-\-c)x''. —h 33. Similar, or Like Quantities, are such as contain the same' letter or letters with the same exponents. DEFINITIONS. 17 Thus, 3o6 and —2ab, BaPb and 5a't, 3a'6 and —^a'b, are similar. Unlike Quantities are such as contain diflferent letters or differeitt puwers of the same letter. Thus, 5a and 3Z>, Zah' and 3a'^6, are unlike or dissim- ilar. Remark. — An exception must be made in those cases where let- ters are taken to represent coefficients. Thus, ax- and bx- are like quantities, when a and 6 are taken as coSfficieuts of a^. 34. The Reciprocal of a quantity is unity divided by that quantity. Thus, The reciprocal of a is — ; of 3, is tj ; of j, is 1-^|=^ ; Hence, Tlie reciprocal of a fraction is the fraction inverted. 35. The same letter, accented, is often used to denote quantities which occupy similar positions in different equa- tions or investigations. Thus, a, a', a", a'", read, a, a prime, n second, a third, and so on. 36. The following signs are also used, for the sake of brevity : oo, a quantity indefinitely great, or infinity. . • . , signifies therefore, or consequently. • . • , signifies since, or because. , .J is used to represent the difierenee between two quan- tities, as c—'d, when it is not known which is the greater. EXERCISES. First, copy each example on the slate or blackboard; and then read it, that is, express it in common language. Second, find the numerical value in each, supposing a=2, b^3, c=4, a;==5, y^6. 2d Bk. 2 18 RAY'S ALGEBRA, SECOND BOOK. ceo— ay 1. 76+x— y. Ans. 20. 2. d'hij-^ixK Ans. —3. 3. c4-aX''^a- Ans. 10. 4. (c-|-a)(c— a.) Ans. 12. 5. — ■ — jJ ^. Ans. l}-,. „ o6(c — a) y—c Ans. 4. Ans. 5. -j/a6y. Ans.O. 9. Find the difference between alix, and a-|-6-|-a:, when a=4, 6=i, x=3 ; and when a^5, i=*7, a:^12. Ans. 11 and 396. 10. Required the values of a''-{~2ab-\-V, and a^ — 2ah-\-¥, when a=7 and Zi=4. Ans. 121 and 9. 11. What is the value of n{n~l) (n— 2) (n— 3), when 7t=4, and when j!=10 ? Ans. 24 and 5040. 12. Find the difference between 6o5c — 2ab, and Qobc^2ab, when rt, i, f, are 3, 5, and 6 respectively. Ans. 492. 13. Find the value of — , when a=5 and 6^3. Ans. 5^. Verify the following-, by giving to each letter any value whatever : 14. a(jn^n')(in — 92)=om' — an^. ,-,■3 )/ 15. ^x'''[-xy-\-y^. TO BE EXPRESSED IN ALGEBRAIC SYMBOLS. 1. Five times a, plus the second power of h. 2. X, plus y divided by 3.3. 3. X plus y, divided by 3-. -t. 3 into X minus n times y, divided by in 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. ADDITION. ANSWERS. 1. 2. ba+h\ ^- ST- 5. 6. a'— a:» ■j/TO -p/jl. m — n 7. -[/(m — n). 19 ADDITION. 37. Addition, in Algebra, is the process of finding the simplest expression for the sum of two or niore 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. 3d. When the quantities are dissimilar, or part similar and part dissimilar. 38. First Case. — Let it be required to find the sum of 3a;'y, bx'y, and /x'y. Here, x'^y is taken, in the first term, 3 times; in operation. the second, 5 times; and in the tliird, 7 times; -f 3x^1/ hence, in all, it is taken 15 times. Since adding -) 5x'''j/ the quantities can not change their character, and -j- 7a:-?/ since each term is positive, their sum is positive. ~ ~ Find the sum of — 3a;'y, — 5.a:'y, and — Va:'^. Here, x-y is taken, in the first term, — 3 times ; in opkration. the second, — 5 times ; and in the third, — 7 times ; — 3x"y hence, in all, it is taken — 15 times. Therefore, to — Eix'h/ add similar quantities having the same sign, — 7x'y —15xhj Utile. — Add the coefficients, and prefix the sum, with the comvion sign, to the literal part. 20 BAY'S ALGEBRA, SECOND BOOK. 39. Second Case.— Let it be required to find the sum of +9«, —5a, +4a, and —2h-\-c, -\-h — a^, and 5i-)-3a-- In writing the quantities, we place, for con- operation. venuncc, those which are similar under each 5a^ — 86-f-c other. — a--\- b The sum in the first column is -f-7a-, and in Sa--|-56 the second, — 2b; there being no term similar . „ ., . 1 -.1 ■. • 7a- — 26+c to C, it IS annexed, with its proper sign. ' 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 similar under each oilier. SUBTRACTION. 21 2. Add the similar quantities hy 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 4ax-|-3%, bax-{-Sbi/, 8aa;-f 66y, and 20ax+hi/. Ans. 3Vaa;-|-18fty. 2. Of 10cz—2ax\ Ibcz—Sax-', 24cz—9ax\ and 3cz — Sax''. Ans. b2cz — 22ax^. 3. Of ox-f—lOy*, —xy-\-by*, 8xy—6y*, and ixy + 2/. Ans. lAxY—9y*. 4. Of a-\-h-\-c-\-d, a-\-l-\-c — d, a-\-h — c-\-d, a — b-\-c -j-d, and — a-\-b-\-c-\-d. Ans. 8a-(-36-|-8c-|-8(i. 5. Of 3(x'— /), Six'—f), and —b(x'—f). Ans. 6 (a;' — y''). 6. Of 10«^& — 12a'6c— 15iV+10, —4a'b + 8a'bc —lOlrc^^i, —3a'b—Ba%c+20Vc*—3, and 2a:%+12a'bc +56V-^2. Ans. ba^b+ba^bc+b. 7. Of a'"—b"+3xP, 2a"— 36"— x*, and a'^+W—xi. Ans. 4a"'+2xi'— x«. 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 RAYS ALGEBRA, SECOND BOOK. 43. First Case. — Let it be required to subtract ia from Ta. It is evident tliat 7 times any quantity, operation. less 4 times that quantity, is equal to 3 times 7a Minuend, the quantity, therefore, 7a less ia is equal 4a Subtrahend, to 3a. 3a Remainder. If it be required to subtract b from o, we operation. can only indicate the operation, by placing a Minuend, the sign minus before the quantity to be sub- b Subtrahend, tractcd. a — b Remainder. 44. Second Case. — Let it be required to subtract b — c from a. If we subtract 6 from a, the result, a — 6, operation. is obviously loo little, for the quantity b a Minuend, ought to be diminished by C before it is taken b — c 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, o — 6, must be increased by C; this g.ivcs, for the true remainder, a — b-\-c. To illustrate the above example by figures, let a=^9, 6^5, and C^3; and let it be required to subtract 5 — 3 from 9. The operation and illustration may be compai-ed, thus; From a From 9 .... t=9 Take 6— c Take 5—3 ... ^2 Rem. a—b-\-o Rem. 9— 5-f 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 ALflEBRAIC QUANTITIES. 1. Wriie the quantities, placing similar tcnna under each other. SUBTRACTION. 23 2. Conceive the signs of all the terms of the stihtrahend to he changed, from -|- to — , or from — to -f , and then pro- ceed hy the rule for algebraic addition. Remark.- — 1. Beginners sliould solve a few examples hy actually changing the signs of Ihe subtrahend. 2. Proof. — Add the remainder and ihe subtrahend, as in arithmetic. 0) (1) From 8a^6-3rrr- .n T„e..me,with f ^a%-Zcx- z^ Take 3ci='i+4ca;— 32= V the Bigns of tUo <^ _3a26_4ca; i 3^2 — I subtrahend j ! Kern, ba^h—1 cx-\-2z' J changed. vRem. ba^h—*J cx^2z' (2) (3) From 5a' — 3mz-(-5?/* From ax' — 3cy — z' Take — 2a^-f 3mz+6/ Take fcx'— 3r/+y Rem. 7a' — Qmz — y Rem. (a — h)x' — f — z^ 4. From 4a— 26-f 3c take 3a+46— c. Ans. a — 66+4c. 5. From 9a:2— 4^+9 ^ake 7x'+5!/— 14. Ans. 2x''— 9.y+23. 6. From 23«/— 7^+lla;' take llxif—by—^x'. Ans. 12a;/— 2y+20x'^ 7. From 12a;+18 take 12a;— 18+y. Ans. 36— y. 8. From x'—f take — 4— y+4a:l Ans. 4— 3x^ 9. From 4ax^-\-hx-\-c take 3a;'— 2x-|-5. Ans. (4a— 3)a;'+(2 + 6)a;+c— 5. 10. From — 17a;'-j-9aa;^— 7a^a;+15a' take — 19a:'+9ox^ — 9a^a;+l7a'. Ans. 2x'+2a^a;— 2a'. 11. From a;'+3a;^+3a;+l take x^—Sx^+Sx~l. Ans. 6a;' -j- 2. 12. From ga^a;'— 13+20ai'a:— 46'"fa;' take 3i"'ca;^ + 9a°'a;'— 6 + 3ai'.r. Ans. 17oi'x— 7i'"ca;'— 7. 13. From 4a'^+2xP~xi take a"— i"+3a:J' and 2a'» — 36"— a;^. Ans. a'^+ib'—xi. 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. Wlif-rc the sign jyhts precedes a parenthesis, or vinculum, it may he amitted iciihout affecting the expression. This is self-evident, as is also the converse, viz. : Any number of terms may he inclosed within a parenthesis preceded by the sign plus, without affecting the value of the expression. Thus, a+(6— e)=n'+6— c; 6+(5— 3)=6+5— 3, and a+6— c -|-d=a+(6-c+d) ; 6+4— 3 + 2=.5^(-l— 3+2). 2d. Where the sign 7ninus j^i'ccedcs a vinculum, it may be omitted if the signs of all the terms within it he changed. For the minus indicates subtraction, which is eifected by changing the signs of all the terms of the cjuantity to be subtracted. Thus, a—(/)—c)=a — b-\-f. a—{x~y-\-z)=a~x^y-z. Sometimes several brackets, or vinculums, are employed in the s;ime expression, all of which may be removed. Thus, a—\a + b—la+b—c—(a—b^ (■)']], =o— |r(+6— [«-|-6— c— a+6— e]}, =a— [rt + 6— rt— 6+c+a— 6+c|, =a—a—b-\-a-\-b~c—a-\-b~c=b~2c. 3d. Any quantity may he inclosed in a parenthesis preceded hy the sign minuf, prodded the signs of all the inclosed terms he changed. This is evident from the preceding principle. Thus, a— 6+c=a— (6— o)=c — (b~a). OBSERVATIONS. 25 This principle often enables us to express the same quan- tity under several different forms. Thus, a — b-\-c-^d=a—{b—c—d], =a-{6-(c+d){. Simplify, as much as possible, the following expressions; 1. (1— 2a;+3x^) + (3 + 2x— a:'). Ans, 4 + 2x1 2. (a-t-c) + (6 + 6— <0 + (d-e+/) + (e— /~9). * Ans a — g. 3. 3 (x^+y) - { (x^+ 2xy+f)—(2xy—x'~f) } . Ans. ^"+2'^. 4. a — (x — a) — [a; — (a — a;)]. 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 Borne 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 gains are positive, his losses are negative; if lati- tude north of the equator is -{-, that south is — j 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 before that hour is — ; if motion in one direction be -\-, motion in an opposite direction is — ^ ; and so on. 418. This relation of the signs gives rise to some important particulars. 1st. The additio-n, to any quantity, of a negative numher, l^roduces 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. 2d The subtraction of a negative quantity produces a greater result than subtracting zero. Thus, 10 10 10 10 10 10 10 3 _2 _1 -1 -2 -3 ^ "S 9 10 11 12 13 Here, subtracting a negative number produces the same result as adding an equal positive numbei-. 40. When two negative quantities are considerfd 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 numerically the greatest; thus, — 3 is numerically greater than — 2. 30. The sum of two positive quantities is always ^rra(f7- than either of them. Thus, +5-|-3=4-8, The sum of two negative quantities, algebraically considered, ia 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, -|-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 — (-f 2a)=-f 3a. The diflFerenoe of two negative quantities is always greater, alge- braically considered, than the minuend. Thus, — 2a from — 5a leaves —3a, or — 5a — (— 2«.)^ — 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 1 A. M., a, thermometer stood at —9° ; at 2 P. M., at +15° ; 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. — The 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 6 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 6 times ; that is, by ba. Hence, ab is equal to ba. In a similar manner, it may be shown that Tlie product of three., or of any number of factors, is the same, in whatever order they are taken,. Thvis, aX^Xc=ahc, cah, lac, or cha, and 2x3x4=4 X2X'^=3x2x4=4x3x2 ; the product in each case being 24. Also, acX 6=6«c', or 6ca; and so on. It also follows from this principle, that When either of the factors of a product is multij)lied, 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. Eemaek. — The distinction between the multiplication of num- bers and of factors should be carefully noticed. Thus, 3iiX2=P4, but 3X2 multiplied by 2 equals 6X2, or 3x4. 28 RAY'S ALGEBRA, SECOND BOOK. 54. In multiplication there are four things to he con- sidered in relation to each term, viz. the cuejficient; 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 3i. To indicate the multiplication, we may write operation. the product thus, 2acX36. But, by Art. 53, this 'lac is the same as 2x3X0^0, and 2x3^6; therefore, 36 the product is 6a6c. Hence, ~ ;~ 6o6>c product. Rule of the Coefficients.- — Multiply together the coefficients of the factors for the coefficient of the product. Rule for the Literal Part. — Aitnex^ to the coefficient all the letters of the factors in alphahelical order. 2. 3«cX56= 15ahc. 3. 2amy_cn^ 2acmn. 4. 5aX4o.T= 2Qaax. 5- 1cyX3yz= 2\cyyz. 56. Of the Exponent— To determine the rule of the exponents, 1. Let it be required to find the product of 2«^ by 3a'. Since 2a-=2o«, and 3a3=3aaa, the product operation. will be laayZaaa, or &aaaaa, which, for the 2a-=2aa sake of brevity, is written 6a'. Hence, we 3a''=3oaa have the following Qa'-=Gaaaaa Rule of the Exponents. — Add the exponents of any letter ill the factors for its exponent in the product. 2 nhXo= a'h 5. a'"Xa''= a"+" 3, x'yXxy= a-y 6. c"+iXf"-'= c^» 4. a'x'zyaxz':^ a^x'z' 1. x'''+PXx"'-^= ^m+n 57. From the two preceding articles, we derive the fol- lowinn; MULTIPLICATION. 29 GENERAL UULE FOR MULTIPLYINa 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 loth, add its exponents. 1. Multiply ic by 2 Ans. hcz. 2. Multiply Zax hy hy Ans. Bahxy. 3. Multiply 4a?/i by Sbn Ans. 12abmn. 4. Multiply ^a'x by *laa?T/ Ans. '6ba^x*i/. 5. Multiply 3a"'a;" by Qa-'x™. . . 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 tn times. The units in a taken m times a-\-b =ma, and the units in b taken m times =mb; m hence, a-\-b taken m times =^maA-'mb. Hence, r 7nCi-\-Tliti 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 n Ans. na;+ny. 3. ax^-{-cz by Sac Ans. Ba'cx''-{-Bac'z. 4. 2a''+S¥ by bab Ans. 10a'b+lbab\ 5. mx-{-ny-{-vz by m'n. Ans. mhix-\-mVy-\-m'nvz. 59, — 1. Required to find the product of a+i by m-\-n. Here, a-\-b is to be taken as many times as there are units in in-\-n, which is evidently as many times as there are units in m, plus as many times as there are units in n. Thus, a+b ma-\-mb=: multiplicand taken m times. na-\-nb= multiplicand taken n times. ma-\-nib-\-na-^nb^ multiplicand taken {m-\-n) times. 30 RAYS ALGEBRA, SECOND BOOK. Hence, when the signs are positive, we have the following Rule for Multiplying^ one Polynomial by Another.— MaUiphj each term of the midtqilicand hy tach term of the niidtiijUir, and add the jO'oducts. 2. Multiply x-^y by a + c. . . Ans. ax-\-aj/-\-cx-]-n/. 3. 2.r+3-; by Sx+2z. . . . Ans. 6x'+13xz+6z\ 4. 2a + c by a + 2r Ans. 2a'+bac+2c\ 5. x--\-xi/-\-T/^ by .T+y. . Ans. a?'+2a;^y+2xy^+y. 6. a^+2((l + V by (( + 1. . Ans. a^-i-Sd'b+Sab'+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 -\-l by a. The quantity 6, taken once, is -j ''; taken twice, is -|-26; taken S times, is +36; and hence, taken CI times, it is -f a&. Hence, the product of two positive quantities is positive; or, more briefly exjuessed, plus multijilied by jihis gives pltis. 2d. Let it be required to find the product of — b by a. The quantity — 6, t.ikcn once, is — 6; taken twice, is — 26; taken 3 times, is — 36; and hence, taken a times, is — ab. Hence, a negative quantity, nniltipliid liy a poyitive quantity, gives a negative product; or, more briefly, minus multiplied by plus gives minus. 3d. Let it be required to multiply h by — a. Since b'X-\-a implies that 6 is to be added a times to 0, by^—a must indicate (Art. 47) that b \sio\xi 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, plm multiplied by minus gives minus. MULTIPLICATION. 31 4th. Let it be required to multiply — b by — a. Reasoning as above, — b subtracted a times from 0, gives -\-ab. Hence, the product of two negative quantities is positive; or, more briefly, minus multiplied by minus gives phts. 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 C — d as many times as there are units in a — 6. This will be done by taking e — d as many times as there are units in a, and then subtracting, from this product, c — d taken as many times as there are units in 6. Thus, c — d a—b ac — ad=c — d taken a times. 6c — bd^c — d taken b times. ac — ad—bc-{-bd. By subtraction, ^c — d taken a — b times. The final result, in the terms, — be and -\-bd, is what it would have been if we had added the partial products, assuming that -\-c multiplied by • — b gives — be, and that — d multiplied by — 6 gives -j-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 Oeneral Rule for the Sigas. — Tlie 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. NFMERICAI> EXAMPLES TO VERIFY THE RULE OF THE SIONS. 1. Multiply "7-4 by 5. Ans. 35-20=15=3x5. 2. 8 + 3 by G- 4. Ans. 48--14— 12=22=11 x2. GENERAL EXAMPLES. 1. Multiply 4rt^— 3ac-f 2 by bax. Ans. 20a''x~-lbahx+10ax. 2. 5a— 2aZ.+10 by —9ah. Ans. -~45a'h + lSa'L'~90ah. 3. 2x + 3.-: by 2.r— 3^. Ans. 4x'-^9z'. 4. 4a^_6a + 9 by 2<,+ 3. Ans. 8a^+27. 5 a — t+f — d by a+i — c — (7. Ans. a'—h'—c''+(P—2ad+2bc. 6. a^+y+z^ by .'■^■+7/^ Ans. a:5+a;y+.T;2c=+a-y+y'+/zl 7. a'+Sa'h + Bah'+h' by a^_3a=i + 3aZ-=-Z.'. Ans. a''—3aV-+8a'l*—h\ 8 12x^— 8a;^y+152-/— 10/ by 8a:+2y. Ans. 36a;*+29a;y— 20/. 9. <:''-{- ax+.r' by +3.T/+/ by Sx-y. Ans. 81rr.*— /. 13. a'+2a'Z- + 2a6^+i' by a''—2a'b + 2alr—h\ Ans. a* — h^. 14. rr^—a^'+a:.'— 3;+l by x' + rr— 1. Ans. x" — a;*+a;'— .r=+2,r— 1. 15. l+.T+.T*+a;5 by 1— a:+.r-— .r' Ans. 1— a:^ IG. .^lultiply together a;— 3, .t+4, .-•— 5, and .t+6. Ans. a;*+2,,-^— 41a;^— 42., + 3d0. 17. a-\-b, a — b, a'+ab+b', and «- — ab+b'. Ans. a^ — 6". MULTIPLICATION. 33 62. Multiplication by Detached Coefficients. — In the multiplication of polynomials, 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^— 2o6-f &' by a-\-h. orEKATioN. 1-2+1 After finding the coSfEcients, it is obvious \-\-\ that a' will be the first term, and 6* the last -. 9, i term; hence, the entire product is a-' — a^b , -^ o\\ — a62-f63. 1_.1_1+1 2. Multiply a?—?,a'h-\-V' by a^—h\ In this example, supposing the powers orERATiON of a to decrease regularly toward the 1 — 3-|-0-|-l left, it is obvious that there is a term l-]-0 — 1 wanting in each factor. These must be 1 — 3_|_0-|-1 supplied by 0. The entire product is — 1_|_3 — Q— 1 a''— 3a46-a362+4a263— 65. i_3_i 1 4_IoZi 3. Multiply 'm?-\-'m?-n-\-mn''-\-n? by m — n. Ans. m* — n*. 4. Multiply l+2»-f3z^-f 42^^+5.^ by 1— 2. Ans. l-j-z-f-z'-f-z^+r.*— 5-^ By this method, let the general examples. Art. 61, from 7 to 14 in- clusive, be solved. REMARKS ON ALGEBRAIC MULTIPLICATION. 63. The degree of the product of any two monomials is equal to the sum of the degrees of the multiplicand and multiplier. Thus, 2a26, which is of the 3d degree, multiplied by 3a&3 of the 4th degree, gives 60^6^, 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, Ait. 61. 6-1. In the multiplication of two polynomials, when the partial products do not contain similar terms, if there be 171 terms in the multiplicand, and n terms in the multiplier, the number of terms in the product will be m\n. Thus, in Example 6, Art. 61, there are 3 terms in the multiplicand, 2 in the multiplier, and 3x2^6 in the product. 63. If the partial products contain similar terms, the number of terms in the reduced product will evidently be less than m.\n; see Examples 7 to 18 inclusive, Art. 61. 66. When the multiplication of two polynomials, indicated by a parenthesis, as (m-\-ri)[2-> — (j), is actually performed, the expres- sion is said to be cj^iandcd, or developed. DIVISION. 67. 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, Diviriion teaches the method of finding- the other. The quantity by which we divide is called the dii-isor; 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 bl=+ab . (+ab^ -+b=+a erefore, - 1 —ab- 1 +('(>- --\-b=—a l—ab- — b=-\-a DIVISION. 35 From the foregoing illustration, we derive the following Rule of the Signs. — Like signs in the iUvisof and divi- dend give jtlus in tlie quotient; unlike signs give minus. TO. The rule of the coefficients, the ride of the exponents^ and the rule of the literal part, may all be derived from the solution of a single example. Eequired to find how often 2a^ is contained in 6a*6. •la- 2 Since diTision 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 Hule of the Coefficients. — Divide the coefficient of the dividend hy 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. — Subtract the exponent of any letter in the divisor from the exponen t of the same letter in the dividend for its exponent in the quotient. 3J. The letter b, which is a factor of (he 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. 71. The preceding rules, taken together, give the fol- lowing 36 KAY'S ALGEBRA, SECOND BOOK. GENERAL RULE FOR DIVIDING ONE MONOMIAL BY ANOTIIEE. 1. Prefix tlic proper sign, on the principle 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 hoth. 4. Amiex any letter found in the dividend hut rwt in the divisor. 1. Divide 4a* by 2o' and by 2. ^OaV by ba%. . . . —28x'yh* by —Ixy^'z. 3. 4. 5. 6. 7. ^SbcrlJc by 5aV S2xyz by — 8.T7/, 42c^m''n by — 3c?)!i(. x"'+'' and x'"~" each by x" ,,m+n by V ,m+p Ans. 2a' and —2a'. . Ans. Qd'b- . . Ans. ix'^f'z^. . . Ans. — Tahc. . Ans. — 4.Z. Ans. — lic^m. Ans. x" and x"'~^". . Ans. v"~f. Note. — In the following examples, the quantities included ■within lie parenthesis are to be considered together, as a single qu.antitj 9. Divide (a+i)' by (n+i)'. . 10. (m—n)' by (m — ny . . 11. 8(a—hyx'i/ by 2(u—h)xy. 12. (a+hx'y'+^ by (^a + hx-)P-' . Ans. (a-\-l). . Ans. (to — ny. Ans. 4(a — hy.r. Ans. (^a+hx;'y. 73. 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 'bund in the dividend. DIVISION. 37 In each of these cases the division is to be indicated hy a fraction. See Art. 119. '7S. 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, 6X9^3=^2X9; or, 6X3=18. '74. Division of Polynomials by Monomials. — In mul- tiplying H polynomial by a monomial, we multiply each term of the multiplicand by the multiplier. Hence, conversely, we have the following RULE rOR 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 -|-6. 2. '6xy-\-2x'y by — xy Ans. — 3 — 2x. 3. 10a-z—lbz'—2bz by 5z. . . . Ans. 2a'— 3z— 5. 4. 3ab-\-12ahx—9a''b by —Sah. Ans. — 1— 4a:+3a. 5. 5a^y — 4:0a:'x''y^-\-2ba*:ry by — bxy. Ans. — x''y''-\-8a^xy — 5a*. 6. 4a6c— 24aZ<'— 32a6d by — 4a6. Ans. — c+6Z-+8(Z. 7. a'"V+a'^+V+a''-% by ah. Ans. a'"~'J^+a'"&-l-a"-'. 8. Sa(x-\-y') + c''(x-\-yy by x+y. Ans. 3a+c'(a;-|-^). 9. (h+c)(^h—cy—{h—c)(b+cy by (6+c)(i— c). Ans. (fc— r)— (6+f)=— 2c. 10. b^c(m-\-n)—hc''(m-\-n) by hc(m-\-n) Ans. b — c. 38 RAY'S ALGEBRA, SECOT!fD 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. Multipliciition, or formation of a product. a''— 5a<6 +2ai6-10a362 -a'6^-|-5a253 a5— 3a'6— Ila362-|-5a263 Division, or deromposition of a product. a'-5a26 a''-3ai6-lla362+5a=63 ft'' -oa*b i.tu+2a<6^na''62+5a263 -j-2ai6-10a-W 2d Kcmaiuder. — O^b-^ha-b'i —a^b^+ba^b^ a^+2ab~b2 Quotient, :id 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 teims, constituting the quotient, we are now to find. Arranging the dividend and divisor according to the decreasing powers of the letter o, it is plain that the division of a'\ tlic 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 (r—5a*b, 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 -\-2a'/> of the 1st remainder, is the product of the l^t term a^ of the divisor by the Ist of the remaining unknown terms of the quotient; theiefore, we shall obtain the 2d term of the required quotient, by dividing -\-2a^b by «■"' ; this gives -\-2ab. Multiplying the divisor by -f 2a6, 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 ; licnce, the division of the 1st term — a^b- of this 2d remainder, by the 1st term cfi of the divisor, must give the 3d term of the quotient, which is found to be ~b-. The remainder zero, shows that the quotient a-~^2ab--b- 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. 76. From the above, we derive the following RULE FOR THE DIVISION OP 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 by this term, and subtract the product from the dioidend. 3. Divide the first term of the remainder by the first term, of the divisor, for the second term of the quotient. Multiply the divisor by this term, and subtract the product from the last remainder. 4. Proceed in the same manner, and if the final remainder is 0, the division is said to be exact. 1. Divide lbx'-\-lQxy — 15/ by 5a; — Sy. OPERATION. lDX^-+l&xy-lby^\ 5x-Sy 15a;2— 9xy Zx^by, Quotient. -\-2bxy—\5y^ -{-25xy—15y^ 2. Divide m'' — )i' by m-\-n. OPERATION. j^2 — fi2 \m-\-n ifffi^ran m — n. Quotient. — ran — v? — mn — v? 3. Divide a;'-f / by x-\-y. OPERATION. x^^^y a;2— a;j/+2/2, Quot. -xhi—xy'^ -xy-+y3 xy^+y^ 40 RAYS ALGEBRA, SECOND BOOK. 4. Divide 7.r>+5a;/+23r'+/ by 3a;j/-j-a;'+/. Arranging the divisor and dividend with reference to X, we have the folluwing: OPERATION. 2j--'-r 'x-ijibxy^+U' I .'-■--'-"■'■.!/ -^'/ 2x'- -Gx->/^2x>/- '±0-11, (jLiotieat. x^ij^5xy^'~if x''-y\?jxy-^ if- 5. Divide ar'+a:'' — 7a;^+5,r5 by a;— a:' Division prrformod, bj' arranging both quantities aLcuitliiig to the asu-itding powers of X. x^-^a:''— 7.r*-j-5z'|x— :c2 — Sar'+o.r'' Division performed, by arranging 'joth quantities according to the dt^cending pOWLTS of X. X ~x^ ox'' 5x-' — 7.i-'-(-a;'-|-a;-| — x--^x x+2x2—'jx\ —5a;* — ox-''-\-±c'^~x, Quotient. — 2x'+ x^ Quotient. -2ari+ 2.1-3 —X-'+X^ —X^'^X- The two quotients above are the same, but diiferently arranged. 6. Divide Gx--\-5xi/ — 4y by Sx-\-ii/. Ans. 2x—ij. 1. .r^— 40a-— 63 by .?■— 7. Ans. .T-4-7.t- + 9. 8. 3A5-fl6/i*A:— 33;t^/l-^+147iV.-' by h^^'Jhl: Ans. 3 /i^— 5 /(-/,--{- 2 7^-|-at'-|-i*. Ans. a'— a'?)H-a^f^— ai^+i*. 22. a'-|-(a_l)x--^(a— l)s'+(a — l)a;* — «5 by a — cc. Ans. a-\-x-\-x''-\-oi^-\-x^. 23. 1— 9x«— 8a;» by 1+23;+cb^ Ans. l—2x+Sx''—4:3f-\-5x*—Qafi-\-1x^—Sx\ 24. l-|-2a; by 1 — 3x to 5 terms in the quotient. Ans. l + 5a;-|-15a:'+45a;^+135a:*-f 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-\-y. l-|-24-lil + l Hence, the cotfficienis of the quotient l-(-l 1^1 are 1 and 1. Also, a;2-=-.x---a;, and 2/^-^«/=2/; -j~l-\-l therefore, the quotient is la;-!-!^/, or x-\-y. +1+1 2. Divide 12«*— 26a'6— Sa^Z-^+lOaZ/'— 86* by 3a^— 2a& -]-b\ 12—26— 8+10— 8|3- 2 + 1 Hence, the coefficients of the 12 — 8+4 4—6—8 quotient are 4—6—8. Also, —18—12+10 a'-^a^=a', and b*^b'^=b'^; —18+12— 6 therefore, the quotient is ia^ ' —24+16-8 — 6a6— 862. —26+16 -8 2d Bk. 4 42 RAY'S ALGEBRA, SECOND BOOK. 3. Divide a^-j-x^ by a^x. l + O + O-t-l IH-I 1+1 1-1-^1 —1 a- — ax-\-x-, Quotient. ^1—1 ^TT+i +1+1 4. Divide vv' — 5??i'')i-|-10?;i'n^ — lOtnhi^ -{- bmn* — n^ by m' — 2mn-\-n''. Ans. m' — '6)n-ii-\-ij>nu'' — n'. 5. Divide <-«— 3(CL'+3a2Z/*— i« by a'—Sa'h-j-Sah'—P. Ans. a^^oa'h-\-Sah'-j-h\ Most of the examples in Art. 76 may be solved by this method. II. ALGEBRAIC THEOREMS, DERIVED FRO.AI 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. TH. Theorem I. — The square of the sum of two quantities is equal to th(f square of the first, plus twice tlic product of the first hy the second, plus tlie square of the second. Let a represent occ of the quantities and b a +6 the other. ' a -\-b Then,o+6=theirsum; and(o+6)X(fl+6), <^'-{- oh or (a+6)2^ the square of their sum. By mul- + a6 + 6- tiplying, we obtain d^-\-2ab-]-b', which proves a-+2o6+62 the theorem. APPLICATION. 1. (2+5)2=4+20+25=49. 2. (27W-|-3nj2_4;,i2_|_io„j„_l_9,j3_ ALGEBRAIC THEOREMS. 43 3. («a;+6?/)2=a2a;2-f2a63;2/+6=2/2. 4. {ax^+Sxzif=a^xi+6aa:^z^+Qx^zfi. 79. Theorem II. — TJw 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 ci' — ab (a—b), or (a— 6)2= the square of their dif- — ab+b^ ference. By multiplying, we obtain a^ — 2o6 q^2 2a6+62 -f 62, which proves the theorem. APPLICATION. 1. (5—3)2=25—30+9=4. 2. {2x—yy^=4x^~4xy-]-y". 3. (3a;— 52)2=9a;2— 30a;z+25z2. 4. {az—3cx)^=a^s^—'6acxz-j-9c^x^. SO. Theorem HI. — 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 6 the other. Then, a-\-b^ their sum, and a— 6= their difference. Multiplying, we obtain a^ — 62^ which proves the theorem. o?—b APPLICATION. 1. (7+4)(7-4)=49-16=33=llx3. 2. (2a;+3/)(2a;-2/)=4a;2— 2/2. 3. (3a2+462)(3a2-462)=9a''-]66*. 4. (3aa;+563/)(3aa;— 56«/)=9a2a;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. a -f 6 a~b o2+ ab - ab- b 44 KAY'S ALGEBRA, SECOND BOOK. Take the fraction j—^. Since we may divide both terms by the same quantity witliout changing the value of the fraction, (Eays Arithmetic, 3d Book, Art. 136), divide first by x^, and then by x'; (Art. 70). Thus, ax-^ ox^ ax^ a. a, ax- a a cix~'^ la a similar manner, it may be shown that -= — , = — = — . ' ■' bx- b 1 xr'^ „ 1 Also, — = = — ^ ^a:-", and a:"^ — -, from which it follows that, x^ 1 X-" ' Tlie recijirocal of a qnatitity is equal to the same quantity with the siyu of its cxpuii<']tl changed. EXAMPLES. a"b b bd-^ <■(/- u--cd' a-~c 2. a'"=' 3. ^=ab-"': 4. a"'-"= 82. Theorem V. — Any quantify, v)hose exponent is 0, is equal to unity. If we divide a^ by n-, and apply the rule for the exponents (Art. 70), we find - ,=a-~-^a"; but, since any quantity is con- tained in itself once, —^1; therefore, a"=l. Similarly, — =a:"'-'»=a;°. But— -=1; therefore, a;"=l, which •" X'" x" ' proves the theorem. By this notation, we may preserve the trace of a letter, which hns a'^b disappeared in division. Thus, — ^=:a^-'6'-'.=a'6"^a. 83. Theorem VI. — T%e difference of the same power of two quantities is always divisible hy the difference of tlie 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, botli as to the exponents and the signs. Thus, (a2— 62)-=-(a— 6)=a+6 ; {a^ — b^)-^{a~b)^a--{-ab-{-b- ; {a*—b')-^(a—b)=n^+a-b^ab-+b"; la'^~b'')^(a—b)~a^-Jf-a^b-\-a^b'^-\~oO-^,- b\ etc. The general and direct proof of this theorem is as follows : Let us divide a" — &"" by a—b. a^^b"'\a-b a'"-^a"'-''b a»"-i 6— 6™=6 (a""-' —6™-' ) a"--! J^^J^ '^ ', Quot. ' a-b In performing this division, we see that the iirst term of the quo- tient is a"^i, and the first remainder, 6(a'"-'— 6"^'). The remainder consists of two factors, b and a""-'— &"*-'. Now, i/"the second of these factors, viz., a"*-' — 6'"-i, is divisible by a — b, then will the quantity a^—b"^ be divisible by a—b. That is. If the difference of the same powers of two quantities is divisible hy the difference of the quantities themselves, then will the difference of the next higher jiowers of the same quan- tities he divisible hy the difference of the quantities. But we have seen that a-—b- is divisible by a—b ; hence, a^— 6' is also divisible by a—b. Again, since a^—b^ is divisible by a— b, it follows that a* — b* is divisible by it, and so on; which proves the theorem generally. 84. Lemma. — In proving the next two theorems, it is necessary to notice, that tlie 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 a'yC — CX — '^= — '^^1 ^^ ^'^ power, is negative. aX — CX — '^X — tt=o^^ the 4th power, is positive ; and so on. 46 KAYS ALGEBRA, SECOND BOOK. 85. Theorem VII. — Tlie difference of the even powers of the same degree of two quantities, is always divisible by the sum of the quantities. If we take the quantities a — b and a"" — 6"*, and put — c instead of b, n — b will become a — ( — c)=a-|-c; and -when m is even, 6"* will become c"', and a" — 6"" will become a"— {-f c™)=a'n — c™ : but Qm — (jm jg always divisible by a — b; Therefore, a" — c"" is always divisible by a-\-C when m is even, which is the theorem. EXA5IPLES. 1. {a-—b^)^{a+b)=a—b. 2. (a<-^b')^{(i + b)=a'-—a-b-\-ab--~/j-: 3. (a'' — b^)^(a-\~b)=a''—a'b-\-a^b-~a'-b^'-\-ab' — b''. 86. Theorem YllL— The sum of the odd powers of the s/niic degree of two quantities, is always divisible by the sum of the quantities. If we take the quantities a — b and a"" — 6"", and put — e instead of b, a—b will become a — (— c)=:a-j-C; and when m is odd, 6"* will b"come — c"', (Art. 84), and OT—V^ will become a'"— (— c'"J ^a"'-(-c"': but am— 6'" is always divisible by a—b; Therefore, a^^c^ is always divisible by a+c when m is odd, which is the theorem. E X .\ M P L E S . 1. ((fl^-}fi)^(a^^ b)=a?—ab.\-b-. S. (a7-(-6')H-(a+6)=a6— a'S+a-ii^— a363-fa=6-t— a6'-|-66. 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, a--\-b-, eC'^b'^, o'-(-6^, etc., are not divisible by a — b. When, in either of the last three theorems, a or b becomes unity, the form of the quotient will be obvious. Thtis, (a''— l)^(a-l)=a< + a'Ha-+«+l- (l+a^)-7-(l-|-0!)=l— a+a^— a^-fa^, etc. FACTORING. 47 FACTORING. 87. 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 II, — 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 ah, 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. 91. A Composite ftuantity 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 BAY'S ALGEBRA, SECOND BOOK. 92. 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 be the ■prime factors of the given quantity. 1. Find the prime factors of 18«6=. Ans. 2x3x3x^.6. 6. 2. Of 28xV-'- • • ■ Ans. 2x2x7x3^.2: ^/.^.z.z. 3. Of 210ax'2/z\ . . Ans. 2x3x5 X7.a.a:,a:..r^.2.z. 03. To separate a polynomial into its factors, wlien one of them is a monomial, S/Tlle. — Divide the given quantity by the greatest monomial ihiil tvilt exactly divide each of its terms. The divisor will he one factor, and the quotient the other. 1. Separate into factors, a-\-ax. . . Ans. a(\-\-x). 2. -.rz^yz Ans. z(a;+^). 3. xhj-\-xy'' Ans. xy(x-\-y'). 4. 6ai=+9a^5c Ans. 3f(^<2?^+3(fc-.) 5. c^bx'y — aVxy'-'^ahcxyz^. Ana. abxy{ax^ — by-^cz'). 94:, 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 tlic middle term ia twice the product of the square roots of tlie 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'-\-2ab+b^-={a+b){a+b); a-~2ab+b-^{a—b)la—b). 2d. Any binomial, which is the difference of two squares, can be separated into factors, one of which is the sum and the other the difference of their roots. (See Art. 80.) Thus, a^—b-=(a-\-b){a—b). FACTORING. 49 3d. Any binomial which is the difference 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.) Thus, x'-y'={x-y){x^+xy+^''). Similarly, x^ — i/^={x—y){x'^+x^y-\-x^y''-lrxy^+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, a*— 6*=(a2+62)(a2— 62)=(a2+62)(a+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, a^+b^={a+b){a^—ab+b^) 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 division : a'+b'={a+VM>+b){a—V2aJ>+b). a^+6*=(a2+T/2.a6+62)(a2— l/2.a6+62). a^+b'^={a'+b^){a^—a'b^+b^)={a'+b')ia'+V3.ai-\-b^) {a^—VS.ai+b'). a'+b'={a*+V^.a''b'+b^){a^—l/i:.a^b'+b^). ai2+6i2=(a*+6*)(a»— a^Ji+ft'). (j2™_|_jz»_Ca«._|_v^a 2 J ? + 6") (a"— 1^2; a ^ 6 ^ +5"). a3m^J3m_(a»_|_5».)(a2»._(j».Jm_|.52»,), Separate the follovt'ing into their simplest factors; 1. c'+2cd+dK 2. a^x*+2ax'y+y'. 3. 25x^y*+20xy'z+4z^ 4. 9x^—6x^z^+z*. 5. 4m^z^ — 4mn'z-\-n^. 6. x^—z^. 7. Qa^a:*— 25. 2d Bk. ^^ 8. 16—a'b^z\ 9. a*-^4. 10. 3^+1. 11. 2/3-1. 12. o3j;3_J3j,3. 13. x^+yK 14. i" — 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 hy inspection. Ob- serve carefully the product resulting from the multiplication of two factors of the form x-\-a, and x^-b. Thus, X^^5x+6=(x~2}{x~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 -|-6. Trinomials to be decomposed into binomial factors. 1. x''-\-Sx-\-2 Ans. (.T;+l)(a;+2) 2. x'— 8x+15 Ans. (a— 3)(a;— 5) 3. x'-x—2 Ans. (x+lXx—2) 4. x'^+.T— 12 Ans. (x— 3)fa;+4) 5. .t'— a— 12 Ans. (a;+3)(a;— 4), 6. x'-\-2x—3b Ans. (x— 5)(a;+7> 05. Examples to be resolved into factors, by first sep- arating the monomial factor, and then applying Arts. 93 and 94. Ex. 1. axr'i/ — ax-if^^ax\j{7? — ■f)=^axy[x-\-y){x — y). 2. Zax^-^-^axy-^-Zay^ Ans. Za(x-\-y)(x-\-y). •18c. 3. 4. 5. 6. 7. 2a;'+4x^— YOz. . 2cx^—\2cx~ 3m'H — 3»!)i' 23fy—2xf. 2a;'-)- 6a;— 8. . . . Ans. 2c(x— 3)(x— 3). .Ans. %mn{m-\-n){m — ?i). Ans. 2xy{3?-^f){x^y){x~y~). . . . Ans. 2(x+4)(a;— 1) . . Ans. 2a:(a--t-7)(a-— 5). Solve the following, by first indicating the operations to be performed, and then canceling common factors. 8. Multiply ix — 12 by 1 — x'', and divide the product by 2+ 2a;. (4.r- 12)(l-.T-)__4(.r-3)fl + y)(l x) 2 T2x "" 2(1 -l-a;) 2(4x-3-a;2)=8a;— 6— 2a;2. =2{a;— 3)(1— X)— GREATEST COMMON DIVISOR. 51 9. Multiply x'-\-2xi/-]-i/' by x — i/, and divide the prod- uct by x'—f. Ans. x-^-y 10. Multiply together 1 — c, 1 — c\ and l-\-c\ and divide the product by l—2c~{-c\ Ans. l+c-fc^-j-c'. 11. Multiply a?—x'~SOx by x^^+llai-l-SO, and divide the product by the product of x' — 36 and a;'+10a;+25. Ans. x. GREATEST COMMON DIVISOR. 96. A Common Divisor, or Common Measure, is any quantity that will exactly divide two or more quantities. Thus, ab is a common divisor of ah' and ahx. Remark.— Two quantities often have more than one common divisor. Thus, a^cx and abdx have three common divisors, a, x, and ax. 97. The Greatest Common Divisor, or Greatest Com- mon Measure of two quantities, is the greatest quantity that will exactly divide each of them. Thus, the greatest common divisor of Qa'hx' and 9a^cxz is Sd'x. 98. Quantities that have a common divisor are said to be commensurable; and those that have no common divisor, incommensurahle. Note. — G.C.D. stands for greatest common divisor. 99. To find the G.C.D. of two or more monomials. 1. Let it be required to find the G.C.D. of the two mono- mials, 14a'cx and 2\a?hx. By separating each quantity into its prime factors, we have l^a?ox=iy%y_aaaox, and 2\a^bx=.Tyi^yi(^abx. 52 RAY'S ALGEBRA, SECOND BOOK. The only factors common to both these quantities, are 7, aa or a^, and x; hence, both can be divided by either of these factors, or by their product, To'x, and by no other quantity; therefore, la^x is their G.C.D. Hence. TO FIND THE GREATEST COMMON DIVISOR OF TWO OR MORE MONOMIALS, Rule. — 1. Resolve the quantities into their prime factors. 2. Multiply together those factors that are common to all the terms, for the greatest common divisor. 2. Find the G.C.D. of 6a'xi/, 9rrV, and ISaVy. OPERATION. Here, 3 is the only numerical factor, 6a^xy ^Sy2(c-xj/ and a and x the only letters common to Ga-'a;^ ^Sy^'Sa^x^ all the quantities. The least powers of Iba-'x'y^ =i3y5a*x''j/^ a and x, are a'^ andx; hence, the G.C.D. is 3a'-x. Find the G.C.D. of the following quantities : 3. 15a6c^ and 2W'cd Ans. 3bc. 4. 4a'6, IQa^c, and I'ia'hc Ans. 2a\ 5. 4(13;^, 20a;yz, and 12^:^2^ . . . Ans. 4x^y'. 6. 12a'^.rV, 18aa;V, 30rt-.»''2, and Gax^z'. 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 rmdtiplc 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. This is evident from Art. 74. GREATEST COMMON DIVISOR. 53 lOl. Let it be required to find the G.C.D. of two poly- nomialii, A and B, of ■wtioli A is tlie greater. If we divide A by B, and there is 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 B)B(Q' E, it is evidently equal to A — BQ. EQ' If, now, there is any common divisor jj RQ'=;K.' 2d Rem. of A and B, it will also divide BQ (Prop. 1st) and A— BQ or R (Prop. A=BQ -fR Since the 2d); or the common divisor must B=EQ''+R' ai^Wc""! 's _ •' ; ' eiiual to the divide A, B, and R, and can not be product of tho (iivisor Ly tlio greater than R. quotient, plus 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 be the greatest common divisor, since no divisor of A, B, and R can be 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=120, and B=35. 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. 104. 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. X0o» From Art, 101, it is evident that the three preceding articles apply also to the successive remainders. 107. 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, talce 553; and ]5.r. iSelting aside x, we find the greatest common measure of 55 and 15 to be 5. Annexing x, we have 5jr. 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'— 2' and x'—xV. 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 (.\rt. 103), and the a:^ — xz'^ \X second quantity becomes x? — z-. Then divide xz- — z^' tlie first by it. or {x — z)z'^ After dividing, wo find that z- is a factor of tlie remainder, but not of x'^ — s-, the next divi- x'^ — z'^\x — z dend. We, therefore, cancel it (Art. 103), and x~ — 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 a?-\-x'z' and x^— xV, The factor x^ is common to both quantities; it is, therefore, a factor of the greatest divisor (Art. 107), and may be talsen 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^ — z^. 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. OPERATTON. ofi-\-z-^'\x^ — z^ x^ — xz'^ \x xz^-^z^ or [x-\-z)z- x'—z' \x+z x'-\-xztx — z — XZ — Z2 . — XZ — 2^ 3. Find the G.C.D. of lOaV— 4a''x— 6a^ and bhx''—llLx +66. By separating the monomial factors, we find 10a^x2—4a2x—6a^=2n-(5x^~2x—S), and 5bx'—Ubx+6b=b[5x'—nx+G). 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. OPEKATION. 5x^—nx+e \bx^—2x--3 5x2— 2a;_3"~il — 9x+9 or — 9(x— 1) 5x2— 2x— 3|x— 1 5x2— 5x ]5x+3 3x— 3 3x— 3 4. Find the G.C.D. of 4a^— 5ay+/, and 3a'— 3a'y In solving this exam- ple, it is necessary, in 3a^- two instances, to multi- ply the dividend, that the coefficient of the first term may be divisible by ^a^y+ ap^ the first term of the di- visor (Art. 104.). 12a22/+ 4aj/2~i6^3 [over.] OPEEATION. 4 12a3— 12a22/^4c/,?/2— 42/3 12a3— 15a22/+3a3/2 4 |3a+3y 56 RAY'S ALGEBRA, SECOND BOOK. We find Idy- is a. 12a-y-\- Aay- — 163/' [brought ovek.] factor of the first re- 12a-y — 15ay^+ Sj/' Diainder, but not of the 19"^- — 192/' first divisor, and hence or 19y^[a — y) can not be a factor of the G. C. D. ; it must, 4a^ — 5ay-{-y^\a—y g.c.d. therefore, be suppressed. 4a- — 4ay \ia—y Hence, —ay-\-y' —ay+y^ TO FIND THE GREATEST COMMON DIVISOR OP TWO POLYNOMIALS, 108. Rule. — 1. Divide the greater polynomial hy the less, and if there is no remainder, the less quantity will be the divisor sought. 2. If there he a remainder, divide the first divisor hy it, and continue to divide the last divisor by the last remainder, •until a divisor is obtained which haves no remainder; this will be the G. CD. of the two given polynomials. Notes. — 1. When the highest power of the leading letter is the same in both, it is immaterial which of the quantities is made the dividend. 2. If both quantities contain a common factor, let it be set aside, as forming a factor of the common divisor, and proceed to find the G.C.D. of the remaining factors, as in Ex. 2. 3. If either quantity contains a factor not found in the other, it may be canceled before commencing the operation, as in Ex. 3. 4. Whenever it is necessary, the dividend may be multiplied by any quantity which will render the first term exactly divisible by the first term of the divisor, as in Ex. 4. 5. If, in any case, the remainder is unity, or does not contain the leading letter, there is no common divisor. 6. To find the G.C.D. of three or more quantities, first find the G.C.D. of two of them; then of that divisor and one of the other quantities, and so on. The last divisor thus found will be the G.C.D. sought. 7. Since the G.C.D. of any two quantities contains all the factors common to both, it may often be found most easily by separating the polynomials into factors. (Arts. 92 to 95.) LEAST COMMON MULTIPLE. 57 Knd the G.C.D. in the following quantities : 1. bx'' — 2x — 3 and fix- — lla;-[-6. . . . Ans. x — 1. 2. 9x^—4 and 9cr.^— 15.T— 14 Ans. 3x+2. 3. a}—ah—\2V and a^+5a^;+6^)^ . . Ans. a-)-36. 4. a* — x* and a?-\-a}x — ax' — x' Ans. a} — x'. 5. x'— 5x=+13a;— 9 and x'— 2x^+4x— 3. Ans. x— 1. 6. 21x'— 26x^+8x and 6x^— x— 2. . . Ans. 3x— 2. 7. x<+2x=4-9 and 7x'— llx2+15x+9. Ans. x^— 2x+3. 8. x^+5x-f 4, X24-2.X— 8, and x^+7x+12. Ans. x+4. 9. 2¥—].0aV^%a^h and ^a*—Za¥-\-ZdV—%a?h. Ans. a — h. 10. x*-\-a}x''-\-a*' and x*-f-ax' — a'x — a*. Ans. x'-^-ax-^-a}. 11. x* — -px^-|-(2 — l)x^-|-l'* — 2 ^'^'^ ^^ — 2*'~I~(P — 1)^^ -\-qx — p. Ans. x' — 1. LEAST COMMON MULTIPLE. 109. A Multiple of a quantity is any quantity that contains it exactly. Thus, 6 is a multiple of 2 or of 3 ; and ah is a multiple of a or of & ; also, a(h — c) is a multiple of a or (h — c). 110. 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 20xy, of 2x and 5y. 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 ; lOxy, of 2x and by. Note. — L.C.M. stands for least common multiple. 113. To find the L.C.AL of two or more quantities. n X nx mnz X X X mz 1 1 mz 58 RAY'S ALGEBRA, SECOND BOOK. It is evident that the L.C.M. of two or more quantities contains ail the prime factors of each of the quantities once, and does not contain any prime factor besides. Thus, the L.C.M. of ah and he must contain the factors a, I, c, and no other factor. Assuming the principle above stated, let us find the L.C.M. of mx, nx, and ni'iiz. Arranging the quantities as in the OPERATION. margin, we see that m. is a, prime factor Tn\mx nx m^m common to two of them. It must, there- fore, even if found in only one of the quantities, be a factor of the L.C.M.; and as it €an occur but once in the L.C.M., my(ny^xx>n~—in'-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 »l, n, andz; also, mz, otherwise it would not contain all the prime factors found in one of the quantities. Hence, my(nXxXmz^=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 hy any prime factor that will exactly divide tico 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 (xactly divide two or more of the quantities. 3. Multiply the divisors and the quantities in the last line together, and the product will he 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, take if the greatest number of times it occurs in cither of the quantities. ALGEBEAIC 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. CD., the quotient will he their L. CM. 1. Find the L.C.M. of Qa\ 9aa:', and 24x*. Ans. V2aV 2. ^2x-y\ '^Qax'y, ba''x(x — y). Ans. 160a'3?y^(a: — y) 3. 3xJ(^6y and 2a;^— 8/ Ans. 6a;^— 24/ 4. a"-)- a;' and a^ — a;^ . . Ans. a* — a?x-\-aa? — x' 5. x—1, x''—l, x—2, and a;'-^4. Ans. a;*— 5x^+4 6. a;^— 1, a;'+l, (x—ly, (a;+l)^ ai'—l, and a^+1 Ans. x'" — a^ — x*4-l 1. 3x'— llx+6, 2x''— tx+S, and 6x'— 7x4-2. (See Art. 113.) Ans. Gx^— 25x^+23x— 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 — h of those parts are taken. 113. The terms proper, improper, simple, compound, and complex, have the same meaning when applied to alg-^braic fractions, as to common numerical fractions. 60 RAY S ALGEBRA, SECOND BOOK. 116. An Entire Algebraic ftuantity is one not ex- pressed under the form of a fraction. liy, A Mixed Quantity is one composed of an entire quantity and a fraction. 118. Proposition. — The value of a fraction is not altered, when hath terms are multiplied or divided hy the same quantity. Let ^^Q. Then, will — ;r=Q. For, since the numerator of a B mB fraction may always be considered a dividend, and the denominator a divisor, if we multiply the numerator or dividend hy any quan- tity, as m, the quotient will be increased r)l 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: mk ,. _, , wi'wi-^A ?n"A ,. „„, A ^B= t^--'- «i)' -B^= Tr= (^'•'- «2)' B- 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 Rule. — Divide both terms of the fraction hy any quantify 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 =^^ — . to its lowest terms. iobcx' lOacx'^ 2acx^ 2ax^ 2a Xabcx^ ' Or, dividing by bcx'^, " Zbcxi ~Sbx^~ 36a;' ^^^' lOacx^ 2a Or, 156ck3 36a;' 10aea;2_ 2ayj>cx^ _ 2a Ibbcx^ ~' ZbxX^cx' ~ Sbx' 2. 3. 4. 5. 6. 12. 3a'6x' a.-c-f-a;' 36a; — ex' 3a^-|-3a6 3d' — 3a6 Ans. 5 Ans. Ans Ans 3ai' a-|-a; a+b a — b' 1 1+x' Ans. — a' 7. 8. 9, 10 nnp — m'p m''p-\-7np'' ' 2ax — iax'' 6ax 5a,''-{-bax cf — x' a:^+2a;— 3 Ans. Ans, Ans. Ans. m.-\-p 1-2.T. ~"3~- 5a a — X x—1 x+2- a;^+5a;+6' x'+l 'x'—x-\-l' 27a;*+63a;^— 12x^—2 8x Ans. Sa'+l 9x'— 4a;" The following examples are to be solved by factoring, but the process requires care and practice. - „ _ , x''-\-(a-{-c)x-i-ac . , 16. Keduce , ;, — ( — ^^- to its lowest terms. x'-\-(b-\-c)x-\-bc x^-\-{a-\-c)x-\-ac=x^-\-ax-\-cx-^ao ^x[x-\-a)-\-c{x-\-a)^x-\-c){x-\-a). Also, a;2+(6+c)a;+6c=(a;+c){a;+6); .-. the fraction becomes (a;+c)(a;-|-a) x-\-a (x+cjjx+b) ~ x+b' , Ans. ^ a/+2bx+2ax+bf /-|-2a;' 62 RAY S ALGEBRA, SECOND BOOK. j^ xM-xy+rry+y ^^^_ a^'+y X* — y* ' x' — y'' -. r, a}+(a-\-l')ax-\-lx' . a-\-x 16. — i— ^ — — Ans. ' , . a* — O'x'' a — ox j^ ax'"—hx"'+' ^^^ g"-' a'ia; — bV 6(a-j-6a;)' 120. Exercises in Division, in whieh the quotient is a fraction, and capable of being reduced : 1. Divide 2aV by bd^x^b Ans. t^. 2. axA-x' by ohx — ex Ans. -r-, — '- ' ■' 6b — c 3. a>-V by a^-b^ Ans. ?!!±^^. •' a-\-b 4. a»— t' by fa— by Ans. + 7 . Case II. — To reduce a Fraction to an Entire or Mixed Quantity. 121. Since the numerator of tbe fraction may be re- garded as a dividend, and the denominator as the divisor, this is merely a case of division. Hence, Rule. — Divide the numerator by the denominator, for the entire part. If there he a remainder, place it over the de- nominator, for the fractional part, and reduce it to its lowest terms. 1. Reduce — ^ '— to an entire or mixed quantity. a' — a.c a^—ax a^—ax a—x ALGEBRAIC FRACTIONS. 63 Reduce the following to entire or mixed quantities : ^- Ans. X . a a O. ;— Ans. a+oH r a— i ^ 'a— 6 A. l+2a: A 1 , c , ISa:^ ^- l=3i ^°^- l+^"+II=3-x- 5- , , Ans. x^ x^ — bx X — 6 „ xV — z'-i-xz — 2 — x-\-l . , , z — 1 x' — 1 x-^1 Case III. — To reduce a Mixed Quantity to the torm or A Fraction. 122. This is, obviously, the reverse of Case II. Hence, we have the following Rule, — 1. Multiply (he entire pari hy the denominator of the fraction. 2. Add the numerator to the product, if the sign of the fraction he plus, or subtract it, if the sign he minus. 3. Place the result over the denominator. Before applying this rule, it is necessary to consider 123. 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^is plus; of 6^, minus; x-\-y 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. 134. 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, ^377-=+^- — ab , If we change the sign of the numerator, we have — — -- ^ —o. -\-ab If we change the sign of the denominator, we have = — o. Hence, 1. Tlie signs of both terms of a fraction may be 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. Tlie signs of either term of a fraction may be changed, ivithont altering its value, if the sign of the fraction be changed at the same time. Thus, ^= = = — l~a—x)=za-\-x. a^x a—x —a-\-x ^ ' , , a2— a;2 — a-A-x'- , a:^—x''- And, a -=o.A ! — —a-\-- — " -a~\-x Applying the above principles, the sign of the fraction may be wade plus, in all cases, if desired. Reduce the following quantities to a fractional form : 1. 2 + 3 and 2— a Ans. M- and I. ' a DO A. a-YxA Ans. — — X x 3. a^ — ax-^x? — — . Ans. a-\-x ' a-\-x ' {a—xf a' 4. 2a-;r+^-^^-^. Ans X X' r a' , ab o. a -} Ans. — -y. a+6 a+6 ALGEBRAIC FRACTIONS. 65 0. a — X — Ans. a+x a-\-x' 7. l-^^H^ Ana ^^•y ^+y ^/- Case rV.— To reduce Fractions of Different Denom- inators TO Equivalent Fractions having A Common Denominator. 125. — 1. Let it be required to reduce — , — , and -, to , . ^ m n r a common aonominator. If we multiply both terms of the first fraction hy nr, of the sec- ond by mr, and of the third by tnn, we have anr bmr , cmn , , and . 'mnr mnr 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 hoth terms of each fraction hy the prod- uct of all the denominators, except its own. Or, 1. Multiply each numerator hy the product of all the denominators except its own, for the new numerators. 2. Multiply all the denominators together for the common denominator. Keduce the fractions in each of the following to a com- mon denominator : „ 1 2 , 3 . yz 2xz Zxy 2. -, -, and - Ans. ^, —, -^ X y 2 a;^z ary« xyz' ah , a'' ^ b^ 3. T- and - Ans. -=- and -j ha ab ab X ., a . a?-irax , ax — a' 4. and Ans. — -! — r- and — x—a x+a x^—a^ x'—a^ 2d Bk. 6 66 RAY'S ALGEBRA, SECOND BOOK. 12G. 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. , ah - c 1. Let it he required to reduce — , — , and ^, to their ' . m run 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 to. But these multi- pliers will evidently be obtained by dividing mnr by m, mn, 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. Dividp the L.C.M. hy the first of the given denominators, and multiply the quotient hy the first of the given numerators ; the product tcill 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 : 2. _^, A, _! Ans. _^ ?^ 3c^ 6xy' 3a;' 2y ' Sa-y' Qxy' Qxy' a+6' a—h' a'—h'' ' a:'—h' ' d'—h' ' d'—h'- ALGEBRAIC FRACTIONS. 67 ^ m — n m-\-n mV (m — n)* {m-\-ny ni'n' m-\-n' m — n ni' — n'' ' ni' — li' ' m' — ii' ' m' — n'' Other exercises will be found in Addition of Fractions. Note . — The two following Articles may be of frequent use. 137. To reduce an entire quantity to the form of a fraction having a given denominator, RulCc — 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 2az to a fraction whose denominator is z'. 2a2» Ans. z' 3. Reduce x-\-y to a fraction whose denominator is x — y. Ans. ^ x—y 128. 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 multiply both terms by the (juu- tienl. 1. Convert | to an equivalent fraction, having 49 for its denominator. Ans. |^. a 5 2. Convert ■= and — to equivalent fractions having; the 3 c 3^p2 ^^^ denominator 9c^ Ans. -7^— - and = — yc' yc' 3. Convert -^,- and — — j- to equivalent fractions hav- • „ ^ "~, . "if A (p+w i<^~^y ins: the denominator a^ — b\ Ans. ^ , /. , -^^ -^ 68 RAY'S ALGEBRA, SECOND BOOK. Case V. ADDITION and subtraction of fractions. 130. — 1. Required to find the value of — , -, and -. add Since in each of these fractions the unit is supposed to he divided into d parts, it is evident that their sum will be expressed by the a-\-b+c ^ fraction — -^ — . 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 mini over the com- 9noH denominator. 130. — 2. Let it be required to subtract — from — . d d 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 denominator. 2. Subtract the rmnieratvr of the subtrahend from the numerator of the minuend, and write the remainder over the common denominator. EXAMPLES IN ADDITION OF FRACTIONS. , , . , a , 3a , .7(1 1. Add - and -n together Ans. -r-r, i 46 "^ 46 a h Z. Add - and - toa;ether. b a 3. Add = and tot!;cther, l-fx 1 — X , „. , , , „c b a , a-\-b:r^-l-cx^ 4. Find the value of -+ --+ -;. .Ans. — r-^ . X X' ay' x" . „. , , , „ 6 , ad — he . u-\-hx 0. Find the value of -+ -j-. — , , . . . . Ans. .Ans. a'+V ah ■ . Ans. 2 \—x'' d d{c-\-dx^ c-\-dx ALGEBRAIC FRACTIONS. 69 6. Find the Talue of 4+ 1+ L. Ans. PJ+^ttl^, au ac he abc 7. Of -^ +-X- Ans. "-;+<. ^- ^^ 4(I+F)+ 4(1-0-)+ 2(T+^)- ^"'- l=i*- 9. Of-P— ^+-^+2=-'- Ans.O. pq pr qr 10 Of . 1 I ^ 'iaXa+xy 4:a\a—xy 2dXa'+x'y , 1 a* — ar 11. Of ~ h - f- - a(a — 6)(a — c) b(h — a!)(i — c) c(c — a)(c — i)' , 1 abc EXAMPLES IN SUBTRACTION OF FRACTIONS. Subtract the second fraction, in the following, from the first : , 5x , 3y , 5ar — Say 1. s- and tI Ans. ,= — -. 2. =- and —r-r Ans. — — =- a — b a-\-o a' — b' 3. -^^ — - and '^— r^ Ans. , , . l>—q " P+q p'—q' . n — 1 , n , 1 — 2™ 4. and =- Ans. — . n n — 1 TO — n 5. = and :; 1 Ans. ^ — -,= — i-j— . 1—x 1 — k" 1 — a^ 1+x R 1 A 1 ^- (z+l)(^+2) ^° {x+l)(x+2Xx+Sy ^'''- (x+l)(*+3)- 70 RAT'S ALGEBRA, SECOND BOOK. tj a . (ad — hc)x . a~^ Ix c c(c-\-dx) ' c-\-dx' „ 1 8m+2n. , 1 3™ — 2re . 12mn o. Ti.is —^r- and ;=-._ p^- Ans. 2"3m— 2ji 2'39»+2)i 9m'— in^- o «+c J 6+c . x-\-c y. ; ~ r and , — — . A. (a — h)(x — a) (a — h)(x — h)' ' (x — a)(x — 6)' Find the value 1 r. /->n 4»i — Sn m-\-Bn , 2n . m 10. Of „y^ K—Q7T ^+1 • • • • Ans. o(l — n) 6(1 — n) 1 — 71 ah ac he 1- -n Ans. 0. Ans. 1. a:+3 12. of^i^ - ^'~^'y y ^+y x^y—f 13. Ofj-j-2^^^^-2^^^. . .Ans. Case VI.— MULTIPLICATION OF FRACTIONS. 131. — 1. Required to find the product of - by -. o d Here, a8 in arithmetic, we take the part of t-, which is expressed 1 \ " a a by -J. and then multiply by C. Thus, the -= part of =- is -n, and c a . ac , ^ ,,_, „ a c ac times -iTjis j-i (Art. 118). Hence, -rX-i=-i— .■ bd bd^ ' ' b-^d bd a Or thus, T- and -j=ab-'^ and cd-l (Art. 81). Multiplying, we (tc have ab-^cd—^^ ,— ,. Hence, 6d ' S.ule. — 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 ; 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. Th„. _gg!_x.^(g+^)'_ 2a^X(a+6)(a+6) a+b 'a^—b^'^ 4a^b {a+b)(a—b)4a'b ~2b{a^- Find the products of the fractions in the following : T 3.1; 4.T , 8a'b , c^d . x' bed 1. ^ by ^ and — by g-,. . . Ans. - and ^. x^ a ,x . a*~x* ■ "■— — > -+- Ans. a X a a'x 3. I_^and2+^ Ans. J^ 4- iUFpJ and j-^ Ans. j^,. rr'+3x+2 a;'+5a;+4 a;+2 ^- a;^+2x+l '^"'^ 5^+7^+12 ^°^- iqiB- 6 1^ «'— ^' 6c-|-&a; 4a;(a+a;) 36y' c''— a;^' o?—ax °^" %y{c~xy I- — i — J Tt } r^ Ans. — -^ a>+!/ a—o (a;— ^)" x—y 8. x-\-l+- by x—l+~ Ans. a;^-f 1 + 4. XX x' „ 4a , 3x 26 3a! , 8a6 , „ , 9.i;^ 6x Ab -^ da; 4a 9a;2 8a6 ,Q pr+(jJg+g?-)a;+gV ps+(yi!— gs)a;— gte ' p — ga; p+ga; Ans. rs-\-(rt-)rqs)x-\-qtx^. Find the value »-«'(i+5)(->;')-(^s)f3-')- Ans ^+2a^ ad fee ■ 72 BAY'S ALGEBRA, SECOND BOOK. Case VII.— DIVISION OF FRACTIONS. 133. — 1. Required to find the quotient of j by -z. a 1 ad Here, as in arithmetic, the quotient of -7- by -1 is -7-, and the quo- ,. , „ a . . 1 c . ad , ad tient of ^ by c times — „ or -,, is -;- dividea by C, or y-— . b ^ a d' b ■' ' be a c Or thus, -J- and -,= (Art. 81) ab-^ and cd-'. Dividing, we have afr-' ad „ —3-,=-=—. Hence, cd" 1 be Rule. — Invert the divisor, and proceed as in 'middplication 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- tious. Required, in their simplest forms, the quotients , „„ aZ»^t' a^Vc . c^ii 1. Of — ; Ans. -f. xy xy a'x 2. Of -J—-; T Ans. ^ : a-\- c a — t) a — c _ ^„ .t' — d'x ax — ((' . x'-\-ax' 3. Of — - — ■ -; Ans. 7- . a' X a 5. Of tt^ "•-'S+f A»s. 1. x' — y X — y r. ^„ a' — X* a''x-\-a^ n+x 6. Of --- ^ — T— „ -^ , , ■ . Ans. {a'+ax+x'). V.Of(jl^^+j^J-.(ll,-I^^). . Ans.l. 8. Of 3ar ALGEBRAIC FRACTIONS. 1x 2a;— 2 • .j:— 1" . Ans Ans. 73 3 a-— 1 X — 5' 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. 6 «+- Thus, ^— ^ is tlie same as to divide aA — by vx . ' n ^c •' r m r acr-^br ac-\-b mr^n ac-\-b X: Or, the following method, obviously true,' will generally be found more convenient. Multiply both terms of the complex fraction hy the product of the denom,inators, or hy their L. CM. acr-\-br cnir — en' Thus, in the above, multiplying by cr, we have, at once. Solve the following examples by both methods : 1. 2x- Ans. _ x—1 a c ili Ans fK^'^+^c) ^- e g- ^°^- hd{eh-fgy f h 2d Bk. 3. a-(-l a — 1 , -f -1. Ans. — TT— . a-\-l a—1 2a a+h+ a+b+t 1 a+1 b' . Ans. 7* 74 RAY'S ALGEBRA, SECOND BOOK. Resolution of Fractions into Series. IS^. 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 ^ .,-1-, etc., any term may he 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 I .r 1-J X 1 — 2.t:+2.(;--2.r'-i , etc. It is evident that the laio of this series is, that each term, after the second, is equal to the preceding term, multiplied by —X. llesolvo the followinp; fractions into infinite series 1 2 !+'■ .2 1- _,-2_J_r-— , ,.6_|_,.8_ -, etc., to infinity. 3. 1 /•" --1 + r-r '—;•*+! i-"-^,-'- ~r^- -'■'"+, etc. 1— / ■ + 4. a4-l - = =1- a a- etc. 1S3. Miscellaneous Propositions in Fractions. — The answer to some general question, that is, the solution to a literal equation (ef. Arts. 162-1G5), may happen to he a fraction: e. g., wo may have k^t. 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 chan-\ R; Or, 15-2=5 \-9,; or, 13=13. SIJIPLE EQUATIONS. 85 6. Find the value of x, in the equation ab bo 1st step, abcx — cx—ac=abcd-\-ax. 2d step, abcx—cx — ax=abcd-^-ac. Factoring, (abc—o—a) x=ac(bd-\-\). 3d step, ^^ae(6d+l)_ abc—c—a 153. 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 memhcr to its simplest form. 4. Divide Loth sides hy the coefficient of the unkwmn quantity. Find the value of the unknown quantity in the following: 1st step, ]8a;+42— 8a;+28+231=21a;— 84; 2d step, 18a;— 8a;-21:r=— 231— 42— 28— 84; 3d step, — lla:=— 385, x=&o. 3x35+7 2X35-7 35-4 Vekification, j-j gj |-2j= — ^> 8-3+2J=7f, 7S=7|. 8. 5(a;+l)— 2=3(a;+5) Ans. x^Q. 9. 3(a;— 2)+4=4C3— ^) Ans. x=^. 10. 5—3(4— a-.)+4(3—2x)^0 Ans. x^l. 86 RAYS ALGEBRA, SECOND BOOK 11 fj-:^ = _+7 Xus. x=12. 1^- 2 + 5^4 + H3 -An. .=10. ^'■l+l~^-^ ^^^-^^^ , , 3j;+1 2x ^„ , :c — 1 . -, . 2 o b 15- -2- = ^r +^r- ^ns. .^=7/,. 16. 5.r— ^^^+l=3x+^ + 7. . Ans. x=8. ^, 7x+9 3a:+l 9.<-13 24!}— 9.C , „ 17. -^ ^= 4 ^-^— .Ans...= 9. 18. l(2x—10)—^\iSx-iO)=.lb—l{b1—x). Ans. a:=17. 19. i(i+^x)-]i2.c-]) = ]l Ans. a-=;. 20. 3'x{28-(j+2l)}=3^x{2]+^|. Alls. j-=^4. 21. K'^-5;)-iUl-3,0-.^-,v( 5..- ^"'' ). Ans. a;=ill. 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.^ + 3 ^3.r-6 ., ^3.T+22 2 t 2.C — ■' 9 3.r-l ^6 + :r, 3..-9 9, . 3.r + 9 . 23. — '-^ =-^=2j-| -=-. Ans. .T=o. x-\-2 4 i2 ' .^■+7 24. i.r + 2.r— „=:3.c— 2c Ans. X 25. a'x+b^^!rx + a^ Ans. x h—V a-\-b ■ 26. ax-lf-h'^a'+hx Ans. x^--(-a;)£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 oihcer 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 - — s — is the X s quantity seized, the value of which is 198—162=36 shillings. rr-10 198 „^ .: — ^ — X — =o6; from which the answer is readily found. 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 difference of the first and second from 100. Eequired the numbers. Ans. 28, 98, and 30. 19. What number is that, whose half, third, and fourth purts, 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 number is that, to which if its third and fourth parts bo 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 al)0ve 40 as it is now below it. Ans. 16. 23. Wliat 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 ii certain number be multiplied by 4, and 20 bo added to the product the sum will be 32. What is the number? Aus. 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. Aus. 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. 2*7. 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 Ibrfeit 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 thi-ee o'clock will the hour and minute hands of a watch be together ? Ans. 2h. 10m. oij'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 a:= 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, x — 10= distance hour hand has gone after 2 o'clock. x= 12(a:— 10)=12x— 120; lla;=120; and a;=1012 min., or the hands are together 10 min. 54-6- sec. after 2 o'clock. 31. The hour and minute hand of a clock are together at noon ; when arc they next together ? Ans. Ih. 5tt niin. 94 RAY'S ALGEBRA, SECOND BOOK. 32. At what time between 8 and 9 o'clock arc the hour and minute hauds of a watch opposite to each other? Aus. 8 h. 10] 6 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 move than its half; 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 sura 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 (fg 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 As age? Ans. 88. 3(. 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/,, 14]1. 39. A can do a piece of work in S days, B in 6 days, and C in 9 days. Find the time in which all together cau perform it. Ans. Ij'^ days. SIMPLE EQUATIONS. 95 Let Xr= the required number of days. Then, in one day, A can 1 do J , B y and C ^, and all three - of the whole work. Hence, HHi=:J- 40. If A docs a piece of work in 10 days, wtioli A and B can do together in 7 days, how long would it take B to do it alone? Aus. 23] 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 ;;niong five per- sons — A, B, C, D, and E. B received $10 les-s 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, $2Y ; 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 Sj 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, 9G RAY S ALGEBRA, SECOND BOOK. and the chaise to twice the price of the horse and harness. What was the cost of each ? Aus. 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 bod}', and his body weighs as much as liis head and his tail. AVhat 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 j'ou buy 100 apples and pears for 94 cents, how many of each will you have? Ans. (5 apples and 25 pears. 52. Suppose that for every 8 sheep a farmer keeps, he should plow an acre of land, and allow one aero 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 ho ride in a stage which travels 12 miles an hour, so as to return home in time, walking back at the r^te 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 | of a lb.? Ans. 135 ibs. 55. A mass of copper and tin weighs 80 lbs.; and for every Y lbs. of copper, there are 3 lbs. of tin. How much copper must be added to the muss, that for every 11 lbs. of 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 i 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. 155. 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 can only become known when they arc 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 by Substitution. 2d. Elimination by Comparison. 3d. Elimination by Addition and Subtraction. 156. 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. 2<\ 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:H-32/=33, (1) 4a:-|-52/=59. (2) From (1), by transposing Zy and dividing by 2, we have Substituting this value of X, instead of X in (2), we have 66^6j/+5j/=59; 3-^3 3-3X7^6 The following is the general form to which two equations of the first degree, containing two unknow'n quantities, may always be reduced. The signs of the known quantities, a, b, c, etc., may be either plus or minus. axA-by=c, (1) From (1), by transposing by, and dividing by a, we have a Substituting this value of x in (2), we have (!'!_■ — n'by-\-nb'ii—ni-'; [ab' — a'b )y=ac' — a 'r ; ae'—rt'e y~ ab'~a'b' / ac'—a'e \ f^h!i^_ e-o\ ab'-a'h /_ (ib'c-n'he—ahc^a'bc jiuta:--^^ _ a(ab'—a'b) b'c—bc' = —,-, ,!• Hence, ab' — a'b Rule for Elimination by Substitution. — Find an expres- sion for the value of one of the unknoicn quantities i'n either SIMPLE EQUATIONS. 99 equation, and substitute this value, instead of the same un- knoion quantity, in the other equation; there will thus be •forined a new equation, containing only one unknown quantity. 1. Zx—by= 2, \ Ans. s:=4, i 3. 4a;=64— 3?/, \ Ans. a;=10, 2a;+7^=22. J . 2/=2. 2a;+32/=44. 1 2. 5a;— 3(x— 2/)=13, i A. a;=5, a;— 3/=4. i 2/=l- 5. ax^hy^c — d, 1 n(c— d) ?»(c— d) 2/= »■ 4. 5x^7y=0, \ Ans. x=:7, 4:X+3y=8y+S. i y=5. \ . n(o—d) r. > Ans. x=^ J-!-, 3/=- > an+bm c mx=ny. > ' an-{-bm' an-i-bm' 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 diiFerent 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. 2x+Sy=33, (1) 4x+5w=59. (2) 33— 3.y From (1), by transposing and dividing, we have x= — s — . 59— Sm From (2), by transposing and dividing, we have x= — j — . Placing these values of X equal to each other, 59— 52/_33-33/ ^~ 2 ' 59— by = 66— Sy, by clearing of fractions; 3/^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—^-^^=6. 100 RAY'S ALGEBRA, SECOND BOOK. c—by a &—b'y a' General equations, ax^by=c, (1) a'x-{-b'y=&. (2) From (1), by ti^insposing and dividing, x. From (2), by transposing and dividing, x^- Equating these values of X, c—hy c'--b'y a'c- o'hy- ri r' —ab'y, by clearing of fraction ; {ab' —a'bxy = <'- — "x o, c' — «'a^ From (1 ), y=—fj- ; from (2), y= ^ ■ Equating these values of y, & -a'x (■ — nr b' ~ b ' bc'—a'bx- b'e -al/x {ab'—a'b)x~ h'c—li<:'\ b'c—bc X—~r-, — —7-.. Ilcncc, ab'—a'b ' Rule for Elimination by Comparison. — Find an c.rprcs- sioii for the ca/in nf /lir name ■unI{)ioirn quantity in each of the i/ivin cqnationn, und place these values equal to each other; there vnll tliiix he formed a new equation, containing only one vnkyiown quantity. 1. 3.r-2y= 9, bx-\-iy =37./ Ans. X --':>, y=3. 2. 7x+y=lOy^-7,\Ans. x=10, x+y= 2y+3. ) y= 7. 3. ix-\- Sy=51, 1 Ans. a;=6. 8x—13y= 9. 2/=3. ■i. mx:=vy, x+y=^a. Ans. x= y am ~m~\-n 5. aX']-by=p, \ bq—dp cx-]-cly=Tq.i ' ^/jc—ad' etq -cp ^^ad^^o Remark.— This method is generally to be preferred where the equations are literal, .ind sometimes in other cases. SIMPLE EQUATIONS. 101 158. Elimination by Addition and Subtraction con- sists ia 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. Takiug the same equations as in the preceding articles, 2.r+32/=33, (1) 4a;+52/=59. (2) It is evident if we multiply (1) by 2, that the coi-fficient of a; will be the same in the two equations. 4x+6y=m (3), by multiplying (1) by 2. 4x-\-5y=:b9, (2) brought down. y=z 7, by subtraction. 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;+152/=165, (4) by multiplying (1) by 5. 12t+152/=177, (5) by multiplying (2) by 3. '2a:=l% by subtracting (4) from (5). x= 6. Or, by substitution, from either of the original equations. Thus, From (1) 2a;+3x7=33; 2a:=33— 21=12; 2=6. General equations, ax-\-by=c, (1 ) a'x-^b'y=&. (2) It is evident that we shall render the coefficients of a; the same in both equations, by multiplying (1) by a\ and (2) by a. aa'x-\-a'by=a'c, (3), by multiplying (1) by a'; aa'x-\-ab'y=ac' , (4), by multiplying (2) by a; (ab' — a'b)y^ac' — a'c, by subtracting; ac'— a'c y^ub'—a'b- 102 RAY'S ALGEBRA, SECOND BOOK. The coeflBoients of y in the two equations will evidently become equal by multiplying (1) by 6', and (2) by b. cib'x-\-bb'y~b'c, (5) by multiplying (1) by b' ; a'bx^bb'y=bc', (6) by multiplying (2) by 6; [ab' — a'b)x^b'c—bc', by subtracting; b'c-bo' „ x=-,, ji- Hence, Rule for Elimination by Addition and Subtraction. — 1. Multiply or divide the equations, if iiciyssary, so that one of the miknoicn quantities will have the same coefficiciit in hoth. 2. Tul-c the difference, or 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. Rem ark. — When the coefficients of the quantity to be eliminated are prime to each other, multiply each by the other. A\'hen 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!/=10, ■) Zx-Yly-- !). 1 2. 2.r I 3^=18, \ 3.r^2i/_^ l.J 3. 2.1- 9^=11, \ 3x-Vly=\b.i Ans. x=l, 2/=3. Ans. .T=3, 2/=4. Ans. K=l, 4^+52/ 40 "" 2.T— y x-y, +2)/=i. .5 x-\ay=b, ax— by^^c. A. X- A. a;=i, y= (I' — bi ab — c The following examples may be solved by either of the three methods of elimination : 1. 9x-iy= 8, 13a;-f72/=101, 2)=5, 43/-i(.r+10) =3./ Ans .^=4, y=l. A. x=h, y y 9 10 =1. Ajis. a;=18, 2/=10. SIMPLE EQUATIONS. 103 '5+2/ 2x r 5^=^35. 12+a;' I Ans. x=2. 8.j;-4=9i/, J 6. Ka;+2/)+J(^-2/)=59.- 5a;— 332/=0. , 3a;+42/+3 2a;+7-3/ :5+ y- 10 15 ~ ^ 5 ' 9.!/+ 5a;— 8 x+y 7x-\-G 8. ax—hy, x-^y=c. 11 Ans. a;=5, y^4. Ans. a;=99, 2/=15. Ans. a;=7, 3/=9. ^2/, 1 =e. i 6c . Ans. a;= — — s-, and j/= — —-,. 9. 3ax~-2by=c, a''x-\-b-y=5bc. ]0. — + ~=a, x^y n m , - + -=6. a; 2/ 1 . 116c , c/ 156— a \ Ans, x= nvfl — T? Tna—nti 11. (a2— 62)(5x+32/)=(4a-6)2a6, o65c a-y- a+6 + (a+6+c)6x=62j/-|- (a+26) a6. a/= m b—na' Ans Xz ab ^a+b' y- ab a-b Rem AUK.— Transpose 6^ in (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 RAY'S 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 in to n ; required the numbers. Solution by using one unknown quantify. Let mx^^ the less numliev, and nx= llie greater. Then, iix — iiix^a. a ma , na iiix^ , anil 'tix= 11 -)n n^iu n — m Solution by using; two unknown quantities. Let x^ the less number, and 2/^ the greater. Then, y—x-a, (1) And X : y : : rii : n ; or, jny^^^nx. (2) Since my^nx, we have y= — ; Substituting this value of y in (1), we find as before, ma , na X= , and 1/^ . 2. The hour and minute hands of a watch are opposite at 6 o'clock ; when are they next opposite? Lot x= minute spaces moved over by the hour hand, and y= min- ute spaces moved over by the minute hand. Then, since the minute hand moves 12 times as fast as the liour hand, x:y::l -.12, or y=r2.v. (1) But the minute hand must evidently pass over 60 minutes more than the hour hand ; lience, 2/=.r+60. (2) Substituting, I2.c=a;-|-G0, ll.r=60, a;=5_5-. min. 2/— Oo-A_ min. =1 h., 5i m. Hence, the hands are next opposite at 5J>- m. past 7. In a similar manner the period of coincidence of the 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 liind, 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 10x3x5; 456=100x4+10x5h 6. Let a;= the tens' digit, and 2/= the units' digit. Then, 10a;-|-2/= the number. And 10y-{-X:= the number when the digits are reversed. Also, ^^^^=7. ''^^^-'> From these equations we readily find x^8, and y^i. 4. A farmer sells to one man 5 sheep and 7 cows for $111, 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 $3.85, 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 R'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. ScGGESTiON.— Let 63;= A's money, and 5^/= B's. 106 RAYS ALGEBRA, SECOND BOOK. 8. Find a fraction, such that if the numerator and de- nominator be each increased by 1, the value is J; but if each be diminished by 1, the value is J. Ans. -|. 9. Find two numbers, such that J of the first exceeds 1 of the seqfiud by 3, and j 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 co.-st 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. When 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 difference of those sums ^16. Ans. 30 and 42, 6 and 10. 14. A boy spends 30 cts. in apples and pears, buyino- his apples at 4 and his pears at 5 for a ct, ; he then finds that half his apples and J of his pears cost 13 cts. How many of each did he buy ? Ans. 72 apples, 60 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 tbo 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 sum 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 2.T— 6, 3a:— 6, and y 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. \ and |J. 21. A purse holds 19 crowns and 6 guineas. Now 4 crowns and 5 guineas fill l| 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 lor the money he had in his purse. If he bought V bu. of rye and laid out the rest of his money in wheat, he would \v:int 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 h'n money, and fill his sack ? Ans. Buy 9 bu. of wheat, and 12 rye. SIMPLE EQUATIONS, INVOLVING THREE OR MORE UN- KNOWN QUANTITIES. 160. Simple equations, involving three ot 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 5.T— 4y+2.:— 48, (1) 3.f+3^-4-=24, (2) 2a,-— 5^+3.3=19, (3) to find X, y, and z. To eliminate z from the first two equations, multiply (1) by 2, and then add this to (") , thus, 10.f-8)/+42= 96, by multiplying (1) by 2, 3:e+3,?/^4-= 24, (2) \Zx~by =120, (5) by adding. To elimin.ite z from equations (1) and (3), multiply (1) by 3, and (3) by 2, and then subtract; thus, 15a;— ]2,y+r,r=;144, by multiplying (1) by 3, 4.r lOy-l-Or- 38, by multiplying (3) by 2, \\x~ 2,y =106, (6) by subtracting. To eliminate y from equations (5) and (6), multiply (5) by 2, and (6) by 5, and then sub'^ract; thus, 55x— 102/=530, by multiplying (6) by 5, 2G.r-lO;/=240, by multiplying (5) by 2, 29.r =290; by subtracting. x--=10. SIMPLE EQUATIONS. 109 U0—2y—We, by substituting 10 for x in (6) ; whence, y=2. 50—8+22=48, by substituting for x and y in (1); wlience, z=3. It is evident tliat the same method may be applied when the num- ber of equations is lour 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 icill be one less. 2d. Combine any one of these nev] equations with each of the others, as before; the number of equations and of unknown quantities will be two less. 3d. Continue this series of operations until a single equation is obtained, with one tmknown quantity, and find its value. 4tli. Substitute this value in the dericed equations, for the values of the other unknown 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=a, (1) x—y^z=b, (2) x-\-y — z=c, (3) to find x, y, and z. By adding the three equations together, and calling a-f 6+c=S, We find a:+2/+z=s. (4) Then, by subtracting (1), (2), and (3), respectively from (4), and dividing by 2, We find . . . x=l(s—a), =J(6+e). 3/=i(s-6), =l{a+c). z=>^[s-c), =l{a^-b). In a similar manner, solve the following examples; 8. x+y+.=22, ^ a;+z+?«=19, a:-\-y-\-u^=\Q, Ans. x=^ 5, z=10, M= 4. no RAY'S ALGEBRA, SECOND BOOK. 4. 2.r— y— 2=12, (1)-) '6y—x—z^lQ, (2) I bz—x—y=^2-i. (8) 3 Put x-\-y^z=s^ and add (1), (2), and (3) successively to the last equation. This gives 3x=S-f-12 (a) 42/=s+16 (6) 63=«+24 (c) Multiplying these by 4, 3, and 2, we have 12a;=4s+ 48 12)/ =38+ 48 - 48 Or, 122 = 12(3;+2/+z)=9s-|-144, by addition. . 12s^9s-144; 3s=144; s= 48. Substituting this value of S in (a), (b), and {(■), we find .r=2n, .{^=16, and J=12. Solve the following by either method of elimination : . Ans. :i:=[, ■ . ■ y=2, . . . --^S. 5. x+i/+z=6, dx-y+2-:=--1, 4x+^,/-z^1, 6. 3,r+4y— 52=32, ■) 4.,~by + 3::=lS. ]- -3^-4.= V] 7. ,r-9y+3.v-10»=21, 2.r+7y— .-a=683, 3,r-fy+52+2M=195, 4.r— 6y— 2z— 9»=516. 8. x-\-ly=10—]z, lix-r.)=2,/-7. Aqs. x^lO, y= 8, 2= 6. Ans. a:=100, .=-13, «=— 50. . Ans. .T=7, 2/=4, c=3. SIMPLE EQUATIONS. Ill 9. 9x—2z-\-n=il, *ly~bz~t=l2, 3^-4M+3fc7, lz-bu^l\. Ans. a;--=5, .=2, Examples to be solved by special methods : Ans. x= 10. - + -=a, X y X z y= a+6- 2 -6+c' 2 6-|-c — d Suggestion. — Subtract (-3) from (2), then combine the resulting equation with (1), to find X and y\ z may be found similarly. 11. — a:+y+«+u=a, X — y-\-z-\-v^b, x-^y—z-\-v=c, x+y^rz—'v=d. Ans. x=),{fi — «), .... 3/=Ks-6), ■ • • • ^=K«-«)> .... ■y=K« — '^)> where s=^(a-|-6-(-c-)-d). PROBLEMS PRODUCING SIMPLE EQUATIONS CONTAINING THREE OR MORE UNKNOWN QUANTITIES. 161. 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 tlie preceding, whose sum is 20, and such that the sum of the first and second is to the sum of the second and third, ll-:> 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 composinn; 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 3j ; 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, i the second, and | the third, shall together make 46 ; J the first, i the second, and I the third, shall to- gether make 35 ; and j the first, i the second, and i the third, shall together make 28'. Ans. 12, 60, and 80. 6. The sura of three numbers, taken two and two, are a, h, and c. What are the numbers? Ans. l(a + h—c), l(a-\-c—l), and l,(h-\-c—a). 7. A person has four casks, the second of which being filled from the first, leaves the first i 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 .j^^ of it. But the first will fill the third and fourth and have fifteen quarts remaining. How many quarts does each hold? Ans. 140, GO, 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 cngago- mcnt, in which the slain were one fourtli 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, 6/0 sailors. SIMPLE EQUATIONS. 113 V. SUPPLEMENT TO SIMPLE EQUATIONS. Remark. — The principles employed in algebraic equations may be variously applied. We may, for example, by their aid demon- strate several of the theorems in fractions. Thus, to prove that ^-^=^, (Art. 118); put 9=^. ,1 ma Then, bq=^a, B.wimbq^'ma; . . q=- — .. a , ma ma a Hence, since O^t, and q^ — 1, . . — =- ^ -r. ' ^6 mb mb b To prove that ^X^ = ^> (Art. 131); put p=| and 9=^. Then, bp=a, and dq^^C. Multiplying the last two equations, member by member, «C , . , , , We have oapq^^ao; .-. pq=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. IGS. 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 Eule is a formula expressed in ordinary language. 2d Bk. 10 114 KAY'S 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, n, and p. Let mx, nx, and px, represent the required parts. Then, mx^nx-\-px^a, And .... X^= ; from which we obtain, rn-; n ^-p ma na , pa mx=- , nx= , and px=- rn-\-n^p m-^n^p m^n-j-p) This formula, expressed in words, gives the following Rule for Dividing a Given Ifumber into Parts having to each other a Given Ratio. — Multiply the given number hy each term of llic ratii/s res^peeiively, and divide the prod- ucts by 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 same ratio as the numbers 5, Y, and 11. Ans. 15, 21, and 83. 3. Divide 38.1 into four parts, having to each other the same ratio as the fractional numbers J, J, ], and i. Ans. i5, 10, 1%, 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 difFerenee b. Required the numbers. at, a b A. Greater, ^ + 2 ' ^^^^^ 2~2' 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, 4*7° 6' 17" 7. The difference of two numbers is a, and the greater is to the less as m to n. Find the numbers. , ma na Ans. , . m — 71 TO — II 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, =- J" ; 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 h, and the third exceed the second by c. a — 26 — c a-\-h — c a-\-b-\-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 to, and the fourth divided by m, shall be aD equal to each other. m,a ma a Tri'a Ans. ; — -1^, — m, - — r^rr,+''''i (1,1+iy^ ' ' (m+1)^^"' (m+1)^' (m+1)' IIG RAYS ALGEBRA, SECOND BOOK. X 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. Aus. 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. o-j-c 16. A person finds that he can row a skifi^ 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. Oiven the sum of two numbers =a, and the quotient of the greater divided by the less ^b. Required the num- bers. , r (^ "-^ Ans. Less =-; — =, greater r=r — z,. i + l' '^ h+1 Tliis gives Uie following simple rule : To find the less number, 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. Kequired the numbers. Ans. 240 and 16. 19. A person distributed a cents among n beggars, giv- ing li cents to some, and c to the rest. How many were thereof each? , n — nc , , nh — a ^ Ans. -; at cts., and -j at c cts. I — c — c 20. A father divided $8500 among *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, 3 daushters. 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. no-i-r n r Ans. _.-' ' , . 1+2 1+2 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 6 days, and B and C in c days: find the time in which each can perform it separately. ■ . . 2ahc . 2ahc _, . 2ahc Ans. A m — -^ , B in -^-— , C in -— da. ac-\-Oc — au au-\-uc — ac aO-{-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 52^'*3, second in 70 {^, 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, b oxen for n months ; and C, c oxen for p months. Re- quired each one's share of the rent. Ans. A's, — ^- — PS,; B's, p-^- — F $ ; and mo-)-}io-(-jjc »na-f-»6-|-_pc ma-\-no-\-pc From fbese 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. ib., h lb. at n shillings, and c lb. at p shillings : what will be its cost per lb. ? . ma-\-nh-\-pc a-\-b-\-c 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 r-owing down and up the stream, also the rate of the current and the rate of rowing. . ~. , 6(^ . he „ Ans. iime down, : ; time up, -. ; rate oi current, c-f-d c-j-rf. -2^;-^; rate of rowing, -^^-^. 30. A vessel sailed icilh the wind and tide 60 miles, and returned tcilh 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, 7A hours ; wind, lOj 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 . a;—- the number Then, 3— x= 7 ; whence, — x=7 — 3 ; or, x = — 4. SIMPLE EQUATIONS. 119 Now, ■ — 4 subtracted from 3, gives a remainder 7 ; 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 (he 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 7 Let the question now be generalized, thus : What number must be subtracted from a, that the re- mainder may he h? Let . a;=r the number. Then, a — x^b ; whence, x=a — 6. While 6 is less than a, the value of X will be positive; and the question will be consistent in an ariihmelital sense. But if 6 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 i ? From this we derive the following important general principles^ 1st. A negative solution indicates some arithmetical incoiv- mtency or absurdity in tlw question from whidi the eqitxdimi was derived. 2d. When a 7wgative solution is obtained, iJie question, to which it is the answer, may be so modified as to be consistent ■with arithmetical 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. 17 and — 8. 120 RAY S 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; ill how many years will the age of the father be 4 times that of the son? (3;= — 4j. III. DISCUSSION OF PROBLEMS. 16S. 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 Jlscmsion of the problem. The various forms which the value of the unknown quantity may assume, are shown in the discus,sion 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. c Then, (a~b)x=o, and x= =-. This result may have five different forms, depending on the different values that may be given to a, b, and c. To express these forms; let A denote, indefinitely, some quantity. I. AVhen b is less than a. In this case, since a — b will be positive, the value of x will be of the form -f A. II. When b is greater than a. In this case, a — b will be negttive, and the value of a;, of the form — A. III. When b is equal to a. In this case, the value of A X is of the form ■^, or, (Art. 136), x=^cc . SIMPLE EQUATIONS. 121 IV. When c is 0, and h either greater or less than a. In this case, the value of x is of the form — , or, ("Art, A ^ 136), a-=0. V. AA'heu h is equal to a, and c is equal to 0. In this case, the value of x is of the form ^, which (Art. 137) 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 COUPLERS. 166. 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 1^««— ««^i^»^««i»»l B and a=AB, the distance between the P two places. Let a;=AP, the distance which the first travels. Then, a — a;=: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. % Therefore, — = the number of hours the -first travels. m fj^ ^ 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=raa — mx\ ma , na X= ; and a — x= m-\-n m^n ma a a Ist. Suppose m,=n; then, x= -- ==7^; and a—x=^; that is, if they travel at the same rate, each travels half the distance. 2d. Suppose n^O; then, x=: — =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 m^n, then the value of x will be greater than that of a — Xj 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 A B i P, and let a;=AB, the distance between the places; x^AP, the distance the first travels; X — a^BP, the distance the second travels. Then, reasoning as in the first case, we have X X -a m" n ' nx = mx — ma ; ma , na x= ; and x— a= . )n~n m — n 1st. If we suppose m greater than n, 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 m, 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 rij/ht of A and B, the negative values must indicate (Art. 47) that the place of meeting is at P', on 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. AVe 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 Imve 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 B¥'=a+x. 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 a;=-Tp=cx, and x — a=-j^= 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, -=: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^m,, and a=0; then, X=^, and x — «=§. But this is the symbol of indetermination, 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 /ii=0; then, x^ — ^a; that is, the first cou- m ' rier travels from A to B, overtaking the second at B. So, if OT:=0, x—a=—a. 124 RAYS ALGEBRA, SECOND COOK. 7th. If we suppose their rate of travel has a given ratio, as 71;= — ■ then, X=- — =2a; that is, the first travels twice the dis- '2, iii tance from A to 15 belore 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 tlie quantities which it contains can not be ob- tained directly from others witli which it is compared. Thus, r!'-L3y=-.19, 2a:^- 5^=33, are equations which are independent of each other, since the one can not be obtained from the other in a direct manner. a-+3!/=19, 2x4-6^=88, arc 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, . rr— 2/=;3, Ry transposition, :r — o-j-?/. If we make 2/=li then, a;=4. If we make 2/=2; then, X—h, and so on; from which it is evident that an unlimited number of values 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 . . . 3" +3?/ -52 --20, x~ 3/-L3c=16, If wc eliminate x. we have, after reducing, 2/— 2z=l ; whence, j/=l-f2z. SIMPLE EQUATIONS. 125 If we make z=l ; then, y=S, and a;=20+5z— 32/=16. If wo make z=2; then, 2/=5, and a:=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 numhcr of unJmown quan- tities exceeds the number of independent equations, the proh- 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-i diminished by the f^ ? Let x=r the number. X X Ux 'Jx ^'""'4 + 6 =20" -IS- Clearing of fractions, 15a;4-103;=33a;— 8a:; or, 25a;^25a;; which will be verified by any value whatever of x. 169. The reverse of the preceding ease requires to be considered ; that is, when the number of equations is greater than the number of unknown quantities. Thus, we may have 2x-\-iy=2Z (1.) 5x-2y= 2 (2.) 5a;+4y=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=V2 (1.) 2x-+ 2/=17 (2.) 3a;+22/=30 (3.) The -values of X and y, found from equations (1) and (2), are a;=5, 2/=7; from (1) and (3), a;=G, and y=Q; and from (2) and (3), a:=4, and 2/^9. 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 ? , vino ^ Ans. a;= -. m — li, 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 ? Ans. — r— mi. from C, or mi. from L. m-\-n m-\-ii Under what circumstances will the boats meet half way between C and L? Under what will they meet at C ? Al 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— 3y=3, ?,x^2y=Vl, 4.r+3y =24 ; to find X and y, and show how many equations are redundant. (Art. 169.) Ans. .t=3, y=ii- 4. Given cc-f 2y=ll, 2x— y=Y, 3a;— y=17, .<-+3^=19; to show that the equations are incompatible. (Art. 169.) SIMPLE EQUATIONS. 137 V. A SIMPLE EQUATION HAS BUT ONE ROOT. ITO. Any simple equation involving only one unkno-wn quantity, (x), may be reduced to the form mx^n ; i'or all the terms containing x may be reduced to one term, and all the known quantities to one term ; whence, ;r= — . m Now, since n divided by m can give but one quotient, we infer that a simple equation has hvt one root; that is, there is but one value that will verify the equation. VI. EXAMPLES INVOLVING THE SECOND POWER OF THE UNKNOWN QUANTITY. 171. 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. (4+x)(x— 5)=(x— 2)^. Performing tlie operations indicated, we have Omitting x'^ on each side, and transposing, we have 3x=2A, or a::=8. (2.^+3)^ 1_^ 1 ^- -2S+r +3x-''+-^- ■ . Ans. a;^l. 3.^--2..+l _ (7..-2)(3.-6) Ans,cc=l,V a. ^ — gj -r,u 55 4, i5±^(3cc-19)=2a=+19 Ans.a;=8. 6cc— 43^ . al^V+x'') ax . Ji hx b <= 128 RAY'S ALGEBRA, SECOND EOO'K. cr'" dx^ . at' — ce Ans. X- a-\-bx e-\-fx cf — bd 7. It is required to find a number which 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; breadth, 9 feet. \L OF POWERS, ROOTS, RADICALS, AXD IXEQUALITIES. I. INVOLUTION, OR FORMATION OF POWERS. 1T2. The Power of a number is the product obtained by multiplying it a certain number of times by itself. Any number is the first poircr of itself. When the number is taken ticice 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 tliird power or cube of the number. Id like manner, the fourth, fifth, 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=3'= 3, is the 1st power of 3. 3X3=3-'= 9, ■' " 2d " " 3. 3X3X3=33= 27, " " 3d " " 3. 3X3X3X3=3'= 81, " " 4tli " " 3. |X^XfX|=(|)^=5S,V " " 4tli " " 4. aX«X«Xa=(a)<=a< " " 4tli " " a. ac2xac2xac2=(ac-)'=aV' " " 3d " ao'^. From the above, we have the following General Eule for Raising any Quantity to any Required Power. — Miilt'tply tlw given quantity hy itself, 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 a= is a-'X«^X«-=«-+^+^=«^^'=«'"'- If the quantity to be involved is positive, any power of that quantity will be positive. If it is negative, the even powers will be positive and the odd powers negative. Thus, — ay^—a^-l-a', and —aX—aX —«=—«'• The 4th power of — a is +«''; the 5th power is — a'; and so on. Hence, we have the following Rule for Involving' a Monomial. — 1. Involve the coeffi- cient by the rule of arithmetic. 2. Multiply the exponents of the literal factors hy the ex- ponent of the required power. 130 RAY'S ALGEBRA, SECOND BOOK. 3. If the quantity he negative, ruake the even powers positive and the odd powers negative. 1. Find the square of Sax'z' Ans. 1ha-x*^. 2. The square of —Wed Ans. QVc'dK 3. The cube of 2x'z Ans. Sa;^. 4. The cube of — Ba'c' Ans. —2la^c\ 5. The fourth power of — 2xz''. . . . Ans. 16a;V. 6. The fifth power of —8aV. . . Ans. —2430'"^'^ 7. The seventh power of — rn'n. . . Ans. — m");'. 8. The square of a^b'" Ans. a'""l"'. 9. The nth power of xy'z'' Ans. a;"^"^"^. 10. The square and the cube of Sa'a;"'+^3^p~'. (1) Ans. rf,ofix""+y-\ (2) ^f joV^+yp-^ 11. The square of , -, Ans. The square of -, Ans 6'z*' 9, 12. The cube of ^ Ans. 3c' ^^"^- 27 c Case II.— To square a Binomial Quantity. The rule for this has already been given, Arts. 78 and 79. 1. Find the square a — x. . . . Ans. a^ — 2ax-\-x'. 2. The- square of a: -|-)/. . . . Ana. x--\-2xy-\-y''. 3. The square of mx — nx''. Ans. m'x'' — 2?)!)ia^+)i'-.r*. 4. The square of fa+^Ji. . . . Ans. ..^a'+Sai+J &l X. a-+5y . x''+10xy-\-2bf 5. The square of -~—^,. . . Ans. — J— ■,— .V -^• rn- — n in — ]l}n-n--\-ir A quantity, consisting of three or four terms, may be squared on the same principle, by reducing it to the form of a binomial, squaring, and completing the operations in- dicated. Thus, a+b—c—a+(b—c). Squaring, we have a'^-\-2a{b~c) -l-(6-c)-=a2+2a6-2ac+62_26c+c2. a+b—c+d=,{a + b)^(c—d). Squaring, (c(+6)2— 2(a+6)(c— d) ^(c— d)2=a2+2a6+62— 2ac— 26c+2arf+26d4-c2— 2ed+d2. FORMATION OF POWERS. 131 1 12 6. Find the square of a; — — 1. A. x' — 2x-\ — --I 1. X x^ X Case III. — To raise a Binomial to the Third Power. By trial, we find the cube of a-\-h to be a?-\-Za''h-\-iaV -{-b^. Hence, Tlie cube of a binomial is equal to the cube of the first term, plus three times the square of the first into the second, phis three times the first into the square of the second, plus the cube of the second. If the quantity is a residual, as a—b, the result will be the sa^e, except that the signs will be alternately plus and minus. A quantity consisting of three or four terms may be cubed in the same manner, by reducing it to the form of a binomial, as explained above in Case 11. Thus, (a-6+c)3=[(a— 6) +c]3=(a— 6)3+3(a-6)2c+3(a— 6) C^-f e', which last may be further expanded. (a+6-c+d)3=[(a+6)— (c-d)]3=(a+6)3— 3(a+6)2(c-c()+, etc. 1. Find the cube of x-|-y. Ans. o^-^Sx'y-]-Sxf-\-y^. 2. The cube of 2x—z. Ans. Sx?—12x''z+6xz'—^. 3. Thecube of 3x-\-2y. Ans. 27x'+b'ix'y-\-3Qxf+8f. „ m — n . m' — 3m^n-j-3?Km^ — n 4. The cube of ^. Ans, — 2ra " m' — 6m^n-\-12mn'' — 8n' 5. The cube of ia— 16. Ans. la'—Wb+lab'—^-^V 6. Involve (a-—)^ A. a^— 3x+- — i=a;^— -3-3(x— -) ^ X X 3? X^ X 7. Involve {f-\-e-'^y. Ans. e'^+3e^+3e-^-j-e-»'=e'^+e-'^+3(e'+c-='). 8. Involve (x-\-:;-\-zy. A. a;'+3x^y+3a;^z4-3a;y^H-6a;^z+3a;2^+/+3^2z+3y2;'+z'. 132 RAY'S ALGEBRA, SECOND BOOK. Case IV. — To raise a Binomial to any Power. Rules for raising a binomial, or residual quantity, to the 4th, 5th, 6th, or to any higher power, may be formed on the same principle as those given under Case II (See Theorems I and II, Art. 78) and Case III. An easier method, however, was discovered by Sir Isaac Newton, which we now proceed to explain. NEWTON'S THEOREM. Let a-\-h be raised to the sixth power by actual multi plication. a + 6 a + 6 rt-'-f- a b a--j-2« 6-J- b- second power of a+6, or (a-(-6)2. a + 6 a"-j-2a-6-|- a b- + a26+ 2a 6^+ 6= a3_j_3f,!^_|_ 3a 52_|_ 63 _ _ _ third power of aJ-6, or (a-f 6)3. a +6 ai_(-3a''6-(- 3a-6-+ a 6^ + o?b^ 3a262+ 3a 63_^6< a'i+4a36-|- 6a262-)- 4a 63+6* . . . . fourth power, or (a+6)<. a -\-b aP\^a'^b^ mw-^ 4a263+ a 6< + a46+ 4a362+ 60=63+ 4a 6<+65 as+5a-'6+10o''62+10a26-+ 5a 6*4 6'' {«+6)^- a +6 a6+5a''6+10a<62+10a36'!+ 5a26<+a6' + a'^6+ 5a*62J-10a363+10a26t+5a6H^'' a''+6a'>6+15a*62_|_20a363 + 15a26-'+6a6''+6'i .... (a+6)6. FORMATION OF POWERS. 133 If we involve a — t, the result will be the same, except that the signs of the terms will he alternately pbis 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 number 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. ^igns of Terms. — If both terms of the binomial are positive, all the terms will be positive. If the second term is negative, the \st, Zd, etc., or the odd terms, will be positive, and the EVEN terms negative. 3d. Exponents. — The exponent of the leading letter is the same, in the first term, as the power to lohich the quan- tity is to he raised, and diminishes by unity, in the sueeeedivg 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+b)<^=a'^^a''b+a^b^-\-a"b^'+a-b*-\-ab'^-\-bf', omitting coefficients. 4th. The Coefficients. — The coefficient of the first and last terms is always unify; 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 RAY'S ALGEBRA, SECOND DOOK. The coefficiriitx of all terms equally 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 7th power, or to expand (x-^-i/y. Literal factors, oc\ x''?/, x')/-. x^y^, x^y^, x-y^, a;*/", y''. The cotfficient of the 1st term is unity; . . 1st term is X^. " " " 2d " " 7 " ;2d " " Ix'y. " 3d " ' 14^ " 3d " " 21x'2/2. Continuing tlius, we have for the complete expansion, o:'^-1x*y^. 2\x'y--\-Sbx'y-^ ■irjx'''y^-\-2lx'^y''+~xy''>-\-y-'. As a second example by the other method, Let it be required to expand (a — i)'. Tlie first term will be a'"'; the second, 6a'b. For the third, multi- ply 6 liy 5, and divide the product by 2, for the coefficient, and annex the literal factors. This gives ISa^fi^ Multiplying 15 by 4 and dividing by 3. we liavc for the next term 20a-^6^. Continuing this process, we find the next term to be 15a-6', the next Qiab\ and the last h''. Giving the proper signs, we have «'■•— 6« '6+150^6=— 20a"6'i-f-15a=6'— ea^HS"- The following additional facts may he noted, and may serve to render the application of the above principles still 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 POWERS. 135 of terms will be even. In the former case, there will be ono middle term, and in the latter Iwo, to the left and right of which the coefficients are the same. Thus, in the above examples, the cotfficients 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 sum of the coefficients, in every case, is equal to 2 raised to the required power of the binomial. Thus, in the above examples, 1+6+15+20+15-1-6+1=64=215, and 1+7+21+35+35+21+7+1=128=2". Expand (o-ft)*. . . . Ans. a*-|-4a='?--|-6o^t^+4a?/'+t*. Expand (x-\-yy- Ans. x^-\-Qa?y-\- 1 5a;*^^+ 20a:y + 1 5xy-|- Qxi/^if. Expand (o — a)* A. cfi — 5a*x-(-10a'a;-^ — 10aW4-5oa;* — x^. Expand (a-j-a;)'- Ans. a«-f8a'a--f28aV-(-56a5a:'-|-'70a*x*+56aV-|- 28aV+8aa'-)-a:«. Expand (n— t)'. Ans. a»— 9a8?)-f36a'Z<'— 84a<'i'+126a=i<— 126aW 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, (2a;3+5a2)<=(2a;3)4+4(2a;3)3(5a2)+6(2a;3)2(5a2)2+4(2a:3) (5a2)3+(5a2)-i=16a:'2_j_i60a;»a2_|_600:£'''ai+1000a:3a6_j^625as. Or, put m=2a:3 and n=5a2. Then, (2a;3+5a2)-<=(OT+r7)«. Then, [■m-{-nY=rii^-\-irrv'"n.-\-&m?'n?-\-imn^-\'n*. Returning to the values of m and n, we have m*={2x^Y=lQx^^, 4m.3n =4x(2a;3)3x 5a2 =4x8a;'X 5a2= 160a;9a2. 6m2n2=6X(2a;3)^X;5o2)^=fiX4a;"5X 25a<= 600a;6a<. 4m ?i3=4x 2a;3 X'5a2)3=4y2a;3xl25a'5=1000a;3a''. n*={5a^Y=62ba». Hence, (2a;3+5a2)4=16a;'2+160a:9a2+600a;6aH100^'a;3a6+625a8. 136 RAVS ALGEBRA, SKCOND 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;^-|-3j/^ to the fifth power, or expand (a;'-|- 3^')°. 2. Expand C2a-+aa-)'. Ans. 8a''-\-12a'x+6a*x''-\-aV. 3. Expand (2a-(-3a;/- Ans. 16o*+96a'a;+216aV+216aa;'+81a;'. 4. Expand Qla — 8hy. Ans. ^'go*— |a'6+-ya'6'— 54ai'-f 816*. 5. Cube a-f 2i.— c. Ans. o'+6a^6— 3aV+86'+12a6^— 126'c— c'+3ac^ -\^Qbc'—12ahc. 6. Expand (a-^b-\-c~dy. Ans. a* -\- 4a'h-l- 6a'V' -f iah' + h* + 4a'c + 12fl^Jc + 12ab'c-\-U'c — ia'd—12a'bd—12ab'd—'ib'd + 6d'c-'-\-12abc'-i-6bV—12a'cd—24abcd~12b'cd ^6ahP-i-12abd'-i-6bhl'-[-4a(^—12a(-'d-j-12acd^ —4a,P-^4bc' — 12bc'd+12bcd''—4bd'-{-c'—4c^d -^6chP—4cd'^d'. In many cnses, 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 a^-|--j=p' — 3^. 2. If two numbers differ liy 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 OP THE SQUARE ROOT OP 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 tli3 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, Wlien 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 POUR, the number of places of figures in the square root will be TWO. 2d Bk. 12 138 RAYS ALGEBRA, SECOND BOOK. In the same manner it may be shown, that when the number of places of figures is more than fmir, and not more than six, the number of places in the square root will be three, and so on. 175. 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. Tiierefore, if we represent the tens by t, and tlie units by w, any number will be represented by t-\-xt, and its square by the square oi t-\-u, or [t-\u)-. (;_|_«)2=r;2-r2^i(+u2=<2+(2i;+M)M. Hence, Tlie square of amy number is composeil of two quantities, one oj which is the square of the tens, and the other licirc the tens jilus the units mnltiplied 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)5-22o 625 1. Required to extract the square root of 625. Since the number consists of three places 625125 of figures, its root will consist of two places, 400| according to the principle established in 20y2=40!225 Art. 174, we, therefore, separate it into two 5 225 periods, as in the margin. 45l Since the square of 2 tens i.^ 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 lens (20) is 400. Subtracting this from 625, the remainder is 225. The remainder, 225, consists of twice the tens plus the units, multiplied by tie units; that is, by the formula, it is (-t-\~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. EXTRACTION OF THE SQUARE ROOT. 139 Dividing 22 by 4 {2t) gives 5 (u) for a quotient. Tliis 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^i-M), 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 45|225 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. |176 In performing this operation, we find the 48311449 square root of the number 590, on the same |l449 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 hito periods oj two places each, beginning at the unit's place. (The left period will often contain but one figure.) 2d. Find 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 tlius increased hy the last figure of the root ; subtract the product fromi the dividend, and to the rcniaindir briny down tlie next period for a new divi- dend. 5th. Double the icholc root already found for a new divi- sor, and continue the operation as before, until all tlie periods are brought doicn. Note — If, in any case, the division cnn not be effected, place a cipher in the root and divisor, and bring down the next period. 176. 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, Lot a and a-|-l. bo two consecutive numbers. Then, (a-j-l)-=o-+2a-j-], the square of the greater. And (a)-~a-, " " " less. Their difference is 2a 4-1. Hence, Tlie difference of the squares of two consecutive numbers is equal to twice the less number, increased by unity. Therefore, when any remainder is less than twice the root already found, plus one, the last figure can not be in- creased. Rc'(|uired the sc[uare root of 1. 2601. . . . Ans. 51. 2. 7225. . . . Ans. 85. 3. 47089. . . Ans. 217. 4. 390625. . Ans. 625. 5. 43046721. Ans. 6561. 6. 49042009. Ans. 7003. 7. 1061326084. A. 32578. 8. 943042681. Ans. 30709. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 177. Since |X 1=2%' *^^^ square root of ^^-, is ? : that is, -^^/^^^^---^^^^'i. Hence, we have the following V 26" EXTRACTION OF THE SQUARE ROOT. 141 Rijle for Extracting the Square Root of a Fraction. — ^Extract 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 2 9. Here, ^^-§?. By canceling the common factor 5, the fraction "becomes 4, the square root of which is S. AVhen botli terms are perfect squares, and contain a common fac- tor, the reduction may be made either before or after the square root is extracted. Tlins n/3fi — fi — 2- or 36_4 nnd t/4 2 Required the square root of 1 JL4_ Ans -8- I ,^ 9 747 Atis 5 7 2 --5 Ans 3 4. .56 111 9 Ati« -37 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'-^la-^-X, is 2a!-(-l ; therefore, there are always la imperfect squares between them. Thus, between the square of 5 (25) and the square of 6 (36), there are 10 (2a=2x5) 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, i/2, -^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 -y,/2. 179. It miglit be supposed, that when the square root of a whole number can not be expressed by a whole number, it might be exactly equal lo 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, -r, in its loiDCsl terms. Then, y c = -t; and c=:T-Ti, by squaring both sides (Art. 148). Now, by supposition, a and b have no common factor; therefore, their Squares, a- and b~, can have no common factor, since to square a number, we merely repeat its factors. Consequently, . , 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 suppod- a t'toa on which it is founded, that is, that -j/C^v-, 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='fg^. Now, the square root of ■'g^ is greater than |, and less than 3; hence, fi, or -, is the square root of 5 to within 1. To iicneralize this explanation, let it be required to ex- tract the square root of a to within a fraction -. 11 Write a (Art. 127) under the form — 5-, and denote the entire part of the square root of an- by r. Then, an- will be comprised between r- and (r-f-l)-, and the square root of — 5- will be comprised r , r+\ '"■ between - and . n n T T-\-\ 1 r But the difference between— and — ■ — 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, and divide the result hy the denominator of the fraction. 1. Find the square root of 3 to within J. . Ans. 1|. 2. Of 10 to within -] Ans. 3. 3. Of 19 to within \ Ans. ^. 4. Of 30 to within y^ Ans. 5.4. 5. Of 75 to within -j-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, TI7ie7i the fraction which determines the degree of approxi- mation is a decimal, add (wo ciphers for each decimal place required; and, after extracting the square root, point off from the right one place of decimals for each two cipiliers 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. Eequircd to find the square root of 4 to within J-. 4 — 4v "J— 28 1—1'' 7 — 4?- The square root || is greater than ^ and less than ^; therefore, S 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 hy the denominator. 144 RAY'S ALGEBRA, SECOND BOOK. 2. Find the square rgot of fy to within j\. Ans. f^ 3. Find the square root of II to within ^L. Ans. j\. 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 ciplieis to both terms. Thus, .3=1^=-,%; .156=J„5jfJl; l,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 cij^hers to the decimal, until the number of decimal places shall be kj^uhI to double the number required in the root. 2. After extracting the root, point off from the right ihe required number of decimal places. 4. Find tlie 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 decimuls 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 fg to five places. y'5=2.23606 + , ^16=4, Vi%=':r ' " ° "+ =.55901+ Or, -5g=.3125, and , .3125=.55901-L. 7. Find the square root of |. . . Ans. .7745S5+. 8. Find the square root of If . . Ans. 1.11803+. 9. Find the square root of 3j. . , Ans. 1.903943+. 10. Find the square root of 11 5. . . Ans, 3.349958+. EXTRACTION OF THE SQUARE ROOT. 145 EXTRACTION OF THE SQUARE ROOT OF ALGE- BRAIC QUANTITIES. EXTRACTION OF THE SQUARE ROOT OP MONOMIALS. 182. To square a monomial, (Art. 172), we square its coefficient, and multiply each exponent by 2. Thus, (37n?i2)2=9m2n''. Therefore, ■^/Qm-n^=3mn-. Hence, we have the following Rule for Extracting the Square Eoot of a Monomial. — Extract the square root of the coefficient as a number, and divide the exponent of each letter hy 2. Since -|-aX+a=+«^, ~a'X^a=-\-a^; Therefore, j/a2=r-|-a, or — a. Hence, The square root of any positive quantity is either phis or minus. This is expressed by writing the double sign before the root. Thus, |/4a^=±2a; 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, -j/ — 4, -j/ — 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. 162:y. Ans. zhixf. 2. 2bmV. Ans. dzbmn. 3. m^a;*_yV. Ans. ztma^yz*. 4. 1024a'bV>.Ans.±32aV^. Since ( ^ ) =5X-5=p; therefore, ^^=I^,=±-^. Hence, To find the square root of a monomial fraction, extract the square root of both terms. 2d Bk. 13* 146 RAY'S ALGEBRA, SECOND BOOK. 5. Find the square root ot -— js- • • • ^ns. ±7y3- 6. Find the square root of -r^-rrr,- ■ Ans. ±^—f-,. EXTRACTIOX OF THE SQUARE ROOT OF POLYNOMIALS. 1S3. 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--b+cy- = a~-r2ab+b^^-2ae+2bf-^c' = a'-+{2a+b)b+(2a +2b+c)c. (a+6+c+rf)- =- a^+2ab+b'-i-2ac~2bc-~c^+2ad+2bd~2cd +rf-'=f(-- i2a^ btb-(2a+2b+c)c+(2a-i-2b^2c^d)d. Or, by calling the successive terms of .i polynomial, r, r', r", r'", and so on, wo shall have (r-f r'-(-r''-)-?-'")-=r--f(2r-Lr')r'-i-(2r -f 2r'-(- r'^)r"-{- (2r -\- 2r' -\- 2r'' A^ r"')r'", wliere the law is mani- fest. In this formula, r, r', 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 J.s equal to the square of the first term — plus ticicc the first term, plus the seeond, mul- tipUfil by the xirond — pZ!(S twice the first and second terms, jtIus the third, multiplied by the third. — plus twice the first, second, and third terms, plus the fourth, multiplied by the fburth, and so on. Heufc, by reversiiip: the operation, we have the following Rule for Extracting the Square Root of a Polyno- mial. — 1st. Arrange the polynomial loith reference to a cer- tain letter. 2d. E.rlracI the square root (f the first term, place the result on the right, and subtract its square from the given quantity. EXTRACTION OF THE SQUARE ROOT. 147 3d. 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 dicisor, and proceed in a similar manner to find the other terms. 1. Find the square root of 4x''y''-\-12x''y-\-9x^ — SOxy^ —20xf+2by*. Arranging the polynomials with reference to y, we have ROOT. 2,by*-20xy3j^4x'y2—30xy^+i2x^y+Qx^\5y^—2xy—3x 252/' 10y^~2xy -'20xy^-{-^x^^ -•lOxy^+ix^y^ lOy^ — ixySx -30xy^-{-12x2y+9x2 -3Qxy^+12x'y+9x' If the preceding example he arranged according to the powers of X, the root found will be 3x+2xy—5yK This is correct also, aa may be shown generally, thus : -^/ {a'-\-2ax+x')=-.±{a+x)=a+x, or —a—x. 2. x'+6ax+9a' Ans. x+Ba. 3. 16x'—4:0xy+2by^ Ans. ix—by. 4. 4a;V— 12x^2+9/ Ans. 2xz—Sy. 5. 49o,'"»-«— 42a«"'-'^+9a»"'+^ . . Ans. 1a^'"-'—Sa*'"+'. 6. l+2x-|-7a;^+6x'+9x* Ans. l+a;+3a;''. 7. 9a*— 12a'6+34a'6''— 20a/;^+25i*. Ans. 3d'—2ab+bb\ 8. a:''+4a;5+10x*+20x'+25a;'+24a:-l-16. Ans. a:'+2x'+3x+4. 9. 9x'^6xy+30xz+Qxt+y' — 10yz~2yt+2bz'+10zt If Ans. '6x — y-{-bz-\-t. 148 RAY S ALGEBRA, SECOND BOOK. 10. :^*-2x^-f^^-| + A- Ans.x'-x+l ^_, 2baV bahc' , c* . bah c' ^1- "4 3^ + 9 ^""'-^^E- T„ 1051a:^ 6x 14x^ , n ,,^ , . H a; „ 12. — g^ -^ ^--_j_9-f49a;*. Ans. 73;^— g+3. 10 a' 9,^' . a I J--^- 77 — ■^+ -. Ans. r . 14. Eeduce the following expression to its simplest form, and extract the square root ; {a—hy—2(a'+h')(a-hy-{^2(a'+h'). Ans. a'-\-l'. 15. Find the square root of 1—x- to five terms. , -, x'' x^ 3" ox' Ans. l-^__g__-^-_,etc. 16. Find the first five terms of the square root of a;^-)-a^ , , a' a* a" 5a' ^'''- "+2:.-8:g5+ife5-T28:^+' ''"■ 184. The following remarks will be found useful : 1st. AT) 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-)-62 Is not a perfect square, but if we add to it, or sub- tract from it, 2ab, it becomes the square of a-\-b or of a — 6. 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 sqxnre roots of the extreme terms, and unite iliem by the sign of the second term. Thus, a--\-iax^4x^ is a perfect square, and its square root is, a-\-1x; 4x-+8.T?/-|-9?/2 is not a perfect square, p^j. 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 HOOT OF NUMBERS. 1S3. The Cube, or third power of a number, is the product arising from taking it three times as a factor. (Art. 1V2.) The Cube Root, 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, 64, 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 tliree places. Again, comparing the numbers 10 and 100, we have. Numbers 10, 100; Cubes, iOOO, 1006006. Since the cube of 10 is IOO6, and the cube of 99, which is less than 100, is less than lOOOOOO; 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 i006006, and the cube of 1000 is 1006006006; 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 hegin 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 RAYS ALGEBRA, SECOND BOOK. 187. 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 . . t^ the tens, and u^= the units of a given number. Then, t-\-u^ the number. And (i+w)3= the cube of the number. But {t+uf=fi-^Zf-uyitu^^u^=P+(Zf^^Ztu-^u^)u. Hence, The cube of any number consisting of tens and vnits, 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 units, plus the square of the vnits, all three multiplied by the un its. 1. Required to extract the cube root of 13824. Separating the number into periods by points, we tind 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 {Zt''--\-Ztu -|-m2)m in the formula. The first term, 3<^, of this fovmuLi, is sometimes termed the trial divisor, as it is used to find the unit's figure u. If the remaining terms were only Zt^u, we could readily find u by dividing by Zt"^ ; but if we divide by 3^^, we may obtain a figure too large, on account of omitting the terms Ztu-\-ifi, 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 ns 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^M+M2=1200+240+16=145e. tu 13824|24 8 zr- =1200 5824 Ztu= 240 m2= 16 1456 5824 EXTRACTION OF THE CUBE ROOT. 151 Multiplying lliis by 4 [u), the product is 5SL>4, wliicli, subtracted from tlie first remainder, leaves zero (0), and shows that 24 is the exact cube root required. In cubing tlie tens, it is customary to omit the ciphers; but in tal£ing three times the square of the tens, also in taking three times the product of the tens by the units, it is best to write ciphers in the vacant orders. 2. Required to find the cube root of 44361864. Jilu 44361864[354 27 3^2=2700 37i<= 450 i^= 25 3175 173G1 15875 3(/i+^)2=36750() 3(h+t)u^ 4200 m2= 16 371716 1486864 1486864 After separating the number into periods, we find the cube root (35) of 44361 on the" same principles as in the preceding example. Then, considering 35 (10/i-|-^) as so many tens, we find the unit's figure (4), as in the preceding example. In dividing by the trial divi- sor 27, to find the second fig- ure (5), we first obtain 6, but tliis is found by trial to be too large. From the preceding, we derive the following Rule for the Extraction of the Cube Root of Kum- bers. — 1st. Separate the given nuniher into periods of three places each, beginning at the units pilace. (The left period will often contain but one or two figures.) 2d. Find the greatest cube in the left period, and place its root on the right, as in division. Subtract the cube of the root from the left period, and to the remainder bring down the next period for a dividend. 3d. Square the root already fovnd, and multiply it by 3 for a trial divisor. Find how many times this divisor is con- tained in (he dividend, omitting the iinit^s and ten's figures, and write the result in the root. Add together, the trial divi- sor with two ciphers annexed ; three times the product of the last figure of the root by the rest, with one cipher annexed; 1.52 RAYS ALGEBRA, SECOND BOOK. and the square of the last figure; the sum will he the complete dtiisor. 4th. Multiply the complete dioisor by the last figure of the root, and suhlrait the product from the dividend, and to the ri maindcr hriny down the next period for a new dividend, and so proceed until all tlie periods are brought down. Extract the cube root of the following numbers : 3. 12167. . . Ans. 23. I 1. 127263527. Ans. 503. 4. 39304. . . Ans. 34. 5. 493039.. Ans. 79. 6. 2097152. Ans. 128. 8. 403583419. Ans. 739. 9. 158252632929. Ans. 5409. By a process of reasoning similar to that given in Art. 177, we deduce the following Rule for Extracting the Cube Boot of a Fraction. — Rrduee the fraction, if necessary, to its lowest terms, and extract the cube root of both terms. 10. Find the cube root of /A Ans. i. 11. Fiud the cube root oi' T^^'f'^. . . . Ans. |. 188. A Perfect Cube is a number whose cube root can be exactly ascertained; as, 8, 27, 64, etc. An Imperfect Cube is a number whose cube root can not be exactly ascertained ; as, 2, 3, 4, etc. It may be shown, by a course of reasoning precisely similar to that employed in Art. 179, that the cube root of an imperfect cube can not be a fraction. APPROXIMATE CUBE ROOTS. 189. To illustrate the method of finding the approxi- mate cube root of an imperfect cube, let it be required to find the cube root of 6 to within J . 6= ■yj'l. Now, the cube root of 3S4 is greater than 7 and less than 8; there- fore the cube root of %'' is greater than % and less than §• hence, I is the cube root of 6 to within less than l 4 ■* EXTRACTION OF THE CUBE ROOT. 153 To generalize this method, let it he required to extract the cube root of a number a, to within a fraction -. n Let r be the root of the greatest cube contained in ari?; then, — ir IS comprised between -~^ and — L ' ■ hence, its cube root is n" ■ 71-* n-^ T T-\-\ comprised between - and : and since (he difference of these n n \ r I fractions is — ; therefore, - is the cube root of a to within -. Hence, n n n Rule for Extracting the Cube Root of a Whole Nniii- ber to within a Given Fraction. — Multiply the given rmm- hrr hy the cube of the denominator of the fraction which determines the degree of approximation ; extract the cube root of this product to the nearest unit, and divide the result by the denominator of the fraction. 2. Find the cube root of 5 to within X. . . Ans. 1?. 3. Find the cube root of 10 to within ^. . . Ans. 2J. Since the cube of 10 is 1000, the cube of 100, 1000000, and so on, the number of ciphers in the cube of the denom- inator of a decimal fraction is equal to three times the number in the denominator itself. Therefore, When the fraction which determines the degree of approxi- mation is a decimal, add three ciphers fur each decimal place required; and after extracting the root, point off from the right one place of decimals for each three ciphers added. 4. Find the cube root of 2 to five places. A. 1.25992. 5. Find the cube root of 3'7 to six places. A. 3.332222. By adding ciphers to both terms, any decimal, or whole number and decimal, may be written in the form of a 154 RAYS ALGEBRA, SECOND BOOK. fraction, having its denominator a perfect cube ; thus, .2=-,\%%, .25=,^,,%, 6.4=f-^oo. and so on. Therefore, to find the cube root, Annex ciphers to the given decimal, until the member of decimal places shall he equal to three times the mimbcr re- quired in thf. root. Extract the root, and point off from the right the required number of decimal places. 6. Find the cube root of .4 to four places. Ans. .7368. 7. Find the cube root of 34.3 to six places. Ans. 3.249112. To find the cube root of a fraction or a mixed number, first reduce the fraction to a decimal. 8. Find the cube root of %. . . . Ans. .82207+. 9. Find the cube root of 5jg|. . . Ans. 1.816 + . 10. Divide the cube root of '^Alliiis j^y t}jg square root 3-768 of the square root of 8.3521. Ans. .25. EXTRACTION OF THE CUBE ROOT OF ALGE- BRAIC QUANTITIES. EXTRACTION OF THE CUBE ROOT OF MONOMIALS. 190. If we cube, for example, 2((,t-, we have (2ax-)' = 8(('.f'' ; that is, we cube the coefiicient, and multiply the exponent of each letter by 3. Hence, conversely, we have the following Rule for Extracting the Cube Root of a Monomial. — Extract the cube root of the coefficient, and divide the expo- nent of each letter by 3. Find the cube root of the followinc Blononiials : 1. %xh\ . . Ans. 2xz\ 2. 27a;y^ . . Ans. ixh/. -64r('»i''. Ans. — 4cam''. "'+^'x\ Ans. a"'+'x\ EXTRACTION OF THE CUBE ROOT. 155 c3- / a \^ a^ ,'va? a „ Since, ^ ^ ^ =-p; therefore, -y;^._ = -^. Hence, To find the cnhe root of a monomial fraction, extract the cube root of both terms. 5. Find the cube root of 7=^^^- Ans — ., 27 x^ 3x 6. Find the cube root of — =- ^' . . Ans. — -p — —. lAom''n^ bmn' EXTRACTION OF THE CUBE ROOT OF POLYNOJIIALS. 101. To investigate a rule for extracting the cube root of polynomials, let us first examine the relation that exists between a polynomial and its cube. {a+b+cf={{a+b) + c}^ = {a+by'+l3(a + b)- + S{a+b]o ■(a+6+c+rf)3=f(a+6+c)+rfi3=(a+5-fc)3+ j3(a-i-6+c)2+3(a+6+c)d-fo;-'jcZ. Hence, the cube of a polynomial is formed according to the fol- lowing law: The cube of a polynomial is equnl to the cube of the first term — phis three times the square of the first term, plus three times the product of the first term by the second, plus the square of the second, all 'hree multiplied by the second — p>^"^ three times the square of the first two terms, plus three times the product of the first two terms by the third, plus the sqvare of the third, all three midtiplied by the third, and so on. By reversing this law, we derive the following Rule for Extracting the Cube Root of a Polynomial. — 1st. Arrange the polynomial with reference to a certain letter. 2d. Extract the cube root of the first term for the first term of the root, and subtract its cube from the given polynomial. 156 BAY'S ALGEBRA, SECOND BOOK. 3d. Tahc thrre times the square of the first term of the root, and call it a trial divisor for finding each of the remaining terms of the root. Find how oftin the trial divisor is con- tainiil in the first tirm of the nnieiinder ; this tcill givr the srrimd term aj the root. Then form a complete divisor hy adding together three times the square of the first term of the root, plus tJtrer times the product of the first term hi/ the sec- ond, plus the square of the second. Multiply these l>y the second term of the root, and subtract the product fromi the firxt remiiiiider. 4th. Again, fiml how often the trial divisor is contained in t]te first term of the remainder ; this uiU give the third term of the root. Then form a complete divisor as hcfore, hi/ add- ing tngellicr three times the square of the first and second terms, plus three times the product of the first and second terms hi/ the third, p>hts the square of the third. MultipAy these hy the third term of the root, and subtract the product from the last remainder. 5th. Con/iiiue thus till all the terms of the root are found. 1. Find the cube root of j-«--6.r'+12.r'+3(r'.r*— 8.c' 1 2„V-\- 1 ■2a'.r'-\- 3aV-— Go^x+a". 2J5_63-''+12a;'<+3a2a;'i— &E-''— 12oV4-12a->--f3c(i3;2_Ga',r+a6 3,r 1—12.1-- 'i2x-+3a-x--^ea~x+a')-\-3a-x^—V2a'-X''^12a-x- I -\-3a ■> renijiintk-r and subtrahend aro each writti'n ' -\-oCf-X i.-CC-X'- l-Cl-X- in two lines. | -^3a''x-—Qa*x-\-(i'\ Vi'e first extract the cube root of x'', which gives X- for the first term of the required root. Then, 3 times the square of this, ^Sx*, constitutes the trial ihvi.ior for finding the remaining terms. Dividing — Gx^ by Sz*, gives — '2x, the second tei'm of the root. We then form the complete divisor by adding together 3(:r-)--] 3 (jx2x— 2a:) + (— 2.'')'— Sj' — 6.r3-f4j;-. Multiplying this by —2x, EXTRACTION OF THE CUBE ROOT. 157 and subtracting, the first term of tlie second remainder is -\-3a'^x^, which divided by the trial divisor, gives -j-a^, for the third term of the root, and so on. SECOND M"ETHOD. 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, (a-\-l-^cy, etc. 1. Arrange the polyjioniial, as in the j^revious rule. 2. Extract the cube root of the first term, etc., as before. 3. Find 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. Ciibe 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. a;S la;2_2a;+a2 Sx^l — 6a;"'-l-12ar', etc., 1st remainder. a;n — 6a;'+12a:<— Sa;', cube of x''-—'lx. 3a;^| Sa^x*—12a^x^, etc., 2d remainder. a''— 6a:''+12a;4+3a2a;<— 8x3— 12a2a;=+12a2a;2+3aia;2— 6a^a;+a5 We first extract the cube root of afi, and find it x'^. Cubing this, subtracting, and dividing the first term of the remainder by 3a;'', we obtain — 2a; for the second term of the root. Cubing x^ — 2a;, writ- ing it below, and subtracting, we have the second remainder. Dividing the first term of this remainder again by Zx*. we obtain a2 for the third term of the root. The cube of x-—ix-\-cfi being equal to the given polynomial, the work is finished. 158 KAYS 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. 'I'he process of cubing the root may be conducted by Newton's Theorem, as explained in Art. 172. Find the cube root 2. Of o'-f 24a'i + 192ai'+5125'. Anb. aH- 8&. 3. Of 8a-'— 84rY%r+294(«^— 343.x^ Ans. 2a— Tr. 4. Of o«— 6o54-15a«— 20«='+15a-'— 6a+l. Ans o- — 2a-\-l. 5. Of .'(■-— 9.i--^+3!:t.<'—9U.c^+loG.:f^—144.T+64. Ans. ./■- — 3.'-|-4. 6. Of (n + ly''x^ — 6caP(,i-\-l}*"x- + 12cV(a-\-ly'' 3.,_8rV. Ans. {a+ly"-x—2ca'>. 7. Fiud the first three terms of the cube root of 1 — x. X x^ Ans. 1 — H — ~Q — , etc. IV EXTRACTION OF THE FOURTH ROOT, SIXTH ROOT, N''" ROOT, Etc. 193. The fourth root of a number is one of four equal factors, into which the number may be resolved ; and, in general, the »"" root of a number is one of the n equal fac- tors into which the number may be resolved. When 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 hi/ extracting the roots of more simple degrees. To explain thi?, we remark that (a^y=a^^^—a^-, and in gen- eral («"')"=a'"^"=«""'- Hence, The n"* power of the m"" power of a nnmhrr is equal to the nm"' poicer of tlie iiiimher. Reciprocally, the mn"' root of a nnmhrr, is equal to the n"" root of the m"' rout of that numher ; that is, EXTRACTION OF THE CUBE ROOT, 159 From this, it follows that (/a=:'\ i/a; and ■^a='\ ^-^a; in like manner pa^\i y a, and so on. 1. Find the 4th root of 65536. Ans. 16. 2. Find the 4th root of 13107.9601. . . 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. SaxK 6. Find the 4th root of a* -|- 4o'Z>x -f 6a'bV -f iab^x^ -\-b'x*. Ans. a-\-bx. 7. Find the 4th root of a;^ — 4x6-f lOx* — 16a;'^ + 19 . Ans. a;^ — 1-\ — . a;' 8. Findthe6throotofa«+-^— 6/a^+- j-l-is/ a^+-\ —20. . 1 Ans. a . a 193. It has been shown already (Arts. 182, 183) that the square root of a monomial, or a polynomial, may be preceded either by the sign -j- or — . We shall now ex- plain the law in regard to the roots generally. If we take the successive powers of -j-a and — a, we have +a, +a2, +a3 -\-a' —a, +a2 — a3, -[-a\ . . . -i-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 €ve7y odd root of a monomial must have the same sign as the m.onomiaI itself. Thus, f^8a^=-\-2a, f-8a^=—2a, {/-~S2a^«=—2a^. 2d. That an even root of a positive monomial viay be either positive or negative. Thus, f 81a46s=±3a62, ^64ai2=±2a2. 100 RAYS ALGEBRA, SECOND BOOK. 3d. That cvenj even root of a negative monomial is im- possible; since no quantity raised to a power of an even degree can o- nent of each letter hy n. Rules for the extraction of any root of a numerical quantity, or algebraic polj'nomial, may be formed on the same principle as is that of the cube root, (.-Vrt. 191.) Thus, since {a- b)* =a<-4«-6-6a-6-'+4a6=-|-6< = a'-^-(4a3_|-6a26-|-4a62 + b^)b. {a+b)'--^a-'+{5a^+10a^b+10a-b-+5ab^+b*)b, etc. The trial divisor for the fourth root would be of the form 4r/^', or four times the third power of the first term of the root, and the complete divisor of the form, 4a^-^CM"b-\-Aab^-\-b^. For the fifth root, the trial and complete divisors would be of the forms, 5a* and 5a'- 10a^b-\-10a-b^^5ab^-\-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 ho of the form 4a-'', for the 4th root, 5a* for the 5th root, no"-' for the «"• root, or, in general, n times the (n — l)"" power of the first term of the root. Remark.— In the following examples, find the root of the numeral coefficient by inspection. It is unnecessary 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 — 32a*a;^<'. 2. The 6th root of l29b^c'\ . The 7th root of 128,r y 4. 5. 6. 7. Extract ^/a*"b"c'" The 8th root of 6561a^b'^. . The 9th root of — 512a:V«. . The 10th root of 102U">z^. 8. Extract the 5th root of 32afi- T-lOx— 1. -80: 161 Ans. — 2ax''. Ans. ±36c'. . Ans. 2x1/''. Ans. ±Sah\ Ans. — 2x::''. Ans. ±2628. . Ans. a*lic^. .r,*+80x'— 40x^ Ans. 2x— 1. V. RADICAL QUANTITIES. Note. — These quantities are generally called siirds by English writers ; while the French more properly term them radicals, from the Latin word radix, a root. 193. A Rational ftuantity is either not affected by the' radical sign, or the root indicated can be exactly as- certained ; thus, 2, a, ^/4, and f^S are rational quan- tities. A Radical ftuantity is one aifected by a radical sign, but whose indicated root can not be exactly expressed in numbers; thus, |/5^2. 23606797 nearly. 196. From Art. 193 it is evident that when a mono- mial is a perfect power of the n"" degree, its numeral coeffi- cient is a perfect power of that degree, and the exponent of each letter is divisible by n. Thus, ia- is a perfect square, while 6a^ is not; and 8a'' is a per- fect cube, while 6a'', 8a^, 7a''', 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. Thus, 3 4 l/a^ may be written a^, and f^a* may be written a^ ; and, in gen- m oral, the n"' root of the ?n"' power of a is either {/a"', or a" . 2d Bk. 14 162 EAT'S ALGEBRA, SECOND BOOK. Since a is the same as a', (Art 19,) the square root of a may be expressed thus, a ; the cube root thus, « ; and the »"' root 1 thus, a". Hence, the following expressions are equiTalent; ■/a and a , fa and a* _ 1 ^a and a". I Also, fa- and a , I/' a" and a» Hence, TVic numerator of the fractional exponent denotes the power of the qiiantift/, and the denominator the root to he extracted. 107. Theorem. — Ani/ quantity affected uith a fractional exponent, may be tranf erred from one terra of a fraction to the other, if, at the banie time, the sign of its exponent he changed. This proposition has already been established (Art. 81) when the exponent is integral. It is also true when the exponent is frac- tional, as we shall now prove. Let it be required to extract the cube root of — . a' As —=a-- (Art. 81); therefore, a -s^f «^-. But, \.'— ,r. JL and f'7F==a-f. (Arts. 190, 194.) ^ "- at Therefore, -^ = o-f. an 1 "! In like manner, generally, — j, = o ■ a» lOS. The Coefficient of the radical is the quantity which stands before the radical sii;n. Thus, in the expressions a^b, and Sj^-'c, the quantities ct and 2 are called coefficients. Radicals are said to be of the same degree "when they have the same index ; as, a' and 5^, or -^a- and f^b'K RADICALS. 163 Similar radicals have the same index, and the same quantity under the radical sign ; as, ay^b and cyZ ; ^fcC' and bf/a\ Before entering into a discussion of the general subject of radicals, it is important to observe that, A radical quantity is raised to a power equal to the index of its root, by simply rejecting the radical sign with its index. Thus, the square of -^a is a, the cube of ^a is a, the square of |/3 is 3, the n"i power of 'J/a is a. etc. In other words, /aXi/a=«, f ax fax fa— a, etc. This is evident from the definition of u, root, (Art. 173.) REDUCTION OF RADICALS. Case I. — To reduce Radicals to their Simplest Form. 199. 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 oj the product of two or more factors is equal to the product of the square roots of those factors. That is, y ab=-^/ aXV ^ t which is thus proved; Squaring both members of this equation, we have, (Art. 198,) ab^axb, or ab=ab. Now, since the equation is true after both sides are squared, it was true before, (Art. 148, Ax. 6,) or ■\/ab=yaX\/^- By this principle, 1/36=^/4x9=2x3; /T44=/9xT6=3X4; y'8=y'4x2=v/4Xi/2=2y'2. Hence, we have the following Rule for the Reduction of a Radical of the Second Degree to its Simplest Form. — 1st. Separate the quantity to he reduced into two parts, one of which shall contain all the factors that are perfect squares, and the other the remain- ing factors. 104 RAYS ALGEBRA, SECOND BOOK. 2d. Extract the square root of the perfect square, and pre- fix it as a coejficieiil to the other part placed under the radical sign. To determine -whether any numeral contains a factor that 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 the following examples : 1. 1,12, 1 18, /45, v/32, -,/50a^ y72a^b^. Ans. 2, 3, 3v 5, 3| 5, 4, 2, 5a, 2ci, 6ab^2b. 2. 1 245, ,'448, ,810, ,/5U76-c2, , ISO.io'^-'. Ans. 7,/5, 8,/7, 9/10, 136c/36, lOa^fi/S. In a similnr manner, jiolynomials may sometimes to simplified. Tims, , |3((^ -6o2c-^3c((--)=, 3a(a-— 2ac-pC-)=(a — e)y'3a. 3. y/(a^—a''b), ,'f(x-'- 6aa:+9a, ■^/ (x- -2/'^){x-\-y). Ans. a, (a— 6), (a;— 3),, a, (^+2/)v/(^— 2/)- To reduce a fractional radical to its simplest form, 1st. Render the dcnnminator of the fraction a perfect square hy multiplying or dividing loth terms hy the same quantity. 2d. Sijxr.-ate into two factors, one of which is a perfect square. 3d. Kr'ricI the square root of this factor, and irrile it as a coefficient to the other factor placed under the radical sign. 4. Eeduce | j?, and ■« L , to their simplest forms. " a b ]ab '1 _ , /I — j- 1 ,— =■ \b~\b>^ir\b^=\u^'''''-'\i.--.v<^i>=jy<^b. Ans. i,/2, l/u; |,/ll, j/3, 3/30, 1/10. RADICALS. 165 „ p jSa ja^ / Zxjf- \\ ^"'•^V^' asv^rs^, I^V^, il^fe^s/)^- 200. To reduce radicals of aoy degree to the most simple form. The principle of Art. 199 is, evidently, applicable to radicals of any degree. Thus, 1. Reduce 1^54 to its most simple form. f'54=f27x2=f27xf2=3^;2. Similarly, f |=^|x|X|=#'if=rjVXl8^i# 18- Heduce each of the following to its simplest form : 2. ^40, ^81c*, f \'i&a^(^\ fl62m^n^, ^144. Ans. 2^ 5, 3c,f3c, 'ia^of2(?, Smnfenw^, 2{/9. 3- fl n fl fl vl I I vl Ans. -,3^, Af6, If 36, if 15, 1(54, If 768. '{.32. 4. f/162, f.3888, ^S'la^b^, ^729a6. Ans. 3f/2, 6^3, 2abp2ab\ 3af/3a. 301. The ran"' root of any quantity may be simplified when it is a complete power of the m"' or n"' degree, as shown, (Art. 192.) V v/9a2=,/ Thus, {/9a2=-y y9a2=f 3a, Also, ■^a2_2a6+62=\/ f a2— 2a6+62=f o^. Reduce each of the following to its simplest form 1. f 36a2c2, {/Simhi*, ^ia^, f/l6a2c<, f 12563. Ans. f 6ac, 3ref m,' f 2a, f 4ac2, y'55. 166 RAY'S ALGEBRA, SECOND BOOK. Case II. — To reduce a Rational Quantity to the FORM OF A Radical. 202. If we square a, and then extract the square root of the square, the result is evidently a. 2 3 Tbat is, a^y a-^a^. In like manner, a^=-^a'^^a^, and gen- m erally, «='{' cT-^a'"- Hence, Rule for reducing a Rational Quantity to the form of a Radical. — Raixc the quantity to a pov:cr corresponding to the given root, and write it -under the radical sign. 1. Reduce 6 to the form of the square root. Ans. -i/SG. 2. — 2 to the form of the cube root. Ads. -^ — 8. 3. Sax to the form of the square root. Ans. yQd'x'. 4. m — n to the form of the square root. Ans. |/m-' — 2mn-\-n''. Similarly, a coefficient may be passed under the radical sign. Thus 2,/3=j/4> ,'§=,/ 12. Generally, a" 5=y'o^X'T''^=v'a™^. 5. Express 5^ Y, and a-j ''^, entirel}^ under the radical sij^n. Ans. ^/175, and -y^a'b. 6. Pass the eoclEcicnt of the quantity 1f/b, under the radical sign. Ans. ■j^40. Case III, — To reduce Radicals having Different Indices to Equivalent Radicals havino a Common Index. 303. 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 •^~^, and ^'6b, or (2a) ^ and (3&)* to quantities of equal value, having the same index. f 36 = (36) 4 =(36) '"1z=if/ (36)3='|/2763. Hence, Rxile.^7?erf«ce 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 he raised, and the common denominator the index of the root to he extracted. J 1 >2 1. Reduce -j/S and -^2, or 3 and 2 to a common index- Ans. -^27 and ^/i, or 21'' and 4^ 2. f^b and |/4 Ans. ■{/2b and y 64. I 8. a^ and h' Ans. -j/a* and |/6. 4. if a, //5&, and -^6^. Ans. ^^, '1^6256*, and ^2167^ 5. -j/^, f/^, and f'^. Ans. \^^, v'a", and \y^». 2 3 J 6. Reduce 3^, 2'*, and 5- to a common index. Ans. 3"r^^ 2t'2, 5"^-, or \>' 6561, \5'5T2, y 15625. ADDITION AND SUBTRACTION OF RADICALS. S04. Required to find the sum of S^a and b-^a. It 13 evident that 3 times and 5 times any quantity, must make 8 times that quantity; therefore, 3fa-\-5^a=S^a. But, if it were required to iind the sum of 3|/a and 5^'o, since ■j/a and fa are different quantities, we can only indicate their addi- tion; thus, 3^a-{-5^a. 168 RAYS ALGEBRA, SECOND BOOK. Similarly, S^/2+Tj/^—iy'^=^v"^- But 3,/5"and 4, 3 -^V^'Hi,/ 3. _ So also 3^/5 and 4(5/5-3, 6+4f 5. _ Radicals that are not similar, may often be made so; thus, i/12 and , -" are equal to 2) '3 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 Radicals. — 1st. Reduce the radicals to their simplest forms, and, if necessary, to a com- mon index. 2d. If the radicals are similar, find the sum of their coefficients, and. prefix it to the common radical; hut if they are not similar, connect them by their proper signs. Rule for the Subtraction of Radicals. — Change the sign of the subtrahend, and jjrocced as in addition of radicals. 1. Find the sum of y 448 and p/lTS. - 44S:^,/(j4 <7= 8,/7 112=, l(j By addition, 12^ 7, Ans. 2. Find the sum of f2i and if 81. . Ans. bfli. 3. Of f/48 and f T(J2. . . . Ans. 5f' G. 4. Of I'TSl^F and y'M^. Ans. (3a'b-{-5ab)]'2ab. 5. Subtract |/T80 from -(/IM. . . . Ans. 3| 5. 6. Subtract fW from f/T35 Ans. fb. Perform the operations indicated in each of the follo-sf- ing: 7_ ^,243^^, 274-y 48. Ans. 16/3. 8. ( 244-, /54—, l»i Ans. ,(1 9. 1/128—2, 50 f, 72 ~, -18. Ans. y 2. RADICALS. 169 10. |/48a62+6/75a+v^3a(a— 96)2 j^na uy'Sa 11. 2yl+^ym+vT^+vJ- Ai>8- Vv/iS- 12. ^128— 1^686— if 16+4^250 Ans. ibfyl 13. 2f r+8f ^ Ans. 3f 2 14. 6^4a2-|-2f 2a+?/8a^. Ans. 9,yM 15. 2v/3-iv/12+4/27— 2/3.3^ Ans. ^V^^ 16. ^re+fST— f^=5l2+f 192— 7f 9 Ans. 10. loF , 1 17, ■ ^j'^ + i^V{(i'f>-ia^b^+4ab>) Ans.g^ab. MULTIPLICATION AND DIVISION OF RADICALS. 205. The rule for the multiplication of radicals ia 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, VaXv^5'=Va5. (See Art. 198.) Hence, (Art. 53,) a'{/bXoVd=aXcX\'bX'l/d=aci/bd. The rule for division is founded on the principle that Tlie quotient of the u"* roots of two quantities is equal to the n'* root of their quotient. iVa nia , . , . , That is, — — ^Mi-; which is thus proved: Raising both sides to the ruh power, we have j- = j-, 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 different indices, reduce them to the sam,e index. Then, I. To Multiply. — Multiply 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 by 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 2y'aZ by Sa-j/oic. 2 ^ab Za^abc 6ay/a262c=6a/a262xc=6aXa&l/c=6a26/c. 2. Divide ^a\/ah by 2|/ac. 4a, cib ia \ab „ |6 „ ]bc 2a j- 2 '^ 2 Vac Vc Vc2 c " 3. Multiply 2^3 by 3i 2. liji 3=2(3)^=2(3)e=2|/P=2|/'9. 3/2=3(2)2=3(2)B=3^23=3"i/"8. Multiplying, . . 6 J '72, Ans. 4. Divide 6i 2 by 3,^2. 6, 2=6f/23=6j/8. (1.) 3,5 2=3," 22=3," 4. (2.) Dividing (1) by (2), we have 2|/2. 5. Multiply 3i'12 by 5, 18. ... Ans. dOi/E. 6. Multiply 4f 12 by 3,r4. .... Ans. 24 f 6. Y. Multiply together Sj 3, 7] ^ and |/2. Ans 140 8. Multiply 3^5 by 4|'''a. . . Ans. 12^^^. 9. Multiply together ^ il if 3, and ^i 5. A. \=' 648000. 10. Multiply together '{7, y 7\ and 'f i?. Ans. '^^a;'. 11. Divide i/40 by ; 2. Ans. 2y'^. 12. Divide 6i/54 by 3v^2. Ans. Gy'S. RADICALS. 171 13. Divide 70f 9 by 7^18 Ans. b^i. 14. Divide ^72 by ^^2. Ans. f3. 15. Divide 4f 9 by 2^3. Ans. 2^W. 16. Divide f 72 by ^3 Ans. y2. 17. Divide ^Uy^jl Ans. ^ Polynomials containing radicals may also be multiplied ; thus, 18. Multiply 3+^/5 by 2— 1/5. 3+ /F 2- 1/5 6+2/F —31/5—5 6— ,/5— 5=1— y/5, Ans. 19. Multiply |/2+l by i/2— 1 Ans. 1. 20. llv/2— 4y'15 by |/6+i/5. Ans. 2|/3— ^/lO. 21. Raise ^/g'+y'S to the 4th power. Ans. 49+20|/6. 22. Multiply a/l2+-i/19 by 4ll2— ^19. Ans. 5. 23. Multiply »'— a;|/2+l by a)2+xi/2+l. Ans. x*+l. 24. (a;^+l)(a;^— a;|/3+l)(x^+X|/3+l). Ans. ai'+l. 206. To reduce a fraction whoso 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 j/6. Thus, ^= ""^v^^ayb yb ybx-^/b b ' Again, if the denominator is -^a, if we multiply both terms by ^a^, the denominator wall become fa'^fa'^^fa^^a. 172 RAY'S ALGEBRA, SECOND BOOK. In like manner, if tlie denominator is i/ct", it will become ra- tional by multiplying it by ")' ' a"'-"- Therefore, When the denominator of the fraction is a monomial, multiply hoth terms by such a factor as will render the ex- ponent 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 — ,/c, the de- nominator will be rational. a Thus, a(h—^'c] ah—a-^ b-~c ' 6+, c (6^, c){6-, c) If the denominator is b — j c, the multiplier will be b-\-y'c. If the denominator is | b---^ r, the multiplier will be yb — j C; and if it 13 I b — I e, the multiplier will be ),- b-\-y/c. If the denominator is of the form i'f'+)/i+^c, it may be rendered rational by two successive multiplications. The first will result in a quantity of the form m — yn, which may be made ra- tional as before. Reduce tlie following fractions to equivalent onea hav- ing rational denominators : 1. 1 8 Ans. r6 o 3, 5"-2,^ 3 6 Ans. f f 0. 4. 8—5/2 3-2/2 t/3+1 2 , o— 1 Ans. ^fl 16. Ans. 4-|-y 2. Ans. 5-1-2,/ 6. Ans. 9-|-|,/10. ^^'^^L^ _ Ans. /6+/2+/5. y6+y-2-/5 +„ EADICALS. 173 1 x-\-^x'''—l x—yx'^—1 , Ans. 2x. yx-^+l—/x-^—l yx'+l+yx^—i Remark. — By the preceding transformations, the process of finding the numerical value of a fractional radical is very much o abridged. Thus, to find the value of ^, we may divide 2 by the 1 ^ 2 2/5 square root of 5, which is 2.2360679+. But — = = --L_, the true y o O value of which is found by multiplying 2.2360679 by 2, and divid- ing the result by 5. Rfeduce each of the following fractions to its simplest form, and find the numerical value of the result : -=' =""1 — S Ans. .894427+, and .707106+. j/5 |/ji 12. " • Ans. 15.745966+. /5— y'd POWERS OF RADICALS. 207. Let it be required to raise -j^Sa to the 3d power. Taking y/3a as a factor three times, we have J '3ax f^X \/~Sa= ^Tfo?. So, "l/ayy aiyCv^oT . . to n factors, =v'a^ Hence, Sule for raising a Radical Quantity to any Power. — Raise the quantity under the radical to the given power, and affect the result with the primitive radical sign. If the quantity have a coefficient, it must also be raised to the given power. Thus, the 4th power of 2fZa^ is l&fWafi. This, by reduction, becomes 16^27a6x3a2=48a2f So^: 174 BAY'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, ( J/ ■Za)'={J ^2af=y/2.a, (Art. 192.) In general, ('^Va)''=y !i/y'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 f 3a to the 4th power, we have f 8YcF—'\ fSla* =-^/'iia, or, dividing 8 by 4, we obtain at once ySa. 1. Raise f^2a to the 4tli power. . . . Ans. 2af^2a. 2. Zf^2aV to the 4th power. . Ans. lQ2ah'-^2ab\ 3. -^^'ac^ to the 2d power. . . . Ans. c^ a. 4. 1 ^ac-' to the 4th power Ans. a'c*. 5. 1 "6c^ to the 3d power Ans. cy 3. 6. -i/x — y to the 3d power. . . Ans. {x — y)\/x — y. ROOTS OF RADICALS. 208. Since "ll y'a^Ty'a (Art. 192), therefore, to ex- tract the roots of radicals, we have the following Rule. — Multiply the index of the radical by the index of the root to be extracted, and leave the quantity under the radi- cal sign unchanged. Thus, the square root of f2a. is ■\ f2d=y-^a. 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, \j i/Scfiis equal (Art. 192) to -^ ^SaS^f/SaT RADICALS. 175 1. Extract the cube root of y'a'b. 2. The 4th root of 16aY^- 3. The square root of ^49a\ 4. The cube root of 64^^80^ 5. The cube root of (rn-\-n)i/m-\-n . Ans. [/a'b. Ans. 2d' f2^. . Ans. -^la. . Ans. 4,^2^. Ans. y'm-\-n. IMAGINARY, OR IMPOSSIBLE QUANTITIES. 309. An imaginary quantity (Arts. 182, 193) is an even root of a negative quantity. Thus, y — a, and ^ — 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), ^"^^Xi/— «=l/— «X— «— |/a2=±a. But, since the square root of any quantity multiplied by the square root itself, must give the original quantity, (Art. 198,) therefore, ,/^aXi/^a=— a. SIO. Every imaginary quantity may he resolved into two factors, one a real quantity, and the other the imaginary ex- pression, |/ — 1, or an expression 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=aX — 1, — 6^=6^X— 1. ''"^ so on. Hence, |/— a2=^a2X— l=v'«^Xl/^T=d=a/^. Since the square root of any quantity, multiplied by the square root itself, must give the original quantity; Therefore, (i/^=T?= ^^-TxV'^ =—^- Also, (/:=ij3=(v/-=Tpx^^r=-v^r=-/^T. (i/-i)^=(i/-if (v-i)'=(-i)(-i)=+i- Attention to this principle will render all the algebraic opera- tions, with imaginary quantities, easily performed. Thus, y'l^S X V^ = >/« X i/=i X V'^X i/^ = -/"Sx (i/=IF=-i/a&. 176 RAY'S ALGEBRA, SECOND BOOK. OPEKATION. If it be required to find the product of ' ^ a-\-b^/^—i by a — by^ — 1, the operation is c — f>^/ — 1 performed as in the margin. a^4-ab\/—[ — a6i/^ + 62 Since a'-\-b- = {a-\-b^/ — l)(ffl — b^/ — 1), any binomial whose terms are positive may be resolved into two factors, one of which is the sum and the other the difference of a real and an imaginary quantity. Thus, m^n={-y/'m-\-yriy^—r){-^/7rv—y''n-/—l). 1. Multiply ^ — ii' by i — b'K . . . Ans. — ah. 2. Find the 3d and 4th powers of a^/ — 1. Aus. — a^■^ — 1, and a*. 3. Multiply 2,-'^ by 8i "-=2: . Ans. — 6^/ 6. 4. Divide 6y ^ by 2^ ^^. . . .Ans. ^^ ¥. 5. Simplify the fraction ::j ;^=^ . . Ans. y^—l- 6. Find the continued product of x-\-a, x-f«^' — 1, a- — (I, and .r—lr-^/ — 1. Ans. x* — a*. 1. or what numlier are 24+Y-i/— I, and 24— t-j/^, the imaginary factors? Ans. 625. VI. THEOEY OP FRACTIONAL EXPONENTS. 211. The rules for integral exponents in multiplica- tion, divi.'iion, involution, and evolution, (Arts. 56, iO, 1'72, and 104.) are equally applicable when the exponents are fractional. Fractional exponents have their origin (Art. 196) 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'l' root of a" is a" . 2 4-"' The forms aS, aS, and a ", may be read a to the power of §, m a to the power of 1 and a to the power of minus — ; or, a expo- m nent |, a exponent |, a exponent — — • MULTIPLICATION AND DIVISION OF QUANTITIES WITH FRACTIONAL EXPONENTS. 212. 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^=fa^='{/aP, a^=^a^=^^ai^, (Art. 205.) But this result is the same as that obtained by adding the expo- nents together. 2 4 2,4 in+12 22 Thus, a3Xa5=a3 s—^is iz^cii". Hence, where the exponents of a quantity are fractional, To Multiply, Rule. — Add the exponents. 2. Let it be required to multiply a ^ by a^. Adding — | and |, we have J-,. Hence, the product is aT2^ or ^p'a. 213. By an explanation similar to that given in the preceding article, we derive the following rule. Where the exponents of a quantity are fractional. To Divide, Rule. — Subtract the exponent of the divisor from the exponent of the dividend. 178 RAYS ALGEBEA, SECOND BOOK. Perform the operations indicated in each of the follow- ing examples : 1 2 _i 3 J I 1. a^Xo.'^, and a 2xa3 Ans. a^, and a^ 2 112 11 , 4. (a3-f a363^_63)(a3— 63) Ans. a— 6 5. (a;4 J/+1/3) (a;?— 2/4). Ans. x^y—y^ l_ 1. 1_ 1_ Tn-^n 6. (a+6)™X(«+*)"X («— ^rXC"— ^)"- Ans. (a2— 62) m» 2 i 1 2, 5 an— gm 7. a;3-4~2;4, and a;'"J/"^a;"j/"' . . Ans. a;T5, and a; '"" 2/"-"* 8. (a?— 64)-j-(a4_64). . . . Ans. a5_|_a?64_|_62 9. (a— 62)^(a4+a262+a46+62.) Ans. a^— 62 POWERS AND ROOTS OF QUANTITIES WITH FRACTIONAL EXPONENTS. S14. Since the m"" power of a quantity is the product of m factors, each equal to the quantity (Art. 172); Therefore, to raise a" to the nith power, we have L L L 51 a"X«"X<^" • • . to ?n factors =0". Hence, to raise a quantity affected with a fractional ex- ponent to any power, Rule. — Multiply the fractional exponent hy (he exponent of the power. Thus, (ai63)<=a263=a263. S13. Conversely, to extract any root of a quantity affected by a fractional exponent, EQUATIONS CONTAINING RADICALS. 179 Rule. — Divide the exponent by the index of the root. _m m *n 1 1 Thus, y'a"=aT~'"=os'*™=ia". 1. Raise aFb^ to the 4th power. . . . Ans. a^b^. ' 1 i 2. Raise — 2x-y^z'* to the 3d, 4th, and 6th powers. 3 3 4 3 Ans. —Sx-1/z'^ ; 16xYz; Mxyz'-'. 3. Find the square of a — (ax — a^y~. 1 Ans. ax — 2a(ax — a")^. 1 4. Find the cube of a^x'' -\-a 'x. \ Ans. ax-^-|-3a^£B-'+3a ^a+a-'rr'. 1 1 5. Find the cube roots of (27a'a;)^ and (27a'.x)^. Ans. bK^x^, or (Sax^y^ ; and (Sax^)K 1 6. Find the square root of bx' — 4a;(5fa;)^-)-4c. Ans. 5M— 2c2. 7. Find the cube root of ^a?—^a'b^-\-6ab^8b'^. Ans. U—2b^. VII. EQUATIONS CONTAINING RADICALS. SIC In the solution of questions containing radicals, the method to be pursued will often depend on the judg- ment of the pupil, as many of them can be solved in dif- ferent ways, and the shortest processes can only be learned from practice. 1st. When the equation to be solved contains only one radical expression, transpose it to one side of the equation and the rational terms to the other; then involve both sides to a power corresponding to the radical sign. 180 RAYS ALGEBRA, SECOND BOOK. 1. Given, f^(a?-{-x) — a^c, to find x. Transposing, f (a^-\-x)=C-\-a; Cubing, a^-\-x—c^J^Sad^-\-Za'^c-\-a^; Whence, x=c^-\5aC'^'da-c. 2d. When a radical expression occurs under the radical sign, the operation of involution must be repeated. 2. Given Ix — -j/l — x=\ — y x, to find x. Squaring, x—^/l — a;=l — 2/3;+ a;; Canceling x on each side, and squaring again, Canceling 1 on each side, transposing, squaring, and reducing. We find, a;=lS. 3d. When there are two or more radical espressions, it is generally preferable to make one of them stand alone before performing the process of involution. 3. Given, i/.T-(-9 — i/a;^l, to find x. Transposing, — y x, we have ^/x^d=l+yx. Squaring each side, a;-|-9^14-2y'a;+X; Canceling X on each side, transposing, and dividing by 2, yx=A; hence, a;=16. In some cases, however, it is preferable, when an equation con- tains two radical expressions, to retain them both on the same side. Thus, the following equation will be cleared of radicals at once, by squaring each side: \ \ x—a / ' \ \ x+a I Wb'—i 4. y'(a;+5)+3=8-,/'i. Ans. a;=4. 5. ^/l + ,/(3+/te)=2 Ans.a;=6. 6. y'x-\-a=^/x+a .Vr.-- x^ ^''''^^'^^ 4 7. ;/2i=3d-|-/2i=3j/a Ans. a:=2a. INEQUALITIES, 8. -/{13+/[7+v/(3+^«)]}=4. 4 9. ^'I^x-^-^/x- ya+a;' 10. i/a+x+yj- = -i/x. 11. /I+13— y/a;— 11=2. 12. Oy x-\-bf/x — Cj/a;=d. 13. 1'- 14. x+a=y'a--rXy' [b'^^x''). y'a;-f2 "^ ^2 yx+y/a 3 ^ ■"• ,/3a;+l--'+ 2 •• 18. |/4a+a:=2y'6+a;— ,/^ ~6^ (/a. 19. ; c _ / 46c v a-\-x ~^ A a — X ~~ "Va^—a^' 20. ^^=^ =^c ■(/a;-|-a — y'x 21. J ,/Ff3— J y/¥S=J2yx. Ans. a; 181 Ans. a;=]. Ans. x=§. a Ans x= a-\-2y a Ans. x=:36, 1 l=Zt 62-4a2 . . Ans. x= Ans. X: 4a ■ Ans. a;=l4i. Ans. x=16a. . Ans. a;: . Ans. a;:=- . Ans. a;^3. {b—af 2a- 6 a(6+c) Ans. a;= b—c • a(c— 1)^ 4c . . . Ans. x=9. 23. ■/(l+a)2+(l— a)a;+/(l— aj2+(l+aja;^2a. Ans. a:=8. VIII. INEQUALITIES. S17. In the discussion of problems, it often becomes necessary to compare quantities that are unequal, and to operate upon them so as to determine the values of the unknown quantities, or to establish certain relations be- tween them. 182 RAY'S ALGEBRA,- SECOND BOOK. In most cases the methods of operating on equations apply to inequalities, but there are some exceptions. 218. In the theory of inequalities, it is convenient to consider negative quantities less than zero. In comparing two negative quantities, that is considered the least which contains the greatest number of units ; thus, 0>— 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. 319. Proposition I. — If the same quantity, or equal quantities, he added to or subtracted Jrom 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, — 1< + 1, and — 9<— 7. Similarly, if a>6, then a-|-c>6-(-c, or a — c>6 — c. Hence, Any quantity may be transposed from one side of an in- eqiiality to the other, if at the same time its sign he changed. 3SO. 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 sense. Thus, if 7^,6, and 5>4; then, 7+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>1 exists in the same sense. 3>2 Second, 10>9 In this case, after subtracting, we find the 8>3 resulting inequality exists in u contrary n^o sense. In general, if a>6 and e>d, then, according to the particular values of a, b, c, and d, we may have a — c^b—d, a — c<6 — d, or a—e=b—d. S21. Proposition III. — If the two members of an in- equality he multiplied or divided by a positive nvmber, the resulting inequality will exist in the same sense. Thus, 8>4 and 8X3>4x3, or 24>12. Also, 8--2>4-i-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>— IX— 2, or 6>2. From this principle we derive S33. Proposition IV. — The signs of all the terms of both members of an inequality may be changed, if at the same time we establish the resulting inequality in a contrary sense. For this is the same as multiplying both members by — 1. 333. Proposition V. — Both members of a positive in- equality may be raised to the same power, or have the same root e'Aracted, and the resulting inequality will exist in the same sense. Thus, 2<3 and 22<32, 23<33; or 4<9, 8<27; and so on. Also, 25>-16, and y'25>y'16, or 5>4; and so on. 184 RAYS 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 io a contrary sense. Thus, 3,, -2, and 32>(— 2)-', or 9>4. But, — 3<-2, and (— 3)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 x^= the number. Then, 5a;+4,.2.r+19, (1) 5a;— 4<-4.r-|-4. (2) 5.r— 2.r>19— 4, from eq. (1) by transposing, 3:r>15, by reducing, a'^S, by dividing both members by 3. hx — 42:<;4-|-4, from eq. (2) by transposing, x<;8, by reducing. Hence, the number is greater than 5 and less than 8, consequently elthei- 6 or 7 will fulfill the conditions. -I. If 4.f— 7<23;4-3, and 3x-fl>13— x, 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+ Ia;<8-|-|a; Ans. a;<36. 5. Show that , ,T'. -> t'^e least, and < the greatest (tec of the fractions, -, -, -, each letter representing a posi- tive quantity. Suppose -,^ to be the greatest, and - the least, of the fractions, " ° ^ Then "■^'^ ° ° ''^° ,« a e a e a V d' 7 ^'''="' b>d' d-a f>d' '^""^ 6=6' d<6' f<-b- EQUATIONS. 185 «>§■ "-'i' ">!• (^'•'■221-) ab ad af , , "'=T' ''<'¥' ®(6-|-d+/)^. (Arts. 219, 220.) a + c + e<:^ (b+d+f)~. (Arts. 219, 220.) o+c+e e a+e+e a 6. It is required to prove that the sum of the squares of any two unequal magijitudes 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-i-62>0. Adding, +2ab to each side (Art. 219), £{2_|_j2--^2a6, which was required to be proved. Most of the inequalities usually met with, are made to depend ultimately upon this principle. I. Which is greater, y'S + i/li or y'3 + 3y^2~? Ans. the former. 8. Given J(a:+2)+-ia;<-i(a;— 4) + 3 and >-^(a:+l)+J, to find X. Ans. x=b. 9. The double of a certain numher 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 -|- the fraction inverted, is greater than 2 ; that is, that j -\--y>2. II. Show that a''-\-b^-\-c'^ab-\-ac-\-bc, unless a=6='--. 12. If x^^a'-\-h'', and y=c^-|-'^^ ''^bich is greater, aj or ac-\-hd? Ans. xi/. 13. Show that a6c>(a-|-6— c)(a-f-c— 6)(&+c— «), un- less a^h^^c. 2d Bk. 16 186 RAY'S ALGEBRA, SECOND BOOK. VII. QUADRATIC EQUATIONS. S24. A ftuadratic Equation, or an equation of the second degree, is one in which the greatest exponent of the unknown quantity is 2 ; as, x''-\-:e^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, xy^a, xy — x — y=^c. S35. Quadratic equations, containing only one unknown quantity, are divided into two classes, pure and (ijfeeted. A Pure ftuadratic Equation is one that contains only the second power of the unknown quantity, and known terms; as, a:'+2=^r47— 4.c^ and ax''-\^h=cx'—d. A pure quadratic equation is also called an incomplete equation of the second degree. An Affected ftuadratic Equation is one that contains both the first and second power of the unknown quantity, and known terms ; as, 5x'-1-Yj;— 34, and ax' — hx''-\-cx — dx^=e—f. An aifeoted quadratic equation is also called a complete equation of the second degree. 226. The general form of a pure equation is ax'=^h. The general form of an affected equation is ax''-\-h.T=^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 .r- may be collected into one term of the form, ax' ; and all the known quantities into another, as h. QUADRATIC EQUATIONS. I87 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. 227. — 1. Let it be required to find the value of x in the equation, lx''—Z-\-f^x'=12l—x\ Clearing of fractions, 4a;2— 36+5a;2=153— 12a;2; Transposing and reducing, 21a;2=189; Dividing, a;2=9; Extracting the square root of both members, a:==b3 ; that is, a;=+3, or x=—Z. Terification. i(-|-3)2— 3+J^(+3)2=12|— (+3)2. 3-3+3|=12|-9; or3|=3i Since the square of — 3 is the same as the square of -(-3, the value x^= — 3, will give the same result as x=-\-Z. 2. Given ax''-\-h::^d-\-cx'', to find the value of x. Transposing, . . ax'^ — cx^^=d—b; Factoring, .... (a — c)x^^d — 6; Dividing, a;2=— ^ ; a—c .=±V''-' =Vi=-c- IFrom the preceding examples, we derive the following Rule for the Solution of a Pure Equation. — Reduce the equation to the form ax^r=b. Divide hy the coefficient of x^, and extract the square root of hoth members. 228. If we solve the equation ax''=zh, we have, a;=±-y/-; that is, a;=+-y/j, and 2;=—^/-. 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 the unknown quantity has two values, or roots, and only two. 2d. That these roots are equal in value, hut have contrary signs. 1. llx'— 44=5.'c'+10 Ans. a;=±3. 2. i(a;'— 12)=jj:'— 1 Ans. a;==±6. 3. (x+2)'=43;+5 Ans. a;=±l ^■l^v + lk^r^' Ans. .=.±.3. .T+7 X — 7 7 . , f, 5. ~-\, --.-=-- — ^r^. . . . Ans. .^=±9. X — 7-c x--\-Tx x~ — To 6. --^+~=c. . . . Ans. a-=^±^i'7-. AVith these principles, a careful consideration of the roots, or values of .r in each of the four different forms, will render the fol- lowing conclusions evident: Isl form, . x--{2pxz:=q. x'=~p-\-^ q 'p-, and x"=^^p — yq-p~. The first root is essentially positive, the second essential!}' ncga^ tive ; and the first is numerically less than the second. 2d form, . X- — "p.r^q. x'-^pi-Vq^p'', and x"=p—Y~(prp2, The first root is essentiallj' positive, the second essentially nega- tive; and the first is numerically greater than the second. 3d form, . , .T--^22J.r^ — q. x'=—p-\-y/ — q- p', nnd x"= — p — \' — q-\-p'^- Both roots are essentially negative, and the first is numerically less than (he second. 4lh form, a-2 — 2px^ — q. 3-'-=_p-|-)/ — q-p>-, and x"—p> — , — q-Vp-- Both rools are essentially positive, and the first is numerically greater than the second. 1336. Wo shall now proceed to show \dien the roots become imaginary, and wlaj. In the third a)id fourth forms, the radical part is y — q-fp'^- Now, when q is greater than p'^, this is essentially negative, and the e.\traction of the root is impossible, (Art. 193.) Hence, QUADRATIC EQUATIONS. 205 When (he known term is negative, and greater than the square of half the coefficient of the first power of x, the roots are imaginary. To show why the rools are imaginary, we must prove that When a number is divided into two equal parts, their product is greater than that of any other two parts into which the number can be divided. Or, as the same principle may be otherwise expressed, Hic product of any two unequal numbers is less than the square of half their sum,. Let 2p represent any number, and let the parts into which it is supposed to be divided, be p~\-z and p — z. The product of these parts is (p+z)(p-z)=p'-z''. Now, this product is eTident]3' the greatest, when z^ is the least; that is, when z^:=0, or z=0. But when z is 0, the parts are p and p, which proves the proposition. Now, it has been shown, (Art. 234, Properties 3d and 4th,) that 2p is equal to the sum of the two roots, and that g is equal to their product. But, when q is greater than p^, we have the product of two numbers greater than the square of half their sum, which, by the preceding principle, is impossible. If, then, any problem furnishes an equation of the form x^±2px= — q, in which q is greater tlian p^, the conditions are incompatible with each other. The following is an example: Let it be required to divide the number 8 into two parts, whose product shall be 18. Let X and 8—x represent the parts. Then, x(8-a;)=18; or x^-8x=—18; Whence, x=i-\-y'~^, or 4—^—2. These expressions for the values of x, show that the problem is impossible, which is obviously true. By the preceding theorem, the greatest product of the parts of 8 is 16. 206 RAY S ALGEBRA, SECOND BOOK. 337. Examination of particular cases. 1st. If, in the third and fourth forms, where q is negative, we suppose q^X>', the radical, ]/ —q-YP't becomes 0, and x^—p iu one, and +p in the other. It is then said, the two roots are equal. In fact, if ■we substitute p- for q, the equation in the 3d form oecomes x---2px--2J-=0. Hence, (x-fp)-, or, (xJj5)(a;+p)=0. The first member is the product of two equal factors, either of which, placed equal to zero, gives the same value for X. A like result is obtained by substituting p- for q in the fourth form. 2d. If, in the general equation, x-~ 2px=q, we suppose q~0, the two values of x reduce to, x=—p fp=0, and x=—p—p:=—2p. In fact, the equation is then of the form ■x-^2px~0, or a-(x-|-2p)=0, wliich can be satisfied only by making a;=0, ora'-)-2p=0; whence, a;r=0, or a;= — '2p. 3J. If, in the general equation, x--\-2px^q, we suppose Sp^O, we liave X'=q; whence, x^±y q. In this case, the two values of x are equal and Imve con- trary signs, real, if g is jiosilive, as in the first and second forms, and imaginary, if q is negative, as in the third and fourth forms. Under this supposition the equation contains only two terms, and belongs to the class treated of in .\rt. 2JS. 4th. If 2p=.0, and <7^0, the equations reduce to a;-=0, and give the two values of x, in all the forms, each equal to 0. 238. There remains a singular case to be examined, which is sometimes met with in the solution of problems producing quadratic equations. To discuss it, take the equation ax''-\-hx:^c. QUADRATIC EQUATIONS. 207 Solving this equation, the values of X are — 6-f-y62+4Sc — 6— /62+4ac 2^ ' *- 25^ If, now, we suppose a=0, these values become __6_|_6_^0 —6—6 -26 That is, one value of X is indeterminate and the other infinite. (Arts. 136, 137.) But if we suppose a^O in the given equation, we have bx=c, and 1=-^. 6 We now propose to show that the indeterminate value is the same as the one last found, and that the infinite value simply expresses an impossibility/. If we multiply both terms of the second member of the equation — 6+v'6^+4ac . , — j — 3 x= ~ , by — 6 — |/62-f4ac, we have _ 62— (62-(-4ac) — 4ac 2a(— 6— J, 6V4acJ 2a{—b—yb'^+iac) Or, by dividing both terms by 2a, and making a=0, _ —2c _— 2c_c ^""— 6— ^P+4ac ~ ^26 ~ 6" c Hence, we see, that the value of x=^, is really t, and arises from having made the factor, 2a, equal to zero. (See Art. 136.) By supposing a=0, the equation ax^-\-bX:=c, reduces to bx=c, an equation of the ^r«( 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=0, 6^0, c=0, the equation would be altogether indeterminate. This is the only case of indetermina- tion OQCurring in quadratic equations. 239. We shall now apply the principles ahove stated, in the discussion of the following 208 RAVS 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// T It is a principle in optics, that the intensity of the same light at different distances, is iiiversclj as the square of the distance. Let a be the distance BC between the two lights. Let 6 be the intensity of the light B at the distance of 1 ft. from B. Let e be the intensity of the light C at the distance of 1 ft. from C. Let P be the point required. Let Br —X- then, CP =a—x. By the principle above slated, since the intensity of the light B at the distance of 1 foot, is 6, at 2, 3, 4, . . feet, it must be /) l> /) .. F, ,r. ■ • ; hence, the intensity of B and of (1 at the 4 9 It) / . ' distance of X and of a — X feet, must be - and . X- (a — xj- But, by the conditions of the problem, these two intensities are cciual. Hence, wo have for the equation of the problem, ^ c , . , , , ("— .r)2 e , = r„ which reduces to , — ==- ; r- M'Ih r- [a^x)- a—x +, e or -,/c , 0' 1^' This gives the following results : 1 . ai a,,~c 1st. x^ — zr -', whence, a — X-~ — ^ ^. yO+y-G 1 ''-hi C o 1 'j' I ^ 1, — ''' 1 C 2d. x= - _ _; whence, rt — x^ L- -. We shall now proceed to discu.s.s these values. QUADRATIC EQUATIONS. 209 I. Let i>c. The first value of X, —^ — -, is positive, and less than a, for ,'6 — = =. IS a pi-oper fraction. Hence, tliis value gives for the point 1 b+yc illuminated equally, n point P situated between B and C. AVe per- ceive, also, that the point P is nearer to C than B ; for, since 6>c, we have y 6-(- y^ 6> y/ 6+ , a, or 2y/byyl}+yc, and . -^ =>2, and, consequently, — ^ =\_ •• l/6+l/e yb+yC^2- This is manifestly correct, for the required point must be nearer the light of less intensity. The corresponding value of a — x, di/c ... a — = is positive, and evidently less than -r. yb+y'o Z T lie second value of X, _/ — =, is positive, and greater than a; yb~yc — — — \' b , , a-,,1) for -,/6>^6-^c; . . —= ->1, and - _ >«. ■^ ^ ^'b—^ c y/b—yo 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 1 6-|/c be, and represents the distance CP', in the opposite direction from C, (Art. 47.) II. Let 6,/b+yb; .-. -^;^-^|/e— ^6 .-. — :"^ r^l; nnd — =^ — ^ , rt. This represents CP", and is the suvi of the dis- ^/o-^'b tances CB and BP'', in the same direction from C as before. III. 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 supjiosed equal, there is no point at any finite distance, except the point P, which is equally illuminated by both. IV. Let h=:c, and a^O. The first system of values of x and a — .T, 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 a=0, 6 not being ^c. All the values of X and a — X reduce to ; 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=0, -.,= 7 to becomes -^ = — s, which can never X- (x—a)^ x^ x" be true except when 6=c, as in Case IV. 23d\ 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. +3, -1-7, 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 X 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 sucli parts that the product shall be 26. Ans. b-\-y — 1, and 5 — |/ — 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. 2158G.j.5-|- miles from the earth, and 24134.5 — " " " moon. Or, 270210.4+ " " " earth, and 30210.4+ " beyond the moon from the earth. This question inyolves 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 X, page 208, calling a=240000, 6=80, and c=l. TRINOMIAL EQUATIONS. 240. A Trinomial Equation is one consisting of three terms, the general form of which is ax'"-\-hx"=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* — 2px^:^q. QUADRATIC EQUATIONS. 213 Completing the square, X* — 2.px--\'p'^=zq^p". x'^—p=y' q-\-pK ,.x=±^p±y/q-^P''- 341. Binomial Surds. — Expressions of the form Aij/B, like the value of x^ just found, or of the form |/A=ty/B, are called Binomial Surds. The first of these forms, viz., A±y/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 ai-ise from involution. If we square 2±y'3, we have 4±4^/3-|-3, which, by reduction, becomes 7±4|/3. Hence, -. I7d=4^3=2±^3. In the same way it may be shown that .^5±2)/6=i/2±^/3. It thus appears that the form A±-[/B may sometimes result from squaring a binomial of the form a±|/6, or ■j/a±j/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 |/B is twice their product. To find the root, therefore, put a;2_[_j,2_A and 2a;2/=i/B^ and pro- ceed to find X and y, the terms of the root. Thus, Extract the square root of . . 7-]-4|/3. Put a;2 +2/2=7 (1), and 2xy=i^Z. Adding, we have x^-[-2xy-|^y'^=l-\■A■^/'Z. Subtracting, we have x2__2:r!/+2/2=7— 4/37 214 RAY'S ALGEBRA, SECOND BOOK. Extracting the root, x+y^Jl+iy'S (2). ^•-.?/=A '-^1 3 (3). (4). By adding and subtracting (1) and (4), we have 2a;-^8 . . x^2 and 2.1/--— 6 . . y=•^■W. Hence, 2— y-3 is the root to be found. 1. Extract the square root of 15-)-6; 6. Ans. 3-\-y 6. 2. Of 34~24i/2 Ans. 4-3^ 2. 3. Of 14±4i/6. Ans. -, 2±2v W. We shall now proceed to demonstrate more fully that the square root of A±| B may always he 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 ^ .T=a — j '5; . . squaring both sides, -,-,-!-, -T X — a-~b , . ... X^n-~'ln, b^b\ ..1 o^ ^ ; that IS, an irrational 111 in quantity is equal to a rational quantity, which is impossible. Theorem II. — Jn any equation of the form x±| y=art 1 b, the rational quan/i/iis on opposite sides are equal, and also the trra/ional quantities. For if X does not =a, let x=a+m; Therefore, a — m — i .i/=-a — /d; .-. m+i/3/=T/6; that is, the value of a quadratic surd is partly rationaj and partly irrational, which has been shown by Th. I, to be impossible; hence, x=a, and 1 .i/=| 6. AYe shall now proceed to find a formula for extracting the square root of A-|-|/B. QCADRATIC EQUATIONS. 215 Assume .... ^A+yii=y'x+yy, A.+y^li=x+y^2^xy, by squaring. By Th. II, CE+3/=A(l); and 2/^=^B(2); Squaring equations (1) and (2), we have a:2+2a;3/+y2=A2 4xy =B; Subtracting, x^—2xy+y2=A^—B; or, {x~yy^=A^—'B. Let A2— B be a perfect square =C^ ; then, C=t/A2— B. Therefore, . . (x—y)^—C^, or x— 2/=C; But, .... a;+2/=A; wv A+C , A— C AVhence, . . . x= — L_ ; and y= — — . — /A+C _ (A— C And . . ^x==tz'\~2~; *"-0. Or, . . . {x^—2axY—6a'{x-—2ax)=iea*. Proceeding with the solution, we find a;=4a, —2a, or adizay'^. 220 RAY'S ALGEBRA, SECOND BOOK. 2. x*—2x'—2x'-\-Sx=10S. Ans. a;=.4, —3, or ^(1±^— 35). 3. X*— 2a;'+a:=30. Ans. x^B, —2, or .i(l±,/_19). 4. .^^'_6a;'+lla;— 6=0. . . . Ads. x=1, 2, or 3. 5. x«— 6.x'+5x^+12a:=60. Ans. x^b, —2, or -^(3±i/-15> Ans. x^b, — 1, or 2±|/5. 7. 43;*+|=4a;'+83. Ans.a;=2, — |, or i.(l±,/=43). X 30 12+Ax _ 1 °- 14 T.-^:'"*" 3x ~2x'^ ^' Ans. x=4, 3, or -i(7±v'69). SIMULTANEOUS QUADRATIC EQUATIONS CONTAINING TWO OR MORE UNKNOWN QUANTITIES. 244. ftuadratic Equations, containing two or more unknown quantities, may be divided into two classes, j^ure 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. 345. Pure equations may in general be reduced to the solution of one of the following forms, or pairs of equa- tions. .1 ^ a;+2/=a 1 ^ x~2/=c ) x'+f=a \ QUADRATIC EQUATIONS. 221 We shall explain the general method of solution in each of these cases. To solve x-\-i/=^a (1), and xy=h (2), we must find X — y. Squaring Eq. (1), . . x^-\-2xy-\-y^=a'^; Multiplying Eq. (2) by 4, ixy =46; Subtracting, . . x-—2xy^y^=d^ — 46, Or, (a;— 2/)2=0(2— 46; Whence, .... a:— 2/=±|/a2— 46; But, x-\-y=a; Adding, and dividing by 2, x^\a±\■^/a'^ — 46. Subtracting, and dividing by 2, 3/=Jaq=J|/a2 — 46. The pair of equations (2) is solved in the same manner, except that in finding x-\-y, we must add 4 times the second equation to the square of the first. The pair ot equations (3) is solved merely by adding and sub- tracting, then dividing by 2 and extracting the square root. 1. Given a::'-(-y=:25, and .-c-f-j/^T, to find x and y. Squaring the 2d Eq., x'^-\-2xy-\-y^= 49; But, a;2 +2/2=25 (1). Subtracting, . ^y =24, (2). Taking (2) from (1), x^—2xy->^y'^= 1 Whence, .... x— 2/=±l (3). But, . . . x+y=l (4). Adding and subtracting (3) and (4), and dividing by 2, a;=4, or 3; and 2/=3, or 4. 2. Given a;^+a;^+/=91(l), and a;+|/^+^=13(2), to find X and y. Divide Eq. (1) by (2), x—^/l^-\-y= 7. (3). But, a:+T/gy+y=13. (2). By subtracting, . . . 2■^/xy=&. Whence l/^=3, and xy=%. (4). 222 RAY'S ALGEBRA, SECOND BOOK. By adding (2) and (3), . . . ^+2/= 10. (5). Squaring, (5), . x^--i^2xy-\-y^-=lQ0; Multiplying (4) by 4, Ajcy = 36; x^~'lxy-\-y~^ 64, . . x—y=±&. But, x-)-2/=10; whence, x=%, 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. J i 3 3 3. Given x'^ -\-y"^Q, and a;''-(-^^^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 a .3 In this example, let X'J^P, andj/S^Q; then, a;4=p3 and 2/*=Q^. The given equations then become, P+Q= 6 (1), p3-( Q3=i26 (2). Dividing Eq, (2) by (1), P=- PQ+Q==21; Squaring Eq. (1), . . P2+2PQ+Q2=36; Subtracting, . 3PQ=15, . . PQ=5. Having P+Q=6, and PQ=5, by the method explained in form (1), we readily find P=5, or I ; and Q=l, or 5. Whence, a:=625, or 1 ; and 3/=!, or 3125. 4. Given (cB— ^)(.'b'— ^')=160 (1), (a;+!/)(a-.^+3/^)=580 (2), to find x and y. x3—a;22/—a:)/2 +2/3=1 60 (1), by multiplying. a;3_|_a;22/-(-a:t/2+2/'=580 (2), " " 2a:=2/+2a;)/2=420 (3), by subtracting. Add (3) to (2), x3+3a;22/4-3x?/2+2/3=1000. Extract cube root, . a;-{-?/=10. From (3) xy{x^y)=2\Q; . .xy=2\. From a;+2/=10, and xy=2\, we readily find a;:=7, or 3; and 2/=3, or 7. QUADRATIC EQUATIONS. 223 Solve the following by the 5. x—y=1, ") 6. a:^+/=13, !:} 7. 2x+3/=7, I 8. a;^— /=:16, I . a:— y=2. j . 9. a;+y= 11,) 10. Y(a»+2/»)=9(a;»-y), x?y — y^x=\.^. 11. x''+a;y=84, .}^f =24. J 12. x'+2/'=152, x° — xy-|-^^=l 9 13. x2+y^-|-x^=208 x+3/ 14. x'— y=7a;y, X } =208, ) =16. I f=.lxy, \ . -y=2. ; . 15. x*+xy+/=9l, ) x^-\- xy -j-^'^13. J 16. X— y=i/x4-j,-'y, x^— ^^=37. 17. J+/^= 5, 3 2 18. x^+y^^ 5, X -|-y ^35. preceding or similar methods ; Ans. x=15, or — 13; ^^13, or — 15. Ans. x:^±3; 3'=±2. Ans. x=2, or | ; y^^, or 4. . . Ans. x=5; . . y=3. Ans. x=7, or 4; y=4, or 7. . . Ans. x=4; . . ^=2. Ans. x=±7;' Ans. x=5, or 3; y=Z, or 5. Ans. x=12, or 4; y= 4, or 12. Ans. x=4, or — 2; 3/=2, or —4. Ans. x^±3, or ±1 ; 3/=±l, or ±3. Ans. x=16, or 9; y^ 9, or 16. Ans. x=16, or 81; 3^=27, or 8. Ans. x:= 8, or 27; y=27, or 8. 224 RAYS ALGEBRA, SECOND BOOK. } : 19. »2+3/2= 4, 20. a^-\-f=3bl, xy= 14. 21. a=+3/= 4,) . . 22. x(y+z)^a, ' y(x+z)=h, z(x+y)=c. Ans. X- -/- Ans. a;=9, or 1 ; y=l, or 9. Ans. a;=7, or 2; 2/^2, or 7. Ans. a;=3, or 1 ; y^l, or 3. ;_|_C— i)(a-)-6— c) 2(6+f— a) )— c)(5+c- J(a-|-c— 6) y^ \ 2(a+c— 6) ' _ / (;,+c— g)(a+c— t) "^^ \ 2(a+?<— c) AFFECTED EQUATIONS. 346. The most general form of quadratic equations, containing two unknown quantities, is ax^-\-hxy-\-cx^dy''-A;-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 : x^+(ay4-&>+cy+rf^+e=0 (1), a:'+(aV+y)a^+cy+'^'3/+«'=0 (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", b — l/=b'^, etc., we have Whence, x= (a"y^b")xJr c"y^-\- d"y + e"=0. _ &'y'^^ d"y^€^' ■ ■ ■ ■ a"y+b ■ QUADRATIC EQUATIONS. 225 As this ralue of X contains y^, that of x^ will evidently con- tain y*, which value of X-, substituted in the first equation, neces- sarily gives rise to an equation of the fourth degree. Hence, The solution of two quadratic equations, containivg two unknoicn quantities, depends upon the solution of an equation of the fourth degree, containing one unknown qiiantity. 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. 247. 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 2/. Sub- stituting this value, for X and x'^ in (2), the new equation will evi- dently contain only y and y'^. 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 valu^ of y in the two equa- tions, we have a a;2+6 te^+e t^x'^=x\a +6 t-\-e P)=d (3), a'x^+b'tx^+c'tix^—x^{a'+b't+&t')=d^ (4). 226 RAY'S ALGEBEA, SECOND BOOK. From eq. (3), we find ... «'= „,.., ,,.2 (5)- d' From eq. (4), we find ... ^''= „,,M^n'f2 (^)- d' '^ a' +b't-^c'P Therefore, d Or, , . . d{a'-\-b't\c't-)=d'{a-\-bt^cfi), a quadratic equation, from which the value of t may he found, (Art. 231 ) and thence X from (5) or (B), and y from the equa- tion y^^tx. 34S. When two quadratic equations are symmetrical with respect to the t»wo unknown quantities ; tliat is, when the two unknown quantities are similarly involved, they may frequently be solved by substituting for the unknown quantities the sum and difference of two others. 1. Given X -\-y =a (1), x^-\-if'=h (2), to find X and y. Let x=S-\-z, and y^=s—z; then, s=^ (3), x^=8^'-\- 5s^z + 1 Os-c- + 1 0.s2z3+ 5S£H -f z\ 7/"=.s''— 5s'i2:+10s%2_10.s«+5s4rt— z''; By substituting the value of s=i^, and reducing, we find Completing the square, we find the value of z; and from (3), that of X and y. S49. 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 I'ollowing is an example; QUADRATIC EQUATIONS. 227 2. Given ^V^. + ^ + ^ = ^ (1), and a;^+y=20 (2), to find x and y. Since y ? + | j =2._|_2+|2; add 2 (o each side of eq. (1), and then J^ to complete the square. Whence, ? + ?=±3-i = 5 or —2. Let -+^=5; tlien, _XA or — =». Whence, xy^=^8, and 2a'2/=16. From the equation a;2-l-?/2— 20, and 2a;2/=16, -we readily find a;=±4, and 2/=±2. x y By taking - -\- - = — |, two other values of X and y may be found. 330. It is often of advantage to consider the sum, difference, product, or quotient of the two unknown quan- tities as a single unknown quantity, and find its value. Thus, in example 9, following, the value of xy should be found from the first equation, and in example 10, the value of -. y Other auxiliaries and expedients may frequently be em- ployed with advantage, but their use can only be learned by experience, judgment, and tact. Note. — In some of the examples all the values of the unknown quantities are not given; those omitted are generally imaginary. 8. x''-\-y''-\-x-\-y=2>ZQ,'\ . . . Ans. a;=15, or — 16; a;2_y_)-a;— ^=150. J . . . y= 9, or —10. 4. a;-|-4y=14, j Ans. a:=2, or —46; /+4.^=23/+ll. 3 y==3, or 15. 228 RAY'S ALGEBRA, SECOND BOOK. 5. 2y— 3x=14, 3:^^+2(3,-11)' 6. x—y=.2, 1. Sx'+ Xt/=1S,\ 4/4-3a;y=54. j 8. x''+x>/=10, I . a:3-+2/=:24. I . 9. 4,-ry=96— a:y, I .4.} x-]-y= X' 85 10. :l + i^ = y y x~y=2. 11. a;y=180— 8*^, ■ a;+3y=ll. =12, Ans. x^= 2, or 11; y=10, or 8f. Ans. x=^b, or 3 • 4 ' ^= =3, or -11. Ans. x=±2, or =t2i. 3 ; ^=±3, or rfiSy S. Ans. x^ y^ ±2 ±3 or ±5^/2 ; or q=4y 2: ns. x=2 y=4 4, 2, or 3±^/21; or 3=1=1/21. • . Ans a;= =5. or iS; y= -3, or -,3^. 12. x+y+^/x-\-y a-'^+/=41. ) 13. x+y+x'-^f=lS, I xy=6. ) 14. x'-\-Sx-\-i/^^S—2x!/ y'-(-33/+^=44. 15. xy+xf=12, 1 . . X +a;/=18. j . . 16. x'-\-f—x—y='J8,\ Ans. a;=9, or 3; . Ans. x=5, or 6; .y=2, or |. . Ans. x=5, or 4; . . ^=4, or 5. Ans. x^3, 2, or — 3=ti/'3"; y=2, 3, or — 3=f=i 3. . . . Ans. x^i, or 16; • • • ^=5, or —7. . . . Ans. a::=2, or 16; y=2, or x-\-i/-\-xy=39. j 18. xy~(x+y)=b4:. Vx+y_ y 17 {x+yy^ x=/+2. y 4v/a y=3, or 9. Ans. a-= 6, or • — 4i ; y=;12, or — 9. Ans. x^Q, or 3; y=:2, or 1. QUADRATIC EQUATIONS. 229 QUESTIONS PRODUCING SIMULTANEOUS QUADRATIC EQUATIONS INVOLVING TWO OR MORE UNKNOWN QUANTITIES. SSI. — 1. There are two numbers, whose sum multi- plied by the less is equal to 4 times the greater, but whose sum multiplied by the greater is equal to 9 times the less. What are the numbers? Ans. 3.6, and 2.4. 2. There is a number consisting of two digits, which being multiplied by the digit in the ten's place, the prod- uct is 46 ; but if the sum of the digits be multiplied by the same digit, the product is only 10. Required the number. Ans. 23. 3. What two numbers are those whose difference multi- plied by the difference of their squares is 32, and whose sum multiplied by the sum of their squares is 272 ? Ans. 5 and 3. 4. The product of two numbers is 10, and the sum of their cubes 133. Required the numbers. Ans. 2 and 5. Note . — The preceding problems may be solved by pure equations. 5. What two numbers are those whose sum multiplied by the greater is 120, and whose difference muliiplied by the less is 16? Ans. 2 and 10. 6. Find two numbers whose sum added to the sum of their squares is 42, and whose product is 15. Ans. 3 and 5. 7. Find two numbers such, that their product added to their sum shall be 47, and their sum taken from the sum of their squares shall leave 62. Ans. 5 and 7. 8. Find two numbers such, that their sum, their product, and the difference of their squares, shall be all equal to each other. Ans. i+^-^/b, and i+iy^b. 230 RAY'S ALGEBRA, SECOND BOOK. 9. Find two numbers whose product is equal to the dif- ference of their squares, and the sum of whose squares is equal to the difference of their cubes. Ans. lyb, and -|(5 + |/'5). 10. A and B gained by trading $100. Half of A's stock was less than B's by $100, and A's gain was tPq of B's stock. Supposing the gains in proportion to the stock, required the stock and gain of each. Ans, A's stock $600, B's $400; A's gain |60, B's $40. 11. The product of two numbers added to their sum is 23 ; and 5 times their sum taken from the sum of their squares leaves 8. Required the numbers. Ans. 2 and ^. 12. There are three numbers, the difference of whose differences is 5 ; their sum is 44, and continued product 1950; find the numbers. Ans. 25, 13, 6. S53. Formulse. — A General Solution to a problem producing a quadratic equation, like one in simple equa- tions, gives rise to a, formula, (Art. 162,) which expressed in ordinary language, furnishes a rule. We shall illustrate the subject by a few examples. Express each of the following formulae in the form of a rule, and -solve the numerical example by it : 1. Investigate a formula for finding two numbers, x and 2/, of which the sum of their squares is s, and difference of the squares d. Ans, x-^ly W^+d) ; y^.i-^, 2(^=7). Example. — Find two numbers such that the sum and difference of their squares are respectively 208 and 80. Ans. 12 and 8. 2. Investigate a formula for finding two numbers, x and 1/, of which the difference is d, and the product p. Ans. x=l,(d+j/d^ + 4p) ; j,=J(-rf+,/rf^+4p). QUADRATIC EQUATIONS. 231 Ex. — A man is 8 years older than his wife, and the product of the numbers expressing the age of each is 2100. How old are they? Ans. Man 50, wife 42. 3. Investigate a formula for finding a number, x, of which the sum of the number and its square root is s. Ans. x=s-|-^— ys+f Ex. — The sum of a number and its square root is 272; what is the number? Ans. 256. 4. The same when the difference of the number x, and its square root is d. Ans. x^d^i,-\-i/ d-\-\. Ex. — Find a number such that if its square root be sub- tracted from it, the remainder will be 132. Ans. 144. 5. Given x-\-y=.s, and xy^p, to find the value of ^^-\-y'i ^-\-ift snd x*-\-y*, in terms of s and p. Ans. x^-fy=s'^' — 2p] a;' -)-^';=s' — Zps ; a*-\-y^^s'^—Aps^^2p''. Ex. — The sum of two numbers is 5, and their prod- uct 6. Required the sum of their squares, of their cubes, and of their fourth powers. Ans. 13, 35, and 97. 333. Special Solutions and Examples. — If an equa- tion can be placed under the form (a;+a)X=0, in which X represents an expression involving x, at least one value of the unknown quantity may be found. For since the equation will be satisfied by making either factor ;=0, we have x-\-a^O, and X=0. Therefore, a;= — a, is one solution of the equation, and the other values of x will be found by solving, if possible, the equa- tion X=0. Thus, the equation x^ — x'^ — 4a;-|-4=0, may be placed under the form (X — 2)[x''-\-x — 2)=0. Hence, a; — 2=0, ora;^-)-2; and, from the other factor, we find a;=-)-l, or — 2. 9 3 r' — =li 232 RAY'S ALGEBRA, SECOND BOOK. Skill in separating such an equation into its factors must be acquired by practice. 2 1. Given .-c— 1=2-1 =, to find x. V -^ 2 2 Since x-l=(yx^l){y7^1) and 2+":=;= -^Ux+I); O Therefore, {y F+l)(/i^l )=-— =(i J>i-1); _ 1 •*- Therefore, , .r+l^rO, and x=(— 1)^=1. o Also, /«^1=-^, by dividing by \/x-\-\. Whence, , .r^-2, or —1; and x~A, or 1. ■' — 3.f=2. (Add 2a: to each side.) Ans. a;= — 1, or 2. I Trauspose ,] and ^. j Ans. ..=-^, or '(lii^OO 4. 2.c='^,r^=l. Ans. a;=l, or J(— l=h^ _7). 5. rr-''— 3.s'+.r+2=0. Ans, a;=2, or ^(1±^ S). 6. a;'=6.r+9. Ans. a:=3, or ^(_3±v — 3)- 7. a;-|-7a;^=22. Ans. a:=8, or 29±7v^^- ar+Ya:^— 22=(2-— 8) + 7(a;^— 2). a;^— 2 is a divisor. 8. a^'+L^a-''— 39z=81. Ans. ar=±3, or 1(— 13±^— 155). An artifice that is frequently employed, consists in adding to each side of the equation, such a number or quantity as will render both sides perfect squares. „ ^. ]2+8i/z ^ - , y. (jriven x^^ — i? — , to find x. X — 5 Clearing of fractions, x'^ — 5a;^12^8j x. Add x-\-^ to each side, and extract the square root. a;-2=±(4+v71-). From which we easily find x=9, 4, or J( — 3dz( —7). QUADRATIC EQUATIONS. 233 10. a:-3^HV^ X Ans. a;=4(7±|/r3), -^(— liy-^F). 493-' , 48 ,„ ^ 6 1 *" " ■ -. Add i X x' Ans. x=-2, —8, or ->(— 3±i/93). 11- ^7 H 5—49=9+-. Add i to each side. 12. a-+-^ 34a:=16. Ans. a;=±2, —8, or —h (1 7'K \ 2 — ^ I to each side. Ans. a;=3, —3', or J(— l±y_251). 9 Divide by ar", and add j— j to each side. (See Ex. 15, Art. 242.) 14. _-|--g— = — ^^-. Ans. 9, —4, or —9. 8x Multiply by 2, and add n^+Si to each side. Ob ^° 841 232 1 15 27x' I IT— -Lf^ ^^- ^'"^ 2x'+-^ - Sx 37^+^- Ans. x^2, —V, or J(-2±i/— 266). Multiply both sides by 3, transpose — ^ and — j, and add 1 to each side to complete the square. We shall now present a few solutions giving examples of other artifices. 16. pr-T'^i'^Cj to find X. (1+xy l+x^=a{l+xY=a{l+4x-\-ex^+4x^-\-x^), (1— a)(l+a;<)=4a(a;+a:^) + 6aa;2. 2d Bk. 20 234 RAY S ALGEBRA, SECOND BOOK. Dividing by x''-, (1— a)^ a:--r^ l"^"*"! ^^x )+""' „ 1 4a / 1 \ 6a ^x- 1— a\ ' X I 1— a / 1 \-' 4a / 1 V 6a - 2+4a \ ^x / 1— a\ a/ 1— a ' 1— a Complete the square, and find the value of x-\ — , which is 2a±|/2(l+a) ^ ^^^^ ^^^^ 2p^ _^^^ ^^ ^^^^^ ^^^ a;=^=hy'i32_i. 17. a;'+''r=y''', and y^+'-'^x'^, to find a; and ^. From 1st equation, y^x lu , a From 2d equation, y=:Xi+l/\ ■'+1' _£- 2;+u a Therefore, a; 4.. =a;i.+y, and — r^ = ; ' ' 4a x-Yy (a;-j-.y)-.— 4«', or x^y=2a; But, x^=yi''. since a;+2/=2o, Therefore, 3;=7/2, and y~-\-y=2a, Whence, y=l{—l±y'8a+T), and x=l{'ia+l=fzy'8a-^l). When two unknown quantities are found in an equation, in tlie form of x-\-i/ and av/, it is generally expedient to put their sum x-\-i/^s, and their product xy^p. 18. Given (x+ij)(xy+l)=.lSxy (1), (x'+/)(xY+l)=208x-f (2), to find x and y. Let x-\-y=S, and xy^p; then, s(p+l)=18;3, (1), and (s2_2;3)(p=4-l)=208p2 (2). From the square of (1), take (2), and after dividing by 'Ip, we have s2+p2-|-l=58p (3). But, . . 2s(p+l)=36p, from(2), And . . 2p= 2p. Adding, . . . (s+p+l)2=96p, s+p-fl=4,'6p; QUADRATIC EQUATIONS. 235 But, .... p+lJ^; 18p Therefore, . . . fi=4^6p— -^ , or s2_4g^6p— _18p; From which, . . s=3j/t)p, or yUp. But, . . . 2'+l=4v'^— s=3v/5p, or ^^. Whence,. . . p=26rfc^675, or 2±^/'^, and s=d=v/{6(26±T/675)j, or d=i/{ 6(2^=1/3)1. Having x-{-y and xy, the values of X and y are easily found, (Art. 246); two of the values are a;=7±4,/^ y=2i^y'3. 19. 2(x+yy+l^(x'+f)(xy+a^+f) (1), x+!/=B (2). Ans. a;=2, 3^=1. 20. l+.3^a(l+.)». Ans. ,^ l+2gl/y=3 . 21. 4-^./x 2a-?=l. a;'' X \ a; Ans. x=ill±v/l— 8a±^2±2(l— 8a)2-)-8a}. 22. a;-)-3^+a;y(a;-|-!/)-|-a!y=85, Ans. a;=6, or 1. xi/+Q<:+]/y+a:y(x+7/)^97. y=l, or 6. 2c+acZ c (a-|-6)#- 240 KAY 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, ii proportion between four. 364. Each of the four quantities in a proportion is 3alled a lerm. The first and last terms are called the ex- tremes; the second and third terms, the means. 263. 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. S66. Three quantities arc 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 e; 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. 367. Proposition I. — In every jjroportion, (he 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- _ , . , , ,. , J 6c ad , ad be Taking oc=aa, we hnd a= — , c=-^, o= — , a=-j-. Hence, Or O G CL RATIO AND PROPORTION. 241 If any three terms of a proportion he given, the remaining term may he found, 1. The first three terms of a proportion are x-\-y, or"- — ^', and X — y; what is the fourth? Ans. x^ — Ixy^y'^. 2 The 1st, 3d, and 4th terms of a proportion are {inx — h)', in'- — m', and m-\-n ; required the 2d. Ans. in — n. 3. The 1st, 2d, and 4th terms of a proportion are — —'—=, a' — 6, and ^ ^ — i-\= i ; required the 3d. a—yh 0+1/6 Ans. 1. This proposition furnishes a more convenient test of proportion- ality tlian the metliod 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 product of tiBO quantities is equal to the product of two others, two of them, may he made the means, and the other two the extremes of a proportion. Let bc = ad. Dividing each of these equals by ac, we have 6c acl ac ac' 6 d Or, - = -. ' a o That is (Art. 263), a : b : : : d. By dividing each of the equals by at), 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 BIc. 21* 242 RAY'S ALGEBRA, SECOND BOOK. Thus, from each of the equations bc=ad, and 3/12^4X9, 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. h: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. 3GO. Proposition III. — If three quantities are in pro- portion, the product of the extremes is eqxial to the square of the mean. If a: b: : b : C; Then, (Art. 267), . . . or^^hh^Jj-. It follows from Art. 268, that the converse of this proposition is also true. Thus, if . . ac=6^, a : b : . b : c. Hence, Jf 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. 270. Proposition IV. — Jf fmr quantilifs ni-r in pro- jiortion, they will he in prrq^urlimi by ALTERNATION; that j's, the first ivill hr to the third as the second to ilu: fourth. Let .... . . a : b : : C : d. b d a o Then, (Art. 263), . Multiply both sides by c, — — r7; c d Divide both sides by 6, — = — . ^ ' ah That is, (Art. 263), . a -. c : : h : d. If ■ ■ . 2:6: : 10:30; theu, 2:10: {6:30. RATIO AND PROPORTION, 243 SYl. Proposition V. — If four quantities are in propor- tion, they will he in proportion by Inversion ; that is, the second will be to the first as the fourth to the third. Let a : b : : : d. Then, (Art. 263), . . . . - = -; d G Inverting the fractions, . • j;=3- That is, (Art. 263), . , b : a : : d : c. If .... 5:10: :6: 12; then, 10:5: ;12; 6. 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 : o. QT^. Proposition VI. — If two sets of proportions have ati antecedent and consequent in the one, equal to an ante- cedent and consequent in the other, the remaining terms will be proportional. Let a . b : : c: d (1), And a:b:.e.f (2); Then will . . . . c : d : : e f. From (1), -=-; from (2), -='-. Hence, -=-; Which gives . . . C : d : : e : f. If 4 : 8 : : 10 : 20 and 4 : 8 : : 6 : 12; then, 10 : 20 : : 6 : 12. 373. 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. a: b : : c : d (1), a + b : b : : b d c+d: d. a c' a ■ b^d' Let Then will . . . From (1) . . . . Inverting the fractions, Adding unity to both members, v-f l^-7+l. a-\-b c-id Reducing to improper fractions, — t — = — j—- 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+6 : a : : c+d : c. S74. Proposition VIII. — If four quanli/ifs arr- hi pro- portion, they will he in proportion hy DIVISION ; tliat is, the diftirini:r of the first and second vjill 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 :: c—d : d. b d From (1), . . -=-. ft c Inverting the fractions, . i: = -i- Subtracting unity from both members, -, — 1^-, — 1. ° •' ' b d _ , . . , . a—b f-d Reducing to improper fractions, . — - — = — - —. This gives (Art. 271), a -b : b : : c—d . d. If IS 6 . ;30:10; then, 18-6:6 . : 30— 10 : 10, or 12 : 6 : 20 : 10. In u, similar manner it may be shown that a — b ; a : ; c — d: c. 373. Proposition. IX. — If four quantities are in pro- portion, the sum of the first and second will be to their difference as the sum of the third and fourth is to tlieir dif- ference. RATIO AND PROPORTIOX. 245 Lei a: b: : c:d (1), Then will . . . .a-{-0:a — b : : C-\-d : c — (/. From (1), by Composition and Division, (Arts. 273, 274,) a-f 6 : b : : c^d : d; And ... . a—b : b : : c — (/ : d. By alternation, . a-f 6 : c-\-d : : b : d; And . . a — b : c —d -. : b : d. From which, (Art. 2712), a+6 : c~\-d : : a—b : c — d. Or, by alternation, a-]-b : a — b : : c-\-d : c— cZ. If 6:2 ; :12:3; then, 6+2 : 6 - 2 : : 12+4:12—4, or8:4::16:8. 376. Proposition X. — If four quantities are in propor- tion, like powers or roots of those qiiantities will also he in proportion. Let a : b : : C : d, Then will .... a" : b" : : c" : rf". From the 1st — = — . Raising each of these equals 6" d" to the n'" power, . . . — = — That is, a" : 6" : : o" : d", Where n may be either a whole number or a fraction. If 2 : 6 : : 10 : 30; then, 22 : 6^ : : 10^ : 302, or 4 ; 36 : : 100 : 900. If 8:27: : 64:216; then, fW: ^27:: ^64: f 216; or 2:3:: 4:6. 277. 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. 246 RAYS ALGEBRA, SECOND BOOK. If 3 : 9 : : 2 : 6, and 5 : 15 : ; 4 ; 12; then, 15 : 135 : : 8 : 72. 2TS. Proposition XII. — In 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. Let . . . (I : b : : C : cl : : m : n, etc. Then a : b : : a+c+wi ; b-\-d^n. Since a : b : : c : d, we have bc=:ad (Art. 267). Since a : b : : m: n, we have bm^=an, Also, ab=zab. The sum of these equal- ities gives .... ab^bc-\'hm^ab-\-ad^an. Factoring, 6(a + c4-m)=a(6+rf-|-?i). This gives (Art. 268), a:b: . a+c+?n : b^d^n. If 5 : 10 : : 2 : 4 : 3 ; 6, etc. ; then, 5 : 10 : : 5+2+3 : 10+4+6, or 5 : 10 : ; 10 ; 20. 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 : ql mq — np Ans. p—q 4. If the ratio of a to i is 2^, what is the ratio of la to i, and of 3a. to 4i? Ans. IJ, and 3g. 5. If the ratio of a to ?> is 13, what is the ratio of a-\-h to 6, and of h — a to a? Ans. g, and |. 6. If the ratio of m to n is -^, what is the ratio of m — n to 6ot, and also to bn! Ans. 14, and 6|. RATIO AND PROPORTION. 247 7. If the ratio of 5^^ — 8a; to 1x — bt/ is 6, what is the ratio of a; to' yl Ans. 7 to 11. 8. What eight proportions aro deducible from the equa- tion ab=a'—x'. Ans. a : a-^.c : : a — x : h, a : a — x : : a-\-x : b, b : a-\-x : : a — x : a, etc. 9. If x''-\-7/^^2ax, what is the ratio of x to ^? Ans. X : y : : y : 2a — x. 10. Four given numbers are represented by a, b, c, d; what quantity added to each will make them proportionals? . be — ad Ans. ; — ^. a — b — c-\-d 11. If four numbers are proportionals, show that there is no number which being added to each, will leave the resulting four numbers proportionals. 12. Find X in terms of ^ from the proportions x:y::o? -.b^, and a : b : : f^c-\-x : -^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 : b : : c : d, 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. 379. 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 35 to 19. 248 BAY'S ALGEBRA, SECOND BOOK. Let X and y denote the required factors; then, a;i/=24, and x^--y' : x-—y Therefore, (Art. 275), Ix-- : 2y^ Or, ... x^ : y' Or, (Art. 270), . . X y 35; 19; 54: 10; 27- 8; 3: 2. From which 2/=j^; then, substituting the value of y in the equEi- tioQ j'y_24, we find X—dizG; hence, y=±4:. 2. Gn ''■c+l+^ -I ^2, to find X. Resolving this equation into a proportion, we have f X+l-^'x^l . ^anJ + f J;— 1 : : 1 : 2; .-. (Art. 275), 2f ;r^l : if x—l : : 3 : 1 ; f^qri . ^ J-1 ::3:1; . a;_|_l : x—\ .'11:1; 2x : 2 ; . 2S : 26 ; Or, . . Or, (Art. 276), (Art. 275), . Whence, . . 52.r^5G, or a;=lJ_. 3. x-[-ij : x~y : : 3 : 1, I . . 4. .r+y : .r ; : 7 : 5, I . . xy-\-y''=-\'2G. ] .-rj/=63. J . Ans. x=A, y=2. Ans. a;=±15, Ans. a;=±9, y=±n. 6. 1 a -\- ^ a' — x' -h Ans 3-= 2ay b 1 "+3;+i a- ~h' . Ans. xz 2ah 8. It is required to find two numbers whose product is 320, and the difference of whose cubes is to the cube of their difference, as 61 is to 1. Ans. 20 and 16. RATIO AND PROPORTION. 249 380. Harmouical Proportion. — Three or four quan- tities are in Harmonical Froportion when the first has the same ratio to the last, that the diflereuce between the first and second has to the diiFerence 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, b, c, d, 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 : b — x ; Therefore, (Art. 267), a(b—x)=x{a—b); Whence, . . x= '2a— b' 2. Find a third harmonical proportional to 3 and 5. Ans. 15. 3. Find a fourth harmonical proportional x, to three eiven numbers, a, b, and c. ac Ans. x= 2a— b' 281. Variation, or, as it i.s 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 vary directly as another, wlen 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 BAY'S ALGEBRA, SECOND BOOK. A oc B, the symbol oc being used instead of varies, or varies as. 282. There are four diiferent kinds of Variation, which are distinguished as follows : I. A tx 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 oi" his wages varies as the number of days in which he works. II. A cc =. 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 oc 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 oc — 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 PROPOKTION. 251 In the following articles, A, B, C, represent corresponding values of any variable quantities, and a, b, c, any other corresponding values of the same quantities. 2S3> If one quantity vary as a second, and tliat second as a third, the first varies as the third. Let A oc B, and B cc C, then shall A cc C. For A : a : : B : 6, and B : t : : C : c; therefore, (Art. 272), A ; a : : C : c ; that is, A cc C. In a similar manner it may be proved that if A cc B, and B oc -=f, that A oc ^j. 284. If each of tioo quantities vary as a third, their siim, or their difference, or the square root of their product, will vary as the third. Let A a C, and B oc C ; then, A±B cc C ; also, |/AB oc C. By the supposition, . . . A : a : ■ C : C : : V : b; Therefore, A:a::B:6; Alternately, (Art. 270), . A : B : : a : 6; By Composition or Division, A±B •. B ■. : a±b : 6; Alternately, A±B : a±6 : : B : 6 : : C : e; That is, A±B oc C. Again, A : a: ; C : e; And B : 6 : . C : e; Therefore, (Art. 277), . . AB : a6 : : C^ : c^; And, (Art. 276), .... |/AB : /a6l : C : c; That is, l/AB"a C. By a similar method of reasoning, the following propo- sitions may be proved: 285. If one quantity 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 mB, or cc — . 252 RAY'S ALGEBRA, SECOND BOOK. 3S6. If one quantity vary as another, any poircr or root of the former will vary as the same jioiei r or root of (he la Iter. Let A oc B ; then, A" oc B", n being integral or frac- tional. 287. If one quantity vary as another, and each of them be multiplied or diiiilcd by any quantity, variable or invari- able., the products or quotients v:itl vary as each other. A B Let A cc B ; then, q\ cc oB, and — oc — . 2 2 2SS. If one quantity vary as tivo others jointly, either of the latter varies as the first directly, and (he other in- versely. A A Let A cc BC ; then, B tx — , and C cc yr-. \j B 3S9. If A vary as B, A i'.s equal to B multiplied by Some constant quantity. Let A cc B ; then, A=n(B. If we know any corresponding values of A and B, the constant quantity m may be found. 200. In general, the simplest method of treating varia- tions, is to convert them into equations. 1. Given that y cc the sum of two quantities, one of which varies as x, and the other as :r, to find the corre- sponding equation. Because one part oc x, lot (his =ma", and the other part oc X-, " '• ^?(.r'-. Therefore, . . 2/- 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 y^r-f-s, where r cc x and s cc — , and if, when x=l, y^6, and when x=^2, y=9, what is the equation between X and y? n n Let r=mx, and s=- .. «=-?na;-|--. a; ■^ ^x But if a;=l, 2/=6, .-. 6=m-f »i; And if a;=2, 2/=0, . . 9=2m4--. 2 Hence, m=4, n-=2, and 2/=4a;+-. 3. If y cc a;, and when x=r2, y=:4ffl ; find the equation between x and y. Ans. j/=2aa;. 4. If y oc -, and when a;=^, 3/=8 ; find the equation X " A between x and y. Ans. y=- ■ 5. If y= the sum of two quantities, one of which varies as X. and the other varies inversely as x^ ; and when x=-\, 2/^6, and when x=^2, y=^b ; find the equation between x and y. i o , 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=^G, 11, 18, respectively; find y in terms of a;. Ans. y^S-\-2x-\-x''. 7. Given that s cc P, when / is constant ; and s cc /, when i is constant; also, 2s=/, when ^=1. Find the equation between /, s, and t. Ans. s=-}^f('. 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=b. 2. The Principles of Variation are extensively applied in mechan- ical philosophy. 254 RAYS ALGEBRA, SECOND BOOK. ARITH:\tETICAL rROGRESSION. 291. An Arithmetical Progression is a series of quan- tities which iuorease or decrease by a common difference. Thus, 1, 3, 5, 1, 9, etc., .,r 12, 9, 6, 3, etc., and a, a-\-d, a-|-2(Z, etc., a, a — d, a — 2t?, etc., are in Arithmeti- cal Progression. The series is said to be iiicrcasinr/ or decreasing, accord- ing as d is positive or negative. 292. To investigate a rule for finding any term of an arithmetical progression, take the following series, in which the first line denotes the number of each term, the second an incrtasing arithmetical series, and the third a decreas- ing arithmetical series. 1 2 3 4 5 a, «4-rf. a +2(1, a+3rf, a4 4f?, etc. a, a~d, a— 2d, a— 3cZ, a—'^d, etc. It is manifest that the coefilcient of d in any term is less by v)iiiy than the number of that term in the series; therefore, the n."* term ^a-\-{n — l)rf. If we designate the n"^ term by I, wc have l=a-\-(n — l)d, when the series is increasing, and l^a—(n—\)d, when the series is decreasing. Hence, Rule for finding Any Term of an Arithmetical Series. — Multiply the common difference hy the number of terms less one; luheti the srn'is is increasi7ig, add this product to the first term; ichen decreasing, subtract it from the first term. The equation l^a-\-{n — l)d, contains four variable quantities, any one of which may be found when the other three are known. ARITIIMKTICAL 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 arithmc(ical series, as the following, and write the same series under it in an inverted order, we have 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^J of 72^36, the sum of the series. To render this method general, let l^ the last term, and write the series both iu u, direct and inverted order. Then, S=a+(a+d) + (a+2d) + (a+3d). . -\-l, And, S=l + {l—d) + (l~2d) + {I— 3d). . . +a. 2s={lJ,a)+(l+a)+{l+a)+(l+a). . . +(?+«), 2S^(^+a) taken as many times as there are terms («) in the series. Hence, .... 2S=(^+fe)/i; S=(;+a)"=r-t^)n. Hence, Rule for finding the Sum of an Arithmetical Series. — Multiply half the sum of (he two extremes by the number of terms. It also appears that The sum of the extremes is equal to the svm of any other two terms equally distant from the extremes. S04. The equations ?=a+(»i — l)c7, and ^:=(a-\-l)-^, lurnish the means of solving this general problem : Knowing any three of the five quantities, a, 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 diiferent 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 itt Arithmetical Progression by the above two formulaj: 0. 10. 11. 12. 13. 14. 15. IG. 17. 18. 19. 20, a, d, n a, d, S n, n, S d, n, S a, d, n a, d, I a, n, I d, n, I a, 71, I a, n, S a, I, S n, I, S a, d, I a, d, S a, I, S d, I, S d, 11, I d, n, S d, I, S n, I, S Ruquireil. =a-\-[n—l)d, =-id±,/|2dS+(a-:irfr}, 2S S (n— l)d '7. I ~> * =i,i[2a + (n-l)d\, l-^u l-—a^ = ln[2.l—{n—\)d}. I — a "n—V 2(S-an) ~ ni^ii—l) ' I- — a" "2S— i!— a' 2(nl—%) I— a +1, _±,/(2a— (i)-+8dS— 2a+c{ " M ' _ 2S _2^+d±;/'(2;+d)^-8dS 'Id =1 — [n—V)d, S (n— l)d ''n 2 ' 2S , ARITHMETICAL PROGRJilSSION. 257 1. Find the IS"" term of the series 3, 7, 11, etc. Ans. 59. Here, a=3, n — 1=14, and d^4. Substituting these values in formula (1), we have ;=3+14x4=3+5G=59. 2. Find the 20"' term of the series 5, 1, —3, etc. Ans. —71. 3. Find the 8"' term of the series |, -^^, J, etc. Ans. -jL 4. Find the 30"' term of the series —27, —20, —13, etc. Ans. 176. 5. Find the «,"' term of 1 + 8 + 5 + 7. Ans. 2n— 1, Of 2 + 2J + 2H Ans. J(«+5) Of 13+12I+12J + . . . . Ans. J(40-^i), 6. Find the sum of l + 2-(- 3+4, etc., to 50 terms From formula (1), we find ^^50. Substituting this in formula (2), we have S=(l-|-50j25=1275, Ans. Or, use formula 5. 7. Of 'J+Y + ^i+' etc., to 16 terms. Ans. 142. 8. Of 12 + 8+4+, etc., to 20 terms. Ans. —520. 9. Of 2+21+2^+, etc., ton terms. Ans.^07 + 11). 10. Of A— 5 — V— , etc., to n terms. A. ^^(13—770- 11. Or hi etc., to n terms. «-l Ans. 2 ■ 12. If a falling hody descends 16^L feet 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. 948}^, 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, o=2, ?=14, and Ji=5. From formula (Ij, we obtain d=3. Hence, the three means will be 5, 8, and 11. To solve this problem generally, let it be required to insert m arithmetical means between a and I. Since there are m terms between a and ?, we shall have n^m-\-2, and formula (1) becomes l:=a -^ {m~ l)d. Hence, tZ= , Therefore, TTie common difference will he equal to tlie difference of the cxtnmes 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, -§, etc., to — i. 17. How many terms of the series 19, lY, 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 term.s 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. 393. 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 J. In general, a, ar, ar', ar', 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 ivill be found by dividing any term, by the term next preceding. S96. 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, w— 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 lz^ar"~'. Hence, 260 RAY'S ALGEBRA, SECOND BOOK. Rule for finding the last Term of a Geometrical Series. — Multiply the first term by the ratio raised tu 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 G'* 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 + 3 + 9 + 27 + 81 (a). Multiplying by the ratio 3, 38=3+94-27+81+2-13 (b). 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+ar-+o?'''. . +ar"-2+ar"-i. Multiplexing this equation by r, we have rS=ar+ar-+<:()'^. . . +a?'''-'-f ar". air^ 1) Subtracting, rS — S=ar"^a; whence, S=— =— • r — 1 Since ;=ar''-', we have rl=zar"; ™. „ ar" —a rl — a Therefore, S= =- = ,-. Hence, Eule for finding the Sum of a Geometrical Series. — Multiply the last term by 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. ^r— a _ 3072X4— 3 Z=3y4'=3072. .. S=+— f:= ;^; =4095, Ans. 298. If the ratio r is less than 1, the progression is decreasing, and the last term I, or ar"~\ is less than u. 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^= , or ^-^i . Therefore, for findina; the sum of 1 — r 1 — r the series, when the progression is decreasing, Rule. — Multiply the last term hy the ratio, subtract the product from the first term, and divide the remainder hy one minus the ratio. 1399. When the series is decreasing, and the number of terms infinite, I is infinitely small, or 0. Therefore, rZ=0, and 8=^ becomes S^^ . Hence, Rnle 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 1-|-2+5+b + . etc. Here, a=l, r— i, and S^^ ^-— — ^=2, Ans. That the sum of an infinite decreasing series may be 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=AB+BC+CD+, etc., in infinitum, =AB+JAB+iAB, etc., in infinitum, =2AB, which agrees with the example. 300. The equations, Z^ar"~', 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. 262 RAY'S ALGEBRA, SECOND BOOK. The following table contains all the Talucs of each unknown quantity, or the equations from which it may be derived : No. ~L 2. 3. 4. 5. 6. 7. 10. 11. 12. 13. 14. 15. 16. Given. 17. 18. 19. 20. a, r, n a, r, S a, n, S r, ?i, S a, r, n a, r, I a, n, I r, n. I r, n, I r, n, S r, I, S n, I, S a, n, S a, !, S n, I, S a, r, S r, !, S Required. ,a+{r-l)S '- F ' ;(S— Z)"-'— a(S— a)"-i=0, .__ (>•— l)Sr "-' s= S: S= a(r"— 1) KIT' rl — a >=P _ lr"—l Jr-VjS r»— 1 ' a—rl~(r—l)S, a(S-a)"-i— «(S--Z)"-i=0. r"— S , S-a _ — r4 =0, S— rt s „ , , ; S— ^ "^s-r =0. ^^_ fog. I -log, a ^ log. r ' Jogr. [a+{r—l)^']—log.a " log. r ' log. I— log. a .. "log. (S—a)—log. (S— ^)+ ' ^ Jog. l-log. [^r-(r-l)S] ^ log. r ' GEOMETRICAL PROGRESSION. 263 By observing, in any particular example, what are given and re- quired, the proper formula 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. Nos. 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 8=^^-=:^", or, (Art. 298,) ±^ (2), are, however, sufficient for the solution of all examples i,i Geometri- cal Progression, and may easily be retained in the memory. 1. Find the S"" term of the series 5, 10, 20, etc. Ans. 640. 2. The '7* term of the series 54, 27, 13^, etc. Ans. ||. 3. The 6'A term of the series 3|, 2i, 1^, etc. Ans. ^. 4. The 7"' term of the series —21, 14, — 9J, etc. Ans. — 4|f. 3>i— a 5. The n."' term of the series J, ^, |, etc. Ans. ;^^j. 6. Find the sum of 1 + 3+9 + , etc., to 9 terms. From (1), Z=1X38=6561. From (2), S=?^^^^^-^^=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 a;— v+— — C+, etc., to n terms. °^' a;+3/l \ x) )' 12. The first term is 4, the last term 12500, and the wumber of terms 6. Required the ratio and the sum of all the terms. Ans. Eatio =5; sunt ^15624. 264 RAY'S ALGEBRA, SECOND BOOK. Find the sum of an infinite number of terms of each of the following series ; 13. Of f + J + i + , etc Ans. j. 14. Of 9+6+4+, etc Ans. 27. 15. Of I-J + 1-, etc Ans. i. 16. Of a+i+- + ^]+, etc Ans. "'^ a a' ' ' a — b 17. The sum of an infinite pcometric 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 geometric mean between 4 and 16. Ans. 8. Here, a=4, 1=1Q, and n=3; or, (Art. 269) the mean =r,, 4^T(x 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^Tn^2; hence, n — l=m+l, and r^^+V — . 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 18122. Ans. 6, 18, 54, 162, 486, 1458, 4374. 301. To find the value of Circulating Decimah; that is, decimals in which one or more figures are continually repeated. In such decimals the ratio is -JL, i __' , etc., accordinff aa 1 I 00 1000' ' ° one, two, or more figures recur. Thus, HARMONICAL PROGRESSION. 265 The part within the parenthesis is an infinite series, having «=T^Vn ■^'"1 '•=Tk- Hence, (Art. 299,) S=^|J,g. Therefore, .253131 .... =Tf5+^|k=M8e=JM§- 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, .... 99008=2506 .-. S=2506. 1. Find the value of .636363. . . . ; Ans. j\. 2. Find the value of .54123123. . . Ans. J|fg-3. 303. Harmonical Progression. — Three or more quan- titles are said to be in Harmonical Progression, when their reciprocals are in arithmetical progression. Thus, 1, 1, ^, 4, etc.; and 1, f, 1, f, etc., are in harmonical progression, because their reciprocals 1, 3, 5, 7, etc.; and 4, 3J, 3, 2J, 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, 6, c, are in harmonical progression, — , j, -, are in arithmetical progression ; therefore, = — . Hence, multiplying by abc, h a c o ac — hc=rah — ac; or c(a — h)=a(h — c). This gives (Art. 268), a : c : : a — h : 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 Ek. 23* 266 RAT'S ALGEBRA, SECOND BOOK. Hence, all problems with respect to numbers in barmon- ical progression, may be solved by inverting tbem, and considering tbe reciprocals as quantities in arithmetical progression. We give, however, below, two formulae of frequent use : 1. Given the first two terms of a harmonical progres- sion, a and h, to find the n"' term. Here, a, b, and I, the first two and n"" terms become (Art. 302), -, ,, and , in formula (l) (Art. 294). Also, d=v ^= — =-• 1 L+(._i)^*=(!^z:llfi^(!^=2)6 I a ' ^ 'ad ab ' Therefore I a ■ ' ab Whence, 1= (n — l)a — (». -'l)b' 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 ??i-(-l=n — 1, we have, as above, T = — \-(n — 1)«> and a= = — ,:= _^ • whence, the arithmetical progression is found; and by inverting it« 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 i. Ans. I and ^. 5. The first term of a harmonic series is i, and the 6'* is j't; ; find the intermediate terms. ^°^- 4' B' B! TO- 6. a, h, c, are in arithmetical proc^ression, and 6, c, d, are in harmonical progression ; prove that a : h : : c : d. AEITHMETIC AND GEOMETRIC PROGRESSION. 267 PROBLEMS IN ARITHMETICAL AND GEOMETRICAL PRO- GRESSION. 304. — 1. The sum of 6 numbers in arithmetical pro- gression is 35, and the sum of their squares. 335 ; find the numbers. Ans. 1, 4, 7, 10, 13. Let x—2y, X — y, x, x-{-y, x-\-2y, he the numbers. 2. There are 4 numbers in arithmetic progression, and the sum of the squares of the extremes is 68, and of the means 52 ; find them. Ans. 2, 4, 6, 8. let x—Zy, x~y, x-\-y, x-\-Zy, be the numbers. Suggestion. — When the number of terms in an arithmetic progression is odd, the common difference siiould be called y, and the middle term X; but when the number of terms is even, the com- mon difference must be 2y, and the two middle terms X — y and x-\~y. 3. The sum of 3 numbers in arithmetical progression is 30, and the sum of their squares 308 ; find them. Ans. 8, 10, 12. 4. There are 4 numbers in arithmetical progression, their sum is 26, and their product 880 ; find them. Ans. 2, 5, 8, 11. 5. There are 3 numbers in geometrical progression, whose sum is 31 ; and the sum of the 1st and 2d . sum of 1st and 3d : : 3 : 13 ; find them. Ans. 1, 5, 25. Let a;=: 1st term and y^ ratio; then, xy and x^- = 2d and 3d terms. 6. The sum of the squares of 3 numbers in arithmetical progression is 83 ; and the square of the mean is greater by 4 than the product of the extremes ; find them. Ans. 3, 5, 7. 7. Find 4 numbers in arithmetical progression, such that the product of the extremes ^27 ; of the means ^35. Ans. 3, 5, 7 9. 268 RAYS ALGEBRA, SECOND BOOK. 8. There are 3 numbers in arithmetical progression, whose sum is 18; but if you multiply the first term by 2, the second by 3, and the third by 6, the products ■will be in geometrical progrcstion ; find them. Ad.s. 3, 6, 9. 9. The Fum of the fourth powers of three suL-ccf-^ivc natural numbers is 9G2 ; find them. Aus. 3, 4, 5. 10. The product of four successive natural numbers is 840; fiud them. Ans. 4, 5, 6, 7. 11. The product of four numbers in arithmetical pro- gression is 280, and the sum of their squares 166; find them. Ans. 1, 4, 7, 10. 12. The sum of 9 numbers in arithmetical progression is 45, and the sum of their squares 285; find them. Ans. 1, 2, 3, etc., to 9. 18. The sum of 7 numbers in arithmetical progression is 35, and the sum of their cubes 1295; find them. Ans. 2, 3, etc., to 8. 14. Prove that whcTi the arithmetical mean of two num- bers is to the geometric mean ; : 5 : 4 ; that one of tliem is 4 times the other. 15. The sum of 3 numbers in geometrical progression is 7 ; and the sum of their reciprocals is ] ; find them. Ans. 1, 2, 4. SuGCE-STiON . — 111 solring difficult jrirobloms in geometrical pro- grcs.sion, it is sometimes preferable to express them hy other forms. — , „ ^- Thiis, for 3 numbers, use X. j .iif, >/, or, .r-, :ri/. y-\ for four, — , r, )/, ■— ; for five, '— , .c-, xu, y-, ' ; for six, '—„ —, X, y, —-, —. In all these cases the product of the first and third of any three, taken consecutively, is equal to the square of the second. To find the ratio in each case, divide any expression by the preceding. 16. There are 4 numbers in geometrical progression, the sum of the first and third is 10, and the sum of the second and fourth is 30; find them. Ans. 1, 3, 9, 27. PERMUTATIONS AND COMBINATIONS. 269 lY. There are 4 numbers in geometrical progression, the sum of the extremes is 35, the sum of the means is 30; find them. Ans. 8, 12, 18, 21. 18. There are 4 numbers in arithmetical progression, which being increased by 2, 4, 8, and 15 respectively, the sums are in geometrical progression; find them. Ans. 6, 8, 10, 12. 19. There are 3 numbers in geometrical progression, whose continued product is 64, and the sum of their cubes 584; find them. Ans. 2, 4, 8. IX. PERMUTATIONS, COMBINATIONS, AND BINOMIAL THEOREM. 303. The Permutations of quantities are the different orders in which they can be arranged. Quantities may be arranged in sets of one and one, two and two, three and three, and so on. Thus, if we have three quantities, a, h, r, we may arrange them in sets of one, of two, or of three, thus : Of one, a, h, c. Of two, ab, ac; ha, he; ea, ch. Of three, abc, ach; hac, hca; cah, cba. 306. To find the number of permutations that can be. formed out of n letters, taken singlj/, taken two together, three together. . . . and r together. Let a, b, c, d, . . Jc, be the n letters; and let P, denote the whole number of permutations where the letters are taken sinffly; Pj the whole number, taken 2 together .... and Pr the number taken r together. 270 RAY'S ALGEBRA, SECOND BOOK. The number of permutations of n letters taken singly, is evi, dently equal to the number of letters; that is, Pi=n. The number of permutations of n letters, taken two together, is n{n — 1). For since there are n quantities, a, 6, c, d, . Tc, if we remoTe <7, there will remain (n — 1) quantities. Writing a before each of these [n — 1) quantities, we shall have ab, ac, ad ... . ak. That is, (n — 1) permutations in which a stands first. In the same manner there are (n — 1) permutations in which b stands first, and so of each of the remaining letters e, d, . k. Or, for n letters, there are mn—l) permutations taken two together. That is. P,=n(n^l). Hence, .er. The niimhrr of permutations of n letters taken two togeth ■is equal to the niinihrr of letters, 'laullipUed by the number less one. For example, if »=4, the number of permutations of a, b, <\ d, taken two together, is 4x3=1^. Thus, ab, ac, ad, \\ bet, be, bd, II ca, c6, cd, II da, db, etc. The number of permutations of n letters, taken three together, is n{n — \)[n -'2]. For if we take yn—l) letters, 6, c, d, . k, the number of permu- tations taken two together, by the last paragraph, is {•»-ln»-2). Let a be placed before each of these permutations; then, there are (n — ])(" — 2) permutations of n letters, taken three together, in which a stands first, ami (11 — l)(n— 2) permutations, in which b stands fir^t ; and so for each of the « letters. Hence, the whole number of permutations of n letters, taken three together, is n{>i —l)(n— 2); That is, V..= ri[it--l](n—2). Hence, The number of pcrmutdtions of n h H,rs taken three to- gether, is equal to the number of letters, midtijiUed by the number less one, multiplied hy the number less two. PERMUTATIONS AND COMBINATIONS. 271 If w=4, the number of permutations of O, 6, c, d, taken three together, is 4x3X2=24. Thus, abc, abd, acb, acd, adb, adc, bac, bad, bca, bed, bda, bdo, cab, cad, cba, cbd, cda, cdb, dab, dac, dba, dbc, dca, deb. Following the same method, we prove that the number of permu- tations of n letters taken four together, is Vi=n{n—\ ) (n— 2) (n— 3). In each of the preceding results, the negative number in the last factor is less by unity, than the number of letters in each permu- tation. Hence, for n things taken r together, P,=n(n— 1)(M— 2) (n—r—1) 306°'. Corollary. — If all the letters be taken together, then r becomes equal to n, and the last factor becomes 1 ; That is, P„=n(»— l)()i— 2) 1. Or, inverting the order of the factors, P„=lX2x3 . . .... (n— l)n. Hence, TTie number of permtttaiions of n letters taken n togetTier, is equal to the product of the natural numbers from 1 up to n. Ex. — The permutations of three letters, a, b, c, taken three together, is 1x2x3—6. 307. If the same letter occur p times, the a-mber of permutations in n letters, taken all together, is 1X2X3 .... (n~l)n 1X2X3. . .p Suppose these p letters to be all different. Then, for any psrtlo- nlar position of the other letters, these p quantities, taken p to- gether, will form (1x2X3 ■ ■ ■ • P) permutations from their interchange with each other; and when these letters are aliie, tbew permutations arc all reduced to one. And 272 RAY'S ALGEBRA, SECOND BOOK. As this is true for every position of the other letters, there 'will be altogether (1x2x3 . ■ p) times fewer permutations when they are alilie than when they are all different. Thus, in the letters A, I, D, there are 1X2X^^6 permutations taken all together, but if I becomes D, then three of these permuta- tions become identical with the remaining three, and the whole number for ADD taken all together, is 1X2X3_ 1x2 ^=3. 307*. Corollary. — In like manner, if the same letter occur p times, another letter q times, a third letter r times, and so on, the number of permutations taken all to- gether, is 1X2X3 (n-l) n (1X2 i>)(lx2 . . 2)(lX2 . 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, ah, ac, be, are the combinations of the letters a, h, c, taken two together; ah and ha, ac and ca, he and cb, though different permutations, forming the same combina- tion. Proposition.- -To fnd the number of combinations that eaii he formed out of n letters, taken singly, taken two to- gether, three together, ... . and r together. Let Cp Cj, . Or denote the number of combinations of n things taken singly, taken two together, . . and taken ;• together. The number of combinations of n letters taken singly, is evi- dently 11] that is, T^ie number of permutations of n letters taken two together, is r,{n—l); but each combination, as ab, admits of (1X2) permuta- PERMUTATIONS AND COMBINATIONS. 273 tioiis, ab, ba; therefore, there are (IX^) times as many permuta- tions as combinations. Hence, W(H-l) '^^— 1X2- Again, in n letters taken iJiree together, the number of permuta- tions is n[n-~l)[rt—2); but each combination of three letters, as abe, admits of IX^X^ permutations; therefore, n(n— l)(/i-2) 2- 1X2X3 " So, for n letters, each of which contains r combinations, n(w— l)(n-2) . . . [n-(r-l)] ""'— 1X2X3 r S09. Intimately connected witli the subject of the pre- ceding articles, is that of the Doctrine of Chances, or the Calculus of Probabilities. This, however, being too abstruse ibr an elementary treatise, is omitted in this work. 1. How many permutations of 2 letters each can be formed out of the letters a, h, c, c1, 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, b, c, rf, 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. SuooESTiON. — Take (he sum of the different permutations. 274 RAYS ALGEBRA, SECOND BOOK. 7. How many different products can be formed ■with any two of the figures 8, 4, 5, IJ ? Ans. 6. 8. The number of permutations of ?! things taken 4 to- gether = 6 times the number taken 3 together ; find n, Ans. 7i=9. 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, ii 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 he posted out of 16? and on how many of these will anjf particular man be on guard? Aus. 1820, and 455. 13. How many changes may be rung with 5 bells ou( of 8, and how many with the whole peal? Ans. 6720, and 40320. 14. Out of 1*7 consonants and 5 vowels, how many worda can be formed, having two consonants and one vowel in each? " Ans. 4080 BINOMIAL THEOREM, WHEN THE EXPONENT IS A POSITIVE INTEGER. SIO. 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 Combinatibus (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^xy, (a+xf\ (a+x)'-, where n or a; 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 multiplication, it appears that {x-\-a)(x-\-b)=zx--\-a i x^ab. +b\ lu like manner, (x-}~a){x-\-b)(x-\-c) =a;3-fa x^-\-ab x+abe. -|-6 -fac -fc +6c Also, {X'\-a){x-\-b)(x~{-c){x-\-d) x-\-abod. x^+a x^-\~ab x'+abc +6 -\-ao +abd +e +ad -\-aod +d +bc -^bed +bd +cd 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 mimher of binomial factors, 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, b, c, etc. ; of the third, the 276 RAV'S ALGEBRA, SECOND BOOK. sum of the products of every two of the quantities a, b, 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 <(uan- tities a, b, c, etc. Suppose, then, this law to hold for the product of n binomial fac- tors a; + a, x-f-b, a;-)-c, . . x+k- so that {x-\-a){x-j-b){x-\-c] {x-\-7e)=x''-+Ax"-^+Bx''-'+Cx"-"-+ .... +K, Where . . . A=a-i-6-fc-|- . . . +/r. B—.ab^ae+acl-\- . . . C=a6c+a6d-|- Etc. =z etc 'K=abcd ... Ic. If we multiply both sides of this equation by a new factor x4-l, we have (x+a){x+b){x+c) . . , {x+7e)[x+l) +i I ^A^l -fBzl .... -fK^. Here, . A+?=a-|6+c+ .. +^'-;7; B--Al--^aO+ac—ad. . . -{-al-\-bl . . -j-Tcl. Etc. = etc. . Kl ^abcd. . H. 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^\ factors are taken; but it has beea. shown, by actual multiplication, to hold up to 4 factors; therefore, it is true for 4-|-l, that is, 5; and if for 5, then for 5-|-l, that is, 6; and so on generally, for any number whatever. Now let . . . 6, c, d, etc., each ^a; Then, A=a-|-a-|-a+a+, etc., to n terms =na. B— a^-|-a-+, etc., =a2 t.aken as many times as-i , is equal to the No. of combinations of n things taken \ = -^,— ,> - two together, which is (Art. 308), J C=a3^-a3-)-, etc., :=a' t.iken as many ^ times as is equal to the No. of combinations ( __n{n~\ )(n~'y]a^ of the things taken three together, which is [ 1 ■ 2 • 3 (Art. 308), '' Etc. = etc. K=aaa to n factors =a" BINOMIAL THEOREM. 277 Also, {x-\-a){x-\-b){x+c) (x+l)—{x+a)''. + ... +0". By changing a; to a and a to x, we have Let a=l ; then, since every power of 1 is 1, (1 +.).=l+..+^).2+ »(»-l)('^^) .3+ +^. Corollary 1. — The sum of the exponents of a and x in each term =^n. Corollary 2. — If either term of the binomial is negative, every odd power of that term will be negative (Art. 193); therefore, the terms which contain the odd powers will be negative. ,-,>,! , n(n—V\ , n{n—\){n — 2") , .-. (\—xY=l—nx-\- "-^ ^ ' x^ ^ I'' -3 — ^^ +' ®'^''- Corollary 3. — Since the last factors, in the fraction which forms the coefficient, are for the 2d term ^, for the „ 1 n — 1 ,. , ^ , m — 2 , . „ dd term — ^ — , tor the 4th term — q— , etc. ; thereiore, lor n — (r— 2) the r"* term they will be ^-=: — - Also, for the exponents of a and x, we have in the 2d term a"-''x, in the 3d term a"-^a:^ in the 4th term a"-^a?; therefore, in the r* term, we shall have o""'"— ■'x''-'. Hence, the general term of the series is nfn-l)(n-2) {n^ r+2) 1-2 -3 (T-\) "" ^"^ ■ 278 RAY S ALGEBRA, SECONB BOOK. This is called the general term, because by making r=2, 3, 4, etc., all the others can be deduced from it. Example. — Required the 5"' term of (a — a;)'. Here, r=5, and 71=7] therefore, the term required Corollary 4. — If n be a positive integer, and r^n-{-2 then, (?i — ''+2) becomes 0, and the («-|-2) term vanishes therefore, the series consists of (;i-|-l) terms altogether that is. The number of (frms is one rjrcaler than the exponent of the power to irhicjb 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 compiire the expansion of (a4 -T)", and (X -«)", we have (a-)-a;)"z-a" -na"-i.r-| -4j— ., 'a"—x--] — -— — ;; 'a"--a:3 -|-, etc. , , 71 In — 1) , , , n(n — l)i;7 -2) , , {x-\-a)''=x" -\-7ix"-^a^- A: — ^— .r"--a--i — ^- — ^-„ — .r"-''n--L,etc. Since the binomials are the same, the series resulting from 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- p.insioD may be taken from the first half. Example. — Expand (a — b)^- Here the number of terms (»+l) is 6; therefore, it is only ncces- Bary to find the coefficients of the first three, thus: a ■ 4 (a—b)^=a'^—5a^b+- — ^a^b'^—lOa-b^+bab*-/,': BINOMIAL THEOREM. 279 Corollary 6. — The sum of the coefficients, where both terms are positive, is always equal to 2". For if we make a;=a^l ; (hen, . (■a;-(-a)"=(l-fl)»=i2». 311. From an inspection of the general expansion of (a-f a;)", 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 be divided by the number of the term, the quotient will be the coefficient of the next term. For examples, see Newton's Theorem, Art. 172. 313. To expand a binomial affected with coefficients or exponents, as (2a^ — 3i')*, 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 — b-\-c to the third power. Let a — b^m, etc., as already explained, Art. 172. 1. Expand (a+6)», («— 6)', and (5— 4a:)*. (1) Ans. a'^-f 8a'6 + 28a''i'+56o^6=+'70a*5*+56a'i5 + 28a^6''+8a6'+?-«. (2) Ans. a'~la%-\-2\a^b'—?,ba'h'-J^Zba'h'-~2\a:'h^ + laV--b\ (3) Ans. 625— 2000^+ 2400a:'— 1280a;=+256x*. » 2. Required the coefficient of a? in the expansion of (x-\-yy. Ans. 210. 3. Find the 5"" term of the expansion of (c^ — d^y^. Ans. 495c'W. ScGGESTiON. — (See Cor. 3, Art. 310.) Instead of o, a;, re, and r, substitute c^, . — dr, 12, and 5. 4. Find the 7"* term of (a'+3ai)». Ans. 61236a'56«. 280 RAYS ALGEBRA, SECOND BOOK. 5. Find the middle term of (a"'+a;'')". A. 924a«'"a^". 6. Find the 8"' term of (!+.<•)"■ ^us. 330.?;'. 7. Expand (Oar- 26c/ )^ Ads. 2iSa'c'SlQa'c*hcl 8. Expand (a-\-2h-~cy. Ans a'^6d'b-\-12ah' + W—Qa'c—12abc—12h'c+Sac^-\-6hc^~c'. 9. Prove that the sum of the coeiEoients of the odd terms of ('i-\-.c-)", is equal to the sum of the coeiEeients of the even terms. X. INDETERMINATE COEFFICIENTS: BINOMIAL THEOREM, GENERAL DEMOxNSTRATION : SUMMATION AND INTERPOLA- TION OF SERIES. 314. Indeterminate Coefficients. — The method of de- veloping aliicbraie expressions, by assuming a series with unknown cnefEoients, and finding the values of the assumed coefficients, is termed the method of Indeterminate Coeffi- cients. It depends on the following THEOREM. If A+B.->;-fCa-'+Da-'+, etc , =A'+B'a-+CV-f DW+, etc., for every possible value of x (A, B, A', B', etc., not contaiBing x, and x being a variable quantity) we shall have A=^A', B=B', C— C, etc.; that is, Till- coefficients of the terms invoicing the same poiC( is of :r ill the lico series, are respectively equal. For, by transposing all the terms into the first member, we have A— A'+(B— B')I^-(C— CO.r--|-{U— D')a,-3-|-, etc., =0. If A — A' is not equal to 0, let it be equal to some quantity p; then, we have (B— B')x-f(C— C')a:H(D— iy)x3-|-, etc., =—p. INDETERMINATE COEFFICIENTS.^ 281 Now, since A and A' are constnnt quantities, their difference, p, must be constant; but — p=(B— B')a;-)-(C — C^)x^-\-, etc., a quan- tity whicli 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 quantity [p) which can express the difference A— A'; or, in other words, A— A'=0 , . A=A'. Hence, (B— B')a;-f-(C— C')a:2-)-(D— D')a:3_|__ etc., =0. By dividing each side by x, we have B-B'+(C— C')a;+(D-D')a:2+, etc., =0. Reasoning as before, we may show that B=B'; and so on, for the remaining coefficients of the like powers of X. Corollary. — If we have an equation of the form A-f-B2:+Ca)2-|-Da;'+Ea;*+, etc., =0, which is true for avy value whatever of x ; then, A:^0, B=n=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-(-Ox-fOX"+Ox'-|-, etc.; then, comparing the cotfficients of the like powers of X, we have A^O, B=0, 0=0, etc. 313. Let it be required to develope =— into a series a-\-bx 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-\-bx, and arrange the terms according to the powers of X; we thus obtain a=Aa-f Ba I x-\-Ca I a;2-|-Da I x^-\-, etc. +A6i -fBfil +C6| 2d Bk. 24 282 RAYS ALGEBRA, SECOND BOOK. But by the preceding theorem and corollary, a=Aa ; hence, A^l ; Ba+A6=0; " B= ; Ca+E6=0; 63 Da+Cb=0; " D= -, etc. Substituting these values in the assumed scries, wc find a ^ b b'- ., b^ , , Z>' , • , 1 X- — .,x- ;X •-{ — -X* etc, the same aa 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 a;, 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 ^ —,, and we assume the oX — X- series to be A-|-Bic-l-C'^"H-I'-f"- Ex^^, etc., we have, after clearing of fractions, l=3Ax+(3B— A)a:2+(3C— E)x3+, 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, S^-l^^ ^""'°S 3^=A+B. + C:r=+,e.c., clearing of fractions, and equating the coefficients of the like powers of X, we find A=-t, B=^, C=^i^, D=gT-, etc. Hence, 1 \ l\ X X? x^ , \ 2-1 a" X x^ 3i^P=x\ 3 + 9 + 27 + 81+' "'"■ ) = ^+9+27 + 8i+'^"'- INDETERMINATE COEFFICIENTS. 283 Or, since the division of 1 by tlie first term of the denominator gives 5-, or 3a;~', we ought to have assumed 5-^^,=Aa:-'+B+Ca;+Da;2+, etc. Again, if we assume ., ^^A-\-Jix-j-Cx^-\-'Dx^-{-, etc.; we shall find the true series to he 1 — 2.i--|-3ar* — 5a;''-f-, etc., the coefficients B, D, F, etc., hecoming zero. 317. Evolution by indeterminate coefficients. Example. — Extract the square root of a'-\-x''. Assume y {a^+x^)=A+Sx+Cx^+'Dx^-\-Ex*-\-, etc. Squaring both sides, we have, a2+x^^=A^+2AJ^x+2AC 1 a;2+2AD I a;3+2AE x*+, etc. +B2 1 +2BC I +2BD +C2 .-. A2=a2, 2AB=0, 2AC+B2=1, 2AD+2BC=0, etc. And, A=a, B=0, C==-, D=0, E=— =-^— , etc. Therefore, l/(«^-f-a;^)=a4-2^— gp+, etc 318. Decomposition of Rational Fractions. — Frac- tions vyhose 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. 5.^ 14 Decompose „ ^ into two other fractions whose x' — bx-\-o denominators shall be the factors of x' — 6x-\-8. Since x^—6x-\-8={x—2)(x—'i), (Art. 284), as.sume 5a;— 14 __A^ , B 3 — id9"r;i x^—6x-\-8 X— 2^x^-4' 284 RAY S ALGEBRA, SECOND BOOK. Reducing the fractions to a common denominator, 5x-U A(a;-4)+B (r -2) Or, oa;— 14=A(a^-4)4-B(a;— 2)=(A-|-B)a;— 4A— 2B. Now, since this equation is true for any value whateTer of x, we may equate the coefficients (Art. 314); this gives A4-B^5; — 4A— 2B=— 14; whence, A=2, and B=3. .5X-14 2 , 3 And X- — 6a;-i-8 x~^2 x—i' By the method of Indeterminate Coefficients, show that 1-1-2 ■• 1. :r^,^ =l + 5a:-f-15.r2+452^-|-. etc. 1 — o.c 2. -^-~^r-^.l + 3x-l-4x:'-]-1x'-{-U.i-*Jr'i-8x'+, etc. 3. -i-i^=l^+2'^x+3'-.t^+4W+5V-(-, etc. :r_ a-^ 3.<-^ 3-5;r' 4. y 1— .t^l—o— 5-4 2-4-6~2 4 b"8~ *^*''" 5. 1 (1+^+^0 = i+'5+:f-ig+. etc. ^ l-J-a; 1 , 2 •7. .<■ — .1-- X 1 — j; .T+l .5 4 {^x^'—i)(x—2)~^3(x—2) 2(x-l) "T 6(a;-f 1)* b a;=— . a BINOMIAL THEOREM. 285 BINOMIAL THEOREM, WHEN THE EXPONENT IS FRACTIONAL OR NEGATIVE. 319. We shall now proceed to prove the truth of the Binomial Theorem general!}' ; that is, to show that ,, ii{n — 11 .,,, , nin—\)(n—2) „ ,,, , (a-l-6)"=a"4-wa"-'6-t-— ^.,-^-«"--6--f ^ . .,\ ^ — ' a''-^b^-\-fi\c. ■whether n be 'integral or fractional, positive or negative. Sinoo a-\-b=a{\-\ — j; Therefore, (a+6)"=a" / 1+- ) »=a"(l-t-a;)", if Hence, if we can find the law of the expansion of (1-j-a;)" we may obtain that of (a-\-b)", by writing - for x, and multiplying by a". We shall therefore prove that, in all cases, (l+a:)"=--l+«a:+J^-^'a:^+ ^ . ^'. 3 -x-+, etc. The proof may be divided into two parts : 1st. To show that (l+a:)"=l+«a;+, etc. 2d. To find the general law of the coefficients. First. — To prove that the coefficient of the second term of the expansion of (1-f a:)" is n, whether n be integral or fractional, positive or negative. Let the index be positive and integral; then, since by multiplica- tion we know that (l-fa:)2=14-2a;-|-, etc., \\\-xf=\J^Zx-\-, etc.; Let us assume that (l-j-:?:)"-'=l-f (n— l)a;-|-, etc. Multiply both sides of this equality by 1+a;; then, (l+a;)"-i(l+a;)={l-(-(»— l)x-|-- etc. Kl+a;); Or, {l-\-xy=^\^nx-\-, etc., by multiplication. 286 RAY'S 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-J-l=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. Next, let n be a fraction =^. . Multiplying both sides by i-j-X, we have M-f-2Ba;+3Ca;2+4rte3+, etc. I =n(l+a;)» + nx+2Bx''+3Cx^+, etc. J =n(l+?ia;+Ba;2-f Ca;'-}-, etc.) By equating the coefficients of the same powers of X, we have 2iii-\-n='n? . . 2B--iin- — «=n(?i— 1). ^_ n(w-l) _ "-" 1-2 ' 3C+2B=Bn .. 3C=B(n— 2); B(n-2) _n(ra— l)(n— 2) 3 ~ 1 ■2- 3 ' Also, 4D+3C=nC .-. 4D=C(n.-3); C(»— 3) __/i(n— l)(n— 2)(n— 3) 1 •2-3-4 and so on for E, F, G, eto. .-. putting - for X, (a+6)»=a"( 1+- )", b , nfn— 1)62 n{n—l)(n—2)b^ =a"+na'-i6+^^a'-26=+?^i^^==^^(^«»-363+, etc. If — 6 be put for 6, then since the odd powers of — b are negative (Art. 193) and the even powers positive, 7^n « n 17. , n{n~l) „ ,., nfw— 11(n— 2) , .,., , (a— &)"=a"— na"-'6-J — ^ — ^a'^^b^ S-^i — J—;, — -a'^^tfi +, eto., which establishes the Binomial Theorem in its most general form. Remark. — From the preceding, corollaries, similar to those in Art. 310, may be drawn, but it is not necessary to repeat them. The following additional proposition is sometimes useful. 288 RAY'S ALGKBRA, SECOND BOOK. 330. To find the numerically/ greatest term in the expan- sion of (tt -{- b)". I. If m is a positive integer : From Cor. 3, Art. 310, it apiicars that the (r-|-l)"' term may he formed from tlie r* by multijilying the latter by —■-, or , — 1 )-, and tliis multiplier diminishes as i increases; \ r Ja wliile it is [greater tlian 1, each term is greater than the preceding; and the v?iiie of r, whiili first makes it less than 1, indicates the greatest term ; that is, the >■"' term will be the greatest when I — ■ 1 I- IS first < 1 or r first > p— \ r /a a~b From the nature of the case, r is necessarily integral ; if — — r- IS fractional, take r = the first integer > — , and the r" a-\-h JO'- ^j^ij m _ in + l)b ^ a+b greater than any other term ; for this can only occur when term will be the greatest. If — --7— is an integer, and we take , then the r'" term = the (r-t-l)'*, and each of these is II. If n is a positive fraction : It — > 1 there is no greatest term, for the series will evidently diver. ic. But if - < 1 the series will have its greatest term (or terms) n wliose jKisition may be ascertained as in I. III. If n be negative, either an integer or a fraction > 1: Tlie multiplier that changes the ?■''' term into the (r-j-l)"*, viz., — ?i— r+1 b , .,, /'v + r—l\h , ., . „ — .- nia\' be written — ( ■ J-, and as tJie numericatly a ' \ r /a greatest term is sought, disregard the sign of the multiplier: then. li + T — 1 b ;is in I, the r'-' term will be the greatest wdien .- is first < 1 T a r- , Mn—1) or r firat > ~. a—b As in I, if ; ■ be a whole number there are two equal terms a — each greater than any other; and, as in II, it — 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 tliis case tiie multiplier ; is < 1 for alt values of r, that is, each term is less than the preceding. Note. — If h is negative, since it is tlie numerical value ot the term tliat is to be considered, we may disregard tlie sign ot b and apply the appropriate one of the preceding rules. [Cf. Todhunter's Algebra, Art. 520.] Examples. — Find the greatest term in each of the following expansions : 1. (2+1)°. Here 6 . 5 . 4 _ 5^ 20000 81 ■ (ii+l)6 _35 a+b ~n -.93.- 1.2.3^ 3" 2. (1+J) I. 3. (l+f)^ 4. (1+^)-". Here^!^'=Ll a — 4 12-13 ]_ 1-2 • 52 "25 5. (l+?)-3. HereM^=5 a — than any other term. 6. (1-AH. r=4 gives the greatest term:= Ans. 2''. Ans. 5"' and 6'". =3 gives the greatest term=: 5"'=6">, and each is greater Ana. 3''''. 331. In the application of the Binomial Theorem, it is merely necessary to take the general formula (a + 6)"^=a"4- no"~^6-f-, etc., and substitute the given quantities in the formula, and then reduce each term to its most simple form Example.— 1. Find the expansion of (l+.r)* Here, o=l, b^x, n=\- {l+x? \(\-\) , j(i-l)(i-2) 1-2 1-2-3 etc. 1 ■ 1 ^l+ia;— 2-7^2:2+2- ^r-g- 2d Bk. 25^ 1-1-3. 1-1-3-5 2 • 4 ■ 6 ■ S' — „.T^+, etc. 290 RAY'S ALGEBRA, SECOND BOOK. Example.- — 2. Develope (1 — x)~^. Here, o=l, 6= — X, n=— J. .-. (l-a;)-*=l-^(-a;)+tzi)|:=^)(_a;)2+, etc.. As the general formula (l^.r)"=l±Ka;-|- .^- — ^x^zh, etc., is more easily retained in the memory, and is less complicated, it is generally most convenient to reduce the quantity to be expanded to this form. Thus : Develope \' a-\-b into a series. Since a^b^al l+_ \ . . y7^PJ=■^/'E{ 1+* U_ Here, ?: — -. n = \- and since (l+.rj»=l-,-»,r+ Yro-* +^T-r.Tr3— ^■'+, etc., •■I ^+5 / -^ +=5+ T~2~V+ 1-2-3 -53+' ''^'=-' 6 _ ]_ 62 1 3 ^,3 ~ +-a i; 4a2 + 2-4-6"a3~ *'"■ Hence, ^a+6^, «(1+,^ _g_^ + _______^+, etc.). 1. =(1— ,r)-'=l+.r+.-r'+.r'+.j* + , etc. 2. ^j-:^^-,=(l-.i-)-'=l+2-^+3x''+4x'+5.T*-h, etc. , 2.r , 3.r^ 4:r' , 5.T* =1 \-— ■-— -f---_ etc;. a a^ a* a^ BINOMIAL THEOREM. 291 a? x^ ba? t. TTi .-«— -L g g g^ , etc. X X 5. v/a^+.^a+2-„-8^3 + Ty;^-128^,+. etc. 6. (a3-^)^^a-g^_y-,_g^3 5a:' 10.^ .X' 24ba" etc. 7. (l + 2x)-=l+a;— Aa;^+-ix'— |a;*+, etc. o /-^; 5- a;'' x* x^ Sx" o. i/a' — x'=a — -71 Tt— ; — T ,. ^ — - „„ , — , etc. ' 2o 8a^ Iba^ 128a' ' Ix Ix^ , 5x' 10, 'X' 9. fa+x^ra(l+^^-^^ + ^-^^^+, etc.). 10. (a'+x3)^=a(l+£3-g^,+ ^^,-, etc.). 11. r9=^8+r=2+|. J-|-l, + l-;-l,-, etc. 12. (a'-x')^=a(l-3-^3-3^,-3^Q^-, etc.). a' , 2x' , 2 ■ 5x« 2 ■ 5 • 8x' Here, — E^!_^ = a3(a3 — x3)"*= a3 X (a^r^i 1- "^' )"*=«' (a3-x3)^ ^ " ' -"V-l)-^-('-S) 333. To find the approximate roots of numbers by the Binomial Theorem. Let N represent any proposed number whose ?i'* root is required ; take a such that a" is the nearest perfect ji'* power 292 RAY'S ALGEBRA, SECOND BOOK. to N so that N^a"±'>, b being small compared witli a, and + or — , according as N> or <«" ; l=t-7; |"=j by -writing — for h in the txeneral formula ; a i 1 ±1 . ^ - 1. "-1 / 1 y±l. "-1 . 2-_-i / A V_ etc i a a" n 'la \ a" I h 'Zn 'Sn \ a" J ' ''' Of this series a few terms only, when 6 is small with regard to a", ■will give the required root to a considerable degree of accuracy. 14. Required the approximate cube root of 128. Here, f 1 .28= --fa: '+3= =5#1+Ti5 i -=l'-3 3 1 ; 3 n 3 \2 1 I25)+3^' 9\ : 12.5 / ' "4- 1 ~5^ 4 1 ... =54 2' 1 KF + 3' '^7 =5+0.04-0.00032+0.0000042— . . =5.0396842. !$23. In the preceding example, we ohtaiu only an approximate value. To determine the limit in the error occasioned b}' neglecting the remaining terms of the series, let R be the true root, and as the terms are alternately positive and negative, put E =a — h-\-c — "'+<;— /-)-5' — h-\-^t^ — ^+, etc., and let K' =a—h-\-c—c7-^e—f, n"=a^b+c~d+e-f+g. Then, since the terms continually decrease, a — b, c — d, e—f, g — h, etc., are all positive, and therefore K', which contains three only of those differences, will be less than R. For the same reason, all the pairs of terms after (/, as — h-\-k, — l-{-m, etc., will be all BINOMIAL THEOREM. 293 negative, and R" will be greater than R; therefore, the true -value of the series lies between R' and R'', or between a— Z)-|-o— d-|-e— /, and a — 6"|-c— (i-|-e— /-|-(y. Hence, The error commiltnl hy the omission of any number of the terms of a converging scries whose signs are alternately posi- tive and negative, is less than the first omitted term. Thus, in the preceding example, had we stopped at the 3d term, the error would have been less than .0000042. 15. Find the b"' root of 35. Ans. 2.036172+ Here, . . . N=35=32+3=25/ 1+4 )• 16. The student may solve the following examples : (1). v'lO =t/9+1 =3.16227 . . . true to 0.00001. (2). ir2'4 =^2T^ =2.88449 . . . true to 0.00001. (3) >/108 =v 128— 20 =1.95204 . . . true to 0.00001. Remark. — The nth root may be extracted by the formula in Art. 321, the number whose root is to be extracted being divided into any two parts whatever. The advantage of the formula in Art. 322 consists in the rapid convergence of its terms. THE DIFFERENTIAL METHOD OF SERIES. 334. The Differential Method is used, 1st, to find the successive diiFerenoes of the terms of a series ; 2d, to find any particular term of the series ; or, 3d, to find the sum of a finite number of its terms. If, in any series, we take the first term from the second, the second from the third, the third from the fourth, and so on, the new series thus formed is called the frst order of differences. 294 RAY'S ALGEBKA, SECOND BOOK. If we proceed with this new series in the same manner, we shall obtain another series, termed the second order of differences. In a similar manner we find the third, fourth, etc., orders of differences. If we have the series, 1 , 8 , 27 , 64 , 12.5 , 216, . . The 1st order of diflFerenoes is 7 , 19 , 37 , 61 , 91 , . . . . The 2d " " " " 12 , 18 , 24 , 30 , . ... The 3d " " " « ,6,6,6, .... 3S3. Problem I. — To find the first term of any order of differences. Lot the series be a, b, c, d, e, ; then, the respective orders of differences are, 1st order, b — a , c — h , d — c , e — d, .... 2d order, c—2b-\-a , d—2c+b , c—2d-{-c, . ... 3d order, d — 3c-\-8b — a, e — 3d-^3c — b, 4th order, e— 4c?+6c— 4i+a. Here, each difference pointed off by commas, though n compound quantity, is called a term. Thus, the first term in the 1st order, is 6 — a; in the second order, c — 2b-\-a, etc. If we denote the first terms in the 1st, 2d, 3d, 4lh, etc., orders of differences by D,, D^, Dj, D^, etc., and invert the order of the letters, we have D,=— a+6; J)2=a—2b+c; 1)3=— 0+36— 3c+d; D.,=a— 46+6c — 4d-{-e, etc. Here, the coeflScients of a, b, C, d, etc., in the «« order of differ- ences, are evidenlly those of the terms of a binomial raised to the n"' power; and their signs are alternately positive and negative. Hence, the first term of the n"" order of differences is , nin^l) n(n—l)(n — 2) , , ^ , . j a—nb-\- -' - c — -n—r, — -d-'r, etc., when n is even, and 1 ' L 1 ' .J ■ o ^ n(n^l) n(n-l)(n-2], ^ v .. • jj SERIES— DIFFERENTIAL METHOD. 205 Corollary. — It is evident from the coeflBcients that when 71^1, the value of D„ has only two terms, for then n — 1=0 ; ■when n=2, this value has only three terms, for then n — 2=0, and so on. 1. Find the first term of the fourth order of differences of the series 1', 2\ 3', 4', 5', . . . or 1, 8, 27, 64, etc Here, n=4 ; hence, take five terms of the first value of D„ and a=l, 6=8, 0=27, d=64, e=125, and 1)^= l-4X8+j^2X27-j^^5x3X^*+lX2x3X4>< 1-32+162—256+125=0, Ans. 2. Find the first term of the second order of differences of the series l^ 2'', S^ 4^ . . . or 1, 4, 9, 16, 25. Ans. 2. 3. What is the first term of the third order of differ- ences of the series 1, 3, 6, 10, 15, etc.? Ans. 0. 4. Required the first term of the fifth order of differ- ences of the series 1, 3, 3'^ 3=, 3*, etc. Ans. 32. 5. Find the first term of the fifth order of differences of the series 1, i, J, i, Jg, etc. Ans. — ^L. 326. Problem II. — To find the w"" term of the series a, h, c, d, e, etc. From the preceding article, we have seen that Dj^ — a-\-b; whence 6=a+Di; D2=a— 26+e; " c=a+2D]+D2; D3=_a+36-3c+d; " d=a+3Di+3D2+D3; D4=a— 46+6C— 4(^+e; " e=a+4Di+6D2+4D3+D4. It is evident from inspection that the coefficients of the n"* terra of the series are the coefficients of the (n — 1) power of a binomial. 296 RAY'S ALGEBRA, SECOND BOOK. Hence, writing n 1 instead of n, in the coefficients of the n"' power of a-)-6, (Art. 319,) the n"' term of the series is (M— l)(?i— 2)^ , ("— l)(n— 2)(?i-3) a+(n-l)Di+ ^-y^^^ ^^+ 1---3 ""^3+ ^'°- 1. Find the 12'* term of the series 1, 3, 6, 10, 15, 21, . . 1 , 3 , 6 , 10 , 15, 2,3,4,5, hence, D,=2; 1,1.1, " D2=l; 0,0, " D„=0; Or, D,, D.,, Dj, etc., may be found from the formula, (Art. 325,) and the succeeding orders of differences are also evidently 0; hence, 12"' term a+(n-l,D,+(?^4K^D3=l + ll\2 + "-^'" .1 =1+22+55=78, Ans. 2. Find the n"" term of the series 2, 6, 12, 20, 30, Proceeding as aboTe, to find the orders of differences, we hare Dl=4, D2=2, andD3=0; hence, n* term =2+(n.— 1)4+^ --— 'x2=n'^-\-n, Ans. 1 * Z From the formula n^+n, or »l(n+l), any term of this series is readily found; thus, the 20"' term =20(20+l)=420. It is also evident that the n'* term of a series can be found ex- actly, only when some order of diflferences is zero. 3. Find the 15"' term and the n"' term of the series 1, 2-', 3^ 4^ . orl,4, 9, 16, .. Ans. 225, and n^ 4 Find the 12"' term of the series 1, 5, 15, 35, 70, 126, etc. Ans. 1365. 5. Find the n"' term of the series 1, 3, 6, 10, etc. . r,(n + l) Ads. K — -. SERIES-DIFFERENTIAL METHOD. 29V 6. Find the 9"' term of the series 2 ■ 5 ■ 7, 4 7-9, 6 -9 -11, 8 11 13, etc. Ans. 8694. 7. What is the jt"' term of the series 1x2, 3x4, 5x6, etc.? Ans. 4h'' — 2^. 327. Problem III.— 7b Jind the sum of n terms of the series a, b, c, d, e, etc. Assume the series 0, a, a-\-b, a-\-h-\-c, a-\-b-{-c-\-d, . . . Subtracting each term from the next succeeding, we have a, b, c, d, e, etc., which is the series whose sum it is proposed to find. Hence, the sum of n terms of the proposed series, which it is now required to find, is the (n-Ll)'* tei'm of the assumed series. It is evident the J?"' order of differences in the given series is equal to the (n-(-l)"' order in the assumed series. Hence, if we compare the quantities in the assumed series, with those of tlie formula for finding the n"' term of a series (Art. 326), we have for a, a for D,, n+lforn, D, for Do, etc. Substituting these values in the formula, we have 0-|-(l+l — 1)0! "t r^2 ^'"^ r^2-3 -'+' ■ Or, . na^ — y--^,-^D,+ 1 .2-3 D=+. etc., which is the sum of n terms of the proposed series. 1. Find the sum of n terms of the odd numbers 1, 3, 5, 7, 9, ... Here, a=l, Di=2, D2=0; hence. Sum =na+!^<^^'D,=nxl+^^^^^X2=n+"2-n=n2. 2. Find the sum of n terms of the series V, 2^, 3^, 4' 5' -z ^ 'J , .... 298 RAY'S ALGEBRA, SECOND LOOK. Here, a=l, Di=3, ©2=2, D3=0; hence, n{n—l) n{n~l){n—2\ 3w(ra— 1) Sum =na+ '^ ^ Di+ 1.2.3 ^ 2=n+ — ^ -^ M (n -1 ) (»— 2) _ w(w+l ){2n+l) ■^ 3 6 ■ 3. Find the sum of m terms of the series l-|-3-|-6-f 10 + 15, etc. ^^^ n{n+l){n+2) 6 4. Find the sum of 20 terms of the series 3-fll-f 31 +69+131, etc. Ans. 443S0. 5. Find the sum of 20 terms of the series 1- 2- 3-1-2 3 • 4+3 • 4 • 5+, etc. Ans. 53130. 6. Find the sum of n terms of the series of cube num- bers l»+2>+3»+, etc. Ans. [4»(„+l)]^ 7. Find the sum of 25 terms of the series whose «'" term is n^(3«— 2). Ans. 305825. PILING OF CANNON BALLS AND SHELLS. 338. Balls and shells are usually piled by horizontal courses, either in the form of a pyramid or a wedge ; the base being either an equilateral triangle, or a square, or a rectangle. In the triangle and square, the pile termi- nates in a single ball, but in the rectangle it finishes in a ridge, or single row of balls. 330. To find the number of balls in a triangular pile. A triangular pile, as V — ABC, is formed V of successive horizontal courses of the form of an equilateral triangle, the number on each side decreasing continually by unity from the bottom to the single ball at the top. SEKIES— PILING OF BALLS. 29S> If wo commence at the top, the number of balls in the respective courses will be as follows: 1''. Z'i. 3''. 4'*. S'*. •••• ••••• •• ••• ••• •••• • • •• •• •mo • • •• and so on. Hence, the number of balls in the respective courses is 1, 1-1-2, 1-1-24-3, 1-I-2-I-3-I-4, l+2+3+4-|-5, and so ou; or, 13 6 10 15 Hence, to find the number of balls in a triangular pile, is to find the sum of the series 1, 3, 6, 10, 15, etc., to as many terms (n) as there are balls in one side of the lowest course. By applying the formula (Art. 327) to finding the sum of n terms of the series 1, 3, 6, 10, etc., we have a=l, Di=2, 0^=1, and D3=0. , ^ , nln—i)^ , n(n— l)(n-2)^ . Hence, the formula na-\- ^ — ?^^^^-\ — ^ — ir^-g — ^2 gi^es n3_3n2+2« «(n-^l)(w+2) n+n^—n+ ^ — = ^ (A) 330. To find the nnviber of balls in a square pile. A square pile, as V — EFH, is formed 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: I". 2'*. S'l. 4'*. ••• •••• •• ••• • ••• •• ••• •••• •••• 300 RAY'S ALGEBRA, SECOND BOOK. and so on. Hence, the number of balls in the respective courses is 1=, 2-, 32, 42, 52, etc., or 1, 4, 9, 16, 25, and so on. Therefore, to find the number of balls in a square pile, is to find the sum of the squares of 1, 2, 3, etc., to as many terms (71) as there are balls in one side of the lowest course. But this sum (Ex. 2, pp. 297, 298) is nfn-|-l) (2n-|-l) (B) 331. To find the number of balk in a rectangular pile. A rectangular pile, as EFD BCA, is formed of successive rectangular courses, the num- ber of balls in each of the sides decreasing by unity from the bottom to the single row at the top. i«ll^^reV\%V«A^ If we commence at the top, the number of balls in the breadth of the successive rows is 1, 2, 8, and so on. Also, if m^l denotes the number of balls in the top row, the number in the length of the second row will be »i-(-2, in the third, »?i-|-3, and so on. Hence, the number in the respective courses, commencing with the top, will be l(m-|-l), 2(m+2), 3(m-|-3j, and in the n"' course n(m-|-n). Or, S=l(m+l)+2(TO+2)+8(TO-f8)+ . . J^n[m^n) =OT(1 +2+3+4 . +n)+(12+22+32+42+ +n2); but the sum of n terms of the series in the two parenthoses (Arts. -1) — . Hence, 827, 330,) is "i!^\ and "i!^ 6 mn{n^\) n(n+l)(2n+l) n{n+l) (3m+2ra+l) (C). Here, wi+n represents the number of balls in the length of the lowest course. If we put TO+n=?, we have 3m+2n=8? — n; sub- stituting this for 8771+2/1, in (C), we have SERIES— PILING OF BALLS. 301 It is evident that the number of courses in a triangular or square pile is equal to the number of balls in one side of the base course, and in the rectangular pile to the number of balls in the breadth of the base course. ,332S. Collecting together the results of the three pre- ceding articles, we have I'or the number of balls in a Triangular pile -,n{n-\-l)(n+2) .... (A); Square pile ^n(n+l'){2n+l) .... (B); Rectangular pile 77n(H-i-l)(3? — »+!) • . • (C). In (A) and (B), n denotes the number of courses, or number of balls in the base course. In (C), n denotes the number in the breadth, and I the number in the length, of the base course. The number of balls in an incomple/e pile is evidently found by subtracting the number in the pile which is wanting at the top, from the whole pile considered as com- plete. 1. Find the number of balls in a triangular pile of 15 courses. Ana. 680. Here, n=15. Substituting this value in (A), we find the number ]5(15+l)(15+2) _15Xl6Xl7_^sn Ans ~ 2x3 6 2. Find the number of balls in an incomplete triangular pile of 15 courses, having 21 balls in the upper course. From the illustrations in Art. 329, it is evident that the number of balls in one side of the upper course is 6; therefore, 5 courses have been removed from the pile. From formula (A), we find that the pile, considered as complete, would contain 1540 balls, and that the removed pile contains 35. Hence, 1540—35=1505, the number left. 302 RAYS ALGEBRA, SECOND BOOK. 3. Find the number of balls in a square pile of 15 courses. Ans. 1240. 4. Find the number of balls in a rectangular pile, the length and breadth of the base containing 52 and 34 balls respectively. Ans. 24395. 5. Find the number of balls in an incomplete triangular pile, a side of the base course having 25 balls, and a side of the top 13. Ans. 2561. 6. How many balls in an incomplete triangular pile of 15 courses, haviog 38 balls in a side of the base? Ans. Y580. 7. Find the number of balls in an incomplete square pile, a side of the base course having 44 balls, and a side of the top 22. Ans. 26059. 8. The whole number of balls in the base and top courses of a square pile are 1521 and 169 respectively; how many are in the incomplete pile? Ans. 19890. 9. The number of balls in a complete rectangular pile of 20 courses is 6440 ; how many balls are in its base? Ans. 740. 10 The number of balls in a triangular pile is to the number in a square pile having the same number of balls in the side of the base as 6 to 11 ; required the number in each pile. Ans. 816, and 1496. 11. How many balls are in an incomplete rectangular pile of 8 courses, having 36 balls in the longer side, and ll in the shorter side of the upper course? Ans. 6520. INTERPOLATION OF SERIES. 333. Interpolation is the process of finding interme- diate numbers in mathematical, astronomical, or other tables. Its object is to furnish a shorter method of com- pleting such tables when portions of them have been cal- culated by formula. INTERPOLATION OF SERIES. 303 Thus, if the logarithms of 5, 6, and 8, are respectively 0.6989, 0.7'782, and 0.9031, it may be required from these data to find the logarithm of 7. The latter numbers are sometimes called functions of the former, and the former arguments of the functions. As the functions constitute a series, the principle upon which interpolation is founded is explained in Art. 326; that is, certain terms of a series being known, it is re- quired to find the «"■ term. Three cases may arise, which we will now consider. Case I. — When the differences of the functions are pro- portional, or nearly proportional, to the diiferences of the arguments, or the functions are in arithmetical progres- sion. Ex. — Given the Dip of the Sea Horizon at the heights of 86, 89, 92, 95, and 98 feet, viz., 9'08", 9'17", 9'26", 9'36", and 9'45"; required that of 101 feet. Ans. 9'54". Here, the first differences being 9'', or nearly so, we add 9" to 9^45" for the Dip at 101 feet. In all practical examples, there is no common first difference, and it becomes necessary to employ the second, third, etc., differ- ences. If in the series composing the functions, we can obtain an order of differences equal to zero, the interpolation will be exact. In most cases, however, Dj, Dj, etc., do not vanish, but become so small after Dj or Dj that they may be omitted without sensible error. 334. Case II. — When the differences of the functions are not proportional to the differences of the arguments, and the term to be interpolated is one of the equidistant functions. Ex.— Given ^25=2.92401,^26=2 96249, fW=S, ^"29=3.07231, to find the cube root of 28. 304 RAY'S ALGEBRA, SECOND BOOK. la such examples, "when three quantities are given, we may sup- pose D3 to vanish or become very small. We then have (Art. 326) the equation — a-{-3b — 3c-(-£^=0, and any of the quantities a, b, 0, or d, may be found, when the other three are given. Similarly, if the fourth diiferenoes vanish, theu a_46-(-6c— 4rf+e=0. In the above example, four quantities are given to find a fifth; therefore, we have a — 46-(-6c — 4d-\-e=0, where d is the term to be interpolated; hence, 4d = a-|-6c+e — 46 =2.92401 + 18+3.07231 -11.84996=12.14636, where d, or ,^28=3.03669, which is true to .00001. 333. Case III. — When the differences are as in Case 2d, and the term to be interpolated is inlermediate to any two of the functions. Ex.— Having given the logarithms of 102, 103, 104, and 105, let it be required to find the logarithm of 103.55. Taking the formula. Art. 326, put p to represent the distance, in intervals, of the required term (t) from a, the first term of the series, in which c.'\sep=n — 1, since the number o{ intervals is one less than the number of terms. Then, t=a+pV,+Pl^^,+PAP^l^\+, etc. The intervals between the given numbers is always to be consid- ered as until/, and p is to be reckoned in parts of this interval; hence, p will be fractional. Sufficient accur.acy is generally obtained by making use of D^ and D2 only, in the above formula. 1q practice, however, the following is generally adopted: Take the i'^o 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 theu have from the above formula, t=b+p'(d^^szLd'), INFINITE SERIES. Applying this formula to the above example, we have 305 Nos. LogarithmB. 1st Diff. 2d DifT. Mean of 2d Ilifl'. 102 103 104 105 2,0086002 2.0128372 2.0170333 2.0211893 42370 41961 41560 -409 -401 —405 Here, p'=.55, d=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, tlie 1'', 3'', 4"*, and 5"" terms being 3, 15, 30, 55 ; and find the 6'", 7'*, and 8'* terms. Ans. T ; 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"', B"", 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 RAYS 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+^+J+H-..,+3i, + 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, 1-1-2+4+8+16+32+, etc. An Ascending Series is one in which the powers of the leading quantity continually increase ; as, a-\-hx-\-cx^-\-da?-\-. A Descending Series is one in which the powers of the leading quantity continually diminish ; as, a-\-hx'^'^-\-cx~''-\-dx~^-\-, or a-\ h— 7 + -T+- X x' of 337. 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 Indetirminate Coefficients, Arts. 315-7; and, 4th. By the Binomial Thtorcm, 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 -^- + — -|--— ]- the general a 1 1- o 4 term is — , because by making a;=l, 2, 3, etc., each term of the scries is found. Again, in the series 2 ■ 2+2 • 3+2 • 4+2 -5+ the general term is 2(.r+l). INFINITE SERIES. 3O7 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 =; — rs+r; — j I I 2 • 3 ' 3 • 4 + 4^ + 5^+' ''°- Then, 1+^+^+^+, etc., = S-J. Subtracting 2V3 + 3T4 + 4T5 + 5T6+' «t<=v = h Ans. Ex. 2. — Find the sum of the infinite series = — 5 + — e -| -| 1 o o -, etc. ' 5 -7^7 • 9 ^""^ T+H3+4+. «*«•' = S; Tten, 1+1+^+^+ etc., = S -1. 2 2 2 11 Subtracting ^-^ + ^-^ + ^_^+, etc, = 1, and j-^ + 3— ^ + 577+' ^^'^■' = i' ^°^- 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-\-p)' able and p constant. Since ?__?L=^?- ... g =l{g_-j_l n n-\-p n(n-\-p) n[n-\-p) p ^n n-\-p i From which we derive the following 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 11 11 -\-p the result Ijy p for the sum of the series. 1. Kequired the sum of the series ^j — n-j- 6 — r + ^= — s + ! etc., ad infinitum, that is, to infinity. Here, . . q=\, p=2, and n=:l, 3, 5, 7, etc. (fill Put . — , == — 5 + = — s+^-=+, etc. n(n-lp) 1 -3^3 ■ 5 ' 5 • 7^' Then, ?=l+l+^+^+, etc., ad inf. And . -^^=l+l-rH. <=*<=-. '^'li'^f- Subtracting, = — — ■ — -^1 ^' n n-\P 'H"+PJ ? 1 <• ■ — ; — -, — ; = ';^ sum oi given seneis. w(n+P) - The sum of n terms of the same series is found in a manner nearly similar. Thus, ri=i+HU-^ 2^1 ^ 1 i_14-l 1 7(_|_p^3T^sT?- • • • 2;j-1^2k+1 ? 9 __ PI ^l ^ ^ -" and g ^ " Ans n 71+p n(n+p) '2n+l 2n+l n{n-\^p} 2n-}-V 111 2. Find the sum of the series ■p7-Q + £77-0 +o—i+, etc., ad infinitum. Here, q=l, p=l, and n=l, 2, 3, etc. Ans, 1. 3. Find the sum of the above series to n terms. Ans. — pj-. KECtJRRING SERIES. 309 4. Find the sum of the series = — -. -\- ,-, — ^ 4- ^ 1 ^ 1 -4^2 -5^3 -6 '4 • 7 +, etc., ad infinitum. Here, 2=1, and p=3, n=l, 2, 3, etc. Ans. j^. 5. Find the sum of the series :f—= + j,— -(-g-_-j- etc., J. * o ^ ■ 4 o " 5 ad infinitum. Here, 3=1, p=2, and )i=l, 2, 3, etc. Ans. ,J. J. w0(+4) 6. Find the series whose general term is — ; — ;— p, ; also find its sum continued to infinity. 1,1,1,1,, 25 48" A. Series =p-g + ^-^ + g^ -|. ^4., etc., sum = The sums of series may often be found by reducing them, by multiplication or division, to the forms already known. Thus, 7. Find the sum of the series I + s + J+ts+i ®'°-i ^^ infinitum. Ans. 2. Divide by 2 and compare with example 2d. 8. Find the sum of the series t: — ^,4-7; — 5-^ + ?^ — =.-7-.+, 00 0-12 9 1b etc., ad infinitum. (Multiply by 3-4). Ans. J;,- Remark. — There are other methods for the summation of cer- tain classes of series, but they are too complex and extensive for an elementary work. RECURRING SERIES. 339. A Recurring Series is a series so constituted that every term is connected with one or more of the terms which precede it by an invariable law, usually dependent on the operations of addition, subtraction, etc. Thus, in the series l+2a;+3x2-)-5x3+8x*+13a;5 f 2]a:'"'+, etc., the sum of the coefficients of any two consecutive terms is equal 310 RAY'S ALGEBRA, SECOND BOOK. to the coefficient of the next following term; and, by means of tills relation between the coefficients, the series may be extended to any desired number of terms. 340. The particular relation, by means of which the coefficient of any term of the series may be found when the preceding coefficients are known, is called the scale of the coefficients. It is easily seen that it is sufficient to find the successive coefficients in order to determine the series, inasmuch as the desired powers of the variable may be supplied as wanted. Recurring series are of the/?-si order, secoiid order, etc., according to the number of terms in the scale. b b' W Thus, in the series 1 x^ — -,x- -x\ etc., the coefficient of a a- a" ' each term after the first is equal to the preceding coefficient multi- plied by , and the series is said to be of the first order. This, the simplest form of a recurring series, is obviously a series in Geometrical Progression. 341. To find the scale of the coefficients of a recurriiig series. When the series is of the first order, the scale is easily determined, being the ratio of any two consecutive coef- ficients. (Art. 295.) Wlien the series is of the second order, the law of the series depends on two terms, and the scale consists of two parts. Let J) and q represent the two terms of the scale of the coefficients of the recurring series, A+'Bx+C.>:-+'D.r'+Ex^+¥x\etc., RECURRING SERIES. 311 Then, by tlie assumed law of the sei'ies : C=Bp+Aq; (1) D=Cp+liq; (2) E=I>pfCg; etc. (3) The values of p and q may be found by eliminating between any two of these equations. Taking the first two, (Art. 158.) EC— AD , BD— C2 Ex. — Find the scale of the series l-|-2a;+3a;''4-4x'-|-5a;*, etc. Here, A=l, B=2, C=3, D=4, etc. 2X3—1X4 „ , 2X4—32 , ■• P= 2^-1X3 ^^ =^"'^«=25=IX3=-1- Now, by the use of the scale, we may extend the series as far as we please : the 5th coefficient =pX the 4th-|-g'X the 3d^2X4 — 3 =5; the 6th coefficient=2X5— 4=6; the 7th=2X6— 5=7, and as the ascending powers of X are wanted, we have 6x' for the 6th term, 7x^ for the 7th, etc. 34S. In a recurring series of the third order, the law of the series involves three terras, which we will represent by p, q, and r, the series being A-{-Bx-\-Cx^-\-Da?-\-'Ex''-\- Fa^^+Gx*, etc. Then, by the law of the series, D=Cp+Bg+A>-; E=Dp+Cg+Br; F=Ep+Dg+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 mamj terms; for in that ease every term of the scale will take the form f 343. To find the sum of an infinite recurring series whose scale of relation is known. Let A+B.T+C.i:'+D.i;''+E.r', etc., be a recurring series of the second order, p and q being the terms of the scale. Then, . . . A=A; Ba;=Ba;; D.r'=C)W'-f-Bg,r''; etc., ad infinitum. Represent by S the required sum, and add together the corre- sponding members of the preceding equations, observing that Bx-f- Cr-'+Dx^+j etc., =S — A ; then, we have S=A+Bz+(S— A) 2->x+Sqx^ ; .-. S— Sp.-c— Sga;2=A+Ba;— A^AC; Or, ... . s= ^+S'-A;y 1 — px — qx- If we make q=0, (remembering that B=Ap), the formula be- comes S=; , which is, as it ought to be, identical with the 1 — pK' ' ^ ' formula of Art 299. Rejiark. — Every recurring series may be supposed to arise from tlie development of a rational fraction, and tlie summation of sncli a series niny 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 + 3a;+5.r+7.c'+9.i;*, etc. Here, A=l, B=3, C=5, D=7, etc. And, hence, (Art. 341.) p=2, g= — 1. „, „ A-LB)-^A;).r 1+ar— 2.r l+.r J. hen, >5 = ^, —= ; — — T^n ^• ' 1— p.c — (jx- 1 — -Ix+x- (1—xr EEVERSION OF SERIES. 3 13 In each of the following series, find the scale of rela- tion, and the sum (S) of an infinite number of terms : 2. l + 6aj+12x'+48x»-fl20a;*+, etc. Ans.p=l,,=6;S=jl±^, 3. l+2a;+3a;^+4a:'+5a;'+6a;5+, etc. Ans. p=2, 2=-l ; S=--J_. . a abx , abV ah'a? *• c-^+-? ?-+.«*«• Ans. The series is in G. P. pz= • S=r c c+to 5. x-\-x'-\-3?-\-, etc. X Ans. The series is in G. P. p=l ; 8- ~l—x 6. X — x'-\-a^ — .i;*-(-, etc. Ans. The series is in G. P. »= — 1 ; S= ^ . l-\-x 7. l+Sx+bx''+7x'+9x*+, etc. 1+x Ans. J3=2, gz= — 1, S= l—2x-\-x'' 8. l^+2^x+3V+4V+5V+6V+, etc S— -(1-x) Ans. p=3, 3=-3, r=l ; S=^i±^,. EEVERSION OF SERIES. 344. To Eevert 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 1/ by a series involving the powers of X ; thus, y^ax-\-bx^-\-ca:?-\-da^-\-, 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 RAYS ALGEBRA, SECOND BOOK. containing the known quantities a, b, c, d, etc., and the powers of y. To revolt this series, assume x^Ay-\-By--\-Cy^-\-T)y*, etc. (2), in •which the coefficients A, B, C . . . aie undetermined. Find the values of y^, ■ifi, y^ . . . from (1), thus, y-=a''-x'^-^2abx^-\-{Jf-+1ai:)X^-\- . . . y^= a^x"+Ba^l)z*+ .... 3/"= a'x'-i- .... etc. Substituting these Talues in (2), and arranging, we have 0=Aa -1 x+Ab Ba- x^-\~ Ac +2Ba6 + Ca3 x^-\- Ad I a;^+, etc. + Bi^l + 2Bac + 3Ca26 I Since this is true, whatever be the value of x, and the coefficients of X, x^, z^, etc., will each =0, (Art. 314, Cor.), we have Aa— 1 a b =0, . .aJ- a =0, . . B=- =0, . r= A6+Ba^ ~, -^, 2,ir-—ric Ac+2Ba6+Ca3 =0, . . r= ^- - -, - , Arf+Bd2+2Bac+3Ca26+Da<=0, .-. Li=— ^^' ^' ^ ' "/^' '' "*' 1 6 , 2b--ac , a2rf-5aie-f56-' , , Hence, ^^-y^-y^-+ ^, y3 _ »/i + , etc. (..) 345. If the given series has a constant term prefixed, thus, y=a'-\~ax-\-bx''-\-cx'-^dx*-{- assume y — a'=^z. and we have ::=a:r-if-hx^-\-cce'-\-dx*-\-, etc. But this is the same as (1) in the preceding article, except that e stands in the place of y; hence, if z 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 2, the result is =^=jXy-<»-')- ■zAy-0''?+ -^^^{y-o.'f- etc. 346. When the given series contains the odd powers of a;, assume for ,r another series containing the odd powers of y. Thus, if ^=:aa;-j-6a;'-(-ca;°-f-da;'-j- to develope x in terms of)/, assume a=A)/+B/+C/+D/+ ... . Then, by substituting the values of y, y^, etc., derived from the former equation, in the latter, and equating the coefficients to zero, we find 1 6 362— ac aM—8abc~\Vlb' , ''=ay-^^+—^r-y ^0 y'+, «'«■ If both sides of the equation be expressed in a series, as '^y-\-by^+C}fi+i etc^ =a'x-\-¥x^-\-c'x^-\-, etc., find it be required to find y in terms of X, we must assume, as before, j/=Aa;+ft);2-(-Ca;''+D2;<+, 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, h, c, 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 y:^x — x'-fx'- — x'-\- ... to find the value of X in terms of y. Ans. x^:^r/-^y''-\-y'-\~y*-\-, etc. Find the value of rr, in an infinite series in terms of y: 2. When y^x-j-^'+^+i ^tc. Ans. x=y— y+y*— y+/— , etc. 316 RAYS ALGEBRA, SECOND BOOK. 3. When ?/=2a;+3a:'+4a^-|-5a;'+, etc. Ans. x=iy-J^f-\-J^^f—, etc. 4. When y=^l—2x-{-Zx\ Ans. x=-J-(3/-1)+-^G'-1)'-/b6'-1)'+, etc. 5. When y=a;+^a;^+ia:'+3,ijx'+, etc. Ans. a;=y— i/+J^_^y+, etc. 6. When y-(-rt/-f J/-(-f^ . . . =gx-\-hx'-{-kx'^lx'' . XI. CONTINUED FRACTIONS: LOGARITHMS: EXPONENTIAL EQUATIONS: INTEREST, AND ANNUITIES. CONTINUED FRACTIONS. 347. A Continued Fraction is one whose denomina- tor is conticued by being itself a mixi'd ntcniher, and the denominator of the fractional part again continued as be- fore, and so on ; thus, 1 1 1 1' , 1 ' ,1 a+r "H 1- «+ b "^, ,1 "^, , 1 c '+d in which a, b, c, d, etc., are positive whole numbers. Continued fractions are useful in approximating to the values of ratios expressed by large nwmbers, in resolving exponential equations, indeterminate equations of the first degree, etc. CONTINUED FRACTIONS. 317 , 348. To express a rational fraction in the form of a continued fraction. 30 Let it be required to reduce ytk to a continued fraction. If we divide both terms of the fraction by the numerator, we ^+30 7 1 If we omit wk, the denominator will be too small, and =, the value oU o of the fraction, will be too large. 7 Again, if we divide both terms of the fraction ^ by the numer- , , 30 1 ator, we find i-^=^ , ■ 2 14 If we omit =, the value will be expressed by =9f' 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 by stopping at an even reductixm, and neglecting the fractional part, the result is too small. 2 1 Since - = =, we find ' o+l 30 1 . =. 1»< reduction, too great ; 1^^ 5_|_?^ 2con shown, a b c' ' ^ R _ Qc+P CONTINUED FRACTIONS. 319 1 S Let us now take the next integral fraction -j, and let — express the 4"' converging fraction. Then, it is obvious that ^^ will become S 1 ^ by substituting c-f-^, instead of C; hence, q(c4) + p . S _ ^\ ^dl^ _» (Q c+P ) d+Q _ Rc?-|-Q Q'(c+^)+P From this we see that the same rule applies to the 4'* converging fraction, and so on. Hence, for the n'* converging fraction, Multiply 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 n'* 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 2%- The integral fractions are ^, |, i, i, i, |, i. The converging fractions are ^, J, |, /,;, if, JJ-^, ^%\. 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 RAYS ALGEBRA, SECOND BOOK. By a process of reasoning similar to that employed in Art. 850, it may be shown, in a general manner, that The difference between any tico consecutive converging frac- tions is always a fraction having -\-l, 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. 33S. To show that every converging fraction is in its lowest terms; and to find the approximate value of the frac- a tion r- If — and ^ be any two consecutive converging fractions, by Art. 351, -_-=+_ or -— ; that is, AD-BC=±1. Now, if A and B have a common divisor greater tlian 1, it will divide their multiples AD and BC, and their difference ±1, (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 convergenta 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. iS>>3. To express |/N, when N^a^+1, in the form of a continued fraction. ^■'^^l=a+l/;iq^-a=a+;^^^^^^ (Art. 206), 1 1 -=:a-( =a-| a+«+l '«'+l— « 0,1 z,a-\ y Ex. |/r7=i/4-''+l=4 CONTINUED FRACTIONS. 1 321 + 8-|-y-(-, etc., the converg- o -I Q /> r ing fractions to be added to 4, are „, 75-=, cTrgi etc. o 00 o^o 3S4. To convert |/N, where N^a^-j-t, into a con- tinued fraction. Ex. Convert |/ 19, or ^16-)-3, into a continued frac- tion. 1 /19=4+-. Hence, 1 ~v/19-4 3 /19+4 „, 1 3—^+6- Hence, 3£y29±^) _ •19+2 _. , 1 ^19—2^ 15 ~ 5 ~ '^c ^__^_ 5(T/19 +3)_va9+3 1 ^^19-3^ 10 2 "' d' 1 . /19=4+ 2+- l+g+, etc. Proceeding in the same manner, the successive values of a, 6, C, d, 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 i, S, 'J, 4S, §1, etc. <$S3« 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 RAYS ALGEBRA, SECOND BOOK. Ex. 1. — Let ^= ',' o_| , etc., ad infinitum. ^, I r4-x Then, . . . , ^x, or ' -=x. . 1 qr+qx+1 From this equation, the value of X is easily found. 3S6. To find in the form of a conflnnud fraction tlie value of X, icliirh satisfies the iqua/iun ;i^=b. Ex. — Required the value of a; in the equation 10^=^2. By substituting and 1 for x, it appears that x^O and <^1. 1 1 Let . . x=-; then, 10-^=2, or ^^^lO. y Since 23^8, and 2^^16, one of which is less and the other greater than ]0; therefore, 2/>3, and <^4 ; let y=3-\ — ; Then, 2=+==10; Or, . . . 2-.2==10, or 25=. 1^=1.25; Therefore, (1.25)^=^2. Again, it appears that 8>3, and <4; let z=3-f- ; then, W (].25)3+"=.(125)3(1.25r=2 .. (1.25)"= -.j-^=1.024; Therefore, (1.024)"^ 1.25, and by trial M>9 and <10. 1_ 3+- Hence, . ... x^ ^ 3+i+, etc This gives x=l— , Jj-(-, 2|_— 30107 nearly, etc. CONTINUED FRACTIONS. 323 Reduce each of the following to a continued fraction, and find the successive integral and converging fractions : 1. 2. 3. 130 Ans. Integral fractions I, |, i, 1. 421 Convercino- " J _4_ aj jso 130 Ans. Integral fractions i, J, i, 1. 2yl Converging " I |, fi, isfi. 157 Ans. Integral fractions J, i, J, J, i. 1 5 2_1 6 8 1 i) 7 Bl 3 1' 1 3U' 4;i 1! "STS- 972 ,Coc 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 I ' -5- Ifi S-T 90 T27 3900 -^"°- i! 31 J4' 4 5' 104' 2631 35=?' TOSes" 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. 1, |, if, -||, -j^j, etc. 6. When the diameter of a circle is 1, the circumfer- ence is found to be greater than 3.141592(3, and less than 3.1415927 ; required the series of fractions converg- ing to the ratio of the circumference to the diameter. Ans. i, 3^, JOfi, and il|. Show that this last ratio, J4i, is true to within less than three ten-millionths of the circumference. Suggestion. — In examples of this kind the integral fractions, corresponding lo both fractions, sliould be found, and llien 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. Ana 1 7 8 3 1 3 9 6_o_5_ 6 9_4 13 4 9 2 929 -aua. ^, -^-g, 33, y^g, , g , , 37041 JIB d'S' o5g91 t)B40U- 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 eighth until the fifth year, instead of having it on the fourth. 8. Find the least fraction with only two figures in each term, jipproximating to l^il- Ans. ji. 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. -y-, -3-, -Trg-, -J43 , eic. 10. Find a series of fractions converging to y 2. An^ ' 3 7 17 41 ote -'^Q*- T) 21 5' T2) U'j' ^''^■ 11. Show that I 5 is > -58|, and < f6|^. 12. If 8'=32, find a; Ans. =. 13. If 3^=^15, find x Ans. 2.465. LOGARITHMS 357. This method of computation was invented by Lord Kapier, but subsequently much improved by Mr. Henry 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. Thus, a^X «'=«', ^1=«S (a'y=a'', T^'^=«'. If some number, arbitrarily assumed, be taken as a base, then The Logarithm of any numher is the exponent of that power of the base, tvhich 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-'-^N', a-^=W, then, 2, 3, nnd 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)''=1 (10)1=10 (10)2=100 (10)3=1000 (10)*=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 betvfecn 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. 358. 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 mimhcr. Thus, the logarithm of 123 is 2 plus a fraction ; the logarithm of 1234 is 3 plus a fraction, and so on. 359. 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. 1 Log. X. Log. N. Lug. X. Log. 0.000000 26 1,414973 51 1.707570 76 1.880814 2 0.. 301030 27 1.431364 52 1.716003 77 1.886491 ■i 0.477121 28 1.447158 53 1.724276 78 1.892095 4 0.602060 29 1.462398 54 1.732394 79 1.897627 6 0.698970 30 31 1.477121 55 1.740363 80 1.903090 6 0.778151 1.491362 56 1.748188 81 1.908485 7 0.845098 32 1.5115150 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 59 1.770852 84 1.924279 10 I.OOOIIOO 35 36 1.544068 60 1.778151 85 86 1.929419 11 1.041393 1.556303 61 1.786330 1.934498 12 1.1179181 37 1.568202 62 1.792392 87 1.939519 13 1.113913 .i8 1.579784 63 1.799341 88 1.944483 14 1.141)128 39 1.591065 64 1.8116180 89 1.949390 15 1.1751)91 1 41 1.60206 1 65 1.812913 90 91 1.954243 16 1.2114120 1.612784 66 1.819544 1.959041 17 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.2787.54 44 1.643453 69 1.838849 94 1.973128 20 1.301030 45 1.653213 70 71 1.S45098 95 1.977724 21 1.322219 46 1.662758 1.861258 96 1,982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 n 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.3S02U 49 1.690196 74 1,869232 99 1.995635 2j 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. S60. 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), 105=100000, 10^=100, a-'=N .(1), LOGAKITHMS. 327 Multiplying equations (1) and (2) together, we find 10" =10000000, =a^+-''=NN'. But, by the definition of logarithms, 7 and x-\-x' are the loga- rithms of 10000000 and of NN' respectively. Hence, TTie SHTH of the logarithms of two mimhers is equal to the logarithm of their prodxict. 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 numbers for the logarithm, of the product. 361. Taking the same equations, (Art. 360), we have 10^^100000, a-=N . . . (1), 10-^=100, a-'^N' .... (2> Dividing equation (1) by equation (2), we find 103=1000, .... a--^'=^. W But, by the definition of logarithms, 3 and x—x' are the loga- N rithms of 1000 and of j^. Hence, to divide by means of logaritlims. Rule. — From the logarithm of the dividend subtract 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 . . 954243 " " the log. of 6 is . . 0.778151 The sum of these logarithms is . . 1.732394 and tlie 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 3. Find the product of V 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. 363. Resuming equation (1), (Art. 360), we have 10^=100 a'^y. Raising both sides to the 3d and to the m"' power, we find 10'''=1000000, a"".-=N". But, (Art. 367), 6 and mx are the logarithms of 1000000 and of N'" respectively Hence, (o raise a number to any power by means of logarithms, Rule. — Multiply the logarithm of the given number hy the cxjHincnt of the required poicer Jor the logarithm of the poiLcr of the number. 363. Take the same equation 10^=1000000, a'=N. Extracting the 3d and n'* root of both sides, we have X 1 10- =100, . . . a"=N". X 1 But I.Kvl. 357), 2 and - are the logarithms of 100 and of N" ^ n respectively. Hence, to extract any root of a number, Bule. — Divide the logarithm of the given number by the indix 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 V. 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. 3G4. By means of negative exponents, wo can also ex- press the logarithms of fractions less than 1. Thus, in the common system, since (10)-'=Jj5 =1 , therefore, —1 is the log. of .1 ; (10)-'=To'uo =-001 • " Etc., The logarithm of any fraction between one and one-tenth, between one-tenth and one-hundredth, etc., may be expressed thus. Log. (J^)= log. (JbX7)= log. ^5+ log. 7=-l+ log. 7. Log- (i!5)= log- (fkX3)= log- -,U+ log. 3=-2+ log. 3. 2d Bk. 28 —2 ' ' log. of .01 ; —3 ' ' log. of .001 ; —4 ' ' log. of .0001 ; Etc. 330 RAY'S ALGEBRA, SECOND BOOK. It is customary not to perform tlie subtraction indicated, but to unite the logaritlim of tlie numerator to tlie negative cliaracteristic. Tlius, Log. 7 =—1+ log. 7=— 1.845098, or 1845098. Log. 0.03 =-2+ log. 3=— 2.477121,, or 2:477121. Log. 0.004=— 3+ log. 4=— 3.602060, or 3.602060. Hence, the cliaractcristic 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. 363. On the piinoiple above explained, we may deduce the following General Rule for finding from the Tables the Loga- rithms of any Decimal Fraction. — 1. Find the logarithm of the figures composing the decimal as if the fraction were a ivhole number. 2. Prefix the negative characteristic according to the rule given in Art. 364. 366. The following examples, illustrative of the prin- ciples already explained, will afford a useful exercise : 1. Log. (a . 6 . c.c?. . )= log. a-|- log. 6-f log. c-|- log. d. 2. Log. j — 1= log. a-\- log. 6-|- log. c — log. d — log. e. 3. Log. (a"*, b" . c'' . )=»i log. a-\-n log. b-\-p log. c. 4. Log. I I ^m log. a-\-n log. h — p log. c 5. Log. (a' — a:-)= log. \_(a-\-x)(a — a:)]= log. (o.-{-x)-\- log. (a— a;). 6. Log. |/a^ — x'^=:\ log. (a-\-x')-\-i, log. (a — x). LOGARITHMS. 331 V. Log. (a'Xlf^)=3| log. a. 8. Log. ^l-if'=ijlog. (a-a.)-3 log. («+x)j. 367> Let us resume the equation a^^N. 1st. If we make ar^l, we have a'^N^a ; hence, log. ^=1 ; that is, Wliatever he the base of the system, its logarithm in that system is 1. 2d. If we make x^Q, in the equation a*=N, we have o''=N=-l ; hence, log. 1^0 ; that is, Jn any system the logarithm of 1 is 0. 36S. In the equation a"^=N, consider a>l, as in the common and the Naperian systems, and x negative ; we then have a~'=--;=N, and -^^-=0'^ =0, or loor. 0= — oo . a" ' a Hence, the logarithm of 0, in a system whose base 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 ^han 1, the logarithm of is infinite and positive. 3I»9. As the positive and negative characteristics are taken to designate whole numbers and fractions, there re- mains no method of designating negative quantities hy logarithms ; or, as N, in each of the equations a''=:N and o"^^N, is positive, .Negative numbers have no real logarithms. 332 RAYS ALGEBRA, SECOND BOOK. 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 mimhers. For, the logarithm of every composite number is equal to the sum of the logarithms of its factors. Hence, the logarithms of 1, 2, 3, 5, 7, etc., being known, we can find those of 4, 6, 8, etc. Thus, 4=22 6=2x3 8=23 9=32 10=2x5 hence, log. 4=2 log. 2, (Art. 362); " log. 6= log. 2-j- log. 3; " log. 8=3 log. 2; " log. 9=2 log. 3; " 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 < ; and why? Ans. Of 23 different numbers, from 32 to 98. 371* In the common system, the equation a'^^N (Art. 357j becomes 10'=N. If we multiply both sides by 10, we have 10^><10=lO'+"=10N; Also, . . 10-'xlOO=10'XlO^=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 Briggs' system. Thus, the log. of 3 ia 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. 373. If we compare the difiFerent 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 numbers 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, the geometrical mean between 10 and 1000 is ^/lOXlOOO =100, and the arithmetical mean between their logarithms, 1 and 3, is (l+3)-r-2=2. In general, if a^=S, and a^'^S'; then, Log. of /NN' is ?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+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.— 1a.\e 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. 534 RAY'S ALGEBRA, SECOND BOOK. FourtUy.—Ta.ke 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 W (Art. 361). To find the log. of 3, take the numbers 2 and 8.162277, and their logarithms, and proceed as in finding the log. of 5. 373. Logarithmic Series. — The most convenient method of computing logarithms is by means of Series, which we shall now proceed to explain. Let a; be a number whose logarithm is to be expressed in a series, and let us apply the method of Indeterminate Coiifficients (Art. 314). If we assume log. x=A-|-Ba;-(-Ca;2+Da;^-|-, etc., and make ^=0, we have, log. 0:=A=co (Art. 368). Hence, GO :^A, which is absurd. If we assume log. x^AaJ-l-Bx^-l-Ccc^-l , etc., and make a;=0, we have log. 0=0; that is, (Art. 368), oo ^0, which is also absurd. Hence, it is impossible to develope the logarithm of a number in powers of that number. But if we assume Log. {l+x)=Ax+Bx'2-\~Cx^+Dx*+, etc . . (1) and m.akc .1=0, wc have log. 1=:0, which is correct (Art. 367). In like manner, also assume Log. {l+z)=Az+'Bz^-+Cz^+'^z^+, etc. ... (2) Subtracting equation (2) from (1) we get Log. (1-fa;)— log. {l+z)=A{x—z)+B{x-—z^) -|-C(a^5— 03)-f, etc. (3). LOGARITHMS. 335 The second member of this equation is divisible by a,"— 2 Ait. 83); let us reduce the first member to u form in which it shall also be divisible by the siime factor. By Art. 361, Log. {l+x}- log. (1+2)= log. {^l) = log. ( 1 + ^ ). Now, regarding = as a single quantity, we may assume /, x~z\ x—z ^1 X — z \2 I x—z \^' ^"s- (^+r+i)='S— .+Mi+i) +"( IT- )+''='" Substituting this for log. (1+a;) — log. (l-|-z), in equation (3), and dividing both sides by x — z, we obtain . 1 x—z „ (x—zV ^•i+.+^-a+5?+^-(T+iP+>''"=- =A+B(x+z)+C(x^+xz+z^)+, etc. Since this equation is true for all values of X and z, it must bo true when x^z. Making this supposition, we have A.J—- =A+2Ba;+3Ca;2+4Da;3+5Ea;<+, etc.; or, performing the division of 1 by l-{-x, wo have A{1 -x+x^—x^+x*—. . . . )=A4-2Ba:+3Ca;2H-4DxS+. Equating the coeificients of the like powers of X (Art. 314), A=A,B=4 0=1, B=-^. The law of this series is obvious, the coefBoient of the W' term beinff zt=— , according as n is odd or even. ^ n' * '\ A A Hence, log. (l+a;)=Aa; — 'j^x^-\-^x^ — T*'^+ • ■ • • X' x^ X* X-' x'' , ,,, =A(^-2+^-4 + 5"(r+ • • • ) (4) 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 syslem, so that when A is given, the base may be determined ; or, when the base is tnown, A may be determined. If we denote the series iu the parenthesis in equation (4) by x', we may write Log. {\-\-x)=A.x'. Hence, The logarithm of a number consists of (wo factors, one of 'vhich .depends on the 7iumbcr itself, and the other on the base of the system in which the logarithm is taken. That factor which depends 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 sj'stem resulting from such a modulus, is called the Nuperian, or Hyperbolic system. For all values of x above x=l the series (5) diverges, and is, there fure, inapplicable. Designating the logarithms in this system by log'., we have n. /V.2 .7*3 7*4 Log'- (l+^)=i-^-+|-^+. e'c- (5) Thus, if x=0, we find log'. 1=0, as in Art. 367. If we make a;^l, we have Log'. 2=l-i + l-l+i- etc. S74. 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), X x^ x^ X* x^ Log'. (l+a;)=j- ^- + __-^- + __, etc. . . (0). Substituting — X for x, in this equation, we obtain X X^ X^ x^ x^ Log'. (l-a;)=-j -- ,_, _-g _----__ etc. . (6). LOGARITHMS. 337 Subtracting equation (6) from (5), and observing that Log'. (1+a;) — log'. (1 — x)= log'. ( —^ 1, we have L''g'T=^=-'(l + y + ^ + T + ^+- • • )• Since 1±?^1 + ,-?^, letl±?=l+?, .. .=^, 1—x ' 1—x l—x ' z 2z-j-l and log'. 1±^= log'. ( 1+J )= log'. ( ?±^ ) = log'. (2+1) — log'. Z. By substitution, the preceding series becomes Log'. (.+1)- log'. ^=2(2^ + 3^2^3 + 5^^5+ • ■ } ; Log'. (^+l)=log'.2+2{2^ + 3^3 + g^2iT?+. .}(7). 373. 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. SSY) ; therefore, making z=l, 2, 4, 6, etc., we ob- tain the following NAPERIAN, OR HYPERBOLIC LOGARITHMS. Log'. 2=log'. l+2{| + ^+ ^33 + ^^^+ . . 1=0.693147 Log'. 3=log'. 2+2{J + 3-i33 + ^, + ^,4- . . 1=1.098612 Log'. 4=2. log. 2 =1.386294 Log'. 5=log'. 4+2{| + ^3 + ^, + ^-ig,+ . .}=1.609438 Log'. 6=log'. 2+ log'. 3 =1.791759 Log'.7=log'.6+2{j^ + 3^3 + 54,+ . . 1=1.945910 2d Bk. 29* 338 RAY'S 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 give^the result true to seven places of decimals. 376. To ixplain the method of computing common loga- rithms from Naperian logarithms. We have already found (Art. 373, Equation 4), / X X- x^' x< x'' ■ x<'' \ Log.(l+a:)=A(j-^ + ^-_ + ^--+. ..) Denoting the Naperian logarithm by an accent, wc have T , ,, , , , , / X x^ a;3 2-4 x'' x'- \ Log'.(l+.)=A'(j-^ + ___ + ___+. . ) Since the series in the second members are the same, "we have Log. (l+x) : log'. (1+x) : : A . A'. Therefore, Jlie logarithms of the same number, in two different sys- tems, are to each other as the moduli of those systems. But in Napier's system the modulus .\'^1. Therefore, Log. (l-fx)=..V log'. (i.-\-x). Hence, To find the common logarithm of any number, multiply the Naperian logarithm of the number by the modulus of the common system. It nove remains to find the modulus of the common system. From the equation, log. (l-|-a;)^A. log', (l+a^), log. (l^.r) We find A= , I, . - 7 . Hence, log'. (1+a- ■ LOGARITHMS. 339 The modulus of the common system is equal to the common log. of any num,ber divided by the Naperian log. of the same number. But the common logarithm of 10 is 1, and we have calculated the Naperian logarithm of 10, (Art. 375) ; therefore, log. 10 _ 1 _ 4342944 •which is the modulus of the common system. Hence, if N is any number, we have Com. log. N=.4342944X Nap. log. N. On account of the importance of the number A, its value has been calculated with great exactness. It is A=.434294481 90325182765. 377. To calculate the common logarithms of numbers directly. Having found the modulus of the common system, if we multiply both members of equation (7), Art. 374, by A, and recollect that AX Nap. log. N^ com. log. N, the series becomes Log. (3+1)= log. .+2A { 2^ + 3^2^ + 5-p^,+ • } • Or, by changing Z into P, for the sake of distinction, and putting B, C, D, etc., to represent the terms immediately preceding those in which they are used, we have Log. (P+l)= log. P+ JA_ + _^ , ^ 3C 2P+1 ^3(2P+1)2^5(2P+1)2 I 5D 7E 9F ^7(2P+1)2^9(2P+1)2 + 11(2P+1)2^' We shall now exemplify its use in finding the logarithm of 2. Here, P=l, and 2P+1=3. 340 RAYS ALGEBRA, SECOMD BOOK. Log. P = log 1 =.00000000; 2A ^.868.58896 =.28952965; (B.) .WfW -^? -"■«'===^' '^■' .-pSi, --^?-^ ■ ■ ■ ■ — ^ <») _»_ ^5x,~» ^,^,,, „,, sww -^^^IS?^ --««»> <'■) niiW -^"" -~«^ '"■' Therefore, common logarithm of 2 =:i-.301O2999. Exercise. — In a. similar manner let the pupil calculate the common logarithms of 3, 5, 7, and 11. For the results to 6 places of decimals, see the Table, page 326. 378. To find the base of the Napcrian system of loga- rithms. If we designate the base by e, we haye, (Art. 376), Log. e : log', r : : A . A'. But A=.4342944, A'=l, and log'. e=l, (Art. 367); hence, Log. c : 1 : : .4342944 : 1 ; whence, log. C=.4342944. Taking the number of which the logarithm is .4342944, from the table of common logarithms, we find 6^2.71828182. We 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. 341 379. The student may prove the following theoronis : 1. No :5ystein 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 diiference 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 Kule of Single Position is applied to the solution of questions which give rise to an equation of the form aaj=m (1). If we assume x' to be the value of a;, 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 in' : 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 numher to the number required. Example. — What the number, whose third, fourth, and sixth part being added, the sum will be 45 ? Ans. (JO. 3S1. 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 x" for X, we have ax' -\-bz=m^e' (2), ax"-{-b—ni+e" (3). If we subtract equation (1) from (2), and (3) from (2), we have a{x'—x )=e' (4), a{x'—x")^6'—e" (5). From these equations, we easily obtain JO »^ ■// tJj (6). e' — e" e' By subtracting equation (1) from (3), we also find a[x" — x)=e"j and thence, x' — x" x" — X e' — e (7). Hence, (Art. 263), The difference of the errors is to the difference of the two assumed numhers^ 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-Yh^m, 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. j equation, a;"-f-Aa;"-'+Ba;«-2-)-Cx"-3+. . . . +Ta;+V=0, (n), then will the equation he divisible hy x — a. For if Ct is one value of a:, the equation will be verified when a is substituted for X. This gives a"-f Aa"-' + B«"---fCa"-3-f. . . . -fTa+V=0; Or, V=— a"— Aa"-'— Ba"-=— Ca"-3— . . — Ta. Substituting this value of V in the given equation, and arranging the terms according to the same powers of x and a, wo have (a;"— a")+A(a;"-'— a"-i)+B(a;"-2_a"-2)+. . +T(x— a)=0. As (Art. 83) each of the expressions (a;" — a"), (a;"~' — c""'), etc., is divisible by X — a, the given equation is divisible by X — a. GENERAL THEORY OF EQUATIONS. 353 Corollary. — Conversely, if the equation a;"+Ax"->+Ba;"-^+. . . . -(-Ta-fV^O, (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 cdU tlie 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- mainderj call it R, and the quotient Q; then, equation (n) becomes (x— a)Q+R=0. But X — a=0, .•. R^O; that is, there is no remainder on dividing equation (M) by X — a. Illttstration. — In the equation a'-j-cc* — 14.t — 24^0, the roots are — 2, — 3, and 4 ; and the equation is divis- ible by x-\-2, x-|-3, and x — 4. 396. Proposition II. — An equation of the n"' degree has B roots. Let a be a root of the equation a;»+Aa;»-i-|-Ba;"-2-f Ca;''-'+. . . . -|-T3;+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 coeiEcients of the powers of X in the quotient, equation (n) becomes (a;— a)(a;"-i+AiX"-2+Bia;»-3+. . . . -(-Tia;+V,)=0. Hence, a;"-i+A,a;"-2+B,a;"-3_|_ ^Tia;-t^Vi=0. Now, this equation must also have a root, which may be denoted by 6, and is (Art. 395) divisible by X — 6. Hence, (a;— 6)(:c"-2+A2a;"-=-|-B2X''-'+. . . . -|-Toa:+V2)=0. Placing the second factor of this equation equal to zero, taking c, a third root, and dividing by X — c, we sliall 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 BU. 30 354 RAY S ALGEBRA, SECOND BOOK. (x—a)(x—b)(x—c){x—d), . . . (a;— ?)=0, which is satisfied by maliing x=a, b, c, d, . . . . or I; that is, the equation has n roots, a, b, 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;2+47a:— 60=0, is 5, and by dividing it by x—b, the quotient is x^—Tx-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 x^-\-ix—0, we have a:(a;-|-4)=0, whence a;=0, and a;=— 4. (See Art. 253.) 1. One root of the equation x^ — lla;''-(-23a---|-35=:0 is — 1 ; find the equation containing the remaining roots. Ans. a;'— 12,t;+35=0. 2. One root of the equation xf — 9x'-\-26x — 24=0 is 3 ; find the remaining roots. Ans. 2 and 4. 3. Two roots of the equation a;*-|-2a:^— 41a;*— 42a--f 360 =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^ — ^x'^-j-12x — 8=0, is the same as (X — 2)(a;— 2)(a;— 2)=0, or (x — 2)3^0. Hence, the three roots are x=2, x=2, and a;=2. 397. Proposition III. — A'o equation of the n* degree can liave more than n roots. If it be possible let the equation a-"-f Aa;''-i+Bx"-^4-Cx''-^+. . . . -f Ta;+V=0, GENERAL THEORY OF EQUATIONS. 355 besides the n roots, a, 6, c, d, etc., have another root, »■, not identical with either of the roots «, i, o, d, etc. ; then, the equation must be divisible by x — t (Art. 895) ; this gives a;"4-A2;»-i+Ba;"-2-)-, etc., =(a;— r)(a;"-i+A'a;"-2+, etc.,) or (x~a){x—b){x—c). . . (a:— ;)— (x— j-)(a;»-'+A'a;"-^+, etc.) But since r is a value of x, wc h»ve, by substitution, {r--a){r—b){r—e). . [r-l) =(»•— r)(a;"-'+A'a;"-2+, etc.) Now, the second member of this equation is =0, because {r — r)=0 ; but the other side can not be 0, since r is not equal to any of the quantities a, 6, c, etc.; hence, the supposition is absurd that X can have any value other than a, 6, c, d, . I. 398. Proposition IV. — To discover the relations between the coefficients of an equation and its roots. Let x^-.a, a:=6, a;=e, x=d, etc. I Then, By multiplying together the corresponding terms of the last set of equations, we have (x — a){x — b){x — e){x — d)=0. If we perform the actual multiplication of the factors, we find x-\-abcd ' =0. x'—a x3-\-ab x-—abc -6 -\-ac —abd — c -\-ad —aed —d +bc +bd +cd —bed Similarly, in the equation of the n"' degree, 2"+Aa;"-i+Ba;"-2+, etc., ={x—a){x—b){x—c). (x—l)=0. If we perform the multiplication of the n factors, we shall have — a — b — c, etc., =A; ab-\-ac-\-ad, etc., =B; — abc — abd — acd, etc., ^C ; and so on. For the last term ±abcd . . . kl=zY. The ± is prefixed to the last, or absolute term, because the prod- uct — aX — ^X — ''• ■ • • X — ^1 ^'11 ^^ pl^^ <"■ "linus, according as the degree of the equation is even or odd. Hence, 350 RAY S ALGEBRA, SECOND BOOK. 1. The coeffiicient of the second term of any equation is equal to the sum of all the roots, with their signs changed. 2. The coefficient of the third term is equal to the sum of the products of all the roots taL-i'ii two and two. 3. Tlie coefficient of the fourth term is equal to the sum of the products of all the roots taken three and three, with their signs changed. And so on ; and 4. The last term is the product of all the roots, v:ith the sign changed if the degree of the equation is odd. Corollary I. — If any term is v:anting, its coefficient is 0. II. If the 2"^ term is wantim^, the sum of the roots is 0. III. If the S"* term is v:unting, the sum of the products of the roots, taken tico and tn:i> in a product, is 0. IV. If the absolute term is icanting, the product of tlif roots must he 0, and hence one of the roots must he 0. V. Since the last term is the product of all the roots, there- fore, it must he dioisihle hy each of them; that is, every ra- tional root of an equation is u divisor of the last term. exa:\iples illustrating the preceding principles. 1. Form tlie equation -whose roots are 3, 4, and — 5. The equations x=S, x=4, and x=z — 5, give a— 3=0, a:— 4^0, and a;+5=0; hence, {x—3)[x—4:)(x+5)=x"'~-'2x-—23x-\-Q0=0. Here, 3-|-4 — 5=-4-2, coefficient of 2'' terra with contrary sign. 3>, 4+3x~5+4X— 5=— -3, the coefficient of the 3'' term. 3x4X — 5^ — ^^1 'li6 s'g'i of which must be changed, for the last term, because the degree of the equation is odd. 2. What is the equation whose roots are 2, 3, and — 5 ? (See Cor. 2.) Ans. a:='— 19a;+30=0. 3. Form the equation with roots 0, — 1, 2, and — 5. Ans. .r'+4a;^— 7.C-— 10.r=0. 4. Find the equation whose roots are l±i/ 2 and 2±i/3. Ans. X*— 6x^+8x^+2^— 1=0. GENERAL THEOBY OF EQUATIONS. 357 6. What is the 4'* term of the equation whose roots are -2, —1, 1, 3, 4? Ans. 29x'. 300. Proposition V. — No equation having unify for the coeffieient of the first term, and all the other coefficients inte- gers, can have a root equal to a rational fraction. Assume that all the coefficients are integers in the gen- eral equation, x"+Ax«-'+Bis»-^4- _^Ta;+V=0. If possible, let -=-, a fraction in its lowest terms, be a root of this equation ; then, by substituting it for X, reducing the terms to a common denominator, transposing, etc., we shall have '^=— Aa"-i-Ba"-26- — Ta6"--— V6"-'. 6 But, by hypothesis, a and b, and, consequently, a" and b, contain no common factor ; therefore, an irreducible fraction as equal to a series of integers, which is absurd. Hence, the supposition is absurd, and the equation has no fractional root. 400. Proposition VI, — If the signs of the alternate terms of an equation he changed, the signs of all the roots uiU he changed. Let a be a root of the equation a;"+Aa:»-"+Ba;»-^-fCx»-3+. . . . +V=0 (1); Then, a"+Aa"-'+Ba"-^+Ca''-'+. . . . +V=0 (2). Changing the signs of the alternate terms of equation (1), K"— Aa;''-i+B3;"-2_cj;"-3^. . . ±V=0 (3). Substituting — a for X in this equation, we have a"— Aa"-'+Ba"-2— Ca»-3. . . ±V=0 (4). Now, if n be even, the 2i,/ — 1, or ±-1/ 6, 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^hy' — 1, we have in the eq. the quadratic factor, {x— (a-j-iy-^T) } Ix— (a— 6^=1) }=(x— a)^-|-Z)' ; 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 a? — 26fl;-|-60=0 is — 6 ; required the other roots. Ans. 3±i/ — 1. 2. One root of »»— Tx^+lSa;— 3=0, is 2— yl; find the other roots. Ans. 2-|-p/3 and 3. 3. One root of x*— 3x^— 42x— 40=0|s— J (3+^^=^) ; find the other roots. Ans. — ^(3 — y" — 31), 4, and —1. 4. Two roots of x^— 10x*+29x'— lOx^— 62x-|-60=0 are 3 and i/2 ; find the other roots. A. — ]/2, 2, and 5. 402. Proposition VIII. — Descartes' Rule op the Signs. — JSTo 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 -|-a, there is one variation, and one positive root. In the equation x-|-a^O, where the value of x is — a, there is one permanence, and one negative root. In x' — (a4-6)x-|-a&=0, where the values of x are -\-a and -|-^j there are two variations and two positive roots. In x''-\-(a-\-b)x-\-ab=:^(), where the values of x are — a, and — b, there are two perm,anences, and two negative roots. In x' — X — 12=0, where x=-|-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 +2, +3, +4, we shall have .t'- 9x^ -)-26,'r — 24=0, where there are three variations, and ilirre j>nsitivc roots. But if we form an equation whose roots are — 2, -—3, -|-4, we shall have cc'+x' — 14.T— 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 -^-(_ — -|- — _|_-|_-|-, and let a new factor X — a^O, corresponding to a new positive root be introduced, the signs in the partial and final products will be + +- + - + + + + + - + - + + + ±-+- + ±±-. 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 must 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, tlie 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 sincfe 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- aginary. Take, for example, the equation a;' 4-16=0, or a;'±0.-c+16=0. 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_^Ba;+C=0, or ^=±03;^+ Ba;+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 numhers, when siihstituted for (he unknown qvantity in an equation, give results affected with different signs, one root, at least, of this equation lies hetween these numbers. Let the equation, for example, be a? — aj'-j-a; — 8=0. If we substitute 2 for X in this equation, the result is — 2; and if we substitute 3 for a;, the result is +13. It is required to show that there must be one real root, at least, between 2 and 5. The equation may evidently be written thus. Now, in substituting 2 for X, a:'-|-a;=10, and a;2+8=12; Therefore, a;3+a;a;2-)-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 X=z2, it is less than the second, but when 2=3, it is greater. Con- sequently, for some Talue of X between 2 and 3, we must have a;3-f-^— 2:--|-8, and this value of X is, therefore, a real root. In general, suppose X=0 to be a polynomial equation involv- ing X, and that p and g, 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 X^p^ let P — N be negative, or P<[N; and when a;=g, let P — N be positive, or P>N. Now, there must be some value of X between p and g, which renders P=N, 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 \iniiy, it is evident that we have found the integral part of one of tlie roots. 1. Find the integral part of one value of x in the equation a;*— 4a;^-f 3s;^+a;— 5=0. If x='.Z, the value of the expression is — 2 ; but if a:=4, the value is 47. Hence, 3 is the first figure of one root. 2. Required the first figure of one of the roots of the equation x^ — 5.r' — a-.-[-l=0. Ans. 5. TRANSFORMATION OF EQUATIONS. 404. The Transformation of an Equation is the changing of it into anotlier of the same degree, whose roots shall have a specified relation to the roots of the given equation. Thus, in a;«+Ax"-'-|-B.f"'-^ . . -^-T.-f+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 y= — x. If - be substituted for X\ then, 11=-, and the roots of the new equation in ij will be the reciprocals of those of equation (1). TRANSFORMATION OF EQUATIONS. 363 405. 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, h, c, etc., be the roots of the equation x"-JrAx"-'-\-Bx''--. . . . -|-Ta:+V=0. (1). Assume y=z7cx, or K^y. Substituting this vnlue for x, in (1), Hence, 2/"+A7f2/^l+BA;22/"-2. . . . -^-Tk^-^y-^-k^Y^Q. Since y=:kx, the roots of this equation are ka, 7cb, kc, etc. It is evident that this equation may be derived from (Ij; or that the transformation of (1) is effected, by multiplying the successive terms by 1, k, /;:-, A'', etc., and changing X into J/. In the case of division, assume 2/=y, or x=kj/, and substitute. Corollary. — By this transformation an equation may he cleared of fractions, or the coefficient of the first term may be made unity. 1. Let it be required to transform the equation a^-\-lpx''-\-lqx-\-r=0, 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, 6x^+3px^+2qx+er=0. Putting y^6x, or a;=j2/, 6^+3p^2+2g|+6r-=0; Multiplying by 6^, y^+3py^+'^2qy+216r=0. 2. Find the equation whose roots are each 3 times those of the equation x^-j-l^-:' — 4a;-j-3— 0. Ans. y+eSy— 108j/+243=0. 304 KAYS ALGEBRA, SECOND BOOK. 3. Find the equation -whose roots are each 5 times those of the cquaticm x*-\-2x^ — ix — 1 = 0. Ans. /+10y—875j/— 625=0. 4. What is the equation whose roots are each ^ of those „f a;'— 3cr^4-4<-+10=:0? Ans. 4/— 63(^+%+5=0. 5. Transform eq. .f' — 2x''-\'lx — 10=^0, into one having integral coefficients. Ans. if' — 6y-|-3y — 270^0. 406. Proposition II. — To tramform an rqiiation iiila one wlioae roots are greater or less by ainj gtcen quardity than the correspond ing roots of the proposed equation. Let a;''-|-A.r"-' + B.r"-= . . -(-T.yj-f V=0, be an equa- tion whose roots are «, h, c, etc. The relation hefween X and 9/ iviU he expi-esscd by the equation }/=^x±r. As the principle is the same in Ijuth cases, let ^=.i' — /', or x=i/-\-r. Substituting y-\-r for x, vie have (y+»-)"+A(2/+r/'-+B(2/+r)"- +T(2/+»-)+V=0. Developing the different powers of 2/t *" by the Binomial Theorem, and arranging the terms, we have y"+nr 2/"-' + 1-2 + (n— 1)A7- +B -f Ar"-i _|-Br"-2 -V ^=0. Now, since 2/^^ — ''; the values of y in this equation are ((- b — r, c — r, etc. 407. Corollary. — By means of the preceding transfor- mation, we may remove any intermediate term of an equar lion. Thus, to transform an equation into one which shall ■want the second term, r must be assumed so that )!»•+ A=^0. TRANSFORMATION OF EQUATIONS. 365 To take away the third term, put 4«(n' — 'l)r^-\-(n — 1)A)--|- B=0. 1. Transform the- equation :r^ — 7a;-J-7=0 into another whose roots shall bo less by one than the corresponding roots of this equation. Ans. y^-\-3i/'^4i/-\-l=:i0. 2. Find the equation whose roots are less by 3 than those of the equation x* — 3x^ — 15.-i;^-|-49,f — 12:=0. Ans. y+9j/«+12/— 14^=0. 3. Transform eq x' — Gx^-\-Sx — 2=0 into another whose second term shall be absent. Ans. y^ — iy — 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;*+Ba;5+Ca:^+Da:+E=0, (1) and let it be required to transform it into another, whose roots shall be less by r; then, y=x — r and x^i/-\-r. By substituting y-\-r, instead of X, we have By developing the powers of y-\^'r, and arranging, as in Art. 40G, the transformed equation will take the form A2/HBi2/''+C,2/2+D,i/+Ei=0, (2) where A is evidently the same as in (1), while Bj, Cj, D,, and E,, are unknown quantities to be determined. For y, substitute its value X — /■, and equation (2) becomes A(x—ry+Bi(x~-rf+Ci{x—rf+'Di{x-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 B,, C(, etc, let (3) or (1) be divided by z — r, and the quotient will be A{x—rf+'B^{x—ry+C^{x—r)+^D^, with the remainder Ej ; hence, Ej is determined. 366 KAY'S ALGEBRA, SECOND BOOK. Divide this quotient by x—r, and tlie next quotient will be A(a;-r)-'+Ci(.!--r) + Ci, with a remainder D,; hence, Di is determined. (Jontinuing the division by x—r, we obtain 0, and Bi, 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 a;' — 7x-f7=0 into another, whose roots shall be less by 1 than the corresponding roots of this equation. Here, y=x — 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 as the highest power of X in the proposed equation. x~ -l)x^~7x+7{x^+x- x^—x^ 1st m +.r-'_7a; x'' — X -6 lot. x—l)x-+x—e{x+2 X^ X 2d qliot -I-2X--6 2a:— 2 — 6a;+7 —Gx+6 1st rem. =+1 2d rem. = — 4 a;— l)a;+2(l, .idouot, x—1 3d rem. — 4"*^ Since the successive remainders are -{-S, — 4, and -|-1, we have A^l, Bi^+3, Ci= — 4, and D] = -|-l. Hence, the transformed equation is y^-\-3y-~4:y-\-\=0. 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 CoL'fficients (Art. 7*7). To explain the process, we shall first divide 5a;* — 12a;'-j-3a;'+4.o — 5 by x — 2, by detached coefficients. TRANSFORMATION OF EQUATIONS. 367 Divisor. Quotieut. l—2j5— 12+3+4— 5(5-2— 1+2, 5-10 or 5.c'i— 2x-'— x+2 -1+2 By changing the sign of the second term of tlie divi- sor, and adding each partial product, except the Jirsi term, which, being always the same as tlie first term of each divi- dend, may be omitted, the op- eration may be represented as in the margin below : Let it be observed that the figures over the stars are the coefficients of the several 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 +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 X, the quotient is 5a;' — 2z- — a;+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 the coefficient of the first term of the divisor. If any term is wanting, its place must be supplied with a zero. -2+3 -2+4 -1+4 -1+2 +2-5 2—4 -1 R 1_|_2)5— 12+3+4— 5(5- ■»+10 -2- —2+3 -:5 4 -1+4 »-2 «+4 410. In application of these principles, 368 RAY'S ALGEBRA, SECOND BOOK. 1. Let it be required to find the equation whose roots f.re less by 1 than those of the equation x^ — Y.-r-f Y. Since the second term is wanting, its place must be supplied with 0. The divisor is X — 1; hence, we divide by -\-l. OPERATK)N BY SYNTHETIC DIVISION. +1) 1 ±0 —7 +7 +1 +1 -6 +1 —6 +1 .-. +1=- 1" R. J-1 +2 ^2 —4 .-. —4= 2'' R. +3 .. +3= SiR, Hence, the required coefficients are 1, -|-3, — 4, and -]-l. . . y"-{'iy~ — 42/-(-l = is the transformed equation required. 2. Transform the equation 5a-*+28a;'-)-51a:^+32,r— 1 = 0, into another having its roots greater by 2 than those of the given equation. Here, M=a;-^2; hence, we divide by — 2, thus, —2) 5 +28 -1-51 +32 —1 _10 —36 —30 —4 +18 +15 +2—5 . . —5= 1" R. —10 —16 + 2 _ 8 — 1 + 4 . . +4= 2.' R. -10 + 4 - 2 - + 3 .-. +3= Z-i R. -10 —12 .-. -12= 4"' R. Hence, A=5, C,=— 12, C,----4 3, Di=4 4, and E,=— 5 .-. the transformed equation is 5?/' — l-jj" -y'ijj" --iy — 5—0. 3. Find the equation whose roots are less by 1.7 than those of the equation a' — 2.c- + 3x — 4=:0. TRANSFORMATION OF EQUATIONS. 369 If we transform this equation into another whose roots arc less by 1, the resulting equation is y-^-\~y'^-\-2y~2=Q. We may then transform this into another whose roots are less by .7, or the whole oper.ation may be performed at once, as follows ; .7) 1 —2 +1.7 +3 .51 —4 +4.233 — .3 +1.7 +14 +1.7 + 2.49 +2.38 + 4.S7 . + .233 . . +.233= 1" R. . +4.87= 2'i R. +3.1 .-. 3.1= Srf R. Hence, the equation is 2/'+3.l2/2+4.872/+.233=0. 4. Find the equation whose roots are each less by 3 than the roots of a?—2lx~S6^0. Ans. ^^^+9/— 90=0. 5. Required the equation whose roots are less by 5 than those of the equation a'— 18a.-^— 32a;--)-l7a;-|-9=0. Ans. i/*+2f~lb2f—llbSi/— 2831^0. 6. Required the equation whose roots are less by 1.2 than those of the equation x^—6x*-\-1 Aa^+1 .92.r'-~11 .S'J2x —.79232=0. Ans. 2/^— 7^+2)/— 8=0. Transform the following; equations into others wanting the 2d term. (See Art. ■ioY.) 7. x^— Ga'.'+Ta;— 2=0. Ans. ?/'— 5y— 4=0. x° ox'+12.r+-[9=0. Ans. y-f 2V=0. Transform the following equations into others wantin;. the 3d term : 9. a;'— 6a;^+-9a;— 20=0. Ans. f +3^—20^0, or y— 3/— 16=0. 10. x'— 4a;''+5a;— 2=0. Ans. y° — ^y^=0, or y'A-'if — ii*7=^' 370 RAY'S ALGEBRA, SECOND BOOK. 411. Proposition III. — To determine the law of Derived Polynomials. Let X represent the general equation of the b"' degree ; that is, X=x"-|-Acr"- + Ba;"-^ . . . -)-Ta;+V=0. If we substitute x-{-h for X, and put Xi to represent the new value of X, we have X,=(a:+/i.)"+A(x+A)"-'+B(a;+/0"-H, etc., and if we expand the different powers of x-\-h by the binomial theorem, we have X^^^ -fAa;"-' +Ba;"-2 +, etc. -{- nx"-'- -\-(n—l)kx'-- +(?i— 2)Ba;»-3 +, etc. li-\- n(n—l)x"-' + (ra— l)()i— 2)Aa;"-3 -j-(n—2){n—3)Bx"-' -f-, etc. 7i= J-T^>+. 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 =a;" + A;r"-'-f Ba;"-2+, etc., X' ~nx"-^ + {n—\)Ax"---\-(n—2)Bx''-^+, etc., X"=))(» — l).r"---(- ()!.—!)()! 2|Aa;"-3-|-, etc.. Etc., etc. By substituting these in the development of X,, we have X,=X + X'7i+j^7(Hj-r^'''+. etc. The expressions X', X", X'", etc., are called dcn'ird polynomials of X, or thrived 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 third, and so on. It is easily seen that X' may be derived from X, X" from X', etc., by multiplying each term by the exponent of x in that term, and diminishing the exponent by unity. TRANSFORMATION OF EQUATIONS. 371 412. Corollary.— If we transpose X, we have Xj— X X" =X7t-(-^j — T>h''-\-, etc. Now, it is evident that h may be taken so small that the sign of the mm X7i-|--^ /t'^^-l-, etc., will be the same as the sign of the first term X7i. Eor, since X'7i+^X"7i-+, etc., =A(X'+JX"/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 the transformed equation in Art. 406, with the development of Xj in Art. 411, it is easily seen that Xj may be considered the transformed equation, y 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' — 7a;+Y=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- tion Xi=X+X'^+=^^^2_|_^^ — j^As^^ etc., we have X^=(x^—7x -|-7)+(3a:2 — 7)l + (6a:)-J^+ ^ =xiJr^x?—ix+\, in which the value of X is equal to that of x in the given equation diminished by 1. By this method, solve the examples in Art. 410. 372 RAYS ALGEBRA, SECOND BOOK. EQUAL ROOTS. 414. To determine the equal roots of an equation. We have already seen (Art. 390, Keni.) tliat 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 3(.f — 2/-=^0. Hence, we see that if .any equation contains the same factor takem tlirpr times, its fiist derived polynomial will contain the same factor taken twice; this last factor is, therefore, a common divisor of the given equation, and its iirst derived polynomiaL In general, if we have an equation X=0, containing the factors (.r — a)"'(X — b)", its first derived polynomial will contain the fac- tors m(x — a)^~hl(x — 6)"~' ; that is, the greatest common divisor of the given equation, and its first derived polynomial, will be (X — a)°'~'(a;— 6)"-', and the given equation will have 7n roots, each equal to a, and n roots, eacli equal to b Therefore, to determine whether an equation has equal roots, Find the greatest common diiisor hetiveen the equation and its first dirici'd jiojynom i) it has two 'coots equal to o, and two equal to h, and so on. 1. Given the equation .r'' — .(;- — 8.r-)-12=i0, to determine ■whether it has e(|ual roots, and if so, to find them. We have for the first derived polynomial (Art. 411), ox- — 2.r — S. The G.C.D. of this and the given eqtiation (Art. 108) is x — 2. Hence, .T -2=0. and x=-\-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 lo 2, it must be divis- ible by (x—2){x—2), or (x-2)-'. (Art. 395). Whence, x3—x--Sx+12=(X^2)\x+3)^0, and 3-+3=0, or .t = — 3. Hence, ivhcn 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. .t'— 2.r^— 15.r+36=0. . . . Ans. 3, 3, —4. 3. x*_9.r=+4.r+l'2=0. . . Ads. 2, 2, —1, —3. 4. a;^—6a;=+12a;-^— 102+3=0. Ans. 1, 1, 1, 3. 5. x'—1x'-\-9x'-\-21x—b4=0. Ans, a:=3, 3, 3, —2. 6. x*+2.r'— 3x2— 4^+4=0. Ans. —2, —2, +1, +1. 1. a:*— 12.r'-f50a;^— 84.i'+49=0. A.3±;/2, 3±v 2^. 8. af—2x'+dx^—1x'+8x—S=.-.0. Ans. 1, 1, 1, —\±ly—n. 9. x^-\~8x'—6x*—6x'-\-9x'+Sx~'i=^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-\-l. Proceeding, we may, 1st, find the common measure of tliis 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 — o)^, etc. By tlielst method, we find x^ — x^—x-\-l={x — l)(x^ — l)=(a;— 1)2 (.T-)-l); by the 2d, (x — 1)^ is a factor of the original equation; hence, [X — 1)- is a factor of x^ — x-—x-\-\. 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 RAYS ALGEBRA, SECOND BOOK. Its characteristic is, that when it, or any number jrreater thun it, is substituted for x in the equation, the result is 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 a-, 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. — rAe greatest negative coefficient, ■increased hy unity, is greater than the greatest root of the equation. Take the general equation a;»-|-Aa;"-i-|-Ba;''-' .... -|-T.t-|-V=0, and suppose A to be the greatest negative coefficient. The reasoning will not be affected if we suppose all the coefficients to be negative, and each equal to A. It is required to find what number substituted for x will make ;c">A(a;"-i-|-x"^"-|-a^"-3. . . . _|_a:-)-l). By Art. 297 the sum in purentlicsis is =- ; hence, we must a— 1 „^ . / a;"— 1 \ A.r" A havea;">A( ^ 1, or a;"> ,. \x" But if x"= ^, we find a;=A-|-l; . . A-fl substiluteil for x will X — 1 ' ' „ A.r" , A.r" A render x'= -. .and, conseqnentlv, a"'> , -. X — ] ' ' .r— 1 .1-1 By considering all the cotfficients 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 he less. 417. Proposition II. — If icc tahe the greatest negatice coeffieient, extract a root of it vhose index is equal to the number of tenns preceding the first negative term, and in- crease it hy unity, the result will he greater titan 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 coefiicient; then, any value of x which makes x''>C(a;»-'--|-x"-'-' +a;+l) (1) will render the first of the proposed equation >0, or positive ; because this supposes all the coefiioients 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, a:">c( ^_^ ), orX">- X n- Cx"~''+^ Cx"-''+^ this inequality will be true if x"^ = — , or ^ = — : ' •' x-1 -^ x^l ' or, by multiplying both members by X — 1, and dividing by »;"-''+', when (a;— l)a;'--i=C, or >C (2). Buta;— 1 is <«, and .-. (a;— l)''-'C; Ora;— l = i'/C, or >vTn Or x=l + v iT, or >! + { CL Find superior limits of the roots of the following equa- tions : 1. a;*— 5a;'-f37x'— 3.x + 39=0. Here, C=5, and r=l .-. 1+^^0=1+5x^6, Ans. 2. x'+'7x*~12x'—49x'+b2x—lS^0. Here, l+^C^l-\-f49^1-\-1=.S, Ans. 3. a-^+llx'— 25a;— 67=0. By supposing the second term +0a;', we have r^3 ; hence, the limit is 1+^^6*7, or 6. 4. 3a:'— 2a;^— llx+4=0. Dividing by 3, x'— fcc^— Va;+|=0. Here, the limit is 1 + Vi or 5. 370 RAY S ALGEBRA, SECOND BOOK. 418. To determine the inferior limit to the negative roots, change the signs of tlie 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 lo the inferior limit of the roots of the proposed equation. 419. Proposition III. — //' the real roots of an equation, taken in tlie order of tlicir magnitude, he a, b, c, d, etc., a being greater tlian b, b grmter than c, enid so on; then, if a series of numbers, a', b', c', d', etc., in which ;i' is greater than a, b' a nundjcr between a and b, c' a number between b and e, and so on, be sidtstitutedfor x in tlie prajiused equa- tion, the results will be alternately positice and negative. The first member of the proposed equation is equivalent to {.r—a)(x—b)(x—e')(.x—d). . =.0. Substituting for X the proposed series of numbers a', b', c', etc., we obtiiin the following results: {a' —a)(a'—b){a' — c)i«' — d), etc. . =+ product, since all the factors are -j-. (6' — «)(&' — ^)[b' — C)(6'— d), etc. . . . ^ — product, since only one factor is — . (ej'—a\ic' — b){e' — c)(o'--(}). etc. . =+ product, since two factors are — , and the rest +. {d'—ei][d' b)(d' — c)(cl'—d), etc. . r^ — product, since an odd number of factors is — , and so on. Corollary 1. — If two numbers be successively substituted for .r, in any equation, and give results with contrary signs, there must bo one, three, fee, or some odd number of roots between these numbers. Corollary 2. — If two numbers, substituted for a-, 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 q, and every quantity greater than q, 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^ — 422_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 X, any number less than 5, and greater than 2, the result is negative. Putting x=2, the result is zero. Substituting for X, 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 descendiug scale. STURM'S THEOREM. 430. To find the number of real and imaginary roots of an equation. In 1834, 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- nation of all the real roots of an equation can, with cer- tainty, be determined. This theorem we shall now proceed to explain. Let X=x»-f Ax-'-'-f Ba:''-^ _|-Ta;+V==0, be any equation of the re"" 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 RAYS ALGEBRA, SECOND BOOK. Let the first derived function of X (Art. 411) be denoted by Xj. Divide X by Xi until the remainder X|)X (Qi is of a lower degree with respect to x XjQi than the divisor, and call this remain- ,- ,, q1_ ^ der — X^; that is, let the remainder, wiih lis sign changed, be denoted by X^. ^:)Xi (Q2 Divide Xi by X2 in the same man- X0Q2 ncr, and so on, as in the margin, de- y^ XoQo^ X., noting the successive remainders, with their signs changed, by X3, X^, etc., " "'I' - (^'^ until we arrive at a i-emainder ^vhich -^^^M?. does not contain X, which must always Xi X3Q3^ X, 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 remainder, having its sijrn changed, be called X,._j,j. In these divisions, we may, to avoid fractions, cither multiply or divide the dividends and divisors by ^Tiy positive number, as this will not affect the signs of the functions X, Xj, Xo, etc. By this operation, we ol)tain the series of quantities X, Xi, X,, X3. . . X,+i (1). Each member of this series is of a lower degree with respect to .r than the preceding, and the last does not contain X. Call X the primitive function, and Xi, Xo, etc., auxiliary functions. 431. Lemma I. — Tico C077secnt!vc functions, Xi, X,, /()/• example, can not both cauhJi for the savie value of x. From the process by which Xi, Xo, etc., are obtained, we have the following equations : X =X,Qi-X, (1) X, =X,Q;-Xo (1^) X. - XiQ.;- X.| (3) X,_i=X,.Q,-X,+i. . . . ^r). If possible, let Xi^O, and Xj^O; then, by eq. (2) we have X..^0; hence, by cq. (3) we have X4^0; and proceeding in the same way, we shall find X-^0, X5=0, and finally Xr-|^] = 0. But this is impossible, since X^-i-i does not contain X, and therefore can not vanish for any value of x. THEUKEM OF STURM. 379 43S. Lemma II. — If one of the auxiliary functions van- ishes for any particular value of x, the two adjacent functions must have contrary signs for the same value of x. Let us suppose tliat X3=0, when x—a; then, because Xo— X3Q3 — X4, and X3=0; therefore, X2=— X4; that is, Xj and X4 have contrary signs. 433. 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-|-h, then will the number of variations and perma- nences, when a — h and a-(-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 Xo or X^ can vanish for the same value of X; and since when X3 vanishes, Xj and X4 have contrary signs, (Art. 422); therefore, the substitution of a for x in X2, X3, X4, must give Xj , X3 , X4 , or Xj, X3, X4. + - , - + And since h is taken so small that no root either of X2^0, or S.^=0, lies between a — h and Ct-|-''', 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 -f- or — by the substitution of OS — h and a-l"" ^"^ ^t there will be one variation and one permanence. Thus, we shall have either X2 , X3 , X4 , or X_. , X3 , X^, + ± - - ± + So that no alteration in the number of variations and perma- nences can be made in passing from a — h to a-]-h. 424. Lemma IV.— If a is a root of the equation X=0, then the series of functions X, Xj, X,, etc., will lose one variation of signs in passing from a — h to a-(-h ; li being taken so small that no root of the function X,=0 lies between a — h and a-l-h. 380 RAY S ALGEBRA, SECOND BOOK. For X substitute a-(-/t in tlie equation X^O, 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+A'/t-|-LA"A=+, etc. But, since a is a root of the eq. X^O, we shall have A^O, \\'hile K' can not be 0, since the eq. X^O has no equiil roots. Hence, H=A'7i-|-JA'"7!2+, etc., =7i(A'-|-JA''A-|-, etc)- Now, A may be talien so small that the quantity within the paren- thesis shall have the same sign as its first term A', (since .V ex- presses the first derived function of X, corresponding to X^, in Art. 4121; therefore, the sign of X, when a:=a^n, will be the same as the sign of Xj. If wo substitute a — A for X in the equation X^O, and denote the result by H', we then have, by changing A into — A, in the expres- sion for H, n'=- A| A'— l.VA-f , etc). Now, it is evident that for very small values of A, the sign of 11' ■will depend upon the first term — A'A, and, consequently, will be contrary to that of A'. Hence, when X^Q — A, there is a variation of signs in the first two terms of the series X, Xj ; and when x=a-\-hj there is a continuation of the same sign. Therefore, one variation is lost in pnssmg from x^^a — li 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 Btill be lost in passing from a— A to a-|-A. 425. Sturm's Theorem. — If any tico numlers, p and q, (p Jicing less than q) he siihslil;ili:d for x in the scrips of functions X, X], X2, etc., the substitution of p for x giving k farialions, and that of q for x, gicing k' variations ; tJun, k — k' uill he the rxact number of real roots of the ri^uatiou X=0, lohich He. hetnven p and q. Lot us suppose that — oo is substituted for x, and sup- pose that u. continually iticreases and passes through all degrees of magnitude till it becomes 0, and finally reaches -|- gc. THEOREM OF STURM. 381 Now, it is evident, that so long as X, with its minus sign, is less than any of the roots of X=0, 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 one variation of signs is always lost. Since, then, a, variation is always lost whenever the value of X passes through u, root of the primitive function X=0, and since a, variation can not be lost in any other vpay, nor can one be ever in- troduced, it follows that the excess of the number of variations given by a;==p, above that given by x^q (p the signs are — + — — , 2 variations, .-. k=2. x=-\- 00 the signs are + + + — , 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^O when a;=3; 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'> — 36x-' + 46x—lb=0. Ans. Three, One be- tween and 1, one between 1 and 2, one between 2 and 3. 5. x' — Sx'' — 4.'r-)-ll=;0. Ans. Three. One between — 2 and — 1, one between 1 and 2, one between 3 and 4. Ans. One between 2 and 3. 384 RAYS ALGEBRA, SECOND BOOK. 7. a-'— 15,r— 22=0. Ans. Three. One root is —2, one between — 2 J and — 2^, one between 4 and 5. 8. x* — 4:x^ — 3a;-f23=0. Ans. Two. One between 2 and 3, and one between 3 and 4. 9. ,fi_2.,3— 7a;^+10.r+10=0. Ans. Four. The limits are (-8, -2); (0, -1); (2, 3); (2, 3). 10. .r'— 10./'+(J.f+l = 0. Ans. Five. The limits are (-4, -3); (-1, 0); (-1, 0;; (0, 1); (3, 4). XIII. RESOLUTION OF NUMERICAL EQUATIOXS. 42S. In the preceding articles we have demonstrated the most important propositions in the theory of cijuations, 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, cither exactly or approsi- 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. R.VTIOX.VL ROOTS. 429. Proposition I. — To determine the integral roots of an equation. If a be an integral root of the equation A.r'-j-Bx'-|-C.T' -fD.(;-fE = 0, we ^hall have A,('-|-Ba=-)-C.(- + D«-f E=0 ; E therefore, -=—An'~Ba--~C>i~D. RESOLUTION OF NUMERICAL EQUATIONS. 385 Now, since the second member of the last equation is evidently a whole number, E is divisible by a. Put -=E'; transpose D to the first member, and divide by a; this gives E'+D ■ (J ==— Aa2— Ba— C; .-. a is also a divisor of E'+D. Tut -1__=D', transpose C, and divide by a; this gives D'+C •= — ha — B; .-. a IS a divisor of D'-j-C. Again, put ' =C', transpose B, divide by a, and — 15=— A. (I a C'-l-B Lastly, making ' =B', and transposing A, we have B'+A=0. If, then, all these conditions are satisfied, a is » 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 by a, and add to the quotient the coefficient of x\ Proceed in this manner unto the first term,, and if Sl he a root, all these quotients will he 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 thp 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). 2d Bk. 33* 386 RAY S ALGEBRA, SECOND BOOK. 1. Find the rational roots of the equation ar+3a;'— 4a;— 12=0. Here, by Art. 417, no positive root can exceed l-j-j' 111, or 4, and the limit of the negative roots is l-|-3— 4. It is also found, by trial, that -|-1 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 Quotients Add —4 Quotients Add +3 Quotients Add +1 + 2, +3, +4, -2, -3 , -4 -6 , -4, -3 , -f6 , +4 , +3 -10 , -8 , -7 , +2 , -0 , -1 - 5 , » , * , — 1 , » - 2 , +2 , +3 , - 1 -1,-1, , 0,0, Since — 8, — 7, and — 1, are not divisible by +3, +4, and — 4, we proceed no further with these divisors, as it is evident that they are not roots of the equation. The roots are -\-2, — 2, and — 3. Find the roots of the following equations : 2. a^— •7.r=+36=0 Ans. 3, 6, and —2. When any term is wanting, as the 3d term in this exapiple, 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 af the equation by division, and proceed as before. 3. a^—6x' +11x^6=0 Ans. 1, 2, 3. 4. x'+a;'— 43;— 4=0 Ans. 2, —1, —2. 5. x"— 3x^— 46x— 72=0. . . . Ans. 9, —2, —4. 6. x^— 5x'— 18a;+72=0. . . . Ans. 3, 6, —4. 7. x*—10x»+35x^— 5021+24=0. Ans. 1, 2, 3, 4. Ans. 3, 4, -3, - -5. A) QS. 2,2,- _2. Ans. — 5, l±l - ^1. A ns. 4, 5, ±v '8. Ans A Ans. S, ns. I ±y2. HORNERS METHOD OF APPROXIMATION. 387 8. a;*+4a;'—a:^— 16a:— 12=0. Ans. 2, —1, —2, —3. 9. x«— 4a;'— 19x'+46x+120=0. Ans. 4, 5, —2, —3. 10. a;*— 27x'-f 14.x+120=0. Ans. 3, 4, —2, —5. 11. x«-|-a;»—29a:^— 9.^+180=0. 12. a;'— 2x»— 4a;+8=0. 13. x»+3x'— 8a;+10=0. 14. a;«— 9.x'+17.c^+2Ya;— 60=0. 15. 2a;»— 3x^+2x— 3=0. 16. 3x'— 2a;^— 6»+4=0. 17. 8.x»— 26x^+llic+10=0. 18. 6a;*— 25a:»+26a;^4-4x— 8=0. Ans. 2, 2, f, —1. 19. a;«— 9x»+ya;-^+ya:— V=0. Ans. f, |, 3±3i/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. 430. Horner's Method of Approximation. — The prin- ciple of this method depends 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 he "found, be P!B"-t-Qa;"-' -j_Tx+V=0. Suppose a to be the integral part of the root required, and r, S, t . . the decimal digits taken in order, so that a;=a-)-?--|-s-)-<. . . 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 P2/"+Q'y"-i. . . . +Ty+V'=0 be the transformed equa- tion ; then, the value_of y is the decimal r-\-s-^t. . . . . ; and siiioe 388 RAY S ALGEBRA, SECOND BOOK. this root is contained between and 1, we may easily find its first digit T, Again, let the joots of this equation be diminished by r, and let the transformed equation be P3"+Q"z"-l _|_T"«-)-V"=0. Now, the Talue of z in this equation is s~\-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 s is in the second place of decimals, 2-, z^ . . will be small, and we may generally find S from the equation T"z+V''=0; or, s=—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 bo 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, f, 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 sufficient accuracy by dividing V" by T". 433. 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 x' — 4a; — 10. 768649^0. We readily find that x must be greater than 5, and less than 6; 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.) a 5) 1-4 —10.768649 +5 + 5 + 1 — 5.768649 +5 +6 1st Trans, eq. . 3/-+% — 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 y=.8, and deduct 2/^^.64 from 5.7, and then divide by 6, we see that?/ 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 _ .328649 7.6 2d Trans, eq. . . «2._|_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, 2^, being now so small that it may be neglected. Thus, .328-=-7.6=.04=g. We next diminish the roots of the last equation by .04. s .04) 1 +7.6 —.328649 .04 .3056 +7.64 .023049 .04 +7.68 3d Trans, eq z'^ h 7 68«' —.023049=0. 390 RAYS ALGEBRA, SECOND BOOK. Here e' is nearly .023H-7.68=.003=i(. By diminishing the roots of the last equation by .003, we have t .003) 1 +7,68 —.023049 .003 .023049 +7.683 .0 The remainder being zero, shoTvg 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, -\--y/2, — y- 2, which gives ■t' — Z.r- — 2a;-f 6^0. Let it now be required to find, by Horner's method, the root which lies between 1 and 2 ; that is, |, 2. One root lies between 1 and 2; hence, a=l, 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 O -4 ^■^ +1 — 1 —1 - 5 \ r=— T' +1 ' Hence, .?r±0^-— 5,/y+2=r0, is the first transformed equation. By dividing the absolute term 2 by 6, the trial divisor or coefficient of 2/, we find r=.4, and proceed to transform the equation so as to diminish its roots by .4. HORNER'S METHOD OF APPROXIMATION. 391 .4) 1 ±0 —5 +2 .4 .16 —1.936 .4 —4.84 + .064 .4 + .32 V" .064 „, .4 ^.52 T" 4.52 This gives 23+1.2zZ—4.52z+. 064=0, for tlie 2d transformed equation ; and for S, the next figure of the root, .01 . Transform this equation so as to diminish its roots by .01. s .01) 1 +1.2 4.52 +.064 .01 .0121 —.045079 1.21 —4.5079 +.018921 .01 .0122 1.22 —4.4957 •01 V" .0189 ... t= — -, = -,-jj^=.004 1.23 1 4.495 This gives 2'3+1.232'2— 4.49572'+ .018921=0, for the 3d trans- formed equation; and for the next figure of the root t^.OOi. Transform this equation so as to diminish its roots by .004. t .004) 1 +1.23 .004 1.234 .004 —4.4957 + .004936 +.018921 —.017963056 -4.490764 + .004952 —4.485812 .000957944 1.238 .004 1.242 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;=1.4142135. 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, wlien the root is required only to tive or six places of decimals, use about this number in the operation. 3. Given a;* — 8r'-\-14x'-\-'ix—8=^0, to find a value of x. OPERATION. 5.236068) —8 5 +14 —15 — 1 10 9 35 -44 2.44 +4 —5 —1 46 «44 9.288 —8 —5 —3 5 '—13 10.6576 2 5 -— 2.3424 1.93880241 7 5 53.288 9.784 »— .40359759 .39905490 *I2 .2 46.44 2.48 *63.072 1.554747 *■— .00454269 .00400954 12.2 48.92 2.52 64.626747 1.566321 -_ .00053315 12.4 .2 »51.44 .3849 -66.193068 .31608 12.6 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 HORNERS METIIOl) OF APPROXIMATION. 303 As the root is found only to six decimal places, carry the true divisor for the third figure (6) to five decimal places. This divisor is 66.50915, which, multiplied by .006, gives eight decimal places; and the dividend ought to be carried thus far, to make the figure in the sixth decimal place of the root correct. The divisor, 66.825, for the fifth figure of the root, requires to be carried only to three decimal places, for the product of this number by .00006 gives eight decimal places, as it ought to do. So the divisor for the last figure (8) of the root would require to be carried only to two decimal places. The numbers in the preceding columns require to be carried to still fewer places, as will readily be perceived. The last three figures of the root may be obtained merely by division; thus, .00454269-=-66.82571 =.000068, nearly. Observe that where decimals are omitted, we always take the figure next to the omitted places, lo the nearest unit. Thus, .07752 is nearer .08 than .07; therefore, the former is taken. 434. Horner's method may be applied to equations of any degree, and is the most elegant method of approxima- tion yet discovered. It may be expressed by the following Rule. — 1. Find, hy trial, or hy Sturm's theorem, the in- tegral part of the required root. 2. Transform the equation (Art. 410) into another whose roots shall he those of the proposed equation, diminished by the part of the roof already found. 3. Witli, the absolute term in the first transformed equation for a dividend, and the eo'efficient of x for a divisor, find the first decimal figure of the root. 4. Transform the last equation into another whose roots shall be diminished by the pari of the root already found, and from the first two terms of this equation, find the second figure of the root. 5. Continue this process till the root is found to the required degree of accuracy. 6. To find the negative roots, change the signs of the alter- nate terms, and proceed as for a positive root. 394 RAY S ALGEBRA, SECOND BOOK. Remauks. — 1. If any figure, found by trial, is too great or too small, it will be made manifest in the next transformation. (See Art. 431.) i!. After finding three figures of the root, the next three may gen- erally be obtained by dividing the absolute term by the coefBcient ol X. Find at least one value of x in each of the following: 1. a:^-)-5a;— 12.24=0. . . 2. x'+12x—SbA02b=0. 3. 4a:^— 28a;— 61.25=0. . 4. 8a;-^— 120a;+394.875=0. 5. 5a;'— 7.4x-16.08=0. . 6. a;'_6.ti+6=0 7. a-'+4a-^— 9.1-— 57.623625z=:.0 8. 2.,'— 50a;+32. 994306=0. 9. 2.^_^4a:2_5.^_20=0. . . . 10. x^—2x—b=0. . . . 11. a-.'+10.-^-24a-— 240=0. 12. .T«— 8a;'+20.a;'— 15.x+.5=0. 13. a;*— 59a:-'+840=0. . 14. 2.r<+5a;'+4.c^+3a;=8002. 15. .r'+2.j;*+3.f'+4a^'^+5.o=54821. A. a-=8. 414455. Ans. a;^1.8. . Ans. r-=2.45. Ans. 3;=8.75. . Ans. .T=10.125. . Ans :r=2.68. Ans. a:=4.73205. Ans. a;^3.45. Ans. ;r=4.63. Ans. .,=2.23608. Ans. a;=2. 0945515. Aus. .r=4.8989795. Ans. a-=1.284724. Ans. :r=4.8989795. Ans. a;=7 335554. 43>>. Tu t.v/racl the roots of numbers hi/ Horner's Method. This is only a particular case of the solution of the equation a;"=N, or a;" — X=0 ; an equation of the jt"' de- gree, in which all the terms are wanting except the first and last. In performing the operation, observe that the successive integral figures have the same relation to each other that the successive decimal places have in the previous examples. In extracting any root, point off the given number into periods, as in the operation by the common rule. APPROXIMATION BY DOUBLE POSITION. 395 For an example, let it be required to find the cube root of 1291l81[b; that is. one root of the equation «'— 13977875=0. 235) 1 12977875 2 4 8 T 4977 2 8 4167 ~4 *12 810875 189 810875 "so 1389 3 198 ~63 »1587 3 3475 66 162175 3 -69 5 695 The reason for placing the figures as they are in the successive columns, will be readily understood by using the numbers 200 and 30, instead of 2 and 3. By the same method find 2. The cube root of 34012224. . . . Ans. 324. 3. The cube root of 9 Ans. 2 080084, 4. The cube root of 30. . . . Ans. 3.107233, 5. The fifth root of 68641485507. . . Ans, 147. APPROXIMATION BY DOUBLE POSITION. 436. Double Position furnishes one of the most use- ful methods of approximating to the roots of equations. It has the advantage of being applicable, whether the equa- tion is fractional, radical, or exponential, or to any other form of function. 390 RAV'S ALGEBRA, SECOND BOOK. Let X^O, represent any equation ; and suppose that a and h, substituted for x, 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 h for x, in the equation Xn=0. Let x==.a~\~h, and h==:a-{-k; then, if we substitute a-\-h and a-\-k for :r, in the equation X=0, we shall have X==A-f A'7i+^A";i'+, etc. B=A+A7,-+|A"A-^+, etc. Here, A', A.", etc., are the derived functions of A (Art. 411). Now, if h and Ic 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'fc " Whence, B— A : X — V : : X'h : k'Ti : 1; : Ti; Or, . . B— A . ;;- : : X— A . /i, (Art. 270); Or, B— A : b—a X— A /(, since li—b—a. Hence, we have the following Rule. — Find, hj] trial, tico numbers which, substituted for X, gice one a result too small, and the other too great. Then say, Ai< the difference of the results is to the difference of the suppositions, so is the difference Letircoi the true and the first result, to the correction to be added to the first sujy- 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 with 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 difference of the suppositions 0.1, and carry the 2d quotient to ia-o places, and so on. NEWTON'S METHOD OF APPROXIMATION. 397 1. Given a-'-|-a"--f-a-=100, to find x. Here, x lies between 4 and 6. Substitute those two numbers for X in tlie given equation, and the result is as follows: 64 . . . x" 125 16 . . X- . ... 25 4 . . X . .... 5 84 . results . 155 156 5 . . lUO 84 ... 4 . . 84 71 : ^ -. : ~ Te 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 a;=4.264. Assuming 3;=:4.264 and 4.265, and continuing, we obtain a:=4.2644299, nearly. Find one root of each of the following equations: 2. a;»-|-30x=420. ..... Ans. a-^^G.lVOlOS. 3. 144.r'— 973.T=319 Ans. .-i;=2.75. 4. a;'+10.«-^+5a:=2600. . . Ans. a— 11.00679. 5. 2j^+3x^— 4a:=10 Ans. a:=l. 62482. 6. a^*_a;'+2x''+.r=4 Ans. a-=1.14699. 7. #-7.r'+4.'c-^+i -10.-rc2a;— 1)=28. A. a-=4.51066. 437. Newton's Method of Approximation.— This 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-\-y fora: in the given equa- tion, and it will be of this form A+A'y-f iA'y+^A"y+, etc., =0 (Art, 411), where A, A', A", etc., are what the proposed equation, the first derived polynomial, etc., become when x=a. 398 RAYS ALGEBRA, SECOND BOOK. From this equation, by transposing and dividing, A k" K'" We find y=— ^f—\-yy-—\—rjy''' — , etc.; and since y is <0.1, y~ will be <0.01, 2/3<0.001, and so on. Therefore, if the sum of the terms containing 2/^, 2/^ etc., be less tlian .01, we shall, in neglecting them, obtain u, value of y within A A .01 of the truth. Putting 2/= — -^, we have a;==a — — ;. This will A A differ from the true value of X by less than .01. Now, put b for this approximate value of X, and let x=.h^z-, we have then as before B+B'3+JB''z2+^B''"^34._ etc., =0; and as z is supposed to be less than .01, 2- will be ^.0001. If then, we neglect the terms containing 2-, 2^, etc., we shall obtain a probiible 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 a-' — 2r —5=0. Ans. a;=.2.09455149. CARDAN'S RULE FOR SOLVING CUBIC EQUATIONS. 438. In its most general form, a cubic' equation may be represented by a? -\-iix''- -\-qx-\-r=^^ ; but as we can always take away the s-econJ term, (Art. 407,) wc will suppose, to avoid fractions, that it is reduced to the form a-'+3(2a;+2r^0. Assume x~y-[z, and the equation becomes Now, since we have two unknown quantities in this equation, and have made only one supposition respecting them, we are at liberty CARUA.NS SOLUTION OF CUBIC EQUATIONS. 399 to make another. Let, therefore, 2/z= — q. Substituting, wc have y^-\^z^-{-2r=Q; hut since yz=—q, z?—~- y o3 Hence, 2/3_X__|_2r=0; Whence, 2/'^ — T-\-y'r^-\~q^. And similarly, . . z^= — r — \/^''-\-q^; the radical heing positive in one, and negative in the other, by reason of the relation yz— — q. And since a;=2/-|-z, we have x=f{-r+^r2+q«)+f{-r-y/r^'+q^). This formula will give one of the roots. The others m.iy be found by reducing the equation (Art. 396, Cor. 1). 439. If r'^+g" be negative, that is, if r«-f2"<0, the values of x become appareDtly imaginary when they are actually real, and we shall now show that Cai-dan's Method of Solution does not extend to those 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, i.nd b—y'Sc, in which, if 3c be positive, the roots are rsal, and if 3e be negative, they are imaginary; and because the second term of the equation is 0, we have (Art. 398), 0=a+(6+,/3c) + (&— i/3c)=a+26; 35=aX26+62— 3c=— 362— 3e; 2r=— a(62— 3c)=263-66c. Hence, we have r2+5»=(63_36c)2_(j2_4,c)'=— 96''c+6?)2c2— c' =._c(362— c)2 . . /r2+53=(3&2_c)y'35. 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 0=0, the roots are a, b, and 6; hence, Cardan's Rule is appli- cable to equatioua 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 a;'— ly.f— 4:=0. By substituting y-\-z for a;, we have And, since . . 3yz=:15, 2/'+z3 — 4^=0. From the solution of these equations, we obtain 2/3=2-|-l] j/^1. By actual multiplication, we iind that 2/=2-f|-' — 1; likewise, 2^=2— 11 y/^, and z=2—^/^^l. Hence, a;=2/+z=(2+/3I)-l- (2—^/^=4. By dividing the given equation by x — 4, we find the other two roots are x=~2-\-y'3, and —2 — j/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 by Cardan's rule. When the equation contains the second term, remove it (Art. 40Y), and reduce the equation to the form a:^-\-Sqx-}-2r=0. Then, x^f{-r+-^/r":+^3) + f(—r—y r2+^), 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 (.\rt. 396, Cor. 1). RECIPROCAL OR RECURRING EQUATIONS. 401 1. Solve the equation v»-j-3t)^-(-9v— 13==0. Substituting x—1 for V (Art. 407), we have x^+6x—20=0. Comparing this with the equation x^-]-Sqx-{-2.r=^0, we find 9=2, »"= — 10 ; hence, a;=f (10+^008)+ f (10— /I087=2.732-.732=2. Whence, v=x — 1=2 — 1=1. The other two roots are easily found to be — l±3y/— 1. 2. a;»— 9a;+28=0. . . . Ans. x=— 4, 2±v^=8. 3. x^+6a;— 2=0. . . Ans. a;=f^— /2=.32V48. 4. x'— 6x''+13a— 10=0. Ans. x=2, 2±y'^=T. 5. a:'— 9a;^+6x— 2=0 Ans. a;=8.306674. 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. 443. 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-)-Ba;"-2. . . . +Sa;2+Ta;-(-V=0, be a recurring equation ; that is, one that is satisfied by the substitution of the re- ciprocal of X for X. This gives \^ , A ^B , S , T ,, + ^,,-1+ y.n-2 + a;2 ' a:"*" ' and multiplying by X", l+Aa;-fCa:2 -|-Sa:"-2+Ta;"-i + Vx"=0, which proves the proposition. 2d Bk. 34 402 RAY'S ALGEBRA, SECOND BOOK. Such equations are called Recurring Equations, from the forms of iheir coejicients ; and Reciprocal Equaticms, from the forms of their roots. Proposition II. — A recurring equation of an odd degree, lias one of its roots equal to -|-1, when the siijiis of the like coefficients are different, hut equal to — 1, when their signs are alike. Since every power of -|-1 is positive ; when the signs of the like coefficients are different, if we substitute -|-1 for X the corresponding terms will destroy each other. When 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 reduced one degree lower by dividing by X — 1, or x-\-l. Proposition III. — A recurring equation of an even degree, whose like coiiffeients have opposite signs, is divisible by x^ — 1, and tlmrefore two of its roots are --\-l, aiid — 1. Lets;2"+Aa;2»-i+Ba;2"-2 — Ba;^— Aa;— 1=0, be an equa- tion of the kind specified. It may evidently be arranged thus, and be divisible by x^-^l (.\rt. 83). (xS"— l) + Aa;(a;2'»-2_l)_j_Ba;2(a;2'^— 1)+, etc. . . .=0. Corollary. — An equation of this forxn 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 lialf that degree, when tlie signs of Hie corresponding tenns are alike. For, 3:51— Ax^n-l-l-Bx^n-z— , ete -1-Bx2— A2:-|-l-=0, by RECURRING EQUATIONS. 403 dividing by x", and collecting the pairs of terma equi-distant from the extremes, becomes of the form ( ^"+S^ ) -"" ( ^"-'+^ ) +e( ^"-'+1^, )-, etc., =0. Let a:-^ — ^c; then, a;2-|-— ^s- — 2, by squaring; also, ( -'+-i ) = ( -^+h )-( -+l ) =(^^-2)-.; and generally ( a;»+^„ "j = (^ x'-' +-^^^ ),t-( x-'+-^^^ ). 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* — 5a;'-|-6a;^ — 5x-\-1^0, to find x. Here a;2-5a;+6- | + 1 = 0, Let a;-| — =s; then, z^ — 5z-|-4=0, and z=4, or 1 ; Putting x-{^— equal to each of these values of z, we obtain for the four values of X, 2±i/3lind ^(Idzi/— 3). The second of these values is the reciprocal of the first, and the fourth of the third, as may be shown, thus : 1 1 2-,/ 3 2-t/3 _ 2+^3-2+^3^2-/3 4-3 EXAMPLES IN RECURRING EQUATIONS. 1. .-r*— 10a;'+26x'— 10x+-l=0. Ans. x=3±2v'2, 2±v/3. 2. x*—^x'-{-2x'—^x+1^0. Ans. x=2, i, iy'^. 404 RAYS ALGEBRA, SECOND BOOK. S. x'—3x'+Bx—l=0. Ans. a:=±l, l(S±yFj. 4. a'— Ila;*+l7a:'+l7a;^— lla;+l=0. Ans x~ 1 ^ + 1 " ^-1/^ S+y-S S-y/S 5. 4a;«— 24a:5+57.r'— 73a:'+57a:^-24a;+4=0. Ans. ^=2, ^ 2, ^, ^±^, -^1^. BINOMIAL EQUATIONS. 443. Binomial equations are those of the form 3/"±A=0. Let . ... '[.'.\=a; th.it is, A=a"; Then 2/"±a''=0. Let . . . y^ax; then, a"a;"±a"=0, Or, . . . a;" ±1=0, which is a recurring equation. 444. — I. The roots of the equation a;''±1^0, are all unequal ; for the first derived polynomial hx"~', evidently lias no divisor in common with a;"±l, and therefore there are no equal roots (Art. 414). II. — If n he even, the equation a;" — 1=0, or a;"^l, has two real roots, -)-l and — 1, and no more, because no other real number can, by its involution, produce 1. By dividing x" — 1=0 by (.■j'-(-l)(a' — l)=j'-' — 1, we have a-"-'^-|- .(■"-*+. . . -|-x*+a;''4-l=0, a recurring equation, having n — 2 imaginary roots. BINOMIAL EQUATIONS. 405 For example, the equation a^=l, or a? — 1=:0 divided ^y a;' — 1 gives x*-f a;^-|-l=0; whence, a;= -4{- -l±l/-^ 2 ^}- This gives for the six roots of 1 +1, -1, W^ 2 3.-V= -1+1/ = 2 =3 W^ -1-1 - 2 3. -V= -l-v/= 2 ::! III. — If n be odd, the equation a;" — 1=:0 has only one real root, viz.: -fl ; for -)-l is the only real number of ■which the odd powers are -|-1. Dividing a;" — 1=^0 by x — 1, we have a recurring equation, having n — 1 imaginary roots. For example, the equation a?^]., or x' — 1^0, divided by X — 1, gives x^-\-x-\-\=.0; whence, a;= -^— — . Hence, the three third roots of 1 are -l+V^-"3 -1-^=3 2 ' 2 ■ IV. — If 11 be even, the equation x"-)-l=0, or x"^ — 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 a;«+l=0 (Art. 442), are V. — If n be odd, the equation x"-\-1^0, 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-1^0, divided by x-\-l, gives x' — x +1=0; whence, .=^^. Therefore, the three third roots of — 1, are — 1, i±-Vr-H, and 1=-^. 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, +1/-1, -i/-l. 2. Find the five fifth roots of unity. Ans. 1, J{l 5-1 + 1 (-10-2i/5)}, .;, 5-l-i/(-10-2, 5)}, -i{v5+l+v(-10+2, 5)!, -i{l/5+l-/(-10 + 2i^5)l. THE END.