IN MEMORIAM FLORIAN CAJORl | Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofalgebrOOclarrich >. @.-4V^ ELEMENTS OF ALGEBRA: XMBRACINO ALSO \)\ji/J^ THE THEORY AND APPLICATION OF LOGARITHMS; TOGETHER WITH AN APPENDIX, CONTAINING INFINITE 8ERIK8, THE GENERAL THEORY OF* EQUATIONS, AND THE MOST APPROVED METHODS OF RESOLVING THE HIGHER EQUATIONS. BY REV. DAVIS W. CLARK, A.M., V FBINCIPAIi or AMBIIIA SBKIIIART • • • • • NEW-YORK: Harper dc Brothers, 82 Cliff-Street. 1843. ^^ .^ ^-^'^ dffJL CAJORI Entered, according to Act of Congress, in the year 1843, by Harper & Brothers, In the Clerk's Office of the Southern District of New-York. PREFACE The object of this treatise is to present to the student a full and systematic course of practical and theoretical elementary Algebra. With this object steadily in view, the author has made no effort for the display of mathematical genius, but has assiduously applied himself to the preparation of a text- book in the science. Believing that original discoveries are not best adapted to beginners, he has satisfied himself with the humble vocation of collecting, arranging, and illustrating the ample materials already provided. But it is due to himself to say that these materials have all been re-wrought, and not a few of them re-written several times. It has been a constant endeavour to make everything explicit, and also to exhibit it in the simplest possible form. By this means, the author has been enabled to embrace, within a comparatively small compass, a more comprehensive view of the science than can be found in any text-book on the subject now in use. Among the works which have shared, and still stare most largely in the patronage of the public, isolated parts or sub- ■Jects are treated with great ability and clearness ; but, in some instances, these works are remarkably deficient, so far as con- cerns any methodical arrangement of the subjects introduced, while also other subjects of great importance are omitted alto- gether. That these books force their way into public patron- age is not surprising, when, on the other hand, those treatises which are systematic in the arrangement of topics are, in general, too theoretical and abstract for the convenience or profit of the beginner, or, indeed, of the practical algebraist. f\A A r>CLCl VI PREFACir. In collecting his materials, the author has consulted the most approved writers upon the subject. It would be difficult, if not impossible, to point out the precise amount of his indebt- edness to each ; yet he does not hesitate to acknowledge it, nor has any desire of appearing original led him to remodel these materials. Indeed, this has been done only when it was necessary in order to preserve the unity of the work, or to render the subjects more explicit. In arranging and digesting these materials, however, the author has been fettered by no adopted system. Whatever seemed most appropriate to his general object, and in keeping with the general plan of his ■work, he has freely made use of, at all times having reference to the wants of our schools, and endeavouring to meet them. How far this object has been attained, he now leaves the reader to judge, claiming only for himself that it is a well- meant contribution to elementary education in an important branch of science. For the article on " Roots of Numbers," as well as for other valuable assistance in the preparation of this work, he is in- debted to the Rev. Joseph Cummings, A.B., lecturer on Nat- ural Science in the Amenia Seminary. In the present work, Algebra has not been regarded merely as an introduction to the higher branches of mathematics, but also as a means of unfolding more clearly the principles and theory of common arithmetic. This is an important consid- eration. A great portion of the students in our academies and schools do not pursue the mathematical course beyond alge- bra. Such, aside from the mental discipline acquired in its study, derive their chief advantage from the superior under- standing it gives them of common arithmetic ; and we speak only the common sentiment of the better-informed school- teachers, when we say that few, if any, are properly qualified to teach arithmetic w^ithout a knowledge of algebra. The au- thor has not, however, limited himself to this object, and be- PREFACE. Til lieves the work will fully answer all the necessary requisitions of an introduction to the higher branches of mathematics. The Logic of Algebra is an object that should not be lost sight of in the study ; and in order that the student may be exercised in this, every important principle has been explained and demonstrated. But, at the same time, the explanations have been made simple, and the demonstrations put in such a form (especially in the first part of the work), that they can be easily comprehended by those unaccustomed to the rigid demonstrations of analytical algebra. In the higher depart- ments of mathematics, it is undoubtedly desirable that the for- mality in stating every proposition, and the course of demon- stration required by the precise rules of logic, should be adhe- red to. But in algebra the case is different. The mind of the student must become gradually habituated to the more ab- stract modes of thinking and precise methods of reasoning ; and, as algebra is commenced in so early a part of the course, a certain degree of familiarity, rather than formality, in the statement of propositions and in their proof, becomes not only excusable, but even necessary. The author, however, has studied precision in the statement of propositions, and endeav- oured to make his reasoning explicit. In this way has he en- deavoured to make the theory obvious and satisfactory. Believing that a knowledge of the general principles of algebra can be perfected and permanently secured only by frequent and rigid application, the author has endeavoured, throughout the work, to blend theory and practice. For this purpose, a careful selection of problems and exercises has been made from the most approved authors. In the ninth section the author has given a clear and con- cise view of the theory of Logarithms, and a method of cal- culating common logarithms, or those in general use, so expli- cit, and yet so simple, that the student well versed in propor- Vlll PREFACE. tion and progression may be able to calculate them with ease and facility. As the last three sections treat upon subjects that are sel- dom called into use by the merely practical algebraist, and yet subjects that are indispensable as an introduction to the higher departments of mathematics, they have been thrown into the form of an Appendix. This work was commenced, and has been- carried to its completion, amid the arduous duties incident to the charge of a large and flourishing seminary of learning. Yet labour and care have been bestowed upon every part of it, and that, too, while the author was daily engaged in instructing classes in this interesting and important branch of study ; and if, under these circumstances, he has been able to discover the wants of the student, and adapt his work to meet those wants, he will feel amply compensated for his toil. D. W. Clark. Amenia Seminaiy, March, 1843. CONTENTS. SECTION L Preliminary Remarks (^ 1-11) 13 Definitions (12-18) U Axioms (19) 15 Algebraic Notation (20-75) 16 SECTION IL Addition (^ 76-61) 25 Subtraction (82-87) 29 MultipUcation (88-99) 31 Diviaion (100-116) 38 SECTION m. Algebraic Fractions (^ 117-126) 47 Discussion of Signs (127-133) 48 Reduction of Mixed Numbers (134) .50 Reduction of Improper Fractions (135) 51 Of the Greatest Common Divisor (136-138) 52 To Reduce a Fraction to its Lowest Terms (139) 58 The Least Common Multiple (140) 60 Reduction of Fractions to a Common Denominator (141) . . . .61 Least Common Denominator (142) 63 Addition of Fractions (143-144) 64 Subtraction of Fractions (145-146) 66 Multiplication of Fractions (147-148) 67 Division of Fractions (149-150) 69 SECTION IV. Of Equations (^151-160) 72 Equations of the First Degree involving only one Unknown Quantity (161-168) 74 Problems (169) 81 Equations of the First Degree involving more than one Unknown Quantity (170-173) 90 Elimir»ation by Comparison (174-175) 91 " by SubstituUon (176-177) 92 " by Addition or Subtraction (178-179) 94 Eqaations involving two Unknown Quantities (180) 95 B X CONTENTS. Problems requiring two Unknown Quantities (181) 97 Equations involving three or more Unknown Quantities (182-185) . .102 Problems requiring " " " «' (186-187) . . 10" SECTION V. Generalization of Algebraic Problems (§ 188-195) 110 Demonstration of Theorems (196-206) 114 Algebraic Demonstration of certain Properties of Numbers (207-213) . .119 Reduction of Formulas relating to Simple Interest (214-217) . . .123 " " " " Compound Interest (218-219) . . .124 " •' '* " Fellowship (220-225) . . . .126 Discussion of Equations of the First Degree (226-231) 129 Theory of Negative Quantities (232-234) 131 Explanation of Symbols. Infinity (235-237) 133 " " Infinitesimals (239-240) 134 " " Indetermination (241-244) 135 Inequations (245-250) 136 SECTION VI. Involution and Powers (^ 251-258) . . 140 Binomial Theorem (259-272) 147 Roots of Numbers (273-288) 155 Evolution of Algebraic Quantities (289-295) .176 Calculus of Radicals (296-314) - . . .184 SECTION VII. Of Equations exceeding the First Degree (<^ 315-318) 199 Pure Equations (319) 200 Problems producmg Pure Equations (321) 208 Affected Equations of the Second Degree (322) 210 First Method of completing the Square (327) 211 Second Method of completing the Square (328-330) 213 Particular Cases of Affected Quadratic Equations (331-337) . . . , 214 Equations 218 Problems producing Affected Quadratic Equations 221 Discussion of the General Equation of the Second Degree (338-351) . . 224 SECTION VIII. Ratio (^ 352-358) . 234 Proportion (359-372) 239 Arithmetical Progression (373-379) 246 Geometrical Progression (380-390) 261 SECTION IX. Theory of Logarithms (<5 391-400) 257 Computation of Logarithmic Tables (401-402) 264 Application of Logarithms (403) 265 CONTENTS. XI Multiplication and Division by Logarithms (40i) 2C0 Involution and Evolution by Logarithms (405) 207 Exponential Equations (406-408) 208 Geometrical Series (409) 271 Compound Interest (410) 273 APPENDIX. SECTION X. Permutations, Arrangements, and Combinations (^ 411-418) . . . 277 General Demonstration of the Binomial Theorem (419-431) .... 282 Continued Fractions (432) 288 Infinite Series (433) 294 Expansion of Infinite Series (434-435) 294 Indeterminate Coefficients (430-437) 297 Summation of Infinite Series (438) 300 Recurring Series (439-444) 302 Method of Differences (445-450) 306 Reversion of Series (451) 312 SECTION XL General Properties of Equations (^ 452-453) . 314 Composition of Equations (454-45G) 318 Transformation of Equations (457-402) 320 SECTION XII. BK80LUTI0N OP THE HIGHER EQUATIONS. Cardan's Method of Resolving Cubic Equations (^ 403^00) . . .332 Young's " " " " (467-472) . . .337 Des Cartes' " " Equations of the Fourth Degree (473-476) . 343 Newton's Method of Approximation (477-480) 345 Resolution of Higher Equations by Trial and Error (481-483) . . .347 Young's Method of Resolving Higher Equations (484) 351 Notes 355 NOTE. The ill health of the author while the work was in course of publication has prevented him from devoting that personal attention to a final examination of the proofs that he desired. This he would offer as an apology for any errors that may have been overlooked. ELEMENTS OF ALGEBRA. SECTION I. Preliminary Remarks. — Definitions. — jJxioms. — .Algebraic Method of J^otation. PHELIMINAEY REMARKS. 1. The object of Mathematical Science is to investigate the relations of quantities and the properties of numbers. 2. Quantity, or magnitude, is a general term, embracing everything which admits of increase, diminution, and meas- urement.* Thus, a given weight or bulk, a sum of money, or a number of yards, are quantities. 3. The measurement of quantity is accomplished by means of an assumed unit or standard of measure. This unit must be of the same kind as the quantity. Thus, the measuring unit of money is one dollar ; of a line is one inch, foot, or mile, &c. ; of area is one square inch, foot, or acre, &c. 4. Jfumbers are symbols adopted to facilitate the investi- gation of quantities. They represent a unit or an assem- blao-e of units. Thus, 35, 42, and 64 are numbers j but $35, 42 cwt., and 64 acres, are quantities. 5. Whole numbers, as 4, 6, 15, 30, &c., are called integers. Broken numbers, as i, |, -j^, &c., are called /ramo;w. 6. Any number which can be divided by 2 without pro- ducing a fraction, is called an even number ; and all numbers which cannot be divided by 2 without producing a fraction, are called odd numbers. 7. Numbers are also distinguished into composite and prime * See Note A. 14 ELEMENTS OF ALGEBRA. [sECT. I. numbers. Any number which can be produced by multi- plying two or more numbers together, each of which is greater than a unit^ is called a composite number, as 4, 6, 12, 20, &c. Numbers which admit of no exact divisor except themselves and unity, are called prime numbers, as 1, 2,, 3, 6, 7, 11, (Src. 8. The foundation of mathematical reasoning is laid in Defi' nitions and Axioms. The absolute certainty of its conclu- sions results no less from the exactness of mathematical definitions, and the clearness and simplicity of its first prin- ciples, than from the nature of the subjects about which it is employed. 9. A Definition, when applied to language, is a brief ex- planation of what is meant by a word or phrase. When ap- plied to "a thing, it is an analysis of its parts or an enumera- tion of its principal attributes ; but this analysis or enumer- ation must be sufficiently extensive and definite to distin- guish the thing defined from everything else. Definitions, in mathematics, are used to determine the meaning of the terms, as well as the signs and symbols used. 10. An Axiom is a self-evident truth or proposition. They are said to be self-evident, because, as soon as enunciated, they produce in the mind a force of conviction that cannot be increased by any subsequent' train of reasoning. This conviction is the result of an instantaneous and intuitive perception of the simple relations involved. 11. By a skilful use of the simple elements of mathemati- cal knowledge, furnished by Definitions and Axioms, we are led on through the most complicated processes of mathe- matical investigation. DEFINITIONS. 12. A problem is a question proposed which requires a solution ; and the problem is said to be solved when the value of the unknown quantity, involved in the conditions of the question, is discovered. 13. A theorem is a general truth, which is to be proved by SECT. I.] AXIOMS. 15 a course of mathematical reasoning called a demonstra- tion. 14. A Itmma is a subsidiary truth previously laid down, in order to render the solution of a problem, or the demon- stration of a theorem, more easy. 15. A proposition is a common name, applied indifferently to problems, theorems, and lemmas. 16. A corollary is an obvious consequence, derived from some proposition already demonstrated, without the aid of any other proposition. 17. A scholium is. a remark made on one or several pre- ceding propositions, to point out their connexion, their use, their restriction, or their extension. 18. A hypothesis is a supposition made either in the enun- ciation of a proposition or in the course of demonstration. AXIOMS. 19. The following is a list of mathematical axioms. The list is incomplete, but sufficiently extensive for our present purpose. 1. The whole of a quantity is greater than a part. ' 2. Quantities equal to the same quantity are equal to each other. 3. If to equal numbers equals be added, the sums will be equal. 4. If from equal numbers equals be subtracted, the re- mainders will be equal. 5. If equal numbers be multiplied by equals, the products will be equal. 6. If equal numbers be divided by equals, the quotients will be equal. 7. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. 8. If a quantity be multiplied and divided by a number, its value will not be altered. 9. If equal numbers are involved to equal degrees, their powers will be equal. 16 ELEMENTS OF ALGEBRA. [SECT. I. 10. If corresponding roots of equal numbers be taken, they will be equal. 11. If to unequal numbers equals be added, the greater will give the greater sum. 12. If from unequal numbers equals be, subtracted, the greater will give the greater remainder. 13. If unequal numbers be multiplied by equals, the greater will give the greater product. 14. If unequal numbers be divided by equals, the greater will give the greater quotient. ALGEBRAIC NOTATION. 20. Jllgehra is that branch of mathematical science in which the relations of quantities are investigated, and the value of unknown quantities determined, by means of let- ters and signs.* 21. Quantities^ in Algebra, are represented by the letters of the alphabet as well as by numbers. 22. The first letters, as a, 6, c, &c., are used to represent the known quantities, 'the last letters, as a?, y, &c., are used to represent the unknown quantities. 23. The use of letters to represent quantities is product- ive of several important advantages. 1. A letter may be made to represent the unknown quan- tity whose value is sought, and then be used in the so- lution of the problem as though its value were already determined. 2. The long and tedious processes of arithmetic may be greatly abridged by the introduction of letters, since a single letter may be made to represent any quantity, however great it may be. 3. The several quantities which enter into the calcula- tion are preserved distinct from each other, in all their combinations. 4. The requisite operations may be performed with much more readiness, and with less liability of mistake, with letters than with numbers. * See Note B. SECT. I.] ALGEBRAIC NOTATION. 17 5. The processes of algebra may be used tp demonstrate theorems and general rules, inasmuch as a letter mjay represent every possible value. 24. The relations of quantities, or the operations to be per- formed upon them, are represented by signs. This method of notation presents to the eye, at one view, the conditions of the problem, and at the same time facilitates the reduc- tion of it. 25. Addition is represented by a horizontal and perpen- dicular line mutually bisecting each other, as +. Thus, a-\-b represents that b is to be added to a, and the expres- sion is read " a plus &." 26. Subtraction is indicated by a horizontal line prefixed to the quantity to be subtracted, as — . Thus, a — b repre- sents that 6 is to be subtracted from a, and the expression is read " a minus i." 27. Multiplication is indicated by a sign formed some- thing like a Roman X, as X. Thus, aXb indicates that a is to be multiplied by b. Sometimes the multiplication is indicated by a dot placed between the quantities to be mul- tiplied, as a.b ; or if the quantities are represented by let- ters, the letters may be written one after another, in alpha- betical order, without any sign, as ab. If numerals are to be multiplied, the sign must be expressed. Thus, 4x 10, or 4.10, without the sign, would become 410. 28. Division is indicated in three ways. 1. By connect- ing the divisor to the dividend by a horizontal line with a dot above and another below it, as —. Thus, a-rb indicates that a is to be divided by b. 2. By making the dividend the numerator, and the divisor the denominator of a vulgar fraction, as ^. 3. Or by placing the divisor to the right of the dividend, and drawing a perpendicular line between them, and a horizontal line under the divisor, as a\b. 29-. To indicate that the difference between two quantities is to be taken without determining which is to be subtracted, a sign like the letter s placed horizontally is used, as^ c/t* C 18 ELEMENTS OF ALGEBRA. [sECT. I. Thus, a (/) 5 represents that the difference between a and h is to be taken. 30. Equality between two quantities or sets of quantities is indicated by two horizontal lines, as =r. Thus, a=6 rep- resents that a is equal to Z*, and is read ," a equals 6." 31. An equation is the algebraic expression of two equal quantities connected by the sign of equality. If the alge- braic quantities are known, the expression is called an equality. • 32. Inequality is indicated by two lines forming an angle, like the letter V placed horizontally, the vertex denoting the less of the two quantities, as >. Thus, ayb represents that a is greater than &, and is read " a greater than Z)." 33. An inequation is the algebraic expression of two un- equal quantities connected by the sign of inequality. If the quantities are known, the expression is called an ine- quality. 34i. Proportion is indicated in the same manner as in Com- mon Arithmetic. Thus, aib'.'.cd represents that the four quantities a, Z>, c, and c/ are proportional, and the expression is read " a is to 6 as c is to c?." 35. A coefficient is a numeral figure or a letter prefixed to a quantity to show how many times the quantity is to be taken. Thus, 4a shows that a is to be taken four times, as o+a+a+a=:4a; and ax shows that x is to be taken as many times as there are units in a. 36. When a quantity has no number prefixed to it, 1 is always understood as its coefficient. Thus, a is the same as la. 37. An Mgehraic expression is a quantity or several quan- tities written in algebraic language ; that is, by the aid of letters and signs. 39. An algebraic formula is a general rule or principle stated in algebraic language ; that is, by the aid of letters and signs. 39. A monomial or simple algebraic quantity is one that SECT. I.] ALGEBRAIC NOTATION. 19 may be represented in an'Rlgebraic expression, without the aid of the signs pltts or minus. Thus, a, 3a6, iab', and lab^mx are monomials. Monomials are sometimes called terms. 40. Polynomials^ or compound quantities, are expressions containing two or more simple quantities <;onnected by the signs plus or minus. Thus, a+3a6 and a-\-*Xb — 3c are poly- nomials. 41. A polynomial composed of two terms is called a W- nomial ; of three terms, a trinomial y of four terms, a quadri- nomial. If the two terms of a binomial are connected by the sign minus, it is sometimes called a residual. 42. To indicate that like operations are to be performed upon all the terms of a polynomial, they must be included in a parenthesis, or have a bar or vinculum drawn over them. Thus, a—(b-{-c) indicates that the sum of b and c is to be subtracted from a; and (a-|-6)Xc indicates that the sum of a and b is to be multiplied by c ; and (^a-^b)^c indi- cates that the sum of a and b is to be divided by c. 43. If both multiplicand and multiplier, or dividend and divisor, are polynomials, each should be included in a pa- renthesis, as (a4-6)x(c-}-c/), or (a-|-6)H-(c+c/). And in general, when a sign is prefixed to a parenthesis, it is to be understood as affecting all the terms included in the paren- thesis, taken collectively. 44. Positive or additive quantities are those to which the sign plus is prefixed. J^egative or subtractive quantities are those to which the minus sign is prefixed. When no sign is prefixed to the first term of an algebraic expression, the sign plus is always to be understood. 45. A quantity is said to be ambiguous with regard to its sign when it is affected with the double sign ± . Thus, a±b represents that b is to be added to or subtracted from a; and the expression is read "a plus or minus A." 46. Equal terms affected by unlike signs, in an algebraic expression, cancel each other, and may be rejected from the expression. Thus, 3a — 56+5^=3«, since — 5^ and -f56 cancel each other. 20 ELEMENTS OF ALGEBRA. [sECT. I. 47. Positive and negative quantities sustain opposite rela- tions with respect to addition ; i. e., a negative quantity must be subtracted when a positive quantity would be addp ed, and added when a positive quantity would be subtracted. 48. The numbers which are multiplied together to form a composite number, are called /ac^'or^. Thus, Wabcx is a composite number, formed by multiplying the factors 11, a, ^, c, and X. 49. A number is said to be resolved into factors when two or more numbers are taken, such that, when multiplied to- gether, their product shall equal the given number. Thus, 54 may be resolved into 6x9, or 3x18, or 2x27. 50. The power of a number is the product arising from the multiplication of the number by itself, till it has been used as a factor a certain number of times. If the number is taken twice as a factor, the product is called the second power ; if three times, the product is called the third power ; if four times, the fourth power, &c.. 51. The index, or exponent, is a figure or letter placed to the right and a little above the number, and is used to show the power to which the number is to be involved. The number is to be used as a factor as many times as there are units in the exponent. When no exponent is expressed, 1 is understood. The first power of a is » - - «, or a\ The second power of a is - - axa, or a^. The third power of a is - - axaxa, or a-^. The fourth power of a is - axaxaxa, or a'^. The mth power of a is ax ax a m times, or a", &c-.. 52. If a polynomial is to be involved, its terms should Be included in a parenthesis, and the exponent placed without the parenthesis to the right. Thus, (a-]-b)- i« the algebraic expression of the second power of the sum of a and b* 53. Involution is finding the powers of numbers. 54. The root of a number is a number which, multiplied into itself till it is taken a certain number of times as a SECT. I.] ALGEBRAIC NOTATION. 21 factor, will produce the given number. The root is called square root, cube root, fourth root, &c., according to the number of times it must be used as a factor to produce the given number. 55. The radical sign, as ^/ , or fractional index, is used to indicate that the root of a number is to be taken. The denominator of the fractional index denotes the root ; and when the radical is used, the figure over the foot of the radical determines the root. Thus, The square root of a is expressed - \/a, or a*. The cube root of a is expressed - y/'a^ or ai. The fourth root of a is expressed - ^ a, or oi The fifth root of a-\-b is expressed, \/a-\-b, or {a-\-b)\, &c. 56. Evolution is finding the roots of algebraic numbers. 57. A power of a root, or root of a power, is a result ob- tained by involving the root of a number, or by extracting the root of a power. Cases of this kind are indicated as follows : The second power of the third root of a is v'a*, or (^a)*i or a!. The third power of the fourth root of o-f-6 is ^(a-|- J)*, or (a+b)i. 58. It should be remarked that the exponent aflfects only the letter over which it is placed. Thus, in the expression abc^, the first powers of a and by and the second power of c, are to be taken. When no coefficient is prefixed to tho radical sign, 1 is always understood as the coefficient. 59. Exponents should not be confounded with coefficients. The exponent indicates that the number is to be used as a factor a certain number of times. Thus, o" represents that a is to be taken six times as a factor, or axaxaxaxaxa =0*. The coefficient indicates that the number is to be used as a term a certain number of times. Thus, 6a repre- sents that a is to be used six times as a term, or a-|-a-|-a-h a4-fl+a=6a. 60. The reciprocal of a quantity is the quotient arising 22 ELEMENTS OF ALGEBRA. [sECT. I. from dividing a unit by that quantity. Thus, the reciprocal of a is 7j of a-\-h is ^ip^ ; and of 4 is J . 61. The reciprocal of a power is the quotient arising from dividing a unit by that power, and is frequently expressed by a negative exponent. Thus, The reciprocal of a^ is - - - ^, or a~^. The reciprocal of 4a^ is - - 4;^, or \a~^. The reciprocal of {a-\-hy is - (^a-\-b)^t or (a+i)~^, &c. 62. Rational quantities are those whose exact value can be expressed in finite terms. Thus, 4a, ^6, and o-|-3Z>, are ra- tional quantities. 63. Irrational quantities^ or surds, are those whose exact value cannot be expressed in finite terms. Thus, since only the approximate value of the square root of 2 can be ob- tained, s/2 is called a surd j also Va is a surd. 64. The measure or divisor of a quantity is thai by which it can be divided without leaving a remainder ; and when a quantity will divide two or more quantities without leaving a remainder, it is called a common measure of those quanti- ties. Thus, la is a measure of 28a, since 7^=4 ; and 3a is a common measure of 12a and 21a, since — =4, and ^1^=7. 65. The multiple of a quantity is that which can be divided by the quantity without leaving a remainder. Thus, 28a is a multiple of 7a, since ^=4, &c. 66. Commensurable quantities are such as have a common measure or divisor. Thus, 12a and 21a are commensurable, because they have a common divisor, 3 or 3a. 67. Incommensurable quantities are such as have no com- mon measure except unity. Thus, 5 and 7, 3a and 10^, are incommensurable quantities. 68. The value of an algebraic expression is the result ob- tained by substituting for the letters their numerical values, and performing the operations indicated by the signs. Thus, the value of 4a— 8Z>, on the supposition that a=i:12 and ^=:5, is 4 X 12—8 X 5=48— 40 = 8. The value of la+s, on the supposition that a=:6 and^= 10, isix6+^l°=3+6=9. SECT. I.] ALGEBRAIC NOTATION. 23 69. The following examples are given for the exercise of the learner. On the supposition that a=6, 6=5, c=4fj d=. 1, and m=10, it is required to find the value of the follow- ing algebraic expressions. 1. a^-\-<2ab+b'=6^+2 . 6 . 5+5^=36 + 60+25=121. 2. 2a'— 3a'6+c»=2.6'— 3.6^5+4»= 3. (a+Z») xa^— 5c(//7i+V' = (6 + 5). 6^— 5 .4 . 1 . 10+^= 4. 4^+a.(2c+c»)— 3w = 5. (4.c'+6').a— (6+3A^).8= 7. 5v/c%^~a*.(3a'— IOot— 7)i— — ^ — = 70. The value of an algebraic expression is not altered by changing the order of the terms or the order of the fac- tors, if its proper sign be prefixed to each. Thus, a+6+c — d is the same in value as — rf+c+ft+a; and aXbXcxd is the same in value as dxcxbxa. For considering the let- ters of the same value as in art. 69, we shall have a+6+ c— . Add ab-^Sy cd~3, 28, and 5crf— 4m + 2. ^ns. ab-\-6cd-\-3b — 4>m, 5. Add babXc'cP, labXc"^, 6abcd, and —3abXc'd^-j-3abcd. Ans. 9ab x c'd^^9abcd. 6. Add 3aa7— 21, 65c-f2, aa;+15— 56c, — 8 + 6ca:— 6c, and llaa;+ 13— 36c. ^ws. 21aa;— 36c+ 1. 7. Add 3m2— 1, Gam— 4m^+8, 7— 9aOT+8, Gm^— 3-f aw, and 4>m^ — am + 12. Ans. ^rr^ — Sam +31. 8. Add \Sx{a^—b^\ —60^0?+ 1262a:, — 10a: (0^—60 +13aX and Sx{a^—V) + 36^0;. ^?i5. 6a;(a2— 6-) + 7a'a:+ 156^3;. 9. Add 4a2+36+2c, — 3a2 + 46 + 8c, Qa^— 76— 10c, and 3a» — 6 + c+aa7. Ans. \Sa — 6+c + oa;. >10. Add 8aa?+2(a?+a)+36, 9aa;+6(a:+o)— 96, and — 7aa;— 8(a;+a)+66+lla:. ^»*. 10aa:+lla?. 11. Add 5a26+12(a— a:)^ 3a''6— 8+9(a— a:)^ 12— 8a26, and --13(a— a:y+3. . Ans. 8(a— a;)-+7. 12. Add 28aXa7+5i/)+21, — 13a''(a:+5y)+18«, — 15a'(a:+ 5y)_8, and —13— 8a. Ans. 10a. > 13. Add72aa:^— Say',— 38aa:^—3ay^+7a3^, 8+ 12a3^^— 6a/ 4, 12— 34aa:'+ baf—^ay\ Ans.—2ay^-^ 20. 14. Add 12a— 13a6+16aa:, 8— 4ffi+2y, — 6a+7a62+12y —24, and 7a6— 16aa:+4m. Ans. 6fl— 6a6+ 14y+7a6'— 16. 15. Add 17a(a:+3a2/)+12a='6V, 8— 18ay— 8a^6V— 7a(a:+ 3a2/), — 4+12a?/— 10a(a:+3a2/)— 4a'6V\and 6ay— 4. Ans. 0. 16. Add Sab's^—Sa'cd, -^7a6V+7a='cc/— 12, 32a6V, and 12-Sa^cd+ab'x\ • Jlns. 29a6V— 40=^0^ gECT. II.] SUBTRACTION. *^, 29 SUBTRACTION. 82. Subtraction is a method of finding the difference be- tween two algebraic quantities or sets of quantities. It is the opposite of Addition. 83. We have already seen that a negative quantity is of an opposite nature to a positive quantity (Art. 4>7), with respect to addition and subtraction : that is, it must be sub- tracted when a positive quantity would be added, and added when a positive quantity would be subtracted. Hence, for subtraction of algebraic quantities, we have the following general RULE. 1. Write the quantity to he subtracted under that from which it is to be taken, placing like terms under each other, 2. Change the signs of all the quantities to be subtracted, or conceive them to be changed^ and then proceed as in addition* EXAMPLES. (1) (2) (3) From Qa—db Uab— 6ac'— 12aj:y Scd?—8axy-i-3bd Take 3a — 4.6 bab~10ac'-— 2axy Ucd^— axy+3bd Ans. 3a— 56 6a6+ 4.ac^— tOaxy — 3cc/^ — laxy 4. From 3a6^— 8aVa:'+26"c take 2a6^+4a^cr'4-3A*c. ^/w. a6^-12a'cr'— 6V. ♦ The principles on which this rule is founded may be stated and de- monstrated as follows I 1. Subtracting a positive quantity will produce the same result; as add- ing an equal negative quantity. Represent the sum of two quantities by - - fl-j-* Taking -f^ away from this expression, there remains a Adding — b to it, we shall have ... a-j.^ — fc=at 2. Subtracting a negative quantity will produce the same result as add- ing an equal positive quantity. Represent the difference of two quantities by - a — h Taking — b away from this expression, there remains a Adding -|-A to it, we shall have - - . a — i-|J-*c=i. 30 ELEMENTS OP ALGEBRA. [sECT. II. 5. From 12ax—19c^b+2abx—a^ take dax—Qc'b—Sabx. ^ns. '^ax—10c%+\0abx—3^ 6. From ^a^b—llcd'^+^y—^aT? take —^a'b—10cd^-\-^y— bax^^2cd\ Ans. la'b—Sax". 7. From ISa'd-^xy-^-d take la^d—xy-^-d-i-hm^—ry. jSns. Qa!^d-{-2xy — hm'^-\-7^y'^, 8. From '7a^bc^—S-\-'7x take 3a^bc'—S—da^-^r. Ans. ^a^he+nx+dx'—r. 9. From l^a'bx^^llb-^c' take \\a'bo^^9b—lc\ Ans. la'bx^+^b—c'. 10. From IGa^iV— 34-48c/a^ take — 4a25V+&a:+12c(/. Ans. '^0a^Wx^—3—bx-\-^Ux—l%cd, 11. From 3a5c— 8a;2/+25H-85take — lla&c+4a;y— 22— 7J. ^;i5. 14ak— 12i:y+4.7+15J. 12. From the sum of 6a?^y — llajj^and 8ar^y+3aa;^ take 4a?^y — ^aar*. ./^W5. lOir^y— 4aar'. 13. From the sum of 15ak4-8ccic — 3 and 24— 8«Z?c+2ccte take the sum of 12aJc — 3cdx — 8 and — 4aJc+cc?a;+16. Ans. — dbc^Vilcdx-\-\3, 14.^ From the difference between 8a& — 12ca; and — 3ab-{- 4iCx take the sum of bab — lex and ab-\-cx. Arts. 5ab — lOcx. 15. From the sum of 4aa;''+2ar'+350, 5aa;'+6a;'+250, and 9aa^+ 12a?=*+ 100, take the sum of 6aa;2+9a:'+432, a3^-{-5x^-^ 328, and 5aa^+a:3+30. Ans. 6ax''+bx''—90. 84. The minus sign, when placed before the marks of pa- renthesis which inchide a polynomial, indicates that each term of the polynomial is to be subtracted, or that the result obtained by reducing the terms, if they are like quantities, is to be subtracted. This is done by removing the marks of parenthesis, and changing the signs of the terms included between them. Thus: 1. 3a— (3c— a?) = 3a— 3c + a?. 2. Sabc — {laic + 3a?— 5) = 3a Jc— 2a Jc— So? + & = a^c— 3a7 -f- J. 3. 4aa? — 3c— 14— (aa?+ 7c — 12):=: 4aa: — 3c— 14— oa?— 7c+ 12r=3aa?— lOc— 2. SECT. II.] MULTIPLICATIOrf, 31 4. 7abc — 13-\-Sabx—(3alc--U-\-9ahx)=labc-^13-^Sabx-^ 3abc-\- 14j— 9aix=4.aic-f 1 — abx. 85. When a number of terms are introduced within the marks of parenthesis, to which the minus sign is prefixed, the signs of the terms should be changed. Thus : 1. 3ax—12a^3b=3ax—(-\-12a+3b). 2. 3abc—Q-\'4'ab+3x=3abc—{6—^ab^3x), 3. lxy—12ab—4y—S—b=lxy—{nab+^y-{-S-\-b). 86. By the above methods, polynomials may be made to undergo a variety of transformations, which are sometimes of great use in algebraic operations. 87. The word ^ddiiion^ as here used, it will be perceived from the foregoing operations, does not always imply in- crease or augmentation, nor does the word Subtraction al- ways imply diminution. Hence the term Reduction has been sometimes employed to express the operations included un- der addition and subtraction. MULTIPLICATION. 88. Multiplication is repeating the multiplicand as many times as there are units in the multiplier. Thus : 1. If a is to be multiplied by by it must be taken as many times as there are units in b, and the expression would become o x i, or ab, 2. If ab is to be multiplied by erf, it must be taken as many times as there are units in cd^ and the expression would become abxcd, or abed. Hence, to multiply letters^ we write them one after the other ^ in alphabetical order. d> If 4a is to be multiplied by 36, it must be taken as many times as there are units in 3h. Thus, 4ax3Z*=4'X<2x3 X 6=4 X 3 X a X i — \^ab. Hence, numerical coefficients must be multiplied together ^ and their product prefixed to the product of the letters. 4. If la" is to be multiplied by 4a', it must be taken as many times as there are units in 4a', and the work may be opressed thus: 7a'x4a'=7aax4aaa=7xaax4x 32 ELEMENTS OF ALGEBRA. [sECT. II. aaa—lx^Xaaxaaa — ^'^a^—^^a^+^. Hence, if the same letter is found in both factors, it is multiplied by adding to- gether its exponents, and their sum is the index of the same letter in the product. 89. With regard to the signs, it should be observed, that if the signs of the two factors are like, the sign of the product will be + j but if their signs are unlike, the sign of the pro- duct will be — . This rule may be illustrated thus: 1. If +« is to be multiplied by -{-b, the multiplication con- sists in repeating -\-a as many times as there are units in -i-b J and, consequently, the product is -\-ab. 2. If — a is to be multiplied by -{-b, the multiplication consists in repeating — a as many times as there are units in -\-b ; and, consequently, the product is — ab. 3. If -^a is to be multiplied by — b, the minus multiplier indicates that the repetitions of -\-a are to be sub- tracted i consequently, the product is — ab. 4. If — a is to be multiplied by — b, the repetitions of — a will be negative ; but the minus multiplier indicates that these repetitions are to be subtracted j conse- quently, the product is -\-ab. 90. It should also be remarked, that if there are more than two factors, an odd number of negative factors will produce — , and an even number -\-. Thus, — ax — bx — c= — abc j for — ax — b=-\-ab, and -{-abx — c= — abc. Again, — aX — bx — ex — d=-\-abcd ; for — ax — h=-]-ab, and -\-abx — c = — abc, and — abcx — d—-\-abcd. 91. The classification of quantities into monomials and polynomials suggests three cases in multiplication, viz. : When the factors are both monomials ; when one is a poly- nomial and the other a monomial j and when both are poly- nomials. CASE I. 92. In this case the factors are monomials, and the signs are like or unlike. SECT II.] MULTIPLICATION. 33 RULE. 1. Multiply together the numejncal coefficients of the facUrrs. 2. To the product of the coefficients annex the product of the letters^ observing that if a letter is contained in both the multl- plican(t and multiplier^ it will be affected mith an exponent in the product equal to the sum of its exponents in the factors. 3. Prefix to the product the sign required by the principle that like signs produce pluSy and unlike signs minus. JVote. — For an illustration of the principles Df this rule, see Arts. 88 and 89. EXAMPLES. (1.) (2.) (3.) (4.) Multiply lOa^bx — laxyz 12t/m'»V —l^a'bc'd'i^ By 3a&V 3abcx — Sd^fmn — Sabx Product lo^l^' ^2Wbcx'yz —96d]fm'n'x* +39o'^Vrf*x' (5) (6) (7) Multiply Sa*h'<^d^a^y* — HoVyz* — ^Sa^z By la'bc'd'xY 12 — l^a'b'cdJ'xyz^ Product 36a'6Vc/Vy' — 204aVyz* +336a'b'cd'xy'2^ 8. Multiply lla^bcdhy 4V, and that product by —1c?b(?. Ans. — 48a«i>V. 15. Multiply — la^x^f by ^ac^xf, and the product by — 4a- xf. Ans. +64aVy'. 16. Multiply — lOJc* by — 3^>*cV, and that product by — 4tVxy. Ans. — 120Z;'cVy. 17. Multiply the product of ISr'y and 5a:^y- by the product of xY and 3xy. Ans. 270a?"y\ 18. Multiply IZa'Vcd' by 10a«ft»a:. E 34 ELEMENTS OF ALGEBRA. [SECT. II. 19. Multiply ^la'b'c'd'x by —12adnxK 20. Multiply — 16aV by —16. 21. Multiply 6ada by 3a, and that product by 12a^JV. 22. Multiply — lax by ISaix, and that product by — 12 a'ccy, 23. Multiply —6c^x'z' by — 12a='a?, and that product by —2ax'z\ 24. Multiply the product of Kax and 3axz by the product of 8ak^and 2aV/. 25. Multiply the product of ISaz and 2a^a? by the product of 6aa? and — da'^y^z'^, 26. Multiply the product of — Sax and — 12aa? by the pro- duct of — 4c'— ISxy by 3ax, Ans. 33a%c^x—3^a:^y, 8._ Multiply 42c^— 1 by —4. .y?;w. — 168c^+4. 9. Multiply — 30a'fta?V+ 13 by —So*. ^715. +150a55a;2y— 65a^ 10. Multiply the product of a-\-b and 3c by 8ax. .^?w. 24a^ca:+24a&ca7. 11. Multiply the product of 2a-\-3b — 4c and — 2a by %abdx. Am. —32o?bdx—^>'^a^b^dx-\-^^>a%cdx, 12. Multiply the sum of 3aJH- 10 and a2>— 8 by 6aa?. Ans. 24a'*a?4-12aa?. 13. Multiply the sum of 12aJx — %ad — 3b and %abx — ad-\-b by 3a. Arts. 60a''bx—27a'd—6ab. 14. Multiply the difference between 12a — Ibdx and 8a+ bdx by 6abdx. Ans. 2Wbdx—4,Sab^ by a—b, ^ns. a^—V, 10. Multiply 2a— i by 3a'— 1. .^w*. 6a»— 3a'fr— 2a+J. 11. Multiply 3ai*— 6 by a+4. Ans. Sa^J'- 6a-f-12a&«— 24. 12. Multiply 6a4-46 by 3a— 2Z». Ans, 18a«— 86«. 13. Multiply 7aZ>c^+3xy+l by %a%K Ans. b^a^'c^ -}- 24a'^&'a:y + 8a»**. 14. Multiply 4a'ftr'+ Zed by 3ctir, and that product by 4a -f*. Ans. 48a'k(ir*-f 36ac'(/2a?+ na^bhdx'-\-Uc^d'x. 15. Multiply a-\-b-\-c by 8aZ>, and that product by a-\-b. Ana. 8ff'64- 16a«6'+8a-6c + 8a6'-f-8a6»c. 16. Multiply 2a4-46 by 2a— 46. Ans. ^a^—\U\ 17. Multiply ar'+x^y-f-xy^+y' by x — y. Ans. x* — y*. 18. Multiply x^-faryH-y^ by x^ — 9a'bK Ans. —2abK 15. Divide 620j?y by 4^xYz\ Ans. lb5xy'z-\ 16. Divide —15a'b' by ba'b'c'd. Ans. —3aWc-'d-\ 17. Divide m^a'b-'d' by 12a'6Vc?'. Ans. U4>a-^b'c-^d\ 18. Divide 684a?' by — 12a?-^ Ans. —57a?'. 19. Divide 328007^^^ by 4>0xyz\ Ans. 82a7yV. 20. Divide 62a'6' by 31a-'b-^d\ Ans. 2a'b'd-\ 21. Divide the product of Sa'bd' and laWdx, by 2Sa'bdx, Ans. 2aWd^. 22. Divide the product of ^a'b'c'd'x' and —lOabc'd by —2a' ^►V. An^. -h25ac^(^V. SECT. II.] DIVISION. 41 23. Divide the sum of 12a*AV and 8a^6V by the sum of 3a6V+7aZ>V. J^ns. 2a^bc, 24. Divide the difference of 21a'iV(/-V and 15a^b*c^(Px* by 6a'bcdx, JJns. aU'c^da^. 107. It sometimes occurs in the operations of division, that a letter becomes affected with the exponent 0. We will, therefore, explain that symbol. Thus, a^-ra'=a*~*=:a° ; or, again, a"'-ra"*=a'"^^=a° j but ^=1, and °m=lj there- fore, since a may represent any quantity whatever, and m any exponent whatever, every quantity affected with the ex- ponent is equal to 1. This will also explain why a'^^h'"—arh-'^ (Art. 103) j for, multiplying the divisor and quotient together, &"*xa'"i~'"= CASE II. 108. In this case the dividend is a nolynomial and the divisor a monomial. KULE. 1. Divide each term of the dividend by the divisor^ and the re- sulting quantitieSy connected by their proper signs, will be the quotient. JVote. — That each term of the dividend should be divided by the divisor, is evident from the fact that when a polyno- mial is multiplied by a monomial factor, that factor enters into every term of the polynomial. Thus, (a-^b-\-c)xd= ad+bd+cd; hence, (ac/+W+c(f)^c/-'5^±^:^^+?+?=:a-f i+c. EXAMPLES. (1) (2) (3) Divide 2ab+6bc 12a?'y4-39aV/ 42aVx+ ISflWo?' By 2b 3x'y —Qac'x Quotient a -{-3c 4 4.13ay —la' — 3 Wo?* (4) (5) (6) Divide 12a'b'c—21a''b'c* lDaxy-{.12cd 9aWx—3abc- By 3abc' 3 3ab(^ Quotient ^-^bc-'—lb'c' 5axy+^cd 3aJVx— 1 F 42 ELEMENTS OF ALGEBRA. [sECT. II. 7. Divide lUd'b^c'^Sc^b'c'' by 8a'/>V. ^ns. 9abc—(^, 8. Divide 35dm-\-Udx by Id. Jlns. ^m-\-1x, 9. Divide 4aa:y— 4a+16ac? by 4«. Ans, a?y— l4-4c?. 10. Divide 3aa:='+6:c'+3aa;-15a? by 3x. Ans. 007^4- 2a: + a— 5. 11. Divide 3a'bc-\-l2abx—3a% by 3aJ. c/^7^.s. c+A>x—a, 12. Divide 25a^j£c— 15axa;^4-5aJc by — 5aa:. Ans. — bab-\-3acx — bcx~\ 13. Divide 20ah^+15ab'+10ab-^5a hy 5a. ^?i5. 4&^+3'^+2S+l. 14. Divide the product of 9a%^-\-la'x' and 4aVJ- by 2a^b. Ans. lSa^c''d'-{-Ud'b~'c'd'x'. 15. Divide the product of 12a'^^a;='+3Ja:' and 8a:' by 24a^x^-\-x^. 16. Divide the product of Gaxy-^-lc^bc^dx and 6x'^-\-4>'if by 2aa?. Jgns. lSx'y+21a^c'dx^-{-12y'-{Ua^bc'df, CASE III. 109. In this case the dividend and divisor are both poly- nomials. RULE. 1. Arrange the terms of the dividend, and also those of the divisor, with reference to the power of some letter, so that its ea?- ponents shall diminish from left to right. 2. Divide the first term of the dividend by the first term of the divisor ; the result is the first term of the quotient. 3. Multiply the whole divisor ly this term, and subtract the product from the dividend ; the remainder will form a new divi- dend. 4. Divide the first term of this new dividend by the first term of the divisor ; the result is the - second term of the quotient. Multiply the divisor by this term, and subtract as before. 5. Proceed in this manner till the dividend has been exhausted, or till no term of the remainder is divisible by the first term of the divisor. JVote 1. — In applying the above rule, we find, successively, how often the divisor is contained in parts of the dividend, for the reason that, as the dividend is made up of all its parts. SECT. II. J DIVISION. 43 the divisor is contained in the whole as often as it is con- tained in all its parts. JVb/e 2. — If the first term of the dividend is not divisible by the first term of the divisor, after the terms in each have been arranged, the division is impossible. J^ote 3. — If the dividend is exactly divisible by the divi- sor, the dividend will be completely exhausted, leaving no remainder. But if it is not exactly divisible, the division may be continued till the first term of the remainder is not divisible by the first term of the divisor j the remainder should then be placed over the divisor so as to form a frac- tion. J^ote 4. — It will not in all cases be necessary to bring down all the terms of the dividend to form the first re- mainder. EXAMPLES. 1. Divide a'4-2a6+*' by a-\-b. Dividend, a'+2oZ>+6'|a-f-6^ Divisor. a' 4- ab a+^, Quotient. Proof. (a+6)x(a+6)=a'+2a6+6^ 2. Divide 12a'*6*— 6a^*'H-8o'6'— 4a»6*— 22a«i+5a^by 4a«i»-h 5fl*— 2a'A. Dividend arranged. Divisor arranged. 5a^— 22a«6+12a^ft»— 6a^y— ■4a»y+8a'6^ |5a^-2(r»6+4fl'y Sa"'— 2a%-\- 4a'6' Quotient, a'—4>a'b-{-2b^ •— 20a«6-h 8aV/— 6a^i»— 4aV+8a«6* — 20o^6-h Sab'~'i6a'b' • ♦ -{-lOa^b^-U'b^+Sa'b' lOa^b^—W'b^+Sa^b' Proof. (5(1*— 20^6+ 4a'i>^) xCo*— 4a'JH-4a'J»)=5a'— 22a*'6-f 12a^ J^— 6a*i»— 4a'i*4- Sa'b\ 44 ELEMENTS OF ALGEBRA. [SECT. U, 3. Divide 0=^— 1 by a— 1. Dividend. Divisor, a^—l |g— 1 o' — a^ a^+a+1, Quotient. -fa^— 1 a'^ — a + a— 1 a— 1 4. Divide a® — b^ by a— &. ■i-a'b^—a^b' ~\-a'b'—b' a'h^—d'b^ -{-a'b'—ab' +ab'—b^ ' ab'—b^ 5. Divide a^-^-^a'b-^-^ab^-^-b^ by a-{-b. Ans. ct'^^^ab^y", 6. Divide a='+2a'6+2aZ''+Z'=' by c^^ab^bK Ans. a+b, 7. Divide x'—^x'-\-'2nx—rt by a?— 3„ Ans. a?2_6a'+9. 8. Divide a?^+y^ by x^y. Ans. x'^—xy+y\ 9. Divide 6a=^a?~9aV— a^+4a?* by a^+Sa?^— 3aa?. ^7i5. 2x'^-\-3ax—a^. 10. Divide 6aa7+2a?2/ — Sab — by-\-3ac-{-cy by Sa+y. 11. Divide 2a?^--19a?2+26a?— 16 by a?— 8. Ans. 2a?'— 3a?+2. 12. Divide 3/'+ 1 by y4- 1. ^^^5. y^—f+y^—y+ 1- 13. Divide /—I by y—1. Ans. y'+y'+y'+^+y+l- 14. Divide a?^ — a' by x—a. Ans. x+a. SECT. II.] DIVISION. 45 15. Divide x* — a' by x — a. Jlns: ar*-|-aa?'-fa'x-|-a'. 16. Divide --15a^ + 37a'*c— 29a'tir-— 206V+44.k(ic— 8J^x* by Sa'—bbc+dx. ^ns. — 5a'H-46c— 8fir. 17. Divide 3a*— 8a^6'4-3aV+56'— 36V by a^— ^. Jlns. 3a2— 5d*4-3c». 18. Divide 20a'— 41a*6-|-50a='fr^— 4.5a-6^H-25a6*— 6/»' by 4a' — 5aZ>4-2A-. ^ns. ba'^A^a^b+bab'—Sb^ 19. Divide 9a:«— 46x'-f 95x^+1 50x by x^—4>x—5. Arts, 9a:*— 10x^+5x2— 30a?. 20. Divide 6x*— 96 by 3a:— 6. Ans, 2ar'+4x-+8x+16. 21. Divide 4324-1152Z»VH-576iV by 6+126V. Ans. 72+48^>V. 22. Divide 8aV— 8a''6a?^4-8a''cx^— 11 a^^ +116-^—11^2 by o— 64-c. ./?»5. 8aV— II62. 23. Divide 6x''—5xy—6xy+6ar'y'+15yx='—9xy +10x^3/* H- 15y' by 3x»+ 2xy + 3/. Am. 2x''— 3x '3/-+ 5^. 24. Divide a''+8a'6+28a*'6«+56a'6''+70a*6*+56a^6'+28a»6« +806'+*' by a*-|-4a'6+6a-6^4-4a6»4-6\ ^»5. a*+4a'6+6o^6'+4a6»+6*. It is sometimes desirable to resolve a polynomial into its original factors. The principles of division enable us to do this ; for, having obtained one of the factors by inspection or trial, the other may be obtained by division. Thus, 1. 4a6c-|-4axy-f 4a6(/=i4a(6c+xy+W). 2. 72a'6+4a6c=4fl6(18a+c). 3. 8a-cx— 18acx'+2ac''y— 30aVx=2ac(4ax — 9x»+c*y— 15 flVx). 4. x'+2axH-a'=(x+a)x(x+a). 5. X*— 2ax4-a^=(a:— fl)x(x— a). 6. 42x''y— 28x^y = 7xy6x-4). In the multiplication of compound quantities, when the signs are unlike, some of the terms disappear or are cancel- led in the product. These terms will reappear in the divis- ion, so that the quotient frequently contains more terms than the dividend. Thus, (a«— «ur+x") X (a+6)=a»+a?». 46 ELEMENTS OF ALGEBRA. [sECT. II. But {a'^-\-cc^)-^a+x=za'—ax-]-x\ 110. The division of quantities -may sometimes be carried ad infinitum. In such cases it will be sulSicient to write out a few of the leading terms. Thus, ,, ' 1 \ l—x 1 — X l-{-x-{-x''-\-x^j &c. » -^oc -{-X — x^ ^ +x' -^x^ — x^ 111. In multiplication, the multiplier is always considered a number, but the multiplicand and product may be either numbers or quantities. In division, we have the product and either one of the factors to find the other. Hence, the dividend and divisor may be either numbers or quantities. 112. If the dividend and divisor are both numbers, the quotient will be a number. Thus, 12-r-4=:3. 113. If the dividend is a quantity and the divisor a num- ber, the quotient will be a quantity of the same kind as the dividend. Thus, 12 rods-r4 = 3 rods. 114. If the dividend and divisor are both quantities, the quotient will be a number. Thus, 12 rods-^4 rods — 4. 115. From the nature of division, it is evident that the value of the quotient depends upon both the divisor and dividend. If the dividend be multiplied while the divisor remains the same, the quotient will be multiplied. Thus, ab-T-b=za; but multiplying the dividend by m, abm-i-h:^am. Dividing the dividend, while the divisor remains the same, divides the quotient. Thus, aim^h=am ; but, dividing the dividend by m, ah^h=a. If the divisor be multiplied while the dividend remains SECT. Ill ] ALGEBRAIC FRACTIONS. 47 the same, the quotient will be divided. Thus, abm-rb=am ; but abm-7-b7n=a. Dividing the divisor, while the dividend remains the same, multiplies the quotient. Thus, abm-r-hmziza ; but abm —b=am. 116. The student will observe that there is a striking re- semblance between the division of compound numbers in algebra, and what is termed "long division" in common arithmetic. But this essential difference should be noted ; the several terms are so independent of each other, that af- ter the first term of the quotient has been obtained, and the first remainder brought down for a new dividend, an en- tirely new arrangement of the terms, with reference to a different letter from that first assurfied, may be made in both the divisor and dividend, and the division completed under this new arrangement without afl^ecting the value of the quotient. SECTION III. •Algebraic Fractions. REDUCTION OF ALGEBRAIC FRACTIONS. 117. Algebraic Feactions are perfectly analogous to vul- gar fractions in common arithmetic. They express a part or parts of a whole number, or unity. 118. The denominator shows the number of parts into which the unit is divided j the numerator shows how many of these parts are taken. 119. Every case in division may be expressed in a frac- tional form, the dividend being used as the numerator, and the divisor as the denominator. 120. The denominator and numerator, taken together, are called terms of the fraction. 48 ELEMENTS OF ALGEBRA. [sECT. lU. 121. A pr op kJ" fraction is one whose numerator is less than its denominator. Example, ^Zl. a-\-b 122. An improper fraction is one whose numerator is equal to, or greater than, its denominator. Example, SjI-. a — b 123. A mixed number is an integer or whole number con- nected with a fraction by the sign plus or minus. Exam- ple,.a+-r c 124. A compound fraction is the fraction of a fraction, the simple fractions of which it is composed being connected by the word of Example, _ of -. b d 125. The value of a fraction, is the quotient resulting from the division of the numerator by the denominator. Hence, if the numerator equal the denominator, the value of the fraction is a unit ; if the numerator is less than the denomi- nator, the value is less than a unit j and if the numerator is greater than the denominator, the value is greater than a unit. 126. The principles involved in the reduction of Algebraic Fractions are the same as those applied in arithmetic. It will, however, be necessary to trace out those operations in accordance with the method of notation adopted in Algebra. CASE I. DISCUSSION OF SIGNS. 127. The sign that is prefixed to the horizontal line drawn between the numerator and denominator, determines whether the value of the fraction is to be added or subtracted. 128. A sign prefixed to one of the terms of the numerator or denominator affects only that term. 129. If the ^\gTi prefixed to the fraction be changed from + to — or from — to +, the value of the fraction will also be changed from + to — , or the contrary. Thus, + — —-{-ab^b — a ; but — — = — ab~b= — a, b b SECT. III. J ALGEBRAIC FRACTIONS. 49 Again, 4-^t^=(a^-hac)-«=4-ft+c; but-f*±^=-. a a ((ai-|-ac)-ra)=— (A+c)=— 6— c. And, ^±Z^=(ab-ac)--a=b^c; hut --±I^=^(iab a a — ac)^a)~ — (6— c)=— A-hc. 130. If the sign prefixed to the several terms of the nuvterator be changed from -f- to — or from — to -|-, the value of the fraction will be changed accordingly. Thus, ±^=-f-a5-f-a=:6; but 11^=— a6-^a=-^. a a Again, e^±ff=(aA-f ac)--a=A+c,- but =±=^=(-afr- OL a ac)-r-a=—b—c. And, — —Z^ =--((a5— ac)H-a)=— (J— c)= — h-{-c; but a a 131. If the sign prefixed to the several terms of the denomina" tor be changed from + to — or from — to +» the value of the fraction will be changed accordingly. Thus, —=ab-i-a=bi but — =ab-r- — a= — b, a —a Again, ±tJ^=b+ci hat'tt^^-b-c. a — a a — fl 4-c)=+fr— c. 132. If any two of the above changes are made, the value of the fraction will not be altered. 1. If the signs before the fraction and also before the several terms of the numerator be changed from -f- to — or from — to +» the value of the fraction will re- main the same. Thus, ^=&; and -3^=— (—06 -r a) =—(—6) =+6. a a 5 6 60 ELEMENTS OF ALGEBRA. [sECT. III. Again, ?*±^=J+c;,and-=:?*=ff=:-(-J_c)=+J a a +c. Also,^^-5-c; and-=±t^=_(-6+c)r=+^^. a a 2. If the signs before the fraction and before the several terms of the denominator be changed from + to — or from — to +, the value of the fraction will remain the same. ^ . Thus, ^=h; and —— =—(ab-^—a)=—{—b) = -j-b. a — a Again, ^^=h+ci and -?^?-'=-((aJ+«c)-r-a)= a — a —(^—b—c) = +b+c. Also, ~^ =b — c; and — ^ ^ — — ((ab — ac)-i- — a) = a — ct ^(—b-\-c)=:-\-b—C. 3. If the signs before the several terms of both nuiperator and denominator be changed from + to — or from — to +, the value of the fraction will remain the same. Thus, — =J; and =:—ab— — a=:-\-b. a — a Again, f^±l'=S+c; and Z:Stz^=+b+c. a — a Also, ?L-=ff=6-c; and -^+'"==+I>-c. a — a 133. Hence, to make a negative fraction positive without altering its value, change the sign before the fraction, and also before all the terms of the numerator, rpi a-\-b , — a — b Thus, — f-.= + — -J- ; c-{-d c-{-a A ■, a—b-\-d , — a-\-b — d^ ^ ' "SM "^ Sabd ' Also,~i?^+=:l?=-2. ' 6 6 CASE II. IS^. To reduce a mixed number to an improper fraction. SECT. III.J^ ALGEBRAIC FRACTIONS. 51 RULE. 1. Make all the fractional parts positive, 2. Multiply the quantity to which the fraction is annexed^ by the denominator of the fraction, and connect the product^ by its proper sign, with the numerator. 3. Under this result write the denominator. EXAMPLES. 1. Reduce ^o'x-f — i^ to an improper fraction. ^ab ^^ Sa'bx-hSa^-i-bx ^ab 2. Reduce 3a+ — 21 — to an improper fraction. Ans. "^-3aa:+x» 3a"— a? 3. Reduce 2x+y — =L to an improper fraction. 2x— y Ans. 2a?— ^ 4. Reduce a — b — — - — to an improper fraction. 5. Reduce 8 — b — to an improper fraction. Ans. 5Q-6^-« 6 6. Reduce 3a +9 — ^ — to an improper fraction. 3+a 3+a CASE m. 135. To reduce an improper fraction to a whole or a mixed number. RTTLE. 1. Divide the numerator by the denominator; the quotient will be the integral part. 52 ELEMENTS OF ALGEBRA. [SECT. III. 2. If there, he a remainder^ write the divisor under it, and con' necty by its proper sign, the fraction so formed to the integral part, EXAMPLES. 1. Keduce to a whole quantity. Ans. 5a^ — 3a^x. 2. Reduce ^ ■" ^ "*" to a mixed quantity. a-\-b Ans. a-\-h a-^-h 3 Reduce 1^^^+^^!^!=:^ to a mixed quantity. 4a6 Ans. c-f2a6^— ??. ^ 26 4. Reduce ^^^^+^^'~"^ to a mixed quantity. ^ws. 2a +1 — — =• 2jc* 5. Reduce to a mixed quantity. ct — ax -{-or Ans. a--f-aa; — — 1 -. ce- — aaj-f-ar 6. Reduce to a whole quantity. a — Ans.(^-\-ab + i^, 7. Reduce" — to a ftiixed quantity. a^ — ab^ Ans. a'+b'— ^"'^^ ^ 8. Reduce ^"'^~^^^'^' ^ to a mixed quantity. 2a^ic^ a^ X aV CASE IV. 136. To find the ^eatest common divisor of two numbers. In order to obtain a general rule for finding the greatest common divisor, we must observe : SECT. III.] ALGEBRAIC FRACTIONS. 53 1. If two numbers are respectively divisible by a third, their sum or difference will also be divisible by the same number. Thus, if a and 6 are each divisible by c, a 4-6 and a — b will also be divisible by c; for, if c is contained in a eight times and in b twice, in a-\-b it will be contained ten times, and in a — b six times. 2. If any number is divisible by another, every multiple of that number will also be divisible by the other. Thus, if a is perfectly divisible by b, 2a, 3a, 4a, or ma will also be divisible by b. 3. Hence, if t\yo numbers are divisible by a third, the dif- ference between the larger and any multiple of the smaller of these numbers must also be divisible by that third number. Thus, if c is contained in a eight times and in b twice, it will be contained in 3b six times, and in a — 3b twice. 4. Also, if the larger of two numbers having a common divisor is divided by the smaller, the remainder will be divisible by the common divisor. 137. From the preceding principles we deduce the follow- ing general rule for finding the greatest common measure. BULB. 1. Divide one of thk given numbers by the other. 2. If there be a remainder , divide the first divisor by this re- mainder, 3. Continue to divide in the same manner till there is no re- mainder ; the last divisor will be the greatest common measure. Jfote 1. — If, in the course of the reduction, one factor is found to be common to all the terms of one of the quanti- ties and not of the other, this factor may be cancelled ; for, since only one of the numbers is divisible by it, it cannot be a factor of the common divisor. J^ote 2. — For a like reason, the dividend may be multi- plied by a factor which does not contain a measure of the divisor. 64 ELEMENTS OF ALGEBRA. [SECT. III. EXABIPLES. 1. Find the greatest common divisor of d^ — a^x-^-daa^ — 3af^ and a^ — 5aa;+4a;2. Finst Division, 4>a^x — aa^ — 3a^ Wx—20ax'-i-16x'' Dividing by 19a;') 19ax^—19x' a — X Second Division* a^ — 6ax-{-4>x^ \a — x a^ — ax a — 4x^ — 4aa:+4a?^ Hence, the greatest common divisor is a — x. 2. Find the greatest common divisor of a^ — ab^ and c^-{- 2ab-\-b\ Dividing a^ — a¥ by a, we obtain a^ — &^. First Division. a^+2ab+ b' \a'-^b^ a' — b' 1 Dividing by 2^') 2ab-i-2b^ a -\- b Second Division* a^—b' \ a~{-b a'^ab a — b —ab—b^ —ab—b^ Hence, the greatest common divisor is a-{-h. SECT. III.] ALGEBRAIC FRACTIONS. 55 3. Find the greatest common divisor of a* — cc*' and 0*4- c^x—ax^—x^. First Division, a* — X* \a^-{-a^x — aa^ — a? o^-\-a'x — a^x^—axr^ a — x — a'x+aV-l-ax* — ap* — a^x — aV-|-ar''+ x* Dividing by 2x^) 2aV —2a?* Second Division, {^-{-a'x—ax'—sc' |a^— ^ fl* — ox* a -f-op a^x — X* 0*0? — x^ Hence, the greatest common divisor is a^ 4. Find the greatest common divisor of a* — b* and o* — ft*. First Division. a^— b' a'— ah' \a^-lP a a -b Second Division. a»— a-'ft (y-^ab^b^ a'b^ab' ab'^y^ ah'— IP Hence, the greatest common divisor is a — b. 55 ELEMENTS OF ALGEBRA. [sect. III. 5. Find the greatest common measure 6( 3a'* — 6a^b-\-5a%^ —6ab'+2b' and 6a'-{-8a'b—llab'-\-2b\ First Division. Multiplying by 2, 3a*— 5a^64- oa^b^— 5aZ>'+ m Qa*—10a'b-\-10aW—10ab^-{- 4Z>''| 6a^4- Sa^b—Uab^ + 2b^ 6a*+ Sa'b~na'b'+ 2ab^ a— 3b — lSa'b-^21a%^—12ab''-{- 4M —18a'b—24>aW+33ab^— 66* Dividing by 5Z>2) 45a'^>'— 45a6^+ 106* 9^2 __ 9^j ^. 262 Second Division. Multiplying by 3, 6a=^+8a'6— lla62+ 26= 18a='+24a^6— 33a62+ 66^ ISa''— 18a'64- 4a62 da"— 9a6+262 2a +146 Multiplying by 3, 42a^6— 37a6^4- ^^ 126a^6— llla62+186=' 1260^6— 126a62 4-286'* Dividing by 56^) 15a6^— 10^ N 3a — . 2^ Third Division. 9a*— 9a6+262|3a— 26 .^ 9a^— 6a6 3a— 6 — 3a6+262 — 3a6+262 rience, the greatest common divisor is 3a — 26. 6. Find the greatest common divisor -of a* — 6* and a* — a*6 — 06^+6^ Jlns. a^—l^. SECT. III.] ALGEBRAIC FRACTIONS. 57 7. Find the greatest common divisor of Scr* — Sa^x+as? — X* and W — 5ax-\-x^. Ans. a — x, 8. Find the greatest common divisor oi W — 2a' — 3a+l and 3a'— 2a— 1. *^ns. a— 1. 9. Find the greatest common divisor of 0^+90^4-270 — 98 and o»-|- 12o— 28. Ans, ar-2. 10. Find the greatest common divisor of 36o'5' — 18a'J* — 27o*^>"+9o*6* and '21a'b^—lSa*b*—9a'lf'. Ans. 9o*^>"— 9o'6«. 138. The greatest common divisor of more, than two num- bers may be obtained by finding, in the first place, the greatest common divisor of two of them, and then of that divisor and the third, and so on. The last divisor thus found will be the greatest common divisor of all the quan- tities. EXAJIPLE. Find the greatest common divisor of o* — J*, a'+2a'J4-2a6^ + &», and a*^a^b'+b\ First Division, o'+2o*&-f 2a^»»+ 6' • |o^— y a" — &» 1 2J). 2o'6+2aA«-f2ft* a* -\- ab-{- l^ Second Division, o'+a'J+o^ o— 6 t^b—ab^—h' Hence, the greatest common divisor of the first two num- bers is a*-{-ab-\-b^, H 58 ELEMENTS OF ALGEBRA. [sECT. HI. Third Division. —a^b+¥ —a^b—a'b^—aW aW+aW-\-b'' a^^J^ab^-^b^ Hence, the greatest common divisor of the three num- bers is a^-\-ab-\-b^. CASE V. 139. To reduce a fraction to its lowest terms. RULE. Divide the two terms of the fraction by their greatest common divisor. JsTote. — To show that the value of the fraction will not be altered by the operation indicated in the preceding rule, we will demonstjate the following theorem : Theor. If both terms of a fraction be divided by the same quantity^ its value will not be altered. Let aim and am represent the numerator and denominator of an algebraic fraction of any assignable value, m repre- senting any whole or fractional number whatever : Then the fraction ^!L^—dbm^am'—b. am Dividing both terms of the fraction by the indefinite num- ber w, and reducing — —ab-^a—h, a Hence (by Ax. 2), ^--=.— ', which was to be demon- am a strated. EXAMPLES. 1. Reduce _- — to its lowest terms. Viiab SECT. III.l ALGEBRAIC FRACTIONS. 59 ■* • The greatest common divisor is 6a; hence, ^^ \%ab -———1-^=-—, which is the simplest fdrm of the fraction. 18aZ>~6a 36 ^ 2. Reduce ill^ to its lowest terms. *dns. ?^ 2\aa* 3 3. Reduce ''"^^^'-'*°'^'f +^'''^^ to its lowest terms. The greatest common divisor is 7a'6V. Ans. _^Z1 Jl_. 4>. Keduce - — __ to its lowest terms. Of, 5 The greatest common divisor is 9aa:. Arts, ^ 5. Reduce 5«^^^4- lOa^ar^ ^^ j^^ 1^^^^^ ^^^^^^ aV+2aV The greatest common divisor is a*a?*+2aj:*. Atis. — . a «' 1? ^,««^*— Sax'— 8aV4- 180*0?— 8aV . , 6. Reduce — —L to its lowest terms. ar*— aj^— 8a«x-f-6a» The greatest common divisor is 3?-\-^lax — 2a'. X — 3a to its loTvi^st tftrms. 2i'+3a^>-}-a» p» Ti 3 2a&' — a'J — a* , . i 7. Reduce — — — to its lowest terms. Am, ±Z±, 6+a b. Reduce ! to its lowest terms. 6aj7 — 8a 2 9. Reduce — ^— to its lowest terms. Arts, ^ . a*+i* 1 10. Reduce ^^+2aV+2a^+a:^ ^^ .^^ j^^^^^ ^^^^^ 5a*+5a'a? 60 ELEMENTS 0^ ALGEBRA. [sECT. III. CASE VI. 140. To find the least common multiple of two or more numbers. The least common multiple of two or more numbers is the least number which can be divided by each of them without a remainder. The reason for the foUoAving rule will be suf- ficiently obvious without farther illustration. RULE. 1. Resolve the numbers into their prime factors. 2. Select all the different factors which occur ^ observing^ when the same factor has different powers, to take the highest power. 3. Multiply together the factors thus selected, and their pro- duct will be the least common multiple. \ EXAMPLES. 1. Find the least common multiple of Sa^cc^y, l^a^b^x^ and IQa^'^cx. Resolving them into their prime factors, 8a Vy = 2^* X a" X 0?^ X y * na%'x=2'xa'xx xb^X^ 16aWcx=2'Xa'xx xb^XC The different factors are 2", a*, x^, y, Z>', 3, and c. Hence, the least common multiple is 2''x3xa''x2>*xcx x^'Xy—4fSa'^b^cx'^y. 2. Find the least common multiple of 12a^J^, IGa'^bc^, and 24a. Jlns. 48a^JV. 3. Find the least common multiple of Sa^b, 5a, 7a^c, 12a', 15a^ ISa^Jc, and 35a*Z'V. Jlns. 1260a«JV. 4. Find the least common multiple of 12a''y+ 12a^by, Gce^i^-^ na'bf^-\-6abY, and 4ay. Resolving the numbers into their prime factors, 12a=y4-12a% =12aya+Z»)=2'x3xa=^xy X(a-\-h) 6ay + l^a'hy^+GabY = Qay\a^-\- %ib-^¥) 4aY =2'X a^xy' The different factors are 2^ 3, a^, /, and {a+bf. t=2 x3xa xy^X(a+hY SECT. III.] ALGEBRAIC FRACTIONS. 6l Hence, the least common multiple is 2*x3xa^Xy'x(a-h bY=l'2aY{a + by. 5. Find the least common multiple of a* — J*, a+J, and 6. Find the least common multiple of a-f ^> ^ — ^> or-\-ah-\- ^, and a»- ah-\-h^, Ans, a^—b\ CASE VII. 141. To reduce fractions to equivalent ones having a com- mon denominator. RULE. 1. Multiply each numerator into all the denominators^ except its own^ for the new numerators. 2. Multiply all the denominators together for the common cfe- nominator. ^ J^ote 1. — It will be perceived that, by the operations indi- cated in the preceding rule, the terms of each fraction are, in effect, multiplied by the product of the othfer denominators, i. e., the numerator and denominator of each fraction are multiplied by the same number. To show that the value of the fractions is not altered by this transformation, it is only necessary to demonstrate the following theorem : Theor. If both terms of a fraction be multiplied by the same number^ the value will not be altered. Let ab and a represent the numerator and denominator of an algebraic fraction of any assignable quantity : Then the fraction —=iab-^a=zb. a Let m represent any whole or fractional number what- ever ; then multiplying the terms of the fraction by fw, we have ^ — =abm-7-am=b. am Hence, since ^=b and ^=i (by Ax. 2), ^=^, a am a am which was to be demonstrated. ' Jfote 2.~Mixed numbers should be reduced to improper 6 62 ELEMENTS OF ALGEBRA. [SECT. III. fractions, and'all the fractions should be made positive be- fore they are reduced to a common denominator. JVb^e 3. — Whole numbers or integers can be put under the form of a fraction by writing 1 for a denominator un- der them, and then be reduced to a common denominator with fractions. EXAMPLES. 1. Reduce -, -, and — to equivalent fractions, having a b d y ' common denominator. axdxy—ady^ first numerator. cxlxy—hcy^ second numerator. xxhxd—hdx, third numerator. bxdxy=^hdy, the common denominator. Hence, the values of the fractions are —^^ — ^, and — . bdy bdy bdy 2. Reduce — , - — , and — to equivalent fractions, having a common denominator, ^ns. i, , and ^. l^cxy llcxy \2cxy 3. Reduce — , — , and __ to equivalent fractions having a ax 36 7c? common denominator. a SUd Ua'dx , Ibabx Jins. , , and . 2labdx ^labdx 21abdx 4}, Reduce — and to equivalent fractions having a 6b 3c common denominator. ,dns. and it — . 15k 156c 5. Reduce -, — I^, and a (or -) to equivalent fractions b c-\-d 1 having a common denominator. ^ ac-{-ad 3bx — 26 . abc-\-abd bc-\-bd^ bc-\-bd ' bc-\-bd 6. Reduce ^ , -, and -— J^ to equivalent fractions 4a; 5 c-\-d having a common denominator. SECT. III.] ALGEBRAIC FRACTIONS. 63 J, 25«c4-25af/— 5c— 5rf 12cxH-12dir , 20&x + 4aj?y ^ 20cx-f-20(^ ' 20cx-f20(/a?' 20cx+20^* 7. Reduce — -1- and "^ to equivalent fractions having a 3 a common denominator. Ans, — -— and — — — . 3a 3a 8. Reduce , — -, and — to equivalent fractions hav- 76a? a — 5 ing a common denominator. a 25 ad —I05bcx , 2Sbdx —^bbdx" —Sbbdx' —Sbbdx 2 I IL2 O IL Z? 2 9. Reduce — l!l-, , and — to equivalent frac- 2a a — b 2aH-2a6 tions having a common denominator. Ans ^'— ^^^' 12^-fl2a^^ and 1?^— ?^ 3 -r^ O Oj2 y _j_ 4 10. Reduce and i— to equivalent fractions 4a a-|-a? having a common denominator. jj^ 3ax^+3x'—2a—2x ^^^ Soa?^— 4.ax+ 16a 4a'+4ax 4a*-f4aa; 142. To reduce fractions to their least common denomi- nator. SITLB. 1. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator, 2. Divide the least common denominator by the denominator of each fraction separately, and multiply the quotient by the re- spective numerators, and the products will be the numerators of the fractions required, EXAMPLES. 1. Reduce — - and — — - to their least common denomi- Sjt 4aV nators. ar« =2»xa« 4flV=2'xa?»xaP 64 ELEMENTS OF ALGEBRA. [sECT. III. Hence, the least common multiple is 2^x07^X0^=80^. Then,^^xSa'=:a'xxSa'z=z Sa'x ?.'^'x5a6=2 X6ah=:10ai 4aV )■ new numerators. ^;,..^andJ^. 8aV Sa'x' 2. Reduce — -, Jl, and to their least common de- Qoric ^cLcrxu Sa?^ nommators. Ans. y^^r-.-> tAi and — _-.. 8acV' 8acV 8acV 3, Reduce _ -, , and to their least common a^ — or 4a — 4a? a-\-x denominator. Ans. _i-^_, Mt^, and ?0^^^=20^. - CASE VIII. ADDITION OF FRACTIONS. 143. Theor. If two fractions have a common denominator^ their sum will he equal to the sum of their numerators-divided by the common denominator. Let — and — , represent two fractions whose common de- a a nominator is a ; rpr am .an am-^an , ^ a a a For, '^=m, and 2^=»; therefore, ^+^=m+». a a a a But, ^'^~^^'^={am-^an)^a=m-^n ; a Hence (by Ax. 2), — +— = ^^"'"°^ , which was to be de- ex a a monstrated. 144. Hence, to add fractions, we obtain the followingf general SECT. III.] ALGEBRAIC FRACTIONS. 65 RULE. 1. Reduce the fractions to equivalerit ones, having a common denominator ^ and make them all positive. 2. Jldd all the numerators together, and under their sum write the common denominator. 3. Reduce the resulting fraction to its lowest terms, EXAMPLES. 1. Add toffether — , — , and — • ^ 2A'y la Reducing the fractions to a common denominator^ 3^ 2a 3b_l06a' 2Sa'b 30Z>' b ~b la~10ab lOab TOab Adding the numerators of the reduced fractions, lOba' 2Sa'b 30b' _ l05a'-\-2Sa'b-{-30b' ^ lOablOab lOab lO^b ^' 2. Add together ± and ?5±^. ^^ns. ^^±^b±Uc^^hx^ * 2b a\-b 2ab-\-2l^ 3. Add together ?^+ \ ^^ and 1 Jlns. 1^±IL ^ 3 ' 5 ' 7 105 4. Add together ^±?f , tl^', and -t b3^-\^xy. 5. Add together ^±^ and ?IZ*. ^ns. ^'+^. a—b + 4 a'— 4* 6. Add together ?, ^i^, and ±±=f?. 6 cd bed Jlns ?!£^.^±i 7. Add together 1^^ and ^. Ans. ?^^. 8. Add together -^ and —, jlns. °'+^' . a-f6 a — b a^— ^ 9. Add together _?1, —?, and —1 ^«j 5g^— 3/>>'— 4qx— 4^g 4Mb+W. I 66 ELEMENTS OF ALGEBRA. [sECT. III. 10. Add together -, — ^2^ , and — ?^. a b — 1 fe+1 ab^ 11. Add together 2a+^ and 4^+?^. 5 4 •dns, 6a + — '■ . 20 12. Add together a — — and b-\- . be a . 7 . 2abx — Sea?* be CASE IX. SUBTRACTION OF FRACTIONS. 145. Theor. If two fractions have a common denominator f their difference is equal to the numerators divided by the common de- nominator. Let — and — represent two fractions whose common de- a a nominator is a ; rpi am an am — an ihen, — — — = . a a a T^ am J an .i f am an ror — =m^ and — —n j therefore, — — — =m — n. a a a a ■D , am — an / \ • But, = (am — an) —a— m — n, a Hence (by Ax. 2), — — — =^ ^^, which was to be de- a a a monstrated. 146. Hence, to subtract one fraction from another, we obtain the following general RULE. 1. Reduce the fractions to equivalent ones, having a common denominator^ and make them all positive. 2. Subtract the numerator of the fraction to be subtracted from that of the other fraction^ and under their diffei'ence write the common denominator. SECT." III.] ALGEBRAIC FRACTIONS. 67 EXAMPLES. 1. From ?? subtract if. 3b bd Reducing the fractions to a common denominator, dec. 2a_^c_l0td_12bc_ 10ad—12bc ^^^ 3b 5d~lbbd lbbd~ 15W % From ^J± subtract 1 Jlns, ^^"^^"^^ c y cy 3. From ^1 subtract e!^±^. lb 3ax 76 3aJ7 ~21aAj? 21aZ»a7 "~ ^labx 4. From ^ subtract ?i±i. wf;w. ^^-^^^ 5 x+\ bx+b 5. From subtract . Ans. ^ x—y x-\-y x'—f 6. From ^±? subtract -J_. vfn*. ?JZ±Z% y x^ — 2 a?*y — 2y 7. rrom subtract -. Ans. ax — x^ ax-\-3^ a^—a* 8. From 6a-[.— subtract Sa-\-—, Am. 2a+^^^~^. 0? c ex 9. From 6ot— 1^!±1 subtract f7»4-?. ^^. 5m— ?2^il. 2 5 10 10. From . subtract .^;w. 8a*— 2a*6^+4a'6»— a6« 6*— 4a« CASE X. MULTIPLICATION OF FRACTIONS. 147. Theor. Tht 'product of two fractions is equivalent to the product of the numerators divided by the product of the denomi- nators. Let - and _ represent any two fractions : Then will ^X^=^'. b b' bb'' 68 ELEMENTS OF ALGEBRA. [sECT. III. For, letting v represent the value of tjie first fraction, and v' the value of the second, we shall have -=v, and - =iV'. ' b b' Multiplying the two equalities together (Ax. 5), - X - =v. V. b b' Multiplying the equality -=v by b (Ax. 5), a=bv, b a' Multiplying the equality -=iv' hy b' (Ax. 5), a'—b't/. Multiplying the last two equalities together (Ax. 5), aa'=z bv .b'v'=bb' xvv'. lualitv bv bb' (Ax. G\ bb' Dividing the last equality by bb' (Ax. 6), — =vv\ Hence (by Ax. 2), _x -=^, which was to be demonstra- ted. CoROL. The product of any number of fractions is equiva- lent to the product of the numerators divided by the product of the denominators : ihus, -X — X — = ► ' b b' b" bb'b'^ 148. From the foregoing theorem we infer the following general rule for the multiplication of fractions. *i' ' " RULE. 1. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. 2. Reduce the resulting fraction to its lowest terms. EXAMPLES. 1 i\/r u- 1 3^2, bah n^^ ^a\bah Ibah 1. Multiply — bv . ^ns. — - x -—-=-——,. ^ ^ 46 ^ Ib'd 46 Wd 2Sb'd 2. Multiply ?^±^ by ^. ^ ^ Sax ^ Sdx^ ^^^ 3a'+6 2ac^_6ah+2abc^_3a''c-\-bc^ Sax 3dx' 24>adx' ndx' 3a^ 3. Muhiply — L— by Jins. -!— . ^ ^ a—1 ^ a+1 a'—l SECT. III.] ALGEBRAIC FRACTIONS. 69 ' 4. Multiply ?^^ by -J^-. Jim. ^^ 5. Multiply by — — -. Ans. 1. 6. Multiply ±ZI by ^!±f?. wf^. ^'"^ ex c-\-x c^x-i-ca^ 7. Multiply —, ^, -, and _i_ together. m y c n — 1 Ans. ^i''-^^ cmny — cmy 8. Multiply —, — . and ^ together. jJns. A. 9. Multiply by — - — -. Ans. -- -L- — . 10. Multiply e!^^^ by -Jl— ^ Jlns.?^±^. J^ote 1. — Since every integer can be expressed in the form of a fraction by writing 1 under it for a denominator, it is evi- dent that an integer, or whole number, may be multiplied into a fraction by multiplying the numerator of the fraction by the whole number, while the denominator remains the same. mi ^b a^ h ab Thus, aX_=_x-= — . c 1 c c ^ote 2. — If the denominator is divisible by a whole num- ber, dividing the denominator multiplies the fraction. Ihus, _xc=-_^=_; for, _xc=_-X-=— =-. be bc—e b be be 1 be b CASE XI. DIVISION OF FRACTIONS. 149. Theor. If one fraction be divided by another fraetion, the quotient will be equivalent to the product of the fracticmal dividend multiplied by the fractional divisor inverted. Let - and ~ represent any two fractions : b' Then will ?-?.'=?*:: b b' ab 70 ELEMENTS OF ALGEBRA. [SECT. III. For, letting v represent the value of-, and v' the value of - ; b h' Then, -=v and -=:v': and -^-=v-^v'. b b' ' b ' b' Multiplying the equality -—V by bb' (Ax. 5), ab'=bb'v. , b Multiplying the equality -^=:v' by bb' (Ax. 5), a'b—bb'v'. Dividing the former by the latter of the last two equalities, ab' bb'v ba' bb'v' :V-T-V'. Hence (by Ax. 2), — 1-_=__, which was to bedemonstra- b b' ba' ted. 150. From the foregoing theorem we infer the following general rule for the division of one fraction by another. RULE. 1. Invert the fractional divisor, 2. Then proceed as in multiplication j and the product thus found will be the quotient required, EXAMPLES. 1. Divide ?^ by ^.• 46 ^ 6d' 4A" ■ Q^~Tb 5c' 'WJI~TOb?' 2.Divide?^!±l^by5^. 3a^-{-2 b_^3a_3a' + 2b ^5b_ 15a''b- ^ 10b'' ^^^ 'WTc ' 56 26+c~ 3a 6ab-f3ac~' 3. Divide -^ by -. Jlns. —, 1—a ^ 5 1—a 4. Divide ^-±tl by 1-, ' Ans, _?^1__. e—y^ ^ c—b c'-\-bc-hb' 5. Divide -^^-Z^ by «!±^l Ans. ^±l=a^t a'—2ab-{-b' ^ a—b a a a r\' 'A 3a — 3b , 5a — 5b n 3 6. Divide bv . Jins. -. a-\.d ^ a+d 5 SECT. III.] ALGEBRAIC FRACTIONS. 71 7. Divide -^ by -Ij. Ant. _, '^ .. 8. Divide 5^ by ^-III. ^n.. 1?^. 9. Divide ^ by 5^1^ ^... ^.. 3c* ^ 7 • 6c» 10. Divide 2aV-2c^ . g'+ac+c^ ^;*,. 2(a3+c»). JVo^e 1. — When a fraction is to be divided by a whole number, or a whole number by a fraction, write the whole number in the form of a fraction by making its denominator 1, and then proceed as before. Thus «-c=«-^=^xl=f: b 1 c be A«j b a.b_a^c_ac And, a--= -_= x-=--. c 1 c 1 ^ b J^ote 2. — The reciprocal of a fraction is expressed by the fraction inverted : Thus, the reciprocal of J is -, or 1^ ^: o I b But, l-r-f!=lx-=-; hence, the reciprocal of - is _. baa b a ^ote 3. — If the numerator or denominator of a fraction has a rational coefficient, the expression may be reduced to a simpler form, on the principle that multiplying the denomi- nator of a fraction has the same effect upon its value as di- viding the numerator ; and multiplying the numerator has the same effect as dividing the denominator. Thus, l.¥=|x«=|. 3a-h2A 3a+26 ^ 3a+2^ 24a-hl66' q f_^^^__^ d_ad ' yi' d'~i^c~'Tc 72 ' ELEMENTS OF ALGEBRA. [sECT. IV. SECTION IV. OF EaUATIONS. 151. An EQUATION is the algebraic expression of two equal quantities connected by the sign of equality. 152. The monomial or polynomial quantity which is writ- ten on the left of the sign of equality is called the first mem' ber ; that which is written on the right, the second member. An equation, then, is composed of two members j and each member is composed of one or more terms. 153. The two members of the equation must be composed of quantities of the same kind ; that is, dollars must be put equal to dollars, weight equal to weight, &c. 154. Equations are distinguished into different degrees, ac- cording to the highest power of the unknown quantity. If it involve only the first power of the unknown quantity, it is called an equation of the first degree. If the highest power of the unknown quantity be the second power, it is called an equation of the second degree ; if it be the third power, an equation of the third degree, &c. Thus, x=a is an equation of the first degree. x^=za ) , _ ^ are equations of the second degree. o^-\-x^—a> \ are equations of the third degree. x^-\-3:?-\-x — aj 155. The solution of a problem is the method of discover- ing, by analysis, the value of the unknown quantity involved in the conditions of the problem, and consists of two parts. 1. The translation of the problem from common into algebraic language ; or the expression of its conditions in the form of an equation by means of algehraic symbols. 2. The reduction of the equation to such a form that the un- SECT. IV.] EQUATIONS. 73 known quantity may stand ly itself^ and form one member of the equation^ while the known quantities form the other. 156: No general rule can be given for translating the problem from common into algebraic language ; only that the algebraic expression shall exhibit the same relations, and indicate the same operations as those implied in the original statement of the problem. 157. Proportions may be converted into equations by taking the product of the first and fourth terms for one member, and the product of the second and third for the other. Thus, if a : 6 : : c : 9. Reduce the equation a;+a?4-^=100--2?. Am. x=39. 10. Reduce the equation x+?4-^+^+4-=14'6- 2 4 7 14 .^;i5. a: =56. 168. Combining the principles discussed in the preceding three cases, we have, for the solution of all equations of the first degree involving only one unknown quantity, the fol- lowing general RULE. 1. Char the equation af fractions. 2. Transpose the terms^ so as to bring all the unknown quanti- ties into the first^ and the known into the second member of the equation. 3. Reduce each member to a monomial. 4 Divide the equation by the coefficient of the unknown quan- tity. JVote 1. — If the unknown quantity in the result is negative, change the signs of all the terms in the equation. Thus - - 4a^-5x=9— 12. Reducing - - — x= — 3. ' Changing the signs a?=3. 78 ELEMENTS OF ALGEBRA. [sECT. IV. Jfote 2. — To verify the result obtained by the reduction of an equation, substitute the value obtained for the unknown quantity in the first equation, and see if it satisfies the con- ditions. Thus, substituting 3 for x in the equation above, we have - - 4x3—5x3=9—12. Multiplying factors 12—15 = 9—12. Reducing - - — 3= -^3. EXAMPLES. 1. Reduce the equation, a7+_-|-_= 11. Ans. x^Q. 2. Reduce the equation ^x-{- — 2——x-\--, Ans. a?=i. o 3. Reduce the equation IL_— 2=::1. Ans. x=1, 4. Reduce the equation — -^-j- — —x-{-2. Ans. a?=4^. 11 o 5. Reduce the equation -+-+- = 94. Ans. a;=120. 3 4 5 6. Reduce the equation — + — =07 — 20 — - ^ 4' 10 Ans. a? =800. 7. Reduce the equation ^^^+2&=.?^±i2. 4 6 Ans.x^^-:^^, 9 8. Reduce the equation 8f +^±l=4+a:— 26i. 5 Ans. 07=39. n -D 1 ^1 ^. 2a7— 5 , 19— a? 10a7 — 7 5 9. Reduce the equation + = — ^-. ^ 18 3 9 2 Ans. 07=7. 10. Reduce the equation 2o;— ?±i+15=l?^±?5 o Ans. 07=12. 11. Reduce the equation ^II?+?= 20—^11^. ^ 2 3 2 Ans. 07=18. SECT. IV.] EQUATIONS OP THE FIRST DEGREE. 79 12, Reduce the equation ?^±5— 5=1 ^7W.x=.^-"^ 13. Reduce the equation _^^lt_=ca?4-4a. 4 2a^3c 14. Reduce the equation Sb-^lax=3x-\-4ic — ex 4c -8i ^ns. x= 7a— 3+c 15. Reduce the equation ^— ?H-?=20— -. 1/6 9 3 2 ^»5. a?=24|t. J7 . X a? 16. Reduce the equation -+-— _4-a?=2a?— 43. 4 5 6 ^715. x=60. 17. Reduce the equation 3af+_ _=a7+a. 6-hfr 18. Reduce the equation -+-4--4-?— ?=1. 2 3 4 5 6 19. Reduce the equation 3^-3_3x-4^^ 27+4j? ^ 4 - 3' ' 9 •^»*. a?=9. 20. Reduce the equation 15^±i5-4=?^Ill?— 5. ^ 3j?+6 x—2 Ans. j:=2. 21. Reduce the equation 1^+^+1 2a =10—?!::^. .^,. .._ 115 + 156~180a 23 22. Reduce the equation 31-^ _|, ^5^+8 __7x— 8^^^ 2 13 11 Ans, 07=9. 80 ELEMENTS OF ALGEBRA. [sECT. IV. 23. Reduce the equation 5a?-4_3a:— 7^^i_ 8j:— 1 ^ 6 10 ^ 3 24. Reduce the equation — I — = +_. 36 5a: — 4 4 *dns. a? =8. 25. Reduce the equation !I^±?— 8=::?Zll?5f +4. ^ 3a:— 1 3a:— 1 .>^7i5. a:=:l. o« T? 1 ^u *• 2a?+l 402— 3a: ^ 471— 6a? 26. Reduce the equation — — = 9 — . ^ 29 12 2 »dns. x=12. 27. Reduce the equation 15^15+11^21^9^+15^ ^ 28 6a:+ 14 14 j^ns. 07=7. 28. Reduce the equation 1?±^ : 3a:+6 : : 2 : 5. 5 ^ns. x=S. 29. Reduce the equation 3a?+25a : 9a?+4Z> : : 4 : 10. ^ns. .^ 250a-16& ^ 6 30. Reduce the equation ??±25 . 7_3^ . . jq : 7. jlns. a:=||. 31. Reduce the equation 21-3a?_4^+6^g_5^+l, ^ 3 9 4 jlns. a: =3. 00 T? J *u .• 6a:+8 5a:+3 27— 4a? 3a:+9 32. Reduce the equation 1— — ! — = — — -L_. ^ 11 2 3 2 jSns. x=z6. 33. Reduce the equation a,^^^— 9g__5a:+2^g 2a?+5 ^ 4 6 " 3 12 SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 81 o. T, 1 *u ♦• 7x— 8 , 15x-i-8 o 31— X 34. Keduce the equation -f- ■ — =6X— — - — ^ 11 13 2 Ans. a: =9. 35. Reduce the equation ^"^^ : 1 : : 2a:-f 19 : 3a:--19. ^ 6x— 43 ^ns. x=8. 36. Reduce the equation 5j:+2!^±^^t^9-f 12^-1?. ^715. x = 3. 37. Reduce the equation ^^±^4- ^^i^=—-|-3U. ^ 25 9x-16 5 '^ 38. Reduce the equation ^ns, x=:4. 4x—34_258— 5x ^ 69— X 17~ 3 2 ^«^. x=51. 4x— 2 2x4-11 7— 8x 39. Reduce the equation 2x ^ 13 5 7 *dns» x=7. 40. Reduce the equation 16x4-5 : ^^±i* : : 36x4- 10 : 1. ^ns. x=5. PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE, IN- VOLVING ONLY ONE UNKNOWN QUANTITY. 169. Though no general and definite rule can be given for the translation of a problem into algebraic language, yet the following precepts. may be found useful for this purpose. 1. Let X represent the unknown quantity whose value we wish to determine. 2. Indicate by the aid of algebraic signs the operations that would be necessary in order to verify the answer were the problem already solved. 3. The equation or proportion thus formed may be reduced by the preceding rules. PROBLEMS. 1. Two men, A and B, trade in company and gain $680, of which B has 4 times as much as A. What is the share of each \ L 82 ELEMENTS OP ALGEBRA. 1 ; 5.4 [SEGT. IV. Let x= number of dollars in A's share ; Then 4a7= number of dollars in B's share, And we shall have the equation a?+4a?=z680. Reducing terms - - - 5a:=680. Dividing by coefficient of a? - a?=136, A's share. And 4a?=::544, B's share. Verification - - 136+4x136 = 680. 2. What number is that, the sum of whose third part and fourth part is 7 1 Let 0?= the number : Then -= one third, And -=z one fourth, ^ 4 X X And we shall have the equation _-4-_=7. 3 4 Clearing of fractions, 4a7+3a:=84 : Reducing terms - 7a7i=84: Dividing by 7 - - a?=:12. jSns, Verification - i^+l?=4+3=:7. 3 4 3. Divide $5000 between A, B, C, and D in such a man- ner that A shall have $300 more than B, and B $50 more than C, and C $|200 more than D. W^hat was the share of each ] Let a?= D's share ; , Then 07+200= C's share; And a?+250= B's share ; And a:+550=: A's share. And we shall have the equation a:+a7+200+a7+250+ a:+550 -1^5000: . Transposing - a?+a? + a?+a?=5000— 200— 250-550; Reducing - 4a?=4000; Dividing by 4 - a?=100 \ D's share : a?+ 200= 1200, C's share : a?+ 250= 1250, B's share: , 07+550=1550, A's share. SECT. IV.] EQUATIONS OF TUE FIRST DEGREE. 83 Verification, 1000-f- 1000+200+ 1000+250+ 1000+550= 5000. Or, reducing, 5000=5000. 4. It is required to divide the number 84 into two such parts that the greater shall be to the less as 8 to 5. Let x= the greater part, And 84 — x= the less part. And we have the proportion x : 84 — x : : 8 : 5, Converting the proportion ) ^x=612—Sx, ' into an equation ) Transposing - - 5a:+8x=672. Reducing terms - - 13x=672. Dividing by 13 - - 0^=51/5, greater part. 84— x=32i^, less part. 5. It is required to divide $972 between A and B in such a manner that B may have fths as much as A. Let Xz= A's share, And ^= B's share, 5 And we shall have the equation j?+ — =972. Clearing of fractions - 5x+4x=4860. Reducing terms - - 9a?=:4860. Dividing by 9 - - - a: =540, A's share. *^=432, B's share. 5 6. A man puts out three fifths of his money at 6 per cent, and the remainder at 7 per cent., and at the end of the year receives $4825 interest. How much money had hel Let x= the amount : 3a: Then — = the amount at 6 per cent, 2x And __=r the amount at 7 per cent., 84 ELEMENTS OF ALGEBRA. [sECT. IV. And, multiplying each amount by its rate, we shall have the equation 5 100 5 100 Multiplying factors — + 1^=4825. ^^ ° 500^500 Clearing of fractions 18a?4- 14a? =24- 12500 : Eeducing terms - - 32a?=2412500 : ^ Dividing by 32 - - a7=75390|. ^ns. 7. A can do a piece of work in 8 days,; B can do the same work in 12 days; in what time will they do it if both work together X A will do ith of the work in one day : B will do yLth of the work in one day : Let x=: the time it would take them to do the work which is represented by 1 : Then, in x days A will do - of the work, And in x days B will do — of the work, ^ 12 ' • ' And we shall have the equation -+ — = 1. ^ 8^12 Reducing - - - - a:=i:4f. Ans, 8. A gentleman meeting 5 poor persons, distributed $4,50 among them, giving to the second twice, to the third 7 three times, to the fourth four times, and to the fifth five times as much as to the first. How much did he . give to each 1 Ans. 30, 60, 90, 120, and 150 cents. 9. A man left $11004 to be divided among his widow, two sons, and three daughters, in such a manner that the widow should have twice as much as both the sons, and each son should have as much as the three daughters. What was the share of each 1 Widow's share, $6288, ) Each son's share, $1572, \ ^ns. Each daughter's share, $524. j SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 85 10. What number is that which, being multiplied by 8, the product increased by 10 times the number, and that sum divided by 12, the quotient shall be 4 1 »^ns. 2*. 11. A post is \ in the earth, f in the water, and 13 feet out of the water. What is the length of the post 1 •^716. 35. 12. After paying away ] and 4 of my money, I had $85 left in my purse. How many dollars had I at first \ Jlns. 140. 13. Of a battalion of soldiers.(the officers being included), I are on duty, ^^ sick, f of the remainder are absent, and there are 48 officers. What is the number of per- sons in the battalion 1 ,^7is, 800. 14. In an orchard of fruit-trees, ^ of them bear apples, \ pears, J plums: 7 bear peaches, 3 bear cherries, and 2 quinces. How many trees are there 1 ./^n*. 96. 15. A farmer being asked how many sheep he had, an- swered, he had them in 4 pastures: in the first he had J of the whole number, in the second i, in the third ^, and in the fourth he had 18 sheep. How many had he 1 ^ns. 72. 16. A and B talking of their ages, A says to B, if ^, i, and ^ of my age be added to my age, and 2 years more, the sum will be twice my age. What was his age 1 Jlns. 84. 17. The rent of an estate is this year 8 per cent, greater than it was last. This year it is $1890; what was it last yearl .Ans. $1750. 18. A capitalist receives a yearly income of $2940 ; | of his money being at 4 per cent, interest, and the re- mainder at 5 per cent. How much has he at interest! Ans. $70,000. 19. A cistern, containing 60 gallons of water, has three unequal cocks for discharging it. The largest will empty it in 1 hour, the second in 2 hours, and the third in 3 hours In what time will they empty the cistern if they all run at once 1 Ans, 32,«j minutes, 8 86 ELEMENTS OF ALGEBRA. [sECT. IV. 20. A farmer wishes to mix 90 bushels of provender, con- sisting of rye, barley, and oats, so that the mixture may contain | as much barley as oats, and i as much rye as barley. How much of each must there be in the mix- ture % Ans. 50 bushels of oats^ 30 of barley^ and 10 of rye. 21. A, B, and C trade in company. A puts into their stock $3 as often as B puts in $7 and C $5. They gain $960. What is each man's share of the gain '? Ans. A's $192, B's $448, C's $320. 22. A, B, and C trade in company. A puts in $700, B $450, and C $950. They gained $420. What was the share of each % Ans. A's $140, B's $90, and Cs $190. 23. At a certain election, the whole number of votes was 673. The candidate chosen had a majority of 11. How many voted for each 1 Ans. One 342, the other 331. 24. On canvassing the votes at a certain election, it was found that there was no choice : it was also ascertained that one of the candidates had | of the whole number of votes, the other | of the whole number, and there were 45 scattering votes. What was the whole number of votes 1 Ans. 200. 25. Three men built 780 rods of fence. The first built 9 rods per day, the second 7, the third 5 ; the second worked three times as many days as the first, and the third twice as many days as the second. How many days did each work 1 26. A gentleman bequeathed $65,600 to his wife, two sons, and three daughters. The wife was to have $2000 less than the elder son and $3000 more than the younger son, and the portion of each of the daughters was $3500 less than that of the younger son. What was the share of the wife and each son \ $16,350 elder son's share. $14,350 wife's share. ^^s. $11,350 younger son's share. $7,850 each daughter's share. SECT. IV.] EQUATIONS OP THE FIRST DEGREE. 87 27. A man meeting some beggars, gave each of them 4cf., and had I6d. left j if he had undertaken to give them 6d. apiece, he would have wanted 12f/. more for that pur- pose. How many beggars were there, and how many pence had he 1 28. A boy being sent to market to buy a certain quantity of meat, found that if he bought beef, which was 4>d. per pound, he would lay out all the money he was intrust- ed with ; but if he bought mutton, which was 3^d. per pound, he would have 2 shillings left. How much meat was he sent for 1 29. It is required to divide 85 into two such parts that I of the one added to ^ of the other may make 60. 30. When the price of a bushel of barley wanted but 3d. to be to the price of a bushel of oats as 8 to 5, four bushels of barley and 90. Reduce the equations _+2y=a, and - — ^y=b 4 Jlns. a;=a+Z>, and y-. 5. Reduce the equations --|-^=:8, and - — ^^=1. iL O O At Ans. a:=12, and y=6, 6. Reduce the equation --\-±=:9^ ^nd the proportion x: 2^ - 4 : 3. ^ns. a:= 12, and y=9. 7. Reduce the equations — ±—£.=8 — -, and -llZ — =rll ^ 6 3 2 -\-y. Ans. a?=6, and y=S. Second Method. — By Substitution. 176. This method of elimination rests upon the principle, that if any equivalent expression be substituted, in an equa- tion, for an unknow^n quantity, it will satisfy the conditions of that equation. SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 93 Let US resume the two equations before used, 2x-f3y=36. Finding a value of j:in the 1st equation, x=16 — y: Substituting this value for x in the 2d equation, 2(16— y)4-3y=36; Reducing - - V—^i And x=12. 177. Hence, for the elimination of one or two unknown quantities by substitution, we obtain the following general RULE. 1. Find the value of one of the unknown quantities in one of the equations. 2. In the other equation^ substitute this value for the unknovm quantity itself and then reduce as before. EXAMPLES. 1. Reduce the equations a;+y=13, and x— y=3. Jlns, 37=8, andy=z6, 2. Reduce the equations x — 7=3y— 21, and x-{-l=2y+ H, Ans. a:=:49, andy=^\, 3. Reduce the equations 7a:=8y, and a?— y+^O. Ans. x= 160, and y= 140. 4. Reduce the equations ic4-10=2y, and y-j-10=3x. jlns. x — 6y and y=:S. 5. Reduce the equations ^+8y=194, and l-\-Sx=zl3h o 8 jJns. 07=16, andy — 2^. 6. Reduce the equations 4a7+^=26, and -4-^=6. 2 2 5 Ans. x=4, andy=^20. 7. Reduce the proportions a? : y : ; 3 : 1, and - : 5y— 4: :3:9. Ans. a: =24, and y=S. 8. Reduce the equations ?±?-|-6y=21, and y4-^+5a:=23. 4 3 Ana, x=4,j and y=3^. 94 ELEMENTS OF ALGEBRA. [sECT. IV. Third Method. — By Addition or Subtraction, 178. This method of elimination rests upon the prmciple, that if equals be added to or subtracted from equals, the re- sults will be equal. As the members of an equation are equal quantities, it will follow that if one equation be added to or subtracted from another, the results will be equal. Let us resume the two equations before used, 2a? -I- 33^^ 36. Multiplying the 1st equation by 2 - 2a7H-2y=32 5 Subtracting the 3d from the 2d equation y= 4; Substituting and reducing - - - 3?=: 12. Again, let us take the two equations, 3r+5y=:28, 2a?— 5y= 2. Adding the two equations - - 5a? =30; Dividing by 5 - - - -a? =6j Substituting and reducing - ' y =2. 179. Hence, for the elimination of one or two unknown quantities, by Addition or Subtraction, we have the following general RULE. 1. Multiply or divide the equations in such a manner that the term containing one of the unknown quantities shall be the same in both equations. 2. If the signs of these terms are alike, subtract one equation from the other ; if unlike, add the two equations together. EXABIPLES. 1. Reduce the equations 2x-{-y—16, and 3a: — 3y=6. Ans. x=6, and y=4f. 2. Reduce the equations a?+y=48, and x—y=z32. Ans. a?=40, andy=S. 3. Reduce the equations 5a?— 3yi=9, and 2x-\-by=16. Ans. 07=3, and y—2. SECT. IV.] EQUATIONS OF TUB FIRST DEGREE. 95 4. Reduce the equations 30x-\-^0yz^210, and 50xH-30y= 340. Jins. x=b, and y=3. 5. Reduce the equations 3a?-}-7y=79, and 2y=9-f ?. ^ns. x=z 10, and y=7. 6. Reduce the equations ?fZ:??4-2=7, and ?+?-:=6. y o 5 ^n*. a; 1=24, and y= 10, 7. Reduce the equations ?fZ^4-14=18, and r^+f-f 16 2 3 = 19. ^715. x=i5, and y=2. 8. Reduce the equations ^+^"^^+^^8, and llZ^l-y^ 11. ^ns, x = 6y and y=S, 9. Reduce the equations '^_}0-^ ^9^:10 ^^^ 2y+4 ^ 5 3 4' 3 8 4 10. Reduce the equations _+?l=:6, and -4-^=5|. ,^ns, x= 12, and y = 16. MISCELLANEOUS EQUATIONS OF THE FIRST DEGREE INVOLVING TWO UNKNOWN QUANTITIES. 1. Reduce the equations 8x + 5y=68, and 12x+7y=100. ,^ns. a: =6, and y=4. 2. Reduce the equations 8x4-^=20, and 20j:-|-3y=70. ^dns. x=\ljand y=15. 3. Reduce the equations £±2^— 2y=2, and '^~~^y-\-y= 23 ' — -. ^ns, x= 1 1, and y=l. Ot Q 4. Reduce the equations — Z_-|-y=7, and 5x — 13y=y • ,^ns, x=S, andy = ^. 96 ELEMENTS OF ALGEBRA. [seCT. IV. 5. Reduce the equations ?^2^=?^y±i, and 8-^!=^ 3 5 5 = 6. Jins. a:=13, and y=zS, 6. Reduce the equations ^+?^=— , and ^-\-'?l=^%\. ^ 5 4 20' 4 5 ''^ jlns. a?=i, and yz=:^» 7. Reduce the equations ?+ 7^=99, and ^ + 7a?zzi51. »^ns^ a? =7, a?itZ yz= 14. 8. Reduce the equations ?— 12=^+8, and ^±y+^— 8= ^y~^+27. Jins. a?=60, a«(/ v=40. 4 ^ 9. Reduce the equations ^—12=^+13, and ^±2(_}_^_f_i6 =^~y+27. ^«5. a? =60, awc^ v=20. 4 10. Reduce the equations "^ — — I^i=3v — 5, and ^ ^ 5 4^' 2 4, J, 3 , 4- — ^ — =18 — 5a;. Ans. x=^^ and y—% 6 11. Reducethe equations 4a?+l^Z:?=2y+5+2!^±ii,and 12. Reduce the equations 1 — ^ . +^—y — 16|, and ^^ — 2 =-. ^»5. 07= 10, and y=20. 13. Reduce the equations ??II?l(=a:— 2|, and a?— ?(ZL^=:0. ./^ws. a?=l, andy—Z. 14. Reduce the equations ^ — ^-+5 = 6, and 3'-f-4= — +6. 4 7 5 14 ./?;j5. a: =28, and y =20. 15. Reduce the equations y — 3=- + 5, and ~^^=y — 32 • ^?^5. a?=2, awc?y=9. 19 SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 97 x-l-3 3x ^v 16. Reduce the equations 2y — _I__=7-f- — — ^, and 4x — 4 5 ^Z:y=2^—^^^. Ans. x=5, and y=5. 17. Reduce the equations X— ?yz:?±f =1-1- i5f±iy, and * 11 33 ' 3x-f2y _ y— 5 _ llx+152 _ 3y-H 6 4 12 2 ' Arts, a; =8, andy=9. 18. Reduce the equations ?^?^=18i— i^?^,and 15 * 7 ' 10y+^^~~^^ = 5^+lQx. Ans. x=:10, and y =16. 5 . Reduce the equations g_3j?-h5y i7^5y^^+7 ^^^ 17 ^3 22--6y_5x-7^x4:l _8y+5^ ^^ ^^^ 3 11 6 18 ' ^ 20. Reduce the equations — ~~'" 4- ^~ =4+ — ^^^^ — »and ^ 6 3 2 2j+y 9a:— 7 3y + 9 4r-|-5y ^ a j a — 21^— ^=J^-— ^. ^/w. j;=9, an(/y=4. PROBLEMS REQUIRING TWO UNKNOWN QUANTITIES, AND PRODUCING TWO EQUATIONS OF THE FIRST DEGREE. 181. — 1. A fruiterer sold to one person 6 lemons and 3 oranges for 42 cents, and to another 3 lemons and 8 oranges for 60 cents. What was the price of each 1 Let x= the price of a lemon. And y= the price of an orange, Then we shall have the two equations, 6x-|-3y=42, 3x-h8y=60. Transposing in the Ist equation - - 6a:=42 — 3y; Dividing by 6 ^^42— 3y.. 6 Transposing in the 2d equation - - 3a7=60— 8y; Dividing by 3 ^^60-8y. 9 N 98 ELEMENTS OF ALGEBRA. [sECT. IV. Forming a new equation from the two values of x, 42— 3y_60— Sy —6 3-"' Reducing - - y=6, the price of an orange, And - - - x—4fj the price of a lemon. 2. What fraction is that, to the numerator of which, if 1 be added, its value will be i, but if 1 be added to the de- nominator, its value will he 1% Let x= the numerator. And y= the denominator, Then -= the fraction, y And we shall have the two equations, x-^1 =h —1. y+1 ' Clearing the 1st of fractions - 3x-\-3=y; Clearing the 2d of fractions - - 4>x—y-{-li Dividing the 4th equation by 4 - - rrr=?Li_ ; 4 Substituting the value of a? ) /y+l\ , «_ in the third equation - y^ \ 4 / ~^ ^ Reducing y=15, And ----._ 07=4 : Hence -=tt5 the frac- y tion required. 3. A boy bought 2 apples and 3 oranges for 13 cents; he afterward bought, at the same rate, 3 apples and 5 oranges for 21 cents. What was the price of eachl Let x= the price of an apple, And y= the price of an orange. Then we shall have the two equations, 2:c+3y=13, 3x+53/=:21. SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 99 Multiplying the Ist equation by 3, 6x-f-9y=39; Multiplying the 2d equation by 2, 6a:+10y=4.2j Subtracting the 3d from the 4th - y=3, the price of an orange, And - a?=2, the price of an apple. 4. What fraction is that, to the numerator of which if 4 be added, the value is i, but if 7 be added to its de- nominator, the value is } 1 Ans. y\. 5. A and B have certain sums of money : says A to B, "Give me $15 of your money, and I shall have five times as much as you have left." Says B to A, " Give me $5 of your money, and I shall have exactly as much as you have left." How many dollars had each 1 Ans. Jl had $35, and B $25. 6. There are two numbers, such that 3 times the greater added to ^ the less is equal to 36 ; and if 2 times the greater be subtracted from 6 times the less and the re- mainder divided by 8, the quotient will be 4. What are the numbers 1 Ans, ^ and l\, 7. A person was desirous of relieving a certain number of beggars by giving them 25. 6f/. each, but found that he had not money enough in his pocket by 3*.: he then gave them 2^. each, and had 4s. to spare. How many- shillings had he, and how many beggars did he relieve 1 Ans. 32*. and 14 beggars, 8. A labourer working for a gentleman for 12 days, and having had with him the first 7 days his wife and son, received 745. : he wrought afterward 8 other days, du- ring 5 of which he had with him his wife and son, and he received 505. Required the gain of the labourer per day, and also that of his wife and son. Ans. Husband 55., and the wife and son Is, 9. A purse holds 19 crowns and 6 guineas. Now 4 crowns and 5 guineas fill yy of it. How many will it hold of each % Ana, 21 crovms and 63 guineas. 100 ELEMENTS OF ALGEBRA. [sECT. IV. 10. A farmer, with 28 bushels of barley at 25. 4c/. per bushel, would mix rye at 3s. per bushel^, and wheat at 4>s. per bushel, so that the whole mixture may consist of 100 bushels, and be worth 3^. 4c?. per bushel. How many bushels of rye, and how many of wheat, must he mix with the barley % Ans. 20 of rye and 52 of wheat. 11. A and B speculate with different sums : A gains $150, B loses $50, and now A's stock is to B's as 3 to 2. And if A had lost $50, and B gained $100, then A's stock would have been to B's as 5 to 9. What was the stock of each 1 Ans. A's. $300, and B's $350. 12. A rectangular bowling-green having been measured, it was observed that, if it were 5 feet broader and 4 feet longer, it would contain 116 feet more j but if it were 4 feet broader and 5 feet longer, it would contain 113 feet more. Required the length and breath. Ans. Length 12, and breadth ^ feet. 13. There is a number consisting of two figures, the sec- ond of which is greater than the first j and if the num- ber be divided by the sum of its figures, the quotient is 4 ; but if the figures be inverted, and the number which results be divided by a number greater by 2 than the difference of the figures, the quotient becomes 14. What is the number 1 Ans. 48. 14. A person owes a certain sum to two creditors. At one time he pays them $53, giving to one y^ of the sum due to him, and to the other $3 more than ^ of his debt to him. At a second time he pays them $42, giving to the first ^ of what remains due to him, and to the other A of what is due to him. What were the debts \ Ans. $121 and^m. 15. Some smugglers discovered a cave which would ex- actly hold the cargo of their boat, viz., 13 bales of cat- ton and 33 casks of wine. While they were unloading, a custom-house cutter coming in sight, they sailed away SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 101 with 9 casks and 5 bales, leaving the cave | full. How many bales or casks would it hold 1 Ana. 24 halts or 72 casks. 16. A and B can perform a piece of work in 16 days. They work together 4 days j then A being called off, B is left to finish it, which he does in 36 days more. In what time would each do it separately 1 Ans. A in 24, and B in 48 days. V7, Two loaded wagons were weighed, and their weights were found to be in the ratio of 4 to 5. Parts of their loads, which were in the proportion of 6 to 7, being taken out, their weights were then found to be in the . ratio of 2 to 3 j and the sum of their weights was then 10 tons. What were their weights at first 1 Ans. 16 and 20 tons. 18. There is a cistern, into which water is admitted by three cocks, two of which are of exactly the same di- mensions. When they are all open, j^ of the cistern is filled in 4 hours ; and if one of the equal cocks be stop- ped, ^ of the cistern is filled in lO^. hours. In how many hours would each cock fill the cistern 1 Ans. Each of the equal ones in 32 hours, and the other in 24. 19. A has a capital of $30,000, which he puts out to in- terest at a certain rate per cent., and he owes $20,000, on which he pays a certain rate per cent, interest* The interest which he receives exceeds that which he pays by $800. B has a capital of $35,000, which he puts out to interest at the same rate per cent, that A paysj he also owes $24,000, on which he pays interest at the same rate that A receives. The interest which he re- ceives exceeds that which he pays by $310. What are the two rates of interest 1 Ans. 6 and 5 per cent. 20. A has a certain capital, which he puts out to interest at a certain rate per cent. B has a capital of $10,000 more than A, which he puts out to interest at one per cent, more, and receives $800 more interest than A. ^ ' v-^ 102 ELEMENTS OF ALGEBRA. [sECT. IV. C has a capital of $15,000 more than A, which he puts out at 2 per cent, more, and receives $ 1500 more interest than A. What is the capital of each, and the three rates of interest 1 Jlns. A's capital, $30,000 ; B's, $40,000 ; C'5, $45,000 ; and the rates of interest 4, 5, and & per cent, Ot ELIMINATION WHERE THERE ARE THREE OR MORE EQUATIONS INVOLVING AS MANY UNKNOWN QUANTITIES. 182. In the problems hitherto given, each has contained no more than two unknown quantities, and two independent equations have been sufficient to express the conditions of the question. Other problems, however, may involve three or more unknown quantities j and if they are determinate, their conditions will give rise to as many independent equa- tions as there are unknown quantities. 183. The principles already discussed, and the rules al- ready given for the elimination of one of two unknown quantities, may also be applied where the number exceeds two. Thus, if there be three independent equations involving three unknown quantities, I. From the three equations involving three unknown quanti- ties, deduce two equations involving only two unknown quantities. II. Then from these two deduce one, involving only one un- known quantity. III. Reduce this equation, or find the numerical or literal value of the unknown quantity involved in it : then substitute this value for the unknown quantity itself, in an equation which involves only that and another unknown quantity whose value may thus be found. The value of the remaining unknown quan- tity may be found in a similar manner. 184. If there be four independent equations involving four unknown quantities, I. From the four equations deduce three, involving only three u?iknown quantities. II. Reduce these three equations as before. 185. If there be n independent equations, involving n un- SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 103 known quantities, they maybe reduced in a similar manner. For from the n equations involving n unknown quantities, we may deduce n — 1 equations involving n— 1 unknown quantities ; and from these n — 2 equations involving n — 2 unknown quantities, and so on until only one equation re- mains, involving only one unknown quantity. The value of this being found, the values of all the rest may be determin- ed by substitution, as before. A calculation may often be very much abridged by the exercise of judgment in stating the question, in selecting the equations from which others are to be deduced, in the manner of performing the reduction, in simplifying fractional expressions, in avoiding radical quantities, &c. EXAMPLES 1. Reduce the equations a?-f- y+ Z-. x-\-2y-\-32z U U -■- 2 3^ 4 29' 62 . Jlns. 10 , , From the 1st equation From the 2d equation From the third equation Making the 1st and 2d values ) of X equal - - V ^ Making the 1st and 3d values of X equal From the 7th equation From the 8th equation x=S, 2=12. x=29—y—z; (4) a:=62— 2y— 3z;(5) x=20-?^— ^; (6) J29-y- Making the two values of y equal - - - 33- Keducing Substituting for z its value the 9th equation Substituting for y and z their values in the 4th equation 1 3 2 -z=62— 2y- -3z. (7) ^z=20-^- -I- (8) y=33-2z; (9) (10) 2.=27_|. (11) z=n; (12) y= 33—24 = =9.(13) x=29— 9- 12=8. (14) 104 ELEMENTS OF ALGEBRA. [sect. IV. [2x^^—3z=22) (x=3, 2. Reduce the equations { 4 r t a?-f y4- 2=12. j 6. Reducethe|^^+^y+ ^=*^' ] equations i^" y+^^= ^' r x+ y-J- z-l- w=14,' 7. Reduce the equations 3x+2y-|2r-f w=lb, 2. ^ 2y z_3^. 3 4 5 ^ns. Ans, Am, rx=5, rx=2o, ^y = 12, iz=:32 ' x=2, y=3, 2 = 4, w=i5. 106 ELEMENTS OF ALGEBRA. [sect. IV. 8. Reduce the equations a?+yr-.52, ^ ( x=:20,. 2/+ 2^=82, 3/=:32, z-\-w=iQS^ > Jljf,s,\ z—bO, u-{-x='62.j [ u—12. 9. Reduce the equations a?i/=28, a?2r=2(>, and yz=35. Ans. a:=4i, y— 7, and z=z5. 10. Reduce the equations 2^3^4 124, ^ 3 4 5 ' ^ Ans, L4 5 6' 76. :48, y=120, :^=:240. 11. Reduce the equations a?y=100, ^0=40, and a?2=160. Ans. 07=20, y = 5, z—'^, 12. Reduce the equations ir+100=2/H-2r, y4-100=2a?+ 22;, and 2;+100=i3a?+3y. Ans. a?:=9Jyj 2/=45/y, aw J 2r=63yV 13. Reduce the equations 107+ 32/=: 23,) 4 2/+32r = 31, ^a?+2/+i2r+2w;=35. J Ans. 2 3 4' :62, 14. Reduce the J ^_|_y_j_^_47 equations o ^ o 4 5 6 2a? + 2^—22:= 4y — a?-j-3zi 3w+w: 3a?— yH-3w — w: Ans. 15. Reduce the equations :40, :35, :13, :15, =49. Am. < 16. Reduce the equations [00+ y+ z=: 53,] \ a?+22/-h32:=105, \ i^a? + 32/+4z=134.J a?=6, y=% z=S, a?=24, y-60, ^=120. a? =20, y = 10, z=5, u^l. fa?=24, Ans. J II — ^2=23. SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 107 PROBLEMS PRODUCING THREE OR MORE EQUATIONS, AND REQUIRING AS MANY UNKNOWN QUANTITIES. 1. Three persons divided a sum of money between them in such a mapner that the shares of A and B together amounted to $900 ; the shares of A and C together amounted to $800 j and the shares of B and C to $700. What was the share of each 1 Let a7=A's share, y=B's share, z=C^s share: Then - x+y=900y And - j?+2=&00, And - y+2=700. Reducing, x=500, A's share, y=400, B's share, 2=300, C's share. 2. A man with his wife and son, talking of their ages, said that his age added to that of his son was 16 year& more than that of his wife ; the wife said that her age added to that of her son made 8 years more than that of her husband ; and that all their ages together amount- ed to 88 years. What was the age of each 1 Ans. Husband 40, wife 36, and son 12 years, 3. Three teachers, A, B, and C, speaking of their respect- ive schools, says A to B, "If you will give me 20 of your scholars, my number will then be to the sum of C's and what you will have left, as 4 to 5." Says B to A and C, " If each of you will give me 10 scholars, my number will be to what you will then have as 5 to 4." Says C to A and B, " If you will give me 10 each, I shall have twice as many as both of you." What was the number of scholars each had 1 Ans, A 20, B 30, and C 40 scholars, 4. A cistern is furnished with three pipes, A, B, C. By the pipes A and B it can be filled in 12 minutes, by the pipes B and C in 20 minutes, and by A and C in 15 108 ELEMENTS OF ALGEBRA. [sECT. IV. minutes. In what time will each fill the cistern alone, and in what time will it be filled if all three run to- gether 1 Ans. A 20, B 30, C 60 minutes^ and the three together in 10 minutes. 5. It is required to divide the number 72 into 4 such parts, Uiat if the first part be increased by 5, the second di- minished by 5, the third part multiplied by 5, the fourth part divided by 5, the sum, difference, product, and quotient shall all be equal. Ans. 5, 15, 2, and 50. 6. Find three numbers, such that \ of the first, \ of the sec- ond, and \ of the third shall be equal to 62 ^ i of the first, \ of the second, and ] of the third equal to 47 j and \ of the first, \ of the second, and \ of the third equal to 38. Ans. 24, 60, and 120. 7. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days, how many days will it take each person alone to perform the same work ? Ans. A in 14f|, B in 17-^, -and C in 23/^ days. . 8. A, B, and C sit down to play, each one with a certain number of shillings : A loses to B and C as many shil- lings as each of them has. Next B loses to A and C as many as each of them now has : lastly, C loses to A and B as many as each of them now has. At the close of the game, each of them has 16 shillings. How much did each one gain or lose 1 Ans. A lost 10s., B gained 25., and C Ss. 9. There are two such fractions, that if 3 be added to the numerator of the first, its value is double that of the second ; but if 3 be added to the denominator, their values are ecfiial. Now the sum of the two fractions is 9 times as great as their difference ; and if the numera- tor of their product be increased by 10, its value will be equal to that of the first fraction. What are the fractions'? Ans. j\ and ^. 10. Three brothers purchased an estate for $15,000: the SECT. IV.] EQUATIONS OF THE FIRST DEGREE. 109 first wanted, in order to complete his part of the pay- ment, ^ of the property of the second j the second would have paid his share with the help of ^ of what the first owned ; and the third required, to make the same payment, in addition to what he had, | part of what the first possessed. What was the amount of each one's property 1 Jlns. $3000, $4000, and $4250 respectively. 11. A merchant has 3 ingots, each composed of gold, sil- ver, and copper, in the following proportions, viz., in the first there are 7 ounces of gold, 8 ounces of silver, and 1 ounce of copper to the pound; in the second, there are 5 ounces of gold, 7 ounces of silver, and 4 ounces of copper; and in the third, 2 ounces of gold, 9 ounces of silver, and 5 ounc^ of copper to the pound. What parts must be taken from each in order to compose a fourth ingot, in which there shall be 4} f ounces of gold, 7|| ounces of silver, and 3j^ ounces of copper to the pound 1 Ans, 4 ounces ofgoldj 9 ounces of silver, and 3 ounces of copper. 12. At an election for two members of Congress, three men offer themselves as candidates: the number of vo- ters for the two successful ones are in the ratio of 9 to 8; and if the first had had 7 more, his majoritj'^ over the second would have been to the majority of the second over the third as 12 to 7. Now if the first and third had formed a coalition, and had one more voter, they would each have succeeded by a majority of 7. How many voted for each 1 Jlns. 369, 328, and 300 respectively. 10 110 .ELEMENTS OF ALGEBRA. [sECT. V. SECTION V. Generalization of Algebraic Problems. — Demonstration of Gen- eral Propositions or Theorems. — Properties of J^umhers. — Reduction of Formulas relating to Simple Interest^ Com- pound Interest^ and Fellowship, — Discussion of Equations of the First Degree. — Theory of Negative Quantities. — Ex- planation of Symbols. — Infinity. — Infinitesimal. — Indetermi' nation. — Inequations. GENERALIZATION OF ALGEBRAIC PROBLEMS. 188. The soiution of many questions does not depend upon the particular numbers given in those questions, but will be the same for any other numbers. By generalizing such questions, we are able to deduce a general method or rule for the solution of all questions whose conditions are similar, or which differ from the proposed only in particular numbers which are given. The following instances of generalization will serve to in- troduce the learner into this important branch of Algebra. First General Problem. 189. The sum of two numbers is a, their difference b; it IS required to find the two numbers. Let x= the greater, and y= the less: Then, by the conditions - - x-\-y—a. And - - - s - - X — y:=b ; Adding the two equations - - 2a? =a-\-b, 2 2' Subtracting the 2d from the 1st equation, 2y=a—b; _a b ^~2 2* Hence, since a and b may represent any numbers what- ever, the sum and difference of two quantities being given, SECT, v.] GENERALIZATION. Ill 1. To find the greater, ^dd the half difference to the half sum, 2. To find the less, Subtract the half difference from the half sum. EX^AMPLES. 1. The sum of two numbers is 24, the difference 6 : what are the two numbers 1 Let x= the greater, and y= the less : Then x=^+|=?iH-^= 124-3=15, the greater, JL JL JL JL • And y=-— -=——-= 12—3=9, the less. ^2222 ' 2. The sum of two numbers is 56, their difiTerence 12: what are the numbers \ 3. It is required to divide $860 between two men, so that the first may have $250 more than the second. 4. Two merchants invest in trade $10,000; the sum in- vested by the first exceeds that invested by the second by $1225; what was the sum invested by eachl Second General Problem. 190, The sum of three numbers is a ; the excess of the mean above the least, b ; and the excess of the greatest above the mean, c. Required the three numbers. Let x= the least, y= the mean, and z= the greatest: Then, by the conditions - x-\-y-\-z—a; y—x=b; z-^=c ; Reducing these three equations, x= ^ ? o a-\-b — c 3 a4-*-f-2c J'=-3-' 2 = 3 Hence, since a, i, and c may represent any values what- ever, having given the sum of three numbers, the excess of the mean above the least, and the excess of the greater above the mean : 112 ELEMENTS OF ALGEBRA. [sECT. V. 1. To find the least, Fro^ their sum subtract the sum of twice the mean above the least, and the excess of the greater above the mean, and divide the remainder by 3. 2. To find the mean, To their sum add the excess of the mean above the least, and from the result take the excess of the greatest above the mean, and divide the remainder by 3. 3. To find the greatest, Jldd together the sum of the three numbers, the excess of the mean above the least, and twice the ex- cess of the greatest above the mean, and divide the sum by 3. EXAMPLES. 1. The sum of three numbers is 440 ; the excess of the mean above the least is 40 ; the excess of the greatest above the mean is 60 : vi'hat are the numbers 1 Let X, y, an^ z. represent the numbers. Then x=g=m=^^°-(-^^"+^°)=100, the least number ; a4j_e^440+40-60^^^Q^^ ^^^^^^^^ ^33' _«4.ft+2c^440+40+2x60^200, the greatest 3 3 number. 2. It is required to divide a prize of $973 among 3 men, so that the second shall have $69 more than the first, and the third $43 more than the. second. 3. The sum of three numbers is 15,730 ; the second ex- ceeds the third by 2320, and the first exceeds the sec- ond by 3575 : what are the three numbers 1 Third Genera^ Problem. 191. The sum of 4 numbers is a; the second exceeds the first by b ; the third exceeds the second by c ; the fourth exceeds the third by d. Required the four numbers. EXAMPLE. Find each of the above numbers, on the supposition that a=:3753, Z>=159, c 275, and t/^389. SECT, v.] GENERALIZATION. ] 13 Fourth General Problem. 192. The sum of 2 numbers is a, and if 3 times the first be divided by 2 times the second, the quotient will be b,' Bequired the numbers. EXAMPLE. If a=420 and 6=&, what are the numbers 1 Fifth General Problem, 193. The sum of two numbers is a, and if the first be di- vided by 5 and the second by 2, the sum of the quotient will be b. Required the numbers. EXAMPLE. If a=120 and J=42, what are the numbers! Sixth General Problem, 194. Three men share a certain sum in the following man- ner, viz.: the sum of A's and B*s shares is a; that of A's and C*8, b ; that of B's and C's, c. What is the sum di- vided, and the share of each % EXAMPLES. 1. If a=:$123, 5= $110, and c=$83, what will be the sum, and the share of each 1 Seventh General Problem. ' 195. A person engaged a workman to labour n days'; for each day that he laboured he was to receive a cents, and for each day he was idle he was to pay b cents : at the time of settlement he received c cents. How many days did he la- bour, and how many was he idle \ Let x= number of days he laboured, y= number he was idle ; 114 ELEMENTS OF ALGEBRA. [sECT. V. Then, by the conditions, ax — hy=^c or x=z — L_, a-i-b' 1 an — c and y: a+b JVote. — If the labourer had paid c cents instead of receiv- ing it, the general equations would become, bn — c a!-\-y=n by—ax=zc or 0?: , a-\-b' J an-{-c and y= ! — EXAMPLES. 1. Jf w=48, a=24, b—-12, and c=r504, how many days did he work, and how many was he idle 1 2. A labourer was hired for 75 days : for each day that he wrought he was to receive $3, but for each day that he was idle he was to forfeit $7. At the time of settle- ment he received $125 : how many days did he labour, and how many was he idle 1 3. A man agreed to carry 20 earthen vessels to a certain place on this condition, viz., that for every one deliver- ed safe he should receive 11 cents, and for every one he broke he should forfeit 13 cents : he received 124 cents. How many did he break 1 4. A fisherman, to encourage his son, promises him 9 cents for each throw of the net in which he should take any fish j but the son is to forfeit 5 cents for each un- successful throw. After 37 throws the son receives from the father 235 cents. What was the number of successful and unsuccessful throws of the net 1 DEMONSTRATION OF GENERAL PROPOSITIONS OR THEOREMS. 196. It was remarked in the introductory section of this work, that algebraic symbols might be applied to the dem- onstration of general truths or principles. We will now exhibit a few of these applications. SECT, v.] THEOREMS. 115 First Theorem. 197. The greater of any two numbers is equal to half their sum added to half their difference, and the less is equal to half their sum minus half their difference. Let a and h represent any two numbers, of which a is the greater and h the less j let their sum be represented by 5, and their difl^erence by d: Then a+b=s. And a—b=d; Adding the equations - - - 2a =s-{-d;\ Dividing a =i + ^-j Subtracting the 2d from the 1st equation, 2b=s — d; ^ Dividing - ^ *=|— -. I Second Theorem. 198. The product of the sum and difference of two num- bers is equal to the difl^erence of their squares. Let a, b, 8, and d sustain the same relations as in the p(re- ceding theorem : Then s=a-\-b, And - - - . - d=a — b. Multiplying the two equations, d.s z=(a-\-b)(a — b)=a* — 6*. CoBOL. 1. — Dividing the above equation by d, we have a»— A« Hence, if the difllerence of the squares of two numbers be divided by the diflJerence of the numbers, the quotient will be their sum. CoROL. 2. — Dividing the same equation by Sy we shall have d-. a -I^ 8 Hence, if the difference of the squares of two numbers be divided by the sum of the numbers, the quotient will be their difference. 116 ELEMENTS OF ALGEBRA. [sECT. V. Third Theorem, 199. Four times the product of any two numbers is equal to the squares of their sum, diminished by the square of their difference. Let a, &, 5, and d sustain the same relations as in the pre- ceding theorem : Then ------ a-\-b=s^ a—b=d. Adding the two equations - - 2a—s-\-d; Subtracting the 2d from the 1st - 2b=zs — d; Multiplying the 3d and 4th - - 4}ah=zs^ — dK Fourth Theorem. - 200. The sum of the squares of any two numbers is equal to the square of their difference plus twice their product. Let a, &, and d sustain the same relations as before, and let p represent the product of the two numbers-: Then - a — b=:d, And ------ ab—p; Squaring the members of the 1st ) o a i /2 ^. equation " " " ) Multiplying the 2d equation by 2, 2a& =2p/ Adding the two equations to-) 2_j_r2_-72io gether - - - - j Fifth Theorem, 201. The square of a polynomial expressing the sum of two numbers, is equal to the square of the first term + twice the product of the two terms + the square of the last term. Let s represent the sum, and a-{-b the polynomial: Then s =a -\-b ; Squaring the equation - - s^—a^+'^ab-\-W. Sixth Theorem, 202. The square of a polynomial expressing the differ- SECT, v.] THEOREMS. 117 ence between two numbers, is equal to the square of the first term — twice the product of the two terms -f- the square of the last term. Let d represent the difference, and a—b the polyno- mial : Then d=a—b; Squaring the equation - - (P=a" — 2a&+J* Seventh Theorem. 203. The difference of any two equal powers of different numbers, is always divisible by the difference of the num- bers. Let a and b represent finy iwo numbers, a being greater than b : Then ^II^=a-\-b, a — b And ^II^=(^+ab-\-l^, a — b And ^LZI^^a^^a'b+ab'+k a — b This process may evidently be continued indefinitely; hence we have ^!lll^=a'^'+a™-2x J + a'"-='x&«-f- a^b'^-{-ab^-^-\-b'^K a — b CoROL. If J=l in the above formula, the formula will be- come a"*— 1 =a'^'+a'"-'4-a'"~^4- .... -f o'+a'+a-fl. Eighth Theorem, 204. The difference of two equal powers of different num- bers, is divisible by the sum of the numbers, when tlie expo- nent of the power is an even number. Let a and b sustain the same relations as before : Then - . . ^Ill=a-b, a-\-b 118 ELEMENTS OF ALGEBRA. [sECT. V. And - - - '^^II^=^a^—d?h+ah^—h\ a -\-b And - - - ^^^=a'—a''b+aW—a'}y'-\-ab*—b\ a -j-b Hence we also conclude (letting m represent any even number), - — ^=a^-'-j-d^^xb+ar-^Xb^-^ . . . -j-a'b^'-^-^-ab^-^+b^K a -^b CoROL. If ^=1, the above formula will become «"» 1 t==a^''-'a^-''+a"'-^— .... -^a^—a'-\-a—l. a +1 JSPinth Theorem. 205. The sum of any two equal powers of different num- bers, is divisible by the sum of the numbers, when the expo- nent of the power is an odd number. Let a and b sustain the same relations as before : Then - - '^±^=a'—ab-]-b% a -\-b And - - '!L±^=:a'—a'b+a'b^—ab^+b\ a -{-b And - - '^-±^=a'—a'b-\-a'b'—a'b'+a'b^—a¥+b\ a -j-b Hence we also conclude (letting m represent any odd number), ^-t!!l=a^~'—a^^xh-{-a""^Xb^— .... — a='6— "^-a^Z^'^-^— a -\-b ab^''-\-b^-\ CoROL. If &=!, the above formula will become a +1 Tenth Theorem, 206. If a given number be divided into two parts, and those parts multiplied together, the product will be the greatest possible when the parts are equal. Let n=: the given number, a=. the greater part, b=. the 8BCT. v.] PROPERTIES OF NUMBERS. 119 less part, d= the difference between the parts, and^= the product of the two parts: Then - - ab=p. And (Art. 189) - - - - a =^ +^, n d 2 2 And (Art. 189) - • -' - - J=s— -• Multiplying the last two equations together, ab=— — — . XT n^ d* Hence - P=— — -• ^ 4 4f Now It is evident that p will increase as d diminishes; hence it will be the greatest possible when d=Oy or p=i . DEMONSTRATIONS RELATING TO CERTAIN PROPERTIES OF NUMBERS. 207. We will now apply the principles of Algebra hereto- fore discussed to the demonstration of some singular prop- erties of numbers. Let it first be premised that the local value of the digits increases in a tenfold ratio from right to left, and that any number is equal to the number of units expressed by the digit in the unit's place, -f the number of units expressed by the digit in the ten's place, -|- the number of units ex- pressed by the digit in the hundred's place, -f j &c. Thus, 3756 = 6 + 504-700+3000, and 12,899=9+90+800 +2000+10,000. > First Proposition, 208. If from any number the sum of its digits be subtract- ed, the remainder will be divisible by 9. Let a, 6, c, (/, &c., represent the digits of any number, a being the digit in the unit's place, h the digit in the ten's place, &c. ; also let N=: the number, n= the num- ber of the digits, S= the sum of the digits, and r=10: 120 ELEMENTS OF ALGEBRA. [sECT. V. Then 'N=a-\-br -j-cr" +c?r' +...+a?r"-', And S=za-{-b +c +d +...+a?. N— S= br—b -{-cr'—c +d7^—d + . . . +a?r"-'-— a?, Or N— S== ^>(r— l) + c(r^— l)H-c/(r'— 1)+- • • +x(r"-'— 1), by subtraction. But (Th. 7, Art. 203) r— 1, r^-l, r»— 1+ . . +r"-^— 1, are divisible by r — 1, which is equal to 9 j hence, N — S is also divisible by 9. Example. 327,856 — (3+2+7+8+5 + 6) = 327,825, and .327,825^ 9 ==36,425. Second Proposition, 209. If the sum of the digits of any number be divisible by 9, the number itself is divisible by 9. Let N= any number, and S= the sum of its digits: Then, since S is supposed to be divisible by 9, let S=9m Since N— S is divisible by 9, let 'N—S=9p : Then - - N— S=N— 9w = 9j9; Transposing - - N=z9p-{-9m ; Resolving into factors, N=9(7?-}-m), which is divisi ble by 9 ; consequently, N also is divisible by 9. Example. 5 1,489 ^9=^5721, and (5+1+4+8+9)^9=27 H-9 = 3. ^ Third Proposition. 210. If the sum of the digits of any number be divisible by 3, the number itself is divisible by 3. Let N represent any number, and S the sum of its digits, as before ; and let S = 3m, and N — S=3p : Then - - N— 37^ = 3;? ; Transposing, N=3p + 3?7i, which is evidently divisible by 3. Example. 785,142-^3=261,714, and (7+8+5+1+4+2) -f-3=27H-3=9. Fourth Proposition. 211. If from any number the sum of the digits standing in the odd places be subtracted, and to it the sum of the SECT, v.] PROPERTIES OP NUMBERS. 121 digits Standing in the even places be added, the result will be divisible by 11. Let the number be a+br -f-cr* -{-drf Sec, : Add - - — a+ft — c -fc/, &c. The result is - br-^b +cr'—c ■i-dr^-{-d, &c., Or - - - 6(r+l)-|-c(r'— l)+c/(r'+l), &c. But (Ths. 8 and 9) r'{- 1, r»— 1, r'-f- 1, &:c., are divisible by r-f- 1 ; hence, b{r-\- l)+c(r»— l)+t/(r'4- 1), &c., is divisible by r+1, or 11. Example. (57,937-(7-t-94-5) + (7+3))-Ml =(57,937— 21 H-10)-H 11=57,926-7-11 = 5266. Fifth Proposition. 212. If the sum of the digits standing in the even places in any number be equal to the sum of the digits standing in the odd places, the number is divisible by 11. Let N= the number, S= the sum of the even digits, and $z= the sum of the odd digits : Then, by Prop. 4, N+S— s is divisible by 11. But S— « =0j therefore N is divisible by 11. Example. (137,456+13- 13)^11 = 137,4.56-M1=12,496. Sixth Proposition. 213. Every prime number which will exactly divide the product of two factors, one of which is also a prime num- ber, will divide the other. Let AxB represent the product of two numbers, which is divisible by P ; A being greater than P, and prime with it, or not divisible by it. Then let us endeavour to find the greatest common divi- sor of A and P, representing the successive quotients ^y Qj Q > &c*> ai*d the successive remainders by R, R', 11 Q 122 ELEMENTS OF ALGEBRA. [sECT. V. P)A(Q QP "IR) P (Q' Q'R R') R (Q" Q"R R", &c. Making each dividend equal to the product of its divisor and quotient, we have 1. - - - A=PQ+R; 2. - - - P=RQ'+R'; 3. - - - R=:R'Q"+R", &c. MuUiplying the first equation (1.) by B, AB=PQB + RBj Dividing by P . . - . ^= BQ+^. By hypothesis, _ - produces a whole number ; and since B and Q are whole nuhibers, the product BQ is a whole •p-p number j hence — - is also a whole number. Multiplying the second equation (2.) by B, and dividing by P, we have P>_ BRQ BR^ ■pin We have already shown that -— . produces a whole num- ber ; hence ^ will also produce a whole number. This BR' being the case, _— - must also be a whole number. If this operation is continued till the number which multiplies B R V 1 becomes 1, we shall still have — — _, equal to an entire num- ber 5 therefore B is divisible by P. SECT, v.] REDUCTION OF FORMULAS. 12$ Hence, if a number will exactly divide the product of two numbers, and is prime with one of them, it will divide the other. REDUCTION OP FORMULAS. 214. The processes of generalization which we have no- ticed will suggest some methods of demonstrating formulas or general rules. FORMULAS RELATING TO SIMPLE INTEREST. * 215. It is required to deduce general formulas or rules for the computations relating to simple interest. 216. To present the subject in a general point of view, let us consider the five things that enter into the calculation, viz.. Principal^ Interest, Rate, Time, and Amount. Let p= principal, t= interest, r= rate per cent., t= time, and a= amount. Taking the dollar as unity, r will be a fraction, whose ae- nominator is 100. If the given sum be put at interest for one year, then <=1 j if for a longer period, ^>1 ; if for a shorter period, t^l. The interest of $1 will evidently be proportional to the rate and time jointly, or the interest of $l=rx^ The rate and time being the same, any given principal will be to any other principal as the interest of the former is to the interest of the latter. Hence - - ^\ i ^p : :rxt : i, ox i=pxrxt. By making the necessary transformations, we obtain the four following formulas : 1. - - i=prt, 2 - - p=iL ^ rt 3. . - /=i. pr 4. r=l pt These formulas may be enunciated in the form of general rales. 124 ELEMENTS OF ALGEBRA. [sECT. V. EXAMPLES. 1. What is the interest of $3875,20 for 3 years, at 7 per cent, per annum % 2. What is the interest of $325 for 3 months, at 6 per cent, per annum % 3. The interest of a certain sum is $92,75, the time 3 years and 6 months, and the rate 5 per cent. : what is the suml 4. The interest received for $4070, at 9 per cent., was $91,575 : how long was it at interest 1 5. The sum $367J was put at interest for 6 months, and at the end of the time 'the interest paid was $183,55 • what was the rate per cent. % 217. Since the amount is equal to the principal + the in- terest, or a=p-\-prt, hy making the necessary transforma- tions we shall have 1. - - a=p-{-prt, % - ' p 1+r? 3. - ^ T=^i:2. pt 4. - . ^=^Zf. pr These formulas may also he stated in the form of gen- eral rules. EXAMPLES. 1. If;?=:$895, r=7, and tz=z^, what is the value of a ? 2. If a=$7589,50, r^S, and ^=5', what is the value ofp^ 3. If a=:$820,20, ;)=:$600, and t—^, what is the numer- ical value of r ? 4. If a=$525,86, ;}z=$35,80, r=4, what is the numerical value of ^ ? FORMULAS RELATING TO COMPOUND INTEREST. 218. In Compound Interest the interest is supposed to re- main in the hands of the borrower, and, being added to the SECT, v.] REDUCTION OF FORMULAS. 1S5 principal at the end of each year, forms a part of the princi- pal for the succeeding year. 219. Let p and r sustain the same relations as before, and let a= the amount for the first year, and, consequently, the principal for the second year, a'=: the amount for the sec- ond year, a"= the amount for the third year, a"'= the amount for the fourth year, &c. Then, as $1 will be to any given sum as the amount of $1 for one year is to the amount of that given sum for the same time, we shall have \:p :: 1+r : a, or a =p(l-\-r), And - l:p(l-^r) : : l+r : a', or a' =Xl+r)^ And - 1 :p{l-\-ry : : 1+r : a", or a" =p(l+ry, And ■ 1 :pll-\-ry : : 1+r : a'", or a" =p(l-{'r)\ &c. Let A represent the amount for n years, and we shall have 1 :;)(l+r)'*-' : : 1+r : A, or A=p(l+ry. Hence, by making the necessary transformations, we ob- tain the following formulas :* 1. - - A=p(l+rY, 2. . ^ p=-A.^. (1+rr These formulas may be stated in the form of general rules. EXAMPLES. 1. If p = $3250, 71=8, and r=5, what is the numerical value of Al 2. If A=$30,200, n=20, and rz=6, what is the numerical value of ^ ? 3. If A = $1479,15, p=$1000, and n=6, what is the nu- merical value of r ? ♦ The fourth formula is omitted, since it would involve Logarithms, which are treated of in a subsequent section. Q2 126 EI^BMENTS OF ALGEBRA. [SECT. V. FORMULAS RELATING TO FELLOWSHIP. 220. Two men engage in trade together, and furnish mopey in proportion to the numbers m and n ; they gain a sum represented by g j it is required to deduce formulas for the division of the gain, so that each man shall receive his equitable share. Let X— the share of the first, and y— the share of the second : Then a?+y=g, And - - - a? : y : : 771 : 7i, or my—nx. Reducing these equations, we obtain x=J^^-. •-\-n Hence, to find each man's share of the gain. Multiply his stock hy the whole gain, and divide the product by the whole stock invested. Example. Two merchants, A and B, gained by trading in company $20,480. A's stock was $15,000, B's $18,000: what was each man's share of the gain % 221. Again: suppose three persons engage in trade, and furnish money in proportion to the numbers lUj w, and p ; they gain a sum represented by g j it is required to deduce formulas for the division of the gain as before. Let a?, y, and z represent the respective shares of the three persons: Then we shall have X : y : : m : Uj or my—nx, X : z : : m : Pj OT mz=px. Reducing the above equations, we obtain x=_^ m-{-n-{-p' SECT, v.] REDUCTION OF FORMULAS. 127 Hence, to find each man's share of the gain, Multiply his stock by the whole gain, and divide the product by the whole amount of stock invested. Example. Three merchants, A, B, and C, gained hy tra- ding in company $1100}. A's stock was $1500, B's $1200, and C*s $850: what was each man's share of the gainl 222. As the above formulas contain four things, viz^y whole stock, whole gain, the particular stock whose share of the gain is to be found, and that share of the gain, it is ev- ident that any one of these may be found if the other three be given. Letting S= whole stock, S'= stock whose share of the gain is to be found, g= the whole gain, and g'= share of the gain to be found, and substituting these letters in the preceding formulas, they become 1. - - g'=^. ^ S 2. . - g=^' . S' 3. . - S'=^. g 4. . . S=^X 8" EXAMPLES. 1. Two men, A and B, traded in company, with a joint capital of $8000; they gained $1250. A's stock was $3250 : what was his share of the gain 1 2. Three men. A, B, and C, jointly invest in trade $2725 ; they gain $560, of which A receives as his share $120, 6 receives as his share $160: what was the stock in- vested by each 1 3. Three men, A, B, and C, gain by trading $6000. A's 12S ELEMENTS OF ALGEBRA. [sECT. V. stock was $8000, and he took as his share of the gain $2800 : what was the whole stock invested 1 4. Two men, A and B, invest in trade $3000. A's gain was $250, and his stock $2600 : what was the whole gainl 223. Let us now consider the cases in which the stock of the partners in trade has been invested for different lengths of time. 224. Two men engage in trade together, and furnish money in proportion to the numbers m and n, for the times t and f ; they gain a sum represented by-g: it is required to deduce formulas for the equitable division of the gain. Let X and y represent the respective shares of the gain : Then we shall have nt'x^=zmty, Keducing these equations, mtor X=z^ 2__. y= ^ mt-\-nt' These results may be enunciated in the form of a general rule. 225. Again : suppose three persons invest in trade money in proportion to the numbers m^ tz, and p, for times ^, /', and t'' ; the sum gained is represented by g : it is required to deduce formulas for the equitable division of the gain. Let a?, y, and z represent their respective shares of the gain: then x-\-yi-z=g, nt'x—mty, pt"x=mtz. Reducing these equations, mts: X=z o . mt-^nt'-\-pt" mt-^nt'-i-pt" SECT, v.] DISCUSSION. 129 mt-\-TW -\-pt" These results may also be enunciated in the form of a general rule. Example. A, B, and C enter into partnership. A invests $1200 for 3 years, B $2000 for 2 years and 9 months, C $950 for 4 months. They gain $2400 : what is each man's share of the gain ] DISCUSSION OF EaUATIONS OF THE FIRST DEGRER 226. When a question has been solved in a general mai> ner, that is, by representing the known quantities by letters, it may be proposed to determine what values the unknown quantities will take when particular suppositions are made upon the known quantities. This is called the discussion of that equation. 227. The discussion of the following problem presents nearly all the circumstances that can ever occur in equa- tions of the first degree. * PROBLEM OF THE COURIERS. A courier sent out from a certain place travels in a right line with a velocity expressed by n. After the first courier had travelled a distance, a second was despatched after him, travelling with a velocity expressed by m. At what dis- tance from the starting point will they be together 1 In order to render the conditions of the question more evident, let ED 1 1 1 ( 1 C A B C represent the line upon which the couriers travel, A the starting point, B the point at which the first cou- rier is when the second starts, and C the point at which the second will overtake the first : Let x= AC, and y=BC : Then x — y=a^ R 130 ELEMENTS OF ALGEBRA. [sECT. V. And —=y.. m n Reducing these equations, we have am, J an -, and y: .^. m — n M — n DISCUSSION. I. Let m >n. 228. In this case the values of x and y will be positive, and the solution of the problem will exactly accord with the enunciation j for if the second courier travels -faster than the first, they will evidently meet somewhere in the direction AD, and to the right of B. II. Let OT< n. 229. In this case the values of x and y will be negative. In order to interpret this result, we observe that, the courier from B travelling faster than the courier from A, the inter- val between them must increase continually. It is absurd, therefore, to require that they should meet in the direction AD. The negative values of x and y, then, indicate an ab- surdity in the conditions of the question. To remove this absurdity, we have only to suppose that the two couriers start at the same time from B and A, and travel in the direc- tion BE', in which case the equations will become y—x=za, And -=t m n f^ am J an Or - - - 0?— , and y= , n — m n—m which give the values of x and y positive, and indicate that the couriers will come together at C instead of C. III. Let m—n. 230. In this case the values of x and y become am am /». — — m — n an an m — n SECT, v.] THEORY OF NEGATIVE QUANTITIES. 131 In order to interpret this result, let us return to the ques- tion. If the couriers travel with equal velocities, it is ev- ident that the interval between them must always continue the same, however far they may travel in either direction. Indeed, on the hypothesis w=:w, the conditions of the ques- tion produce And X — y=0, equations which are incompatible with each other. It is therefore absurd to suppose that the couriers will come to- gether on this supposition. IV, Let m=n, and a=0. 231. In this case, Oxm X=:. m — n m — n an _Ox_m_0 jn—n m — n In order to obtain a correct interpretation of this result, it is only necessary to observe that, if the couriers set out each from the same point at the same time, and travel equally fast, there is no particular point at which one can be said to overtake the other, since they will be together, however far, and in whatever direction they may travel. Indeed, on this supposition, the conditions of the problem produce a? — y=0, x—y=0, two dependant or identical equations. The problem is therefore indeterminate, since we have, in fact, but one equation with two unknown quantities. THEORY OF NEGATIVE aUANTITIES. 232. It has already been shown, • 1. That adding a negative quantity is the same as sub- tracting an equal positive quantity. 2. That subtracting a negative quantity is the same as adding an equal positive. 132 ELEMENTS OF ALGEBRA. [sECT. V. 3. If a negative quantity be multiplied or divided by a positive, the result will be negative. 4. If a positive quantity be multiplied or divided by a negative, the result will be negative. 5. If a negative quantity be multiplied or divided by a negative, the result will be positive. 233. We will now proceed to show that, if the conditions of the problem are such as to render the unknown quantity essentially negative, it will appear in the result with the minus sign, although it may have been regarded as positive in the statement of the problem. 1. The length of a certain field is a, and its breadth h: how much must be added to its length that its contents may be c ? Let xz=z the quantity to be added to the length: Then a-{-x=i whole length. Since the area of a field is found by multiplymg its length by its breadth, we have ab-\-lxz=:c. Reducing - - - xz=z- — a. Now, letting a=8, 6=5, and 0=60, the equation becomes a:^^— 8 = 12— 8zzz4. 5 This value of x satisfies the conditions of the problem in the precise sense in which it was stated. Again: letting a=:8, 6=5, and c=30, Then .... a:=r^— 8 = 6— 8=— 2. 5 In order to interpret this negative result, let us return to the original eq^uation : ab-\-lx—c. Substituting - - - 8x5 + 5x— 2=30j Resolving into factors - - 5(8 — 2) = 30. Hence we perceive that, though addition was required by the enunciation, yet it was incompatible with the conditions SECT. ▼.] EXPLANATION OF SYMBOLS. 133 of the question ; and the algebraic result, true to the condi- tions of the question, detects the error in the enunciation, and shows that x is to be subtracted from instead of being added to the length of the side. Thus, ab — bxz=c, Or x=a-~^. o By substitution - - - a?=8--^=8— 6=2. This result answers to the question modified in this man- ner: The length of a certain field is a, and its breadth b: how much must be subtracted from its length that its contents may be c ? 234-. Discuss in like manner the following questions : 2. A father is a years old, and his son b: in how many years will the son be one fourth as old as the father 1 3. A man when he was married was a years old, his wife b : how many years before his marriage was he t\vice as old as she 1 EXPLANATION OF SYMBOLS. Infinity. 235. A mathematical quantity is said to be infinite when it is supposed to be increased beyond any determinate limits. The symbols usually adopted by mathematicians to ex- press such quantities are — and oo, A being used to repre- sent any finite quantity. In order to explain these symbols, let us resume the equa- tions for X and y in the problem of the couriers: am 1 an ,andy=: ; ffi — n tn — n Or, if m=n - - x=—, and y=--. 236. In order to explain these expressions for the values 12 t34 ELEMENTS OF ALGEBRA. [SECT. V. of X and y, we will show how these values are affected by assuming different values for m and n. If 7w = 3, and n—% 3a, and y= =2o. 3—2 " 3—2 If OT=:3, and %= 2,9, icz=?^30a, and w=?l?^=:29a. If»i=3, and»=2,99, a;:zz^ = 300a, and y=?l^=299a. ,01 ^ ,01 IfOT=:3, and7z=2,999, ic=-^=3000a, and y=M??i^=2999a. ,001 ' ^ ,001 If »jzz:3, and 71=2,9999, ir=-?^ = 30 000«, and y=M^^_?.= 29 999a, &c. ,0001 ^ ,0001 ' Hence we infer, that if the difference between m and n be- comes less than any assignable quantity, the values of x and y will be greater than any assignable quantity ; and _ or oo is the proper symbol of infinity. 237. CoROL. Since, by the conditions of the question, a?= y+a, which will continue to be the case when the values of X and y become infinite, we infer that one mathematical in- finity may be greater thdn another. Infinitesimal. 239. A mathematical quantity is called an infinitesimal, or sometimes nothing, when it is supposed to be decreased below any determinate limits. A The symbols used to express such a quantity are — or 0. CO In order to explain these symbols, let us resume again the equations am J an iC== , and y=- w, — n in — n SECT, v.] EXPLANATION OF SYMBOLS. 135 239. A course of reasoning similar to that adopted in the preceding case will show that the values of x and y decrease as the difference between m and n increases. Hence, when that difference becomes greater than any assignable quan- tity, the values of x and y become less than any assignable quantity. That is, x= = — , or the value of x may be expressed m — n 00 by the symbol — or 0. QO And y=— — =^, or the value of y may be expressed by m — n 00 the symbol _ or 0. QO 240. CoROL. Since x=y4-fl, we infer that one infinitesimal may be greater than another. EXPLANATION OF THE SYMBOL OF INDETERMINATION. 0* 241. A quantity is said to be indeterminate when every possible value will satisfy the conditions of the question. 242. The symbol used to express indetermination is -. We have already seen that the equations x= "^ , and m — n , on the hypothesis m — n and a=0, reduce to x= I — n -, and y=-y and also that all possible values of x and y will satisfy the conditions of these two equations. 243. We will, however, add another illustration to this case. I «» Take the expression : if we perform the division, the 1 — X quotient will be 1 j ahd if we make x= 1, there will result U . . . !=£=!=?. l—x 136 ELEMENTS OF ALGEBRA. [sECT. V. 2. - - - h:^=i+x=i=l l—x 3- - - - 1eJ=i+-+-=3=5. drc, ad infin. Hence every possible value will satisfy the conditions — 244. It should, however, be observed, that this symbol does not always imply indetermination. Thus, the expression x— , if a=b, will become or — r But, resolving the terms of the fraction into factors, ^^ (a-b){a' + ah-\-b') _ d'-\-ab-\-b^ (a—bXa+b) ~ a-\-b ' ^ * which, on the supposition a—b^ becomes X — . a-\-a '■la 2 Hence we conclude that the symbol - , in Algebra, some- times indicates the existence of a factor common to the two terms of the fraction, which, in consequence of a particular hypothesis, becomes 0, and reduces the fraction to the form INEaUATIONS OR INEaUALITIES. 245. The principles established respecting equations will in most cases also apply to inequations. As there are some exceptions, we will here illustrate the principal transforma- tions which may be made upon inequations, and then apply those transformations to the determination of the limits of unknown quantities. 246. Two inequations are said to subsist in the same sense when the greater quantity stands at the right in both or at * See Note C. SECT, v.] INEQUATIONS. 137 the left in both, and ia a contrary sense when the greater quantity stands at the right in one and at the left in another. 24-7. 1. We may always add the same quantity to both members of an inequality, or subtract the same quantity from both members, and the inequahty will continue in the same sense. Thus, let - - 2<12 ; adding 6 to both sides, we have - 6+2< 124-6, Or - . - 8<18. Again: let - — 2> — 12: Then - - 6— 2>6— 12, or 4>— 6. CoROL. A term may be transposed from one member of an inequation to the other. 24-8. 2. If we add the corresponding members of two or more inequations subsisting in the same sense, the inequa- tion which results will exist in the same sense of those added. Thus - - - 8>5, And - - - 10>2. Adding - - 19 + 8>2+5, or 18>7. But if we subtract the corresponding members of one in- equation from another subsisting in the same sense, the re- sulting inequation will not always exist in the same sense. Thus, from - - - 4.<7 Subtract - - - - 2<3. There will remain - 4— 2<7— 3, or 2<4.. But if from - - - 9<10 We subtract - - - 6< 8, There will remain - • - 9— 6>10— 8, or 3>2. 249. 3. If both members of an inequation be multiplied or divided by any positive whole number, the resulting ine- quation will exist in the same sense as the inequation mul- tiplied. Thus - - - - 6<10; Muhiplying by 3 - - 18<30. ; Or, again ... - ^<^; Multiplying by 6 - - 2<3. 138 ELEMENTS OF ALGEBRA. [sECT. T. CoROL. An inequation may be freed from fractions in the same manner as an equation. 250. 4. If we multiply or divide the two members of an inequation by a negative quantity, the resulting inequation will subsist in a contrary sense. Thus 6<10; Muhiplying by —3 - - - — 18>— 30. Or, again ... - i— 3. Hence it also follows, that if we change the sign of each term of an inequation, the inequation which results will ex- ist in a sense contrary to the inequation proposed j for this .transformation will be equivalent to multiplying the inequa- tion by — !► EXAMPLES. 1. Find the limit of the value of x in the inequation „ 23^ 2a? , . 3^3 Clearing of fractions, 21a:— 23>2a?+ 15 ; Transposing - - '21a?— 2a?> 15 + 23; Reducing - - 19a?>38; Dividing by nineteen, a?>2. 2. Find the limits pf the value of x in the inequations 14a:+A>i|+230, 7 J^Tote. — To determine both the limits of a?, it is necessary that we have two inequations existing in a contrary sense. These inequations are not combined together like equations, but reduced separately. 3. Find the limits of x in the inequations X X ^7 2a? 5"^¥>5"^'3'' X X 6 X 7 14 5 10* SECT. V ] INEQUATIONS. 139 4. The double of a number diminished by 5 is greater than 25, and triple the number diminished by 7 is less than double the number increased by 13. Required a number that will satisfy the conditions. Let x= the number : then, by the question, we have 2x— 5>25, 3x— 7<2a:+13. Resolving these inequalities, we have ar>15 and a:<20. Any number, therefore, either entire or fractional, compri- sed between 15 and *20, will satisfy the conditions. 5. A shepherd being asked the number of his sheep, re- plied that double their number diminished by 7 is great- er than 29, and triple their number diminished by 5 is less than double their number increased by 16. Requi- red a number that will satisfy the conditions. Resolving the question, we have a:>18 and x<21. Here all the numbers comprised between 18 and 21 will satisfy the inequalities ; but since the nature of the question re- quires that the answer should be an entire number, the num- ber of solutions is limited to two, viz., x=19, 07=20. 6. A market-woman has a number of oranges, such that triple the number increased by 2 exceeds double the number increased by 61, and 5 times the number di- minished by 70 is less than four times the number di- minished by 9. Required a number that will satisfy the conditions. • 7. The sum of two numbers is 32 ; and if the greater be divided by the less, the quotient will be less than 5, but greater than 2. What are the numbers! 8. A boy being asked how many apples he had in his basket, replied, that the sum of three times the number plus half the number diminished by 5, is greater than 16 ; and twice the number diminished by one third of the number plus 2, is less than 22. Required the num- bers that will satisfy these conditions. 140 ELEMENTS OF ALGEBRA. [sECT. V. SECTION VI. ; Involution and Powers, — Of Monomials. — Of Polynomials. — Binomial Theorem. — Evolution and Roots. — Square Root of Jf umbers. — Cuhe Root of JVumbers. — General Method of ob- taining any Root of Jfumbers. — Evolution of Monomials, — Of Polynomials. — Calculus of Radicals. INVOLUTION AND POWERS. 251. Involution is the multiplying anumber by itself till it has been used as a factor as many times as there are units in the exponent. 252. The product thus produced is called the power of that quantity ; and the power is designated ^r^if, second^ third^ fourth^ &c., accordingly as the number has been used once, twice, three times, four times, &c., as a factor. 253. To indicate the involution of a polynomial, or of a monomial composed of several factors, the numbers should be placed within a parenthesis, to the right of which the ex- ponent should be written. INVOLUTION OF MONOMIALS. 254. In order to obtain a general rule for the involution of monomials, let the following proposition be demonstra- ted, viz. : The power of the product of two or more factors is equal to the product of their powers. Let {abf represent the second power of the product of two factors, And a^lf^ the product of the second power of the same fac- tors : Then (abY—a^¥ ; for, by the definition of involution (Art. 251), {abf=zabxab=aaxbb = a'b\ Again: {abY—d^b'^} for {abY—obxabxabx .... taken m times=aaa . . m iimesxbbb . . m times=a"'&"'. SECT. VI.] INVOLUTION OF MONOMIALS. 141 255. Now let it be required to involve Sah^ to the fourth power : (3aby = 3ai« x 3ab^ X3a6'x3aft« = 3x3x3x3x aaaa x bl'b^il'b* = 81a*6'=3*xa'^*x6'^*. 256. The same reasoning will evidently apply to every case of monomials ; hence, for the involution of monomials we have the following general RULE. 1. Involve the coefficient to the required power. 2. Multiply the exponent of each letter by the exponent which denotes the power to which the monomial is to be involved. J^ote 1. — If the number to be involved is positive, all its powers will be positive (Art. 89). If it be negative, the even powers will be positive and the odd powers negative (Art. 90). ^ote 2. — If the given number be fractional, involve both the numerator and denominator. This results from the prin- ciple that the product of fractions is equal to the product of their numerators divided by the product of their denomina- tors (Art. 149). Thus, ('^y=?x-=^. Jfote 3. — The above rule is applicable to numbers having negative exponents, since the negative exponent expresses the reciprocal of a power (Art. 61). Thus, (a~^)^=a~'^^= JVb^e 4. — The fourth power of a number is equal to the square of the second power ; thus, (a)*=o X a X a x a=oa x aa =(a')'. The sixth power is equal to the cube of the second power J thus, a^z=axaxaxaxaxa=aaxaaxaa=(a^f, &;c. EXAMPLES. 1. Required the second power of 8a'6'. •^tis, 640"^". 2. Required the third power of 5x^z, Ans. 125jV. 3. Required the third power of <6dfx, Ans. 2l6a'yr'. 4. Required the fourth power of 2aVc*. Ans, lea'd'^c'*. 142 ELEMENTS OF ALGEBRA. [sECT. VI. 5. Eequired the fifth power of 2a5V. Ans. 32a^6'^a?2^ 6. Required the second power of — 6a^Z>^ Jins. 36a^6'^ 7. Required the third power of — 3aJc^. Ans. — 27a%«. 8. Required the sixth power of \d^h. Ans. i^-^-^c^^W, 9. Required the seventh power of — 2a7^y. Ans. -— 128a;'y. 10. Required the fourth power of — 4^a^2>. Ans. 256a«6^ 11. Required the fourth power of — . Ans. '''' 12. Required the second power of 3a~^. Ans. 9a-*. 13. Required the second power of 1?^. Ans. _5?^*-. U. Required the second power of ??-. Ans ^^^^^ 82a? 6724a?2 15. Required the third power of 6a~^b~\ Ans. n6a-'b-^. 16. Required the fourth power of Sah~^ ' Ans. 4>0%a'b-'^ 17. Required the fourth power of lOx^z"^. Ans 10000a?'V*». 18. Required the fifth power of ^a^xy^. Ans. 1024a^V2/'°. 19. Required the fifth power of — Sabxy. Ans. —U^a'b'xY. 20. Required the eighth power of ba^x.-^ Ans. 390625a'V. 21. Required the fourth power of ^. Ans. i5??^'. xyz x^y^z^ 22. Required the second power of ISyz Ans, 400j ?^ 324yV' ft SECT. VI.] INVOLUTION OF POLYNOMIALS. 143 23. Required the third power of We'd-* ,1ns. 27a-*'^'^ 24. Required the fifth power of Wb^c*dK Ans, 1024a«'J'»c»(i». 25. Required the seventh power of — 2aa:ccP. Ans. --128aVc'(/'\ 26. Required the nth power of _^. Ans. ^ ^ . -. ^ ^ 8cdy S"c"d'Y 27. Required the eighth power of —Wb^. Ans. 65536a'«6^. 28. Required the fourth power of lOx^fz. Ans. lOOOOa^'y^^. 29. Required the sixth power of — 3a^b^c*d-\ Ans. 7•29a'«i'»c"(^*. 30. Required the third power of 12a-^b-^c-^dr*. Ans. 1728a-*6-»c-«(f-'«. INVOLUTION OF POLYNOMIALS. 257. Multiply the polynomial by itself till it has been used as a factor as many times as there are units in the exponent de- noting the power to which it is to be raised; the final product will be the power required. EXAMPLES. 1. Required the first, second, third, fourth, and fifth pow- ers of the binomial a-\-b, (a-\-by=a -i-b ------ 1st power. (a-hJ)*=a'+2a*+6' 2d power. a+b a*+2a'6-h ah" 144 (a+6) ELEMENTS OF ALGEBRA. a+b a'-j-3a'b-\-3aW-\- ah'' a!'b + 3a'b^-\-3ah'+b* (a-\-byz=a'-{-U'b-\-6a%^-^4>ah^-\-b^ a-\-b a'-{-4^a'b-j- 6a'b'-i- 4a'b^+ ab' . Qa'h^+^ab'-l^ (a—by=a'—^a'b^- IQa^b'^—lOa'b^+bah'—b' 3. Required the second power oi a-\-b, < a -\-b a-\-b o^+ ab + ab+b^ a^-^2ab-\-y, Ans, [sect. vi. 3d power. 4th power. 5th power, and fifth pow- 1st power. 2d power. 3d power. 4th power. 5th power. SECT. VI.] INVOLUTION OF POLYNOMIALS. 145 J^ote. — Since a and b may represent any numbers what- ever, we infer the following general principle : The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term. 4. Required the second power of a—b. a—b a—b a"— ab — ab-\-b^ a^—2ab-\-b\ Ans, JhCott. — Since a and b may represent any two numbers, a being greater than 6, we infer the following general princi- ple : The square of d residual is the square of the first term, minus twice the product of the two terms, plus the square of the last term. 5. Required the second pt)wer of 6a4-3A. Ans. 36a'-|-36aJ-h95». 6. Required the second power of la — 2J. Jlns. 49a'— 28a6+4J'. 7. Required the second power of 2a6+3c. Am. 4a'6'-fl2a&c+9c'. 8. Required the second power of babe — 2acd, Ans. 25a'6V— 20a»*c'rf+4aVrf». 9. Required the third power of 2a -|- 36. Ans. 8a'+36a»6-f 54a^-f 276^ 10. Required the third power of 2a— 5&. Ans. 8a'— eOa'/j-hloOaft'— 1256*. 11. Required the second power of a-|-l. ^n*. a«-f-2a-fl. 12. Required the second power of 2a — 1. wf;i«. 4a»— 4a-|-I. 13. Required the third power of a+1. ^n*. a»+3a*+3a+l. 13 T 146 ELEMENTS OF ALGEBRA. [sECT. VI. 3a— 1 14f. Required the second power of b+c Ans, 5^6.^1. 15. Required the third power of a-\-b-\-c. Ans, (i?-\-3(i^'b + 3a'c4- 3a^>'+ 3ac'+ 6a&c+ Wc-i^il&-\-W-\-&, 16. Required the fourth power of 3a+26c. Ans. 81a^+216a='k+216a2iV+96aiV+ 16&V. 17. Required the fifth power of 6a?— 2i. ./?7i5. 7776a7^ — 12960a?^Z>+8640:c^Z>2— 2880a?'^^+480a?&^— 32Z>^ 18. Required the second power of 6a4-2& — 3c. Ans, 36a'+ 24''a6— 36ac+ 4.^>2— 126c+ Qc"; 19. Required the third power of 2a^ — 3x, Ans, 8a«— 36a^T+54aV— 27a:'. la—W 20. Required the second power of 8a?+y * 64a:'-f 16a:y+/* 21. Required the second power of — ± — Ans, 9a?y 9/^2— 246(i+ 16(i^ 6a~^b^ 22. Required the third power of Ans, 3a— 1 216a-'b' 27a^— 27a^+9a— 1 23. Required the fourth power of 4a^6 — 2c^ .^715. 2b6a'^b'—51'ila'b'c^+3S4>a'b'c'—n8a'bc'-\- 16c\ 24. Required the fourth power of ^_. ^^^ 4>096a Vy-^ ^*' ^H^aH^'+4^Tl' 258. Remark. — Any factor may be transferred from the numerator to the denominator, or from the denominator to the numerator of a fraction, by changing the sign of its ex- ponent. SECT. VI.] BINOMIAL THEOREM. 147 1. ?fl'^!!!xx-«=^xirArt.061)=4-. y y y a?^ a^y ' 2/~ 2 y' 2 ^ ~ 2 * CX-' c X-* c c \ x^J c c . 2a-* 2 ^_, 1 2^1 ;, 2i» *• ZT-i=^x^ Xr- 2=c:><-l><^=^-3• 5. ^=exx»=^^i=^-x-=^x4=-^. c c c ' or c c x-* cx~* fi Vcf. BINOMIAL THEOREM. 259. The method of involving polynomials by repeated multiplications is somewhat tedious, especially when high powers arc required. This has led mathematicians to seek for other methods. The most simple method known is the one invented by Sir Isaac Newton, called the Binomial The' orem. Its use is very important and extensive in algebraic operations. 260. Let us take the binomird a-h^, of which a is called the leading quantity, b the following quantity. Involving by the preceding rule, we shall find, (a-hiy=a'4-2a64-i* (a-f6)*=a'+3a«6-f 3a 6»4-^. (a+by=a'-\-Wb-^ 6a^t^-\' ^ab^-\-b*. (a+6)*=a*-f 5a*A-h lOa^A^-f- 10a^6'-h 5a b'+b\ la+by=a*+6a'b+ 15a^A^4-20a'6'-f 15a^i*+ ^a b'+b\ (cf+A)'=a'+7a«64-21a'6*H-35a«6*+35a'6*+21a**'+7aA«+3\ 148 ELEMENTS OF ALGEBRA. [sECT. VI. 261. By observing the several results above, the number of terms will be found to be greater by 1 than the index de- noting the power to which the binomial is to be expanded. Thus, The square has three terms j The cube has four terms ; The fourth power has five terms ; The fifth power has six terms j The sixth power has seven terms ; The seventh power has eight terms ; And if the nih. power of a-\-b were required, the number of terms would be n-\- 1. Hence, if the index of a binomial be a positive whole number, the number of terms will be one greater than the number of units contained in the index. 262. By attending to the exponents of the letters in the above powers, we shall find that they preserve an invariable order. In the square, the exponents \ ^^ ^ ^^® ^' ■^' ^ ' ^ of ^> are 0, 1, 2. In the cube, the exponents ^ of « are 3, 2, 1, ; ^ ^ of 5 are 0, 1, 2, 3. In the fourth power, the exponents \ ^^ ^ ^^® ^' ^' 2, 1, ; ^ "^ ^ of Z> are 0, 1, 2, 3, 4, &c. Two laws are discoverable here : • 1. The sums of the exponents of the two letters in each term are equal, and each sum is equal to the index de- noting the power to which the binomial was to be raised. 2. The exponent of the leading quantity in the first term is the same as the index denoting the power to which the binomial was to be raised, and decreases regularly by 1 J the exponent of the following quantity is 1 in the second term, and increases regularly by 1. 263. If it be required to involve a-^b to the power denoted by n^ the exponents of a would be «, n—l, n—2j n—Sy —3, 2, 1, j SECT. VI.] BINOMIAL THEOREM. 149 Of i, 0, 1, 2, 3, 71—3, n—2y w— 1, n. Or, expressing the letters without the coefficients, b\ 264. The same principle may be applied if the exponents be negative or fractional. Thus, (a-f6)-*-o-»+a-='64-a-'^>'+a-'6'+a-«J'-far'5*4-, &c., ad infin. Also, {a-\-b)^=J-^a~h-{-a~h'-{-a~^b^-\-a~ib'-ha~^!/'-{-, &c., ad infin. It is evident that the above two series will never termi- nate, as a negative or fractional index can never become by the successive subtractions of a unit ; hence, when the index of the binomial is negative or fractional, the number of terms in the series will be infinite. 265. The law of the coefficients is more complicated, though not less remarkable. In the preceding series of powers (Art. 259), the coeffi- cients taken separately are, - - 1, 1. - - 1, 2, 1. 1, 3, 3, 1. 1, 4. 6, 4, 1. - 1, 5, 10, 10, 5, 1. 1, 6, 15, 20, 15, 6, 1. 1, 7, 21, 35, 35, 21, 7, 1. In the first power In the second power In the third power In the fourth power In the fifth power In the sixth power In the seventh power By examining the above series of coefficients, it will be discovered, 1. That the coefficient of the first term is 1. 2. That the coefficient of the second term is the same as the index denoting the power to which the binomial is to be raised. 3. If the coefficient of any term be multiplied by the in- dex of the leading quantity in the same term, and the 150 ELEMENTS OF ALGEBRA. [sECT; VI. product divided by the index of the following quantity- increased by 1, the quotient will be the coefficient of the following term. 266. By recurring to the above series of coefficients, it will be observed that they increase and then decrease in the same ratio, so that the coefficients of terms equally dis- tant from the first and last terms are equal. It is sufficient, then, to find the coefficients oi half the terms ; these, repeat- ed in the inverse order, will give the coefficients for the re- maining terms. 267. By inspecting the coefficients farther, we shall dis- cover that in any power of a+i, the sum of the coefficients is equal to the number 2 raised to that power. Thus, the sum of the coefficients In the second power is - J - - 4=: 2^; In the third power is - - - - 8==2^; In the fourth power is - - - - 16 = 2^; In the fifth power is - - - - 32=2' ; In the sixth power is - - - - 64 = 2^; In the seventh power i^ . - - 128=2', 268. If it be required to involve a-\-l to the power ex- pressed by ra, first, taking the letters and exponents without their coefficients, we shall have Let A, B, C,^ (Sec, represent the coefficients of the several terms in order, excepting the first and the last, which are always 1. A=w, coefficient of the second term. 2 B=^i:^, coefficient of the third term. 2 ' {n''- n){n- 'X) coefficient of the fourth term. 2x3 The same coefficients may be used in the inverse order for the last terms of the indefinite series. Then we shall have, by restoring the coefficients, SECT. VI.] BINOMIAL THEOREM. 151 (fl-f J)"=a"4-Aa'^'6+Ba'^'6»+Ca''-'6* . . . Ca»6'»-»+Ba«J'^«-h 269. We proceed, in the next place, to consider the signs to be prefixed to the several terms produced by the involu- tion of a binomial. When a term is composed of several factors, the sign of the term will evidently depend upon the proper signs of the factors ; if an even number of them be minus, or if none of them be minus, the quantity will be positive ; if an odd number of them be minus, the quantity will be negative (Art. 90). Thus, analyzing the fourth pow- er of a-{-by each term is composed of one numerical and four literal factors, oil plus j and consequently each term will be positive. Thus, lXaXoXaxa=aVthe first term; ^Xaxaxaxb=4^a^by the second term; 6Xaxaxbxb = 6aW^ the third term ; ^Xaxbxbxb =4.aA', the fourth term j lxbxbxbxb=b\ the fifth term. The letters and exponents are a^-j- a^b-^ a-b^-\- ab*-\-b*; The coefficients are - - 1 +4« +6 +4 +1. Compounding the series, (a-f 6)S=a* + 4ja'6H-6a*6*+4aZ>'-f 6*. Again, (a—by=a*-Aa'b+ea^b^—4^ab^-{-b\ lxaxaXaxa=+a*, the first term; 4XaXaXax — b= — i-a'A, the second term; 6xaxax — bx — b=-{-6a^b^, the third term; 4xax — bx — kx — b= — 4a6^, the fourth term; lx—b-\ bx—bx—b= + b\ the fifth term. The letters and exponents are a*— c^b-{- a*b* — ab'^+b* ; The coefficients are - - 1+4 -f6 +4 +1. Compounding the series, (a — by=a'^ — 4a'^-|-6a^6^ — ^ah'^+b*. 270. The signs of the terms are also affected by the sign of the exponent. Let it be required to expand (a+d)"* i 152 ELEMENTS OF ALGEBRA. [SECT. VI. The letters and ^ ^.,^ ^.3^^ ^_,^,_^ a-^^^^ &c., ad infin. ; exponents are S The coefficients ) j _2 ^3 _^^ &c.,adinfin. \ ar'^ — a~^b-{- ar^W — a~^Z>^, &c., ad infin. ; are Multiplying, "" (a4-Z.)-2=ra-2— 2a-^^>+ Sa-^i^— 4a-^6^ &c., ad infin. ; Ur, (^+6) =— — —A- — — — > &c., ad mtin. a^ a^ or a' Again: let it be required to expand (« — &)~^: The letters and exponents are The coefficients ^ ^ _^ _^3 . _^^ &c., ad infin. are J Multiplying, (a_ j)-2^a-2_f_2a-364-3a-^&2+4a-55^ &c., ad infin. ; Or, (a— ^) — -,+-T+— r + -^j ^c., ad infin. a^ or a* a"" 271. The principles of the Binomial Theorem may be stated as follows : I. The exponent of the leading quantity in the first term of the power is the same as the index denoting the power to which the binomial is to be raised^ and decreases regularly by 1. The ex- ponent of the following quantity is 1 in the second term, and in- creases regularly by 1 in the succeeding terms. II. The coefficient of the first term is 1 ; that of the second the same as the power to which the binomial is to be raised ; and universally^ if the coefficient of any term be multiplied by the exponent of the leading quantity in that term, and the product be divided by the exponent of the following quantity -j-1, the result will be the coefficient of the succeeding term. JVote 1. — The learner will find it convenient to obtain the series of the letters and exponents, and the series of coeffi- cients separately, and then compound them by multiplying their corresponding terms, as in the preceding cases. J^ote 2. — The preceding discussions relating to the Bino- mial Theorem will suggest some methods of verifying the work, and also of abridging it. SECT. TI.] BINOMIAL THEOREM. 153 EXAMPLES. 1. Required the fourth power of a+b. Expanding letters, &c., a*-{- a^b-\- a'i'-f ai^-\-b*', Finding coefficients - 1+^ -\-6 -}-4 +1. Compounding, (a-{-bY=a*-\-Wb-^Qa'b^+^aly'i-b', jJns 2. Required the fourth power of a — b. The letters and exponents are a* — c£^b-\- a^l^ — aft* -f 5*5 The coefficients are - - 1 -|-4 +6 +4 +1. Compounding, - {a—by=a*—^(^b^^a^b'—^a}^-\-b\ 3. Required the fifth power of a-\-b and of a — b. 4. Required the sixth power of a+^ and of a — x. 5. Required the seventh power of a:-fy and of x — y. 6. Required the eighth power oi a-\-b and of a — b. 7. Required the eighth power of x — y. 8? Required the fourth power of \-\-a. Expanding the terms - l*-hl'xa+l'Xa*-fl'xa'4-o*; Finding coefficients - 1 +4 +6 4-4 +1. Compounding, and reject- > ^ mg the factor 1 - J 9. Required the fourth power of 3a4-25. Let x=3a, and y = 25: the« (3a+26)*=(x-f y)\ Expanding this last expression, x*+ or'y-f- xy-|- xy^-\-y^l Finding coefficients - - .1+4 -|-6 +4 -fl. Compounding - (a:+yy=x*-t-4ar'y-f-6a:'y* + 4jry'-}-y*. Restoring the values of x and y, (3a + 2Z.y ^ {Zay + 4 x (Sa^ x 25 +6 x {^af x (25)*4-4 X 3ax(25)='-}-(25)*. Involving the terms, (3^+25)* = 81a* + 4x270^x25 + 6 X9a«x45»+4X 3ax 85»+165*. Multiplying factors, (3a+25)* = 81a*+216a'54-216a'5»+96a5»+165*. 10. Required the fifth power of 2cx — 4y. Let a=2cx, and 5=4y : then {lex — 4y)*=(a— 5)*. U 154 ELEMENTS OF ALGEBRA. [sECT. VI. Expanding the terms, a^—.a'b-\- aW— a%'^-\-ah''—b^i Finding coefficients, 1+5 +10 +10 +5 +1. Compounding, {a—Vf=za''—ba'h-\.l(ia?b''—10a%^+bab''—l\ Restoring the values of a and J, (2ca:-4y)^z= 32cV— 320cVy + 1280cV/— 25600^/ + 2560ca??/*— 1024^^ 11. Required the fourth power of a^ ^b^. Let x^a\ and y=Z>^: then {a^+by={x-]-yy. Expanding the terms - a:'*+ a:^y+ a:^ + x'if-\-y'^ : Finding the coefficients - 1+4 +6 +4 +1. Compounding - (a?+2/)''— a?''+4ar'y+6a?y+4a?y'+y*. Restoring the values of x and y, {a^+by=a?-\-^a%^-\-Qa'b'+A>a^b^-\-VK 12. Expand (2^^— 5J)^ 13. Expand {Zabx+yy, 14. Expand L_z=(a+*)-^ {a+bf ^ ^ 15. Expand (a+a:)"^. 16. Expand {a-^b)~K 17. Expand (6a5c — laxyy. 18. Expand (33?^— 4y)^ 19. Expand {c^—Qax)K 20. Expand (3a^— 1)*. 272. The powers of any polynomial whatever may be found by the Binomial Theorem. Take, for example, {a-\-b +c)^ Letting a?=:&+c, we shall have {aJtb-{-cy={a-\-x)\ Expanding - - - (a+a?)^=a^+3a^a?+3aa?^+a7'. Restoring the value of a?, (a + Z>+cy=za^+3a^(^> + c) + 3a(i + c)2+(&+c)^ Expanding and multiplying factors, (a+Z>+cy=a'+3a'^>+3a2c+3a6' + 6ak + 3ac2+&=»+3J'c + 3k2+c^ 2. Required the third power of a — b-\-c. 3. Required the third power of 25c — 3a;+y. SECT. VI.] EVOLUTION AND ROOTS. 165 EVOLUTION AND ROOTS. . Extraction of the Square Root of JVumhers, 273. A power of a number has already been defined to be the result of multiplying the number into itself continually, until the number has been used as a factor as many times as there are units in the exponent denoting the power. The second power of 6:r6 ><6 = 36. The third power or cube of 6::^6 x 6 x 6 = 216. The fifth power of 47=47 x 47 x 47 x 47 x 47=229,345,007. Involution is the method of finding the various powers of numbers. Evolution is the reverse of this: it explains the method of resolving a number into equal factors, called roots. 274. When a number is resolved into two equal factors, one of the factors is called the Square Root ; when resolved into three, the Cube Root j when into four, the Fourth Root, Sec. The first ten numbers are 1,2, 3, 4, 5, 6, 7, 8, 9, 10. And their squares - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. By inspecting this table, it will be perceived that among entire numbers, consisting of one or two figures, there are nine only which are squares of other numbers. The square roots of other numbers, expressed by one or two figures, will be found between two whole numbers differing from each other by unity. Thus, 55, comprised between 49 and 64, has for its square root a number between 7 and 8 ; 78 has for its square root a number between 8 and 9. The num- bers in the second line of the table being the squares of those in the first, the numbers in the first are the square roots of those in the second ; therefore the square root of numbers consisting of one or two figures will readily be found by the table. 275. Let it next be required to find the root of a number consisting of more than two figures. It has already been shown that the square of any binomial, as (a-\-by=za-\-2ab +6*. Every number may be regarded as made up of a cer- 156 ELEMENTS OF ALGEBRA. [sECT. VI. tain number of tens and a certain number of units ; thus, 46 is composed of 4 tens and 6 units, and may be expressed thus, 40 + 6; the square of which may be obtained in the same manner as the square of a-{-b ; thus, 40 + 6 40 + 6 1600+240 240+36 1600+480 + 36=^2116. In this result, as in the Square of the binomial a-{-b, in which a may represent tens and b units, it will be observed there are three parts, viz. : the square of the tens, 40^,= 1600 ; twice the product of the tens by the units, 2x40x6=480; and the square of the units, 6^=36. These three parts will be found in the second power of every number. 276. We next proceed to reverse this process, and find the square root of 2116. As the square of 4 tens, or 40, is 1600, and the square of 5 tens, or 50, is 2500, the root can contain only 4 tens. Subtracting the square of this - - - 2116 Square of 4 tens, or 40 - - - - 1600 516 This remainder contains twice the product of the tens by the units, plus the square of the units. Now, if we double the tens, which gives 80, and divide 516 by 80, the quotient is the figure of the units, or a figure greater than the units. This quotient figure can evidently never be too small, but it may be too large, as 516, besides containing double the product of the tens by the units, may contain tens arising from the square of the units. The figure representing the units can never be greater than 9. 516-^80=6. To ascer- tain whether 6 express the units, we multiply 80 by 6 = 480, and subtract it from 516: the remainder is 36; from this subtract the square of the units 6 x 6 = 36 : the remainder is ; hence 4 tens and six units, or 46, which is the root. SECT. VI.] SQUARE ROOT OF NUMBERS. 16t The operation will stand thus : 2116^ 1600 2116^40-f-6=46, root. 40x2=80)516 480 ""is 6x6= 36 277. The work may be abridged by several modifications. By observing the table of the squares of the numbers 1, 2, 3, 4, &c., it will be perceived that the square of a num- ber consisting of one figure can contain no figure of a high- er denomination than tens. If we annex a cipher to the numbers 1, 2, 3, 4, &c., they become 10, 20, 30, 40, 50, 60, 70,. 80, 90, 100; And their squares are 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000. From which we see that the square of tens will contain no figure of a less denomination than hundreds, nor higher than thousands. When, then, the square root of a number consisting of three or four figures is required, in finding the tens, we may reject the first two figures on the right, as they can in no way influence the result. As the square of hun- dreds can contain no figure of a less denomination than thousands, when the square root of a number consisting of five or six figures is required, in obtaining the hundreds we may reject four figures at the right liand. When, then, the square root of any number is required, we may divide it into periods of two figures each (if a number consist of an odd number of figures, the last period will contain but one figure), and the number of these periods will be the number of figures in the root. Each of these periods, in connexion with the remainder resulting from the operations on the preceding period, may be used independently of the follow- ing periods in obtaining that figure of the root contained in 14 158 ELEMENTS OF ALGEBRA. [sECT. VI. it. In the above example, likewise, in which the square root of 2116 is required, as the product of tens by units ev- idently can contain no figure less than tens, after subtract- ing the square of the tens, the next step, the division, may be as well performed after rejecting the cipher from the right of the tens, and the unit figure from the right of the dividend. Moreover, it will be perceived that, instead of finding first twice the product of the tens by the units, and then the square of the units, we may obtain the sum of both numbers by placing the unit figure at the right of the tens in the divisor, and multiplying the result by the unit figure. With these modifications, the work of extracting the square root of 2116 will stand thus : 2il6]46 16 86)516 516 Find the square of the tens in the first period j subtract, and bring down to the right of the remainder the next peri- od. Divide by twice the tens, rejecting the right-hand fig- ure of the dividend. Place the quotient figure in the root, and at the right of the divisor, and multiply this last num- ber by the quotient figure, and subtract j as there is no re- mainder, 46 is the root. Kequired the square root of 53361. 5336i|231 4 ^43)133 129 461)461 461. ^ns. 231. 278. The same process may be extended to any number, however large. From the preceding operations, the follow- ing rule for the extraction of the second root will be readily inferred : SECT. VI.] SQUARE ROOT OF NUMBERS. 159 RULE. I. Separate the number into periods of two figures each^ be- ginning at the right hand: the left hand period mil often con- tain but one figure, II. Find the greatest square in the first period on the left ; write the root in the place of a quotient in division^ and subtract the second power from the left-hand period. III. Bring down the next period to the right of the remainder for a dividend^ and double the root already found for a divisor. See how many times the divisor is contained in the dividend, ex* elusive of the right-hand figure, and place the result in the root^ and also at the right of the divisor. IV. Multiply the divisor thus augmented by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. V. Douhle the whole root already found for a new divisor ^ and proceed as before, till all the periods are brought down. The root will be doubled if the right-hand figure of the last divisor be doubled. If there is no remainder after all the periods are brought down, the proposed number is a perfect square. If there is a remainder, by the above rule, the root of the greatest square number contained {n the proposed number will be obtained. When the proposed number is not a perfect square, a doubt may arise whether the root found be that of the great- est square contained in the number. This may be deter- mined by the following rule. The square of a-|-l is a'-f-2a -f 1 J whence the square of a quantity greater by unity than a exceeds the square of a by 2a+l ; or, the difference be- tween the squares of two consecutive numbers is equal to twice the less number augmented by unity. Hence the entire part of the root cannot be augment- ed unless the remainder exceed twice the root found plus unity. 160 ELEMENTS OF ALGEBRA. [sECT. VI. Required the square root of 1287135 9 '~ 65)387 325 Now, as 35x2+l=71>62, 35 is the entire part of the root. EXAMPLES. 1. What is the square root of 451,5841 ^ns. 672. 2. What is the square root of 9,186,9611 Jins. 3031. 3. What is the square root of 13,032,1001 ^ns. 3610. 4. What is the square root of 4,543,164,409 1 ^ns. 67,403. 5. What is the square root of 669,420,148,761 1 ^ns. 818,181. 279. From Avhat has been done, it will be perceived that there are many numbers the roots of which are not whole numbers; and although there must be a number which, mul- tiplied into itself, will produce any number whatever, yet these numbers can have no assignable roots, either among whole or fractional numbers. The proof of this depends on the following proposition, which has already been demon- strated (see Art. 213) : Every number, P, which will exactly divide the product, AxB, of two numbers, and which is prime to one of them, will divide the other. The root of an imperfect power evidently cannot be ex- pressed by a whole number, and, to show that it cannot be expressed by a fraction, let c be an imperfect square j if its root can be expressed by a fractional number, let - repre- a sent that fractional number : then we shall have |^ o Or ... - c=_. SECT. VI.] SQUARE ROOT OF NUMBERS. 161 If c be not a perfect square, its root will not be an entire number j that is, a will not be divisible by b ; but it has been demonstrated that if a is not divisible by Z>, axa or a' is not 2 divisible by b or bxb=b'^, whence — cannot be equal to an tr entire number c. All numbers, both entire and fractional, have a common measure with unity ; on this account they are said to be commensurable j and since the ratio of these numbers to unity may always be expressed, they are called rational numbers. The root of a numbeir not a perfect square can have no common measure with unity, as no fraction can be assigned sufficiently small to measure at the same time this root and unity. The roots of such numbers are called incommensu- rable or irrational numbers. They are likewise called surds. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 280. The square root of a fraction may be found by ex- tracting the square root of the numerator and of the denom- inator ; thus, the square root of ^s is |. If the numerator or denominator is not a perfect square, the root of the frac- tion cannot be found exactly, but the root to within less than one of the equal parts of the fraction may readily be found by the following RULE. Multiply both terms of the fraction by the denominator which does not change the value of the fraction ; then extract the square root of the perfect square nearest the value of the numerator^ and place the root of the denominator under it ; this fraction mill be the approximate root. Required the square root of | : multiply both terms by 5, which gives ^^y of which f is the required root exact to within less than |. We might multiply both terms of if by any perfect square, and thus approximate the root mor« nearly. Thus, multiplying by 144, it becomes ||^§, the rotti X 162 ELEMENTS OF ALGEBRA. [sECT. VI. of which is nearest ||. Thus we have the root of f to within less than j^^. The approximate root of a number not a perfect square may be found in a similar manner within a given fraction. Multiply the proposed number by the square of the denomina' tor of the fraction ; then extract the square root of the product to the nearest unit, and divide this root by the denominator of the fraction. This rule may be demonstrated as follows : Let a be the number proposed, of which it is required to find the root'to within less than - : a—-— ; let r be the en- n TV- tire part of the root of the numerator an^ ; ar^ will be com- prised between r^ and (r-|- 1)^ ; consequently, the square root of a will be comprised between those of — and 1_2_jl that r^ n^ is, between - and ^ ^ , whence - will be the root of a to n n n within less than -. n Find the square root of 59 to within less than ■^^' 59'x (12)^=8496. V 8496 = 92. ff, ^ns, 281. The manner of determining the approximate root in decimals is a consequence of the preceding rule. To obtain the square root of a number within J^, yi^, to-Vtt? ^^"> "^6 multiply, by the preceding rule, the number by (10)^, (100/, &c., or, what is the same thing, we add to the right of the number two, four, six, &c., ciphers j then extract the square root of the product to the nearest unit, and divide this root by 10, 100, 1000, &:c. The number of ciphers annexed to the whole number should always be double the number of decimal places re- quired to be found in the root. The roots of decimal frac- tions, whole numbers, and decimals, may be found by the preceding rules. The number of decimals in the proposed 8£CT. YI.] CUBE ROOT OF NUMBERS. 163 number must always be made even by annexing ciphers if necessary. A vulgar fraction may be changed to a decimal fraction before extracting its root, and a mixed number to a whole number and decimal. EXAMPLES. 1. What is the square root of 31,027 to within ,01 1 Ans. 5,57. 2. What is the square root of 0,0100,1 to within ,00001 \ Ans, 0,10004. 3. What is the square root of \\ to withiti ,001 \ \ Ans. 0,886. * 4. What is the square root of 2}^ to within 0,0001 1 Ans. 1,6931. 5. What is the square root of 7 *? Aiis. 2,645+. 6. What is the square root of 4-^ 1 Ans. 2,027+. 7. What is the square root of i^ 1 Ans. 0,8044+. 8. What is the square root of 0,01001 1 Ans. 9. What is the square root of 0,0001234. 1 Ans. 10. What is the square root of 227 to within x^o o7 '^ * Ans. 15,0665. 11. What is the square root of 3,425 to within yiy 1 Ans. 1,85. 12. What is the square root of fjj^f 1 Ansi | J. 13. What is the square root of 11}^ to within ,001 1 Ans. 3,418. 14. What is the Square root of 3 to within ,00000000011 Ans. 1,7320508076. EXTRACTION OF THE CUBE ROOT OF NUMBERS. 282. The third power ^ or cube of a number, is the product arising from multiplying this number into itself till it has been used three times as a factor. The third or cube root is a number which, being raised to the third power, will pro* duce the proposed number. 164 ELEMENTS OF ALGEBRA. [sECT. VI. The first ten numbers being 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Their cubes are, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. The numbers of the first line are the cube roots of the second. By inspecting these lines, we perceive there are but nine perfect cubes among numbers expressed by one, two, or three figures ', the cube root of other numbers consisting of one, two, or three figures, cannot be expressed exactly by means of unity, as may be shown by a process similar to that used in Art. 279. The cube root of an entire number consisting of not more than three figures, may be obtained by merely inspecting the cubes of the first nine numbers. Thus, the cube root of 125 is 5 ; the cube root of 30 is 3 plus a fraction, or within one of 3. To extract the cube root of a number consisting of more than three figures, we present the following RULE. 1. Separate the given number into periods of three figures each, beginning at the right hand : the left-hand period will often contain less than three figures. 2. Find the greatest cube in the left-hand period, and place its root on the right, in the place of a quotient in division. Sub- tract the cube of this figure of the root from the first period, and to the remainder bring down the next period, afid call this num- ber the dividend. 3. Multiply the square of the root just found by 300 for a divisor. Find how many times the divisor is contained in the dividend, and place the quotient for a second figure of the root. Multiply the divisor by this second figure^ and place the product under the dividend. Multiply the former figure or figures of the root by 30, and that product by the square of the last figure, and place the result under the last ; under these two products place the cube of the last figure of the root, and call the sum of the last three numbers the subtrahend. SECT. VI.] CUBE ROOT OF NUMBERS. 165 4. Subtract the subtrahend from the dividend^ and to the re- mainder bring down the next period for a new dividend ; and in finding a divisor and subtrahend^ proceed precisely as before^ and so continue till all the periods have been brought down. We proceed next to the explanation of this tule. The cube of a binomial, as a-\-b=c^+2a'b^'^ab''-\-b\ In the number 45, a may represent tens, and b units, or we max find the cube of 45 writing it 40+5. 45=40-f- 5 . 40-h 5 " 200+ 25 1600+200 (45^=1600+400+ 25 40+ 5 8000+2000+125 64000+16000+1000 (45)»=64000 + 24000 + 3000+ 125=91125 283. On inspecting the above examples, it will be per- ceived that the cube of a number composed of tens and units is equal to the cube of the tens, plus three times the prod- uct of the square of the tens by the units, plus three times the tens by the square of the units, plus the cube of the units. Let it now be required to reverse the above process, and find the cube root of 91125. (40^=64000, (50)^=125000. Hence the cube root is evidently 40 plus a certain number of units. Subtracting the cube of 40, that is, the cube of the tens, there remains 27125, which contains the remainder of the parts above specified. As it is evident that the third power of tens can have no significant figure below the fourth place, in finding the third root of the tens the three figures on the right may be rejected, as they will not influence the result. As the cube of 100 is 1000000, in obtaining the cube root of hundreds in a number consisting of more than six figures, we may reject the first six figures on the right as 166 ELEMENTS OF ALGEBRA. [sECT. VI. not influencing the result ; hence any number of which the root is required may be separated into periods of three fig- ures each, each one of the periods may be used separately in connexion with the remainder resulting from the prece- ding operations, and the number of periods will be the num- ber of figures in the root. The cube of no one of the digits .contains more than three figures. In the above example, rejecting the first three figures on the right, the cube root of the tens found in 91 is 4. Subtracting the cube of this (64), and bringing down the next period, the result of the operation is 27125. This must contain, from what has been said, triple the product of the square of the tens by the units, together with two remaining parts already specified. As the square of tens contains no significant figures less than hundreds, we may reject the two right-hand figures from 27125, and dividing the remainder by three times the square of the tens, Ave should obtain the unit figure. In practice it is found more convenient to use the whole divi- dend, and to annex two ciphers to the divisor, as, instead of multiplying the square of the tens by three, we multiply by 300. For the same reason, instead of multiplying the prod- uct of the square of the units by the tens by 3, we multiply by 30. Dividing 27125 by the square of the tens (16), multiplied by 300, which =4800, the quotient 5 will be the unit figur.e sought ; or it may be too large by 1 or 2, as there may be hundreds arising from the other parts of the root sought : this can only be determined by trial. Having now the tens and the units, and having already subtracted the cube of the tens, we next proceed to subtract the other parts of the cube from the remainder. The square of the tens, multiplied by 300 and by the units, the last figure of the root - - - - =24000 The tens, multiplied by the square of the units and by 30 =3000 The cube of the last figure or units - - - = 125 Sum - - 27125 SECT. VI.] CUBE ROOT OF NUMBERS. • 167 As there is no remainder, 45 is the root. The operation may be exhibited as follows : 91125:45 (4)'= 64 '~~ (4)*= 16) X 300=4800) 27125 4800 X 5= 24000 (5)«x4x 30= 3000 (5)'=5x5x5= 12 5 000 284. Any number, however large, may be considered as composed of units and tens : the process of finding the cube root may therefore be reduced to that of the preceding ex- ample. Required the third root of 9663597. 9663597!213, root. (2)»= 8 ' Di » imr. T" (2)*x 300= 1200 ) 166^ , first dividend. 1200x1= 1200 2x30x(l)'= 60 (1)'= L ^_^ 1261, first subtrahend. (21^x300 = 132300)402597, second dividend. 132300x3= 396900 21x30x(3^)= 5670 (3^)= 27^ 402597, second subtrahend. OOOOW Should the divisor not be contained in the dividend as prepared above, place a cipher in the root, and bring down the next period to form a new dividend. The difference between the cubes of two consecutive whole num- hers is equal to three times the square of the least number^ plus three times this number ^ plus 1. Let a and a-j-1 be two consecutive whole numbers. (a+l)*=tf'+3o'+3a-|-l. (a-f-1)'— a'=3a' + 3a+l. (90)»-(89)*=3 X (89)^+3 X 89+ 1 =24031. In extracting the cube root of any number not a perfect 168 ELEMENTS QF ALGEBRA. [sECT. YI. cube, if any of the remainders are equal to, or exceed three times the square of the root obtained, plus three times this root, plus 1, the last figure of the root is too small, and must be augmented by at least unity. 285. The third root of a fraction is found by extracting the third root of the numerator and denominator. When the denominator is not a perfect third power, we may ob- tain the root approximately by multiplying both terms by the square of the denominator j thus, in obtaining the cube root of ^, we multiply both terms by 49 J the fraction then becomes i|^|, the root of which is nearest ^ accurate to within ^. We might multiply both terms of ^^^ by any perfect cube, and then extract the cube root, and we should approximate still nearer the true rooti By a process similar to that explained in the article on square root, we may ap- proximate the third root of a number not a perfect third power, by converting it into a fraction, the denominator of which is a perfect third power. Thus the approximate root of 15 may be found, putting it under the following form ; 15x1 2=^= 25920 (12f 1728 ' the third root of which is f |, or 2^2 accurate to within less than j^j. The root may be obtained with greater accuracy by using some number greater than 12. In such cases it is most convenient to convert the propo- sed number into a fraction, the denominator of which shall be the third power of 10, 100, 1000, &c. Let it be required to find the third root of 25 to within ,001 ; converting 25 into a decimal, the denominator of which is the third power of 1000, viz., 25,000 000000, the third root of which is 2,920 to within ,001, we have then v^ 25=2,920 accurate to with- in less than ,001. To approximate the third- root of an entire number by means of decimals, we annex to the proposed number three times as many ciphers as there are decimal places required in the root ; we then extract the root of the number thus prepared to within a SECT. VI.] ROOTS OF ANY DEGREE. 160 timV, and point off for decimals as many places as there are deci- tnal figures required i?i the root. If the proposed number contain decimals, beginning at the place of units, separate the number, both to the right and left, into periods of three figures, annexing ciphers, if ne- cessary, to complete the right-hand period in the decimal part. Then extract the root, and point off for decimals in the root as many places as there are periods in the decimal part of the power. The third root of a vulgar fraction may be most readily obtained after converting it first into a decimal fraction. EXAMPLES. 1. What is the cube root of 75686967 1 ^ns. 423. 2. What is the cube root of 128787625 1 ^ns. 505. 3. What is the cube root of 2054.83447701 1 ^ns. 5901. 4. What is the cube root of 52458 1674,6251 Ans, 806,5. 5. What is the cube root of 1003,003001 1 Ans. 10,01. 6. What is the cube root of 0,756058031 \ Ans, 0,911. 7. What is the cube root of 32977340218432 % Ans, 32068. 8. What is the cube root of 473 to within ^V '^- •^^*- ^h 9. What is the cube root of 79 to within ,0001 1 Ans. 4,2908. 10. What is the cube root of 3,00415 to within ,0001 \ Ans. 1,4429. 11. What is the cube root of 0,00101 to within ,01 \ Ans. 0,10. 12. What is the cube root of 0,000003442951 1 Ans. 0.0151. 13. What is the cube root of 6iff f 1 Ans. Iff 14. What is the cube root of iJfH '^ *^^^' f f • TO EXTRACT ANY GIVEN ROOT OF A WHOLE NUMBER. 286. Any root exceeding the third, consisting simply of two and three, as factors, may be found by the preceding rules j thus, the fourth root may be found by extracting the square root twice ; the sixth root by extracting the third 15 Y 170 ELEMENTS OF ALGEBRA. [sECT. VI. root, and then the square root of that ; the twelfth root hy extracting the square root twice, and then th^ third of the last root. Before proceeding to give a rule for the extrac- tion of any root, we subjoin a table of roots and powers. 05 TH 05 ^—i 05 1—1 05 1—1 05 tH 00 CM CO Tff -^ CO ot 00 o t' m o ^ Oi t- rf< -* CO 05 r-( c^ CO O -# lO CO 00 "-^ C^ 00 vo g ^ 00 CO CO 00 00 -* ^ ^. 00 T^ c^ c^ l- t-- t^ r-i CO CO 05 J> rH ^ c< O G<1 1> CO rH CO 1— 1 1> co I- Oi CO ,_( t- 05 CO 1-1 t- 05 tJ) rft ^5 o "* '^ o ^5 '^i* CO S 00 CO in 00 CO c^ CO t- CO T? CO lO T^ in CO o tococococotococococo C0»-lO5t^lOCO'-lO5ir- (MCNt't005C0C0rH rH i> CO 05 05 i> CO tJ) t^ 1> 1> CO c< CO o T? rH O O r-l CO vOiomvovoiniOkOiTiifs ^-HCOrHCOr-lCOrHCO CO lO 00 O CO lO r-l f 05 »0 CO CO 05 t- rH O^ •^CO-^COffiCO-^COrJlCO »-»CO»OC<0500CO'rJ0 rH »o r-l T^ CO VTi Ol Oj rH CO CO "^Jt C< O 0005l>r-lC005i>rHC005 C^OO^C^OOCOQO^ 0< t^ rH lO CO O (M CO 05 05 rH Id O*rJ(C0COCr U •*■» Q> Qi (U o o o o 0) V ^ ^ ^ ^ ^ ^ ^ ^ ^ O O o o o o o o o flH Ph Ph Ph Ph ^ PL, flH Ph •^. TS ^ ■*-* ^ ^ •«-> ^ "5 -B C< CO ^ \a CO t- 00 05 o SECT. VI.] ROOTS OF ANY DEGREE. 171 {a-\-bY=a'-\-ba*b-{- 10a'ft'+ 10a'lr'-^Dab*-^b\ {a-{-by=d'+la'b-\-2la'b'-{-S5a*lf'-^3ba'b*-\-2la'b'-^lab'+b\ In these examples a and b may represent tens and units in any given number, as 47 j and to obtain the root of any given power, we evidently must reverse the process by which the power is obtained from the root. By carefully attending to the preceding explanations, and the different powers of the binomial (a-{-b), the reason of the following rule for extracting any given root of a proposed number will readily be discovered: 1. Divide the number into periods of as many figures each as there are units in the index denoting the root, 2. Find the first figure of the root by trials and subtract its power from the left-hand period^ and to the remainder bring down the first figure of the next period for a dividend. 3. Involve the root to the next inferior power to that which is given, and multiply it by the number denotijig the given power^ and it will be the divisor. 4. Find how many times the divisor is contained in the divi- dend, and the quotient will be another figure of the root, or 1 or 2 too large. 5. Involve the whole root to the given power, and subtract it from the two left-hand periods of the given number; bring down the first figure of the next period to the remainder for a new div' idend, find a new divisor, another figure of the root, and again involving the whole root to the given power, subtract it from the first three left-hand periods. Thus proceed till the whole root is obtained. • Required the fifth root of 36936242722357. 36936242722357|517 5^= 3125 5* X 5 = 3125, first divisor. 5686, first dividend. (51)*= 34502 5251, subtrahend. (51)* X 5 = 33826005, 2d divisor. 243371762, 2d dividend, (517)*= 36936242722357 0000000 172 ELEMENTS OF ALGEBRA. [sECT. VI. 287. The preceding rule may be put in another form, em- bracing the same principles, but more consistent with the method by which we have explained the extraction of the second and third roots. RULE. 1. Separate the numler into periods of as many figures as there are units in the index denoting the rooti 2. Find by trial the root of the first period : this will he the first figure of the root : place this figure to the left^ in a column called FIRST column ; then multiply it by itself and place the product for the first term of a second column. This, multi- plied by the same figure, will give the first term of a third col- umn. Thus continue until the number of columns is one less than the units in the index denoting the root. , Multiply the term in the last column by the same figure, and subtract the product from the first period, and to the remainder bring down the next period, and it will form the first dividend. Jlgain, add this same figure to the term of the first column, multiply the sum by the same figure, and add the product to the term of the second column, which, in turn, must be multiplied- by the same figure, and added to the term of the third column,. and so on till we reach the last column, the term of which will form the first trial divisor. Jlgain, beginning with the first column, repeat the above pro- cess until you reach the column next to the last ; and so continue to do until there are as many terms in the first column as there are units in the index denoting the root, observing in each suc- cessive operation to terminate in the column of the next inferior order. 3. Seek how many times the first trial divisor, when there are annexed to it as many ciphers, less one, as there are units in the index, is contained in the first dividend ; the quotient figure will be the second figure of the root. Then proceed to form a new series by annexing this figure to the last term in the first column ; multiply the result by the last figure, and add it to the last term in the second column, advan- SECT. VI.] ROOTS OF ANY DEGREE. 173 cing the number to be added two places to the right of the other before adding. Multiply this result by the same figure^ and add the product to the last term in the third column, having previ- ously advanced it three places to the right of that term ; proceed in the same manner to the last term^ observing to advance the numbers added to the different columns as many places to the right of the terms as the number expressing the order of the col- umn ; that iSy advancing the terms of the first column one place, those of the tEcpND column two places, Src Multiply the term thus obtained in the last column by the last' figure of the root, and subtract it from the dividend ; to the remainder bring down the next period for a new dividend, and proceed to find a divisor and the third figure of the root in the same manner as the second was obtained. Proceed in the same manner^till all the periods are brought down. If there is still a remainder, the process can be extended by forming periods of ciphers. Required the third root of 103823. Ist col. 4 8 127 1 2+3a') x^;. The same expla- nation will apply, however extended the operations may be. Required the fifth root of 36936242722357. ^ »£L! ' i> t- i>. lO iO lO o CO CO CO c< CN c^ c^ (>* c^ t^ t^ y-i CO CO c< c^ lO I- i> rft -^ C^ 1—1 rH c< c< lO t- t^ C£> CO fN CO CO •CO in GO \a CO 00 ) as c< CO c* Tf. ^ CO wH \a CO c< c< CO CO T— 1 in § CO 1-1 o \ei \Ci c^ c^ CO oi c\ 00 t- CO rH c^ CO -^ CO 00 CO CO as 1—^ '<*< o c^ ' iO \Ci '^ . c-* CO 1-* »-* to CO c;i to ^u Si § CO 00 00 t_* 1^ H^ -^ o to lO *^ CO o ^1 ^^ o 00 *a s> 05 wx 1^ CO »4 1-* a. CO g 1^ 05 Oi o» lO <^ o oo to ^ •^ ^ c 4^ CO to t-^ e CO to CO - •4 CO l{^ o< K) CO H- o 00 Wi ^ g >f^ «o en Oi ^ s ^ CO CO ^^ s^ -^ to CO to • CO a> •(^ C3i K-^- Oi o o o as 00 00 CO o> €fi 05 .»— I 9R CJt h^ s? CO CO 00 o 00 o g Sx 1^ CO c«n I-- CO 00 CO 8 CO CO *^ »!*• <>* CO 00 ^ CO *■ Oi o» CT> -4 Od to o to 00 • 00 00 gj s to to »-» 2» t-^ •-' CJ^ ^ o CJX »-' -i ?>• fjl -^ a> o to p. CO Oi 00 CO CO CO t>^ t-^ •— ' Ot CJ1 c^ Ui *3 -1 00 to to to to to CJ» to to CO H- I-* to ^ ^ o o CO CO it o s to H- CO to 00 *- -i »-^* -? ^ H- h-1 tn fj\ •*»• tf*^ 00 00 l-t l-» CO 1 1 176 ELEMENTS OF ALGEBRA. [sECT. VI. EXAMPLES.' 1. Find the fifth root of 418227202051. Jins. 211. 2. Find the fourth root of 75450765,3376. Ans. 93,2. 3. Find the fifth root of 0,000016850581551. ^7X5.. 0,1 11. 4. Find the fourth root of 25^6,88187761. Ans. 7,09. 5. Find the sixth root of 2985984. Ans. 12. 6. Find the eighth root of 1679616. Ans. 6. 7. Find the seventh root of 2. Ans. 1,10409, nearly. EVOLUTION OF MONOMIALS, * 289. From a previous demonstration (Art. 156), it is evi- dent that the root of the product of two or more factors is equal to the product of the roots. Thus, \/ a^b*c^z= Va^xVtf^xV c\ Again, by the definition of evolution (Art. 274), \/c^=c^', for c'xc^'^c^^^c'; hence v/"?=c^-^^ or ci=c\ And, ^~Sc'=2c^ ', for 2c''x'2c'x2c'=Sc' -, hence ^Sc'zzz ^8"x ^7'=2xc''r-\ or 2(^=^c\. The same reasoning will evidently apply to every case of monomials. Hence, for the evolution of monomials, we have ^the. following general J?:. RULE. t. 1. Extract the required root of the coefficient. 2. Divide the exponent of each literal factor by the number denoting the root, and annex the result to the root of the coeffi- cient. JVote 1. — "With regard to the sign to be prefixed to the root, it is important to observe, * ^ a. An odd root of a number will have the same sign as the number itself. Thus, the cube root of a^, or ^a^=^a, for ax axa—a^\ and the cube root of — a^, ox ^ —a = — a, for — ax — flX —a=—a^, b. The even root of an affirmative number is ambiguous. Thus, the square root of a'', or ^fa=±a; {or a Xa=a\ and , — ax — a=+a ; also, the square root of 16, or \/16=±4, for 4x4=16, and -4x-4= + l6. SECT. VJ.] EVOLUTION OF MONOMIALS. 1T7 c. The even root of a negative number is impossible. Thus, the square root of — a', or V — a*, can be neither +« nor — a, for -\-aX'{-a=-^a^f and — aX — a=z-\-a^. Also, the square root of — 16, or V — 16, can be neither +4 nor — 4. J^ote 2.— The root of a fraction is equal to the root of the numerator divided by the root of the denominator. Thus, v\ -=— : for — X — = 3C-. c ck c* ci c^i c ^ote 3. — The above rule for the evolution of monomials is equally applicable where the exponents are negative. Thus, the square root of a~^, or y/ a~*=ar^ =a~^ ; for a~*=: I; hence, ^a-^=^l=l=a-\ a* V a* or EXAMPLES. 1. Bequired the square root of da*b^, 2. Required the square root of 64a*a?*. Arts. Sa'oc^, 3. Required the cube root of 21a'b\ A?is. 3a'b. 4. Required the fourth root of 16a^a?'y. Ans. 2aVy*« 5. Required the square root of .- — Aits. -^. ^ ^ 9a?y 3a:/ 6. Required the fifth root of 243a'°6^ Ans. Sa'b. 7. Required the fourth root of 16o-^Z>*. Ans. 2a-^b. 8. Required the sixth root of 64a®a:'y. Ans. 2ax^yh, 9. Required the third root of Sa-^b-^c^, Ans. 2ar'b-^c, 10. Required the square root of 196a*6V. Ans. 14a^i~'c*. 11. Required the square root of 784x^2*'. Ans.^Sxi^z^. 12. Required the square root of — J-. ' Ans. — ^. ^ ^ 496V 7k» 13. Required the cube root of —Tta^b^. Ans. ~3a^b^. 14. Required the nih. root of a"J*'c~*". Ans. aV^cr*. 15. Required the fifth root of — 32a'^6'V*. Ans. —1ah^(?. 16. Required the fourth root of Sla^^^V. Ans. 3(iShc^ . 17. Required the cube root of — 64a~^Z»"^c-'^ Ans. — 4a-'J-*(r^. Z 178 ELEMENTS OF ALGEBRA. [sECT. VI. 18. Required the square root of 44 la?^3/V. ^ns.^lx'^yz^, 19. Required the square root of 576a^^>-^c-''c?'^ 20. Required the fifth root of — 243a?-'2/'°;s-»^ Ans. —3x-yz-\ 21. Required the square root of . Ans. — ^^. 290. If all the factors of which the monomial is composed are not complete powers of the same name as the root, it is evident that the root of the entire number cannot be obtain- ed. Still the expression may he simplified by removing that factor which is a complete power of. the same name as the required root, from under the radical sign. This is done on the principle that the root of the product is equal to the product of the roots. Thus, V 8^ = v/4a2x 26== v/4?x \/'2i = 2a V26. And, ^2Ub'=4^Sb'x3a='^Sb^x^3a=2b VSa. And, V6a"Z> =ya" x^b=Va'' xVQb=:a V6b. Hence, to reduce radicals to their simplest forms : 1. Resolve the quantity under the radical sign into two faC' tors, one of which shall be a complete power of the same name as the root. 2. Extract the root of this factor, and multiply it by the co- efficient of the radical f if it has any, and prefix the result to the radical sign under which the factor that is not a complete power will remain. EXAMPLES. Required the simplest form of \/S2a'*b'^c. Ans. U^by/Yc, 2. Required the simplest form of ^/^Sa^¥c'^d, Ans. '7ab^cW'2d, 3. Required the simplest form of v^24a^c^^ Ans. 2ad'VSc, ^. Required the simplest form of \/ b4^a^xy'-^z\ * Ans. 3aSjz^V~^. SECT. VI.] EVOLUTION OF MONOMIALS. 179 5. Required the simplest form of VS2a*b'c. Jlns. QaVV^c, 6. Required the simplest form of a/— ^. V 4fox y V 48x^y V 16x* 3y V 16x* V 3y ^x' V 3y 2^ V 3^* Jlns. ^^f. 7. Required the simplest form of V^a^b—l^a^x. Jlns. 2aN/2A— 3x. v/8a»6- 12a«x= v/4a2 X (2A-3x)= v/4o^ X x/26-3a:=2a V2^3^ 8. Required the simplest form of ^^/.JlfL— Ans. ^l/E 3c \^ d 9. Required the simplest forip of ^24a'c— 32tt'cx. Ans. 2a^3c— 4cx. 10. Required the simplest form of y/c^-^a^b^. Ans. a^l+^. 11. Required the simplest form of \/^Oba^b*c^de. v'405a»6V(/e= v/81a'6Vx v/5^=9a*»Cv/"Wer ^/w. 9a^CN/5a(ie. 12. Required the simplest form of '^QOba'b^c'd^. Ans. lWb^(^ds/bacd. 13. Required the simplest form of v^lOUaVc'd Ans. 13a«6*cv/6aicd 14. Required the simplest form of V —Sd'x. v/-8a«x=v/4a'x-2j;=v/4?Xv/"^^^=2av/^r2i. Ans. 2av/— 2a:. 15. Required the simplest form of v/ — 16. Ans. 4v/^n". 180 * ELEMENTS OF ALGEBRA. [sECT. VI. EVOLUTION OF POLYNOMIALS. 291. We might give rules for the extraction of the differ- ent roots separately, but it will comport better with our purpose to introduce the student at once to a general rule by which we may evolve any root whatever. The reason for the following rule will be sufficiently obvious if we re- cur to the formation of powers by the binomial theorem or by actual multiplication ; and, indeed, the work verifiear itself. • RULE. 1. Arrange the terms according to the powers of one of the letters^ so that the highest power shall stand in the first term, the next highest in the second, Sfc. 2. Find the root of the first term, and place it in the quotient ; then subtract its power from the first term, and Iring down the second term for a dividend. 3. Involve the first term of the root to the next inferior power, and multiply it by the index of the given power for a divisor. Divide the dividend hy this divisor, and the quotient will he the second term of the root. 4-. Involve the terms of the root thus found to the given power, and subtract it from the whole polynomial. Divide the first term of the remainder by the divisor first found ; the quotient will be another term of the root. 5. Proceed in this manner till the power obtained by the invo- lution of the terms of the root is equal to the given polynomial. This will be the case only when the true root is found. EXAMPLES. 1. Kequired the square root of 4a^-{-4ai-f 6^ 4>a^-\-^ab-{-b\ \ 2a-{-b. Ans. 4.a^ 4a) * -\-^'+8l6*. ' 4. Required the cube root of a«— 6a^64-15a*J*— 20a'^>'+ 15a'6*— 6a5»H-6^ Ans, a^—'lab + h''. 5. Required the fifth root of 32a»-80a*a:H-80aV-40aV + lOox* — X*. Ans, 2a — x, 6. Required the fifth root of a' + 5a*6+10a'6»4-10a*J'+ 5aZ>*-f J*. ^ Ans, a + 6. 7. Required the sixth root of a»— 6a«6+15a*^— 20a='6»+ l^a^b^—%ab^-\-b\ Ans, a—b. 292. Remark 1. — The square of a binomial consists of three parts, viz., the square of the first term, twice the prod- uct of the two terms, and the square of the last term. Hence the second power of the simplest polynomial will consist of three terms ; and every trinomial in which, when the terms are arranged, the extremes are complete squares, and the middle term is double the product of the square roots of the extremes, is a perfect square, whose root may be found by the following BI7LE. Take the square roots of the two terms that are complete pow^ erSj and connect them by the sign prefixed to the other term* 16 182 ELEMENTS OF ALGEB*RA. [sECT. VI. 1. sfa"- -i-2ab i-b^ =a +h. 2. Va" —^ah +6^ =a —5. 3. x/a^ +2a +1 =0+1. V 3 9 3 5. x/36y2+36y +9 =6^+3. 6. V^d^ -Qdk +A2 =:3c?-A. 7. v/a'Z>2+2ak(^+c2t/2-a6+cJ. 8. ./^-^ +-^ ..^ -1 293. JRemark 2. — Since the fourth power of a quantity- may be found by squaring the second power, it is evident that the fourth root may be obtained by extracting the square root of the square root. Thus, V^*=\/ V^=y/^=a. And, V^'=\/v~^=V^^=a,Scc. Hence, 1. To obtain the fourth rootj we may extract the square root of the square root. 2. To obtain the sixth root, we may extract the cube root of the square root, Src. • > EXAMPLES. 1. Required the square root and fourth root of 16a'' -j- 96(^b+'21QaW-\-216ab^+Slb\ Ans, 4a'+12a6+95^ and 2a+3J. 2. Required the sixth root of a;'— 12a?^+60a?''— 160ar'+ 240a:'— 192a;+64. *^ns. a?— 2. 3. Required the eighth root of a^-\-^a'b+'^Sa%''+bQa'b^-\' 10a''¥+bQa^b'-{-2'^a%^+^aV^b\ Ans. a+b, 4. Required the ninth root of a^+9a«i+36a'Z»2+84a^5''-{- 126a^6^+126a^55+ 84^=^6^+ 36a2Z>'-h9a5« 4- *'. Ans. a^b. 294. Remark 3. — If the polynomial is not a perfect pow- er, it may sometimes be simplified in the same manner as monomials. ^ SECT. VI.] EVOLUTION OF POLYNOMIALS. 183 1. Required the simplest form of y/5(^-\-l0ab-\-5tl^. 2. Required the simplest form of \/o'6+4a'^-f 4o^' jJns. {a-{-2b)y/ab. 3. Required the simplest form of V2a'+8a*ft+12a*6*4- 4. Required the simplest form of VSa*b—9a^t^-{-9a^l/^ — 3a^. •^ns. (a—b)V3ab. 295. Remark 4. — Roots may also be obtained by the Bino- mial Theorem, since n in the general formula (Art. 268) may be either an integer or a fraction. ' The series produced by the expansion of a binomial, however, will never terminate, since the successive subtractions of unit's from the fraction- al exponent of the leading letter can never reduce that ex- ponent to 0. EXAMPLES. 1. Expand by the Binomial Theorem (a-\-by. The exponents in the successive terms of the result will be as follows : Of a Of J 0, 1, 2, 3, 4, &c. Hence the letters, without their coefficients, will be J^arh+a~h--{-a~^l/'-^a~h\ &c., ad infin. Proceeding as in Art. 271, we shall obtain for the coeffi- cients of the successive terms, *» 3> 2 X — ^"r2= — £4, — 2^X — f "^"=2X6, iisX 5^T-4«=— 8,4^8' ****" Or - 1, i, —I, tV — ihy &^c-» «^ ^^fi^' Hence, by compounding the series of letters and coeffi- cients, we obtain (a+»)*=ai+d*_i2*!+fd^_?±', &c, ad infin. ^ ' 8 8 16 128 ' ' •' 184 ELEMENTS OF ALGEBRA. [sECT. VI. Transferring a, where it is affected with the negative ex- ponent, to the denominator, as in Art. 258, the expression becomes x 1 b b'' V b' Ca-\-by=a^-{- — 1 — — 3-1- 5 — Y, &c., adinjin, 2. Expand (a—b)^. *, The exponents, obtained as in the last example, are as follows : Of/7 1 1 1— 3 ^ 1— 5. S 1— 8 ^f. Of 6 0, 1, ^ 2, 3, &c. Hence the letters, &c., will be ai—a~^b-\-a-^b^—am\ &c. adinjin. The coefficients, obtained as before, are 1, h ix— 1-^2=— 3-e, — 34x— f--3=3^, &0. Or - - l+^—re+Hgj &CC., ad infin. 1 ± A- ^^' 2.5Z>" Hence - ^a-b)^=a^^^^-^^^^j-^^^^^j,6cc., adinjin. EXAMPLES. 1. Expand by the Binomial Theorem (a-^b)^. Diminishing the exponents ^ successively by 1, the expo- nents of a, in the successive terms, are i, — ^, — |, — |, &c. (Art. 271.) The coefficients obtained by the general theorem (Art. 271) are as follows : 1, i, i X -_|-7-2=:— |, — } x— f -^3=yV• 2. Expand (a-^bf. 3. Expand ( I +a:)^. 4. Expand (a+&)^. 5. Expand (a — b)^. 6. Expand (a— a?)"^. 7. Expand b{a^—b)-\ 8. Expand (a^-bx)-^. 9. Expand V2=(l + 1)^ CALCULUS OF RADICALS. 296. A radical quantity is the indicated root of an imper- fect power; as, \/2, \/a, and \/b. Radicals are similar when they are composed of the same numbers or letters, placed under the same radical sign or index. Thus, v/a, 4v/o, and a^a are similar radicals* SBCT. TI.] CALCULUS OF RADICALS. 185 Before entering upon equations of the higher degrees, we will consider some of the transformations that may be made upon algebraic expressions involving radicals. CASE I. 297: To reduce a rational number to the form of a radioed, RULE. 1. Involve the given number to a power of the same name as the root. 2. *^pply the corresponding radical sign or index to the power thw produced, 1. Reduce 3a to the form of the fourth root. 3a involved to the fourth power equals 81o*. Applying the radical sign, 3a=V81a*j or applying the fractional index, 3a=(81a*)*^. 2. Reduce 5c^b to the form of the third root. Jins. Vl25a'^, or {I25a«i»)^. 3. Reduce 2ax'y* to the form of the eighth root. Jlns. 4^2560^3^, or (256a«x*V"A 4. Reduce {a^cx^ to the form of the fourth root. ^ns, ^^ya'cV", or {-^ja'c'x'^)K 6. Reduce _— _ to the form of the third root. Wd-'if Ans V-?Z^!^' or /27aW\.} CASE II. 298. To introduce a rational coefficient under the radical sign or fractional index. We have already seen (Art. 290) that a part of the root may be removed from under radicals of the form Va'^b, Thus, ^~^= Va^Xb= V^x Vb=aV'^. Now, by reversing this process, a Vb=z ;/T" X !t/b= ^~cFxb= yo^. Hence we have the following RULE. 1. Raise the rational coefficient to a power of the same name as the root indicated by the radical sign or fractional index. 186 ELEMENTS OF ALGEBRA. [SECT. VI. 2. Multiply the quantity under the radical sign or index by this power, and place the given radical sign or index over the product, EXAMPLES. 1. In the expression 3a\/26, let the coefficient be introdu- ced under the radical. Za^/2b=: \/~9? X y/2h=:'^Wx2bz= >/l8a^. Ans. 2. In the expression ia^-^Sax, let the coefficient be intro- duced under the radicaL ^ns. y/l92a'^x. 3. In the expression 2a\4:Xyy, let the coefficient be intro- duced under the fractional index. 2a\4xy)^=(8ay x {4xyy = (Sa' X 4xyY={32a^xyY. Ans, 4. In the expression 6v^l3, let the coefficient be introdu- ced under the radical. . Ans. \/2808. 5. In the expression 2al{2db^y, let the rational cofficient be introduced under the fractional index. Ans. (IQa^h^) . 6. In the expression (a + Z>)\/aA, let the rational coefficient be introduced under the radical sign. Ans. Vl^'Wxab='sf a%^2a^y'-\-ah\ 7. In the expression ^ (__£_) , let the rational coeffi- cient be introduced under the fractional index. Ans, ' "*" ^ ' \a%''-\-h'') CASE III. 299. To reduce radicals of different indices to equivalent radi- cals having a common fractional index. RULE. 1. Reduce the indices to a common denominator. 2. Involve each quantity to the power expressed by the numer- ator of the reduced index. 3. Take the root denoted by the denominator, EXAMPLES. 1 1 1. Reduce a^ and b^ to a common index. a*=a'^^^=(a''y^, i i 3 ^ \ Ans, (a^y^, and (b^y^ b^z=b^^=(bY^, SECT. VI.] ADDITION OF RADICALS. 187 2. Reduce 2 and 3 to a common index. _ _ Ans. 8*^, and 9^» 3. Reduce \/^ and v^^ to a common index. vT=(i)*=(i)'=((i)')*=(TV)*- 4.. Reduce ( ^ ] and | ^ ] to a common index. _ ^~.(C)'-(|)'. 5. Reduce v^|, y/\y and ^2 to a common index. __ Ans. i'^l296, ^'V656T, and '-^8. 6. Reduce V^h ^"^^ v'5^ to a common index. Am. V 150^j, and '^151^^ CASE IV. ADDITION OP RADICALS. 300. If the radicals are not similar, and cannot be made 80 by reduction, it is evident that the addition can only be expressed. Thus, s/a-\->/h can be reduced to no simpler form. 301. If the radicals are similar, they may be added by the following' RULE. Add the coefficients^ and to their sum annex the common radical, J^ote. — If the radicals are not similar, they may frequently be made so by reduction. EXAMPLES. 1. Add 4\/ax, 2\/aj:, 5v^ax, and Z^^/ax^ ^\/ax 2s/ax by/ax 3iv'ai lii'/ox. Ans. 14iJ\/ax. 2. Add 3a^^ and 5c^^ Ans. (3a+5c)^|. .J^»mt 188 ELEMENTS OF ALGEBRA. 3. Add\/8andN/32. [sect. VI. v/8 =V4> x2==v/4 X\/2=:2v/2. V32=n/16x2=x/16x%/2-4v/2. 4. Add i/50 and v/l28. 5. Add (SGa^y)^ and (25y)^. 6. Add ^54^ and ^1280^. Jlns. eV2. Ans. 13\/2. ' Ans. (6aH-5)v/y. Ans. 4av/J-y. 7. Addy^^^d y/^. /Q>,2 /9^2 s/ Va' Xj\ = Va' XVj\=: «n/tV- 4a\/yV- 21 24 Ans. (b-^y)Vb^^ Ans. 4(a+x)^ and (4a'^>2+4a%)^. ./?»s. (4+2a5)v/a4-a7» 13. Add v/f", 4v'l2, and 3 s/~J~ Ans. 9v/3. 14. Add ^192 and ^24. Ans. 6^3. 15. Add 3^5^16^ and aby^^. Ans. 9ab^^. CASE V. SUBTRACTION OF RADICALS. 302. If the radicals are not similar, and cannot be made so by reduction, the subtraction can only be expressed. Thus, y/a — Vb can be reduced to no simpler form. 303. If the radicals are similar, the subtraction may be performed by the following SECT. VI.] MULTIPLICATION OF RADICALS. 189 RULE. Subtract the coefficient of the subtrahend from the coefficient of the minuend, and to the difference annex the common radical, J^ote. — If the radicals are not similar, they may frequently be made so by reduction. EXAMPLES. 1. From ^ab^/cd subtract SabVcd, Ans, aby/cd. 2. From V^Oa^ subtract VSa\ V50a^=v/25a^ X 2=i/25a^ x V2=baV2. y/Sa' =s/4>a' x2=v/4a' Xv/2=2av/2. 3aV2. Jlns, 3. From -5^192 subtract 4^24-. jJns. 2^3. 4. From IW^Qa^bc" subtract 3av/24^(r'. JJns. 3Sacs^6bc, 5. From v/| subtract \/^. JJns. j^n/I^. 6. From Vi subtract v/|. ^ns. Jv^3. 7. From 3^^ subtract 1^400. JJns. J^^Bo. 8. From 4a^l250a^ subtract i^640a«. Ans. ISaVlO. 9. From 1^^567 subtract 1^112. Ans. -^^1, 10. From ^^%W subtract fv/Qo'. Am, |a. CASE VI. MULTIPLICATION OF RADICALS. 604. If the quantities are not under the same radical sign, and are not roots of the same letters, the multiplication can only be indicated. But since the product of the roots is equal to the root of the product (Art. 156), and since the product of several factors, composed of the same letters or quantities, is obtained by taking one of the factors affected by an exponent equal to the sum of the exponents of the several factors (Art. 88), that is, a"* x a** =a'*»x «***=«"", we have the following general 190 ELEMENTS OF ALGEBRA. [SECT. VI. RULE. 1. If the, quantities are under the same radical sign or index, multiply thefn like rational quantities, and place the common radical sign over the product. 2. If the qu(fitities are composed of the same letters or num^ bers, and are affected with different fractional exponents; add these exponents. 3. If the radicals have rational coefficients, multiply them, and prefix the product to the product of the radicals. EXAMPLES. 1. Multiply 2ay/3ax by SbVSa^cx. 2a>/3ax X Sb\/3a^cx=Qaby/3ax x 3a^cxz=6ab\/^ci^ca^' 6aby/9a'^cx^=z6aby/9a^x^ X ac=6a5\/9aV X y/ac=6ab x Sax^ac = ISc^bxy/ac, Arts. % Multiply {3af by (3a)*. Ans. ^~^Ma^. {3af=:{3af {3af={3ay 5 1 (3ay=(24^3a'y=z ^24<3a'. 3. Multiply 4n/2 by V6. Ans. 4.5^288. 4. Multiply 5v/5 by 3v^8. Ans. 30x/10. 5. Multiply 2av/aM^ by ^3aV^Tb^ ^ Ans, —ea'ia'-i-b'). 6. Multiply 5a^a+x by UV(a~+xf. Ans. 20ab{a-\-x). 8. Multiply (6a^bcy by (6a'bc)K Ans. (6a'bcyK 9. Multiply a , a , and a together. Ans. a^^. 10. Multiply 7e^l8 by 5^4. Ans. 70^9. 11. Multiply 2a%a+bY by 6ac(a-\-by. Ans. 12a^c(a-{-b)~i^ . 12. Multiply 6aV3a^ by 64^27a^ Ans. dOa'Vb. 13. Multiply 4>^J^ by 3s/8. Ans. 12^2. 14. Multiply 27 V by iVh ^'^^' ^^^Vt- 8£CT. TX.] DIVISION OF RADICALS. IS 15. Multiply 4.^f by ia^J •*"! 16. Multiply a¥ by flM. \dn8,ab^. 17. Multiply 3a-V by 3a**~«. Am. 9. 18. Multiply laJ" by llxK 3 1 1 ^715. 14a*x^. . '3 3 19. Multiply 6a V by 1 la^a:. Ans, %d^ £^ . CASE VII. DIVISION OF RADICALS. 305. If the numbers are not under the same radical sign, nor roots of the same letters, the division can only be indi- cated. 306. But in those two cases it may be performed by the following general RULE. 1. Jf the quantities are under the same radical sign or index, divide them like rational quantities, and place the common radi- cal sign over the quotient. 2. If the quantities are composed of the same letters or num- bers, and affected vrith different fractional exponents, subtract the exponent of the divisor from that of the dividend. 3. If the radicals have rational coefficients, divide the coeffi- cient in the dividend by -that in the divisor. EXAMPLES. 1. Divide 6v/12a^Z> by 3v/3a*. Ans. Wa. V oah 2. Divide Ua^ by 7a^' Ans. 2a^ 14a'-^7a* = 14a^~7a^=2a^-^=2a' ; Or, 14^^H-7^/a=14y^-^7W=2e^a. 3. Divide 4* by 4^. Ans. 4^ 4. Divide 6v/54 by 3v/2. Ans. 6v/3. 5. Divide 4^72 by 2>/18. Ans. Wl> 192 ELEMENTS OF ALGEBRA. [sECT. VI. 6. Divide v/7 by V7. ^;js. V7. 7. Divide 8v/108 by 276. Ans. 12n/2. 8. Divide (a%H'f by /. ^»s. (a&f . 9. Divide v/3 by v/|. .^W5. v/|, or ^v/,6. 10. Divide i^J by i^I. ^7i5. f 4^l2. 11. Divide %ah^hy1ah^. Ans. A^a^h^, 12. Divide 21a&'^' bv 3a*A Am. 7a¥. CASE VIII. INVOLUTION OF RADICALS. 307. Let it be required to involve ^as/W to the second power. By the definition of involution (Art. 25), we shall have (6a\/35^)^=6aN/36^ X 6a\/36^ ; or, performing the multi- plication, (6av/3^f = 6ax/3^^ X 6av/"36^= 36aV"96'^= 36a=^ X 3&^ = 108a2&^ \ 3 Again, let it be required to involve 3a h^ to the third power. (3a¥)='z:r3aVx 3a^^»^X 3a¥=27a^6l 308. The above operation will evidently apply to all cases of monomial radicals. Hence we have the following general RULE. 1. Involve the coefficients to the required power, 2. If the number is under the radical sign, involve it as if it were rational ; over the power place the radical sign, and then reduce the result to its simplest form. 3. If the number to be involved is affected by a fractional exponent^ multiply the exponent of each letter by the index of the required power. EXAMPLES. 1. Required the third power of ^a^y. Ans. 'Ula^y. 2. Required the second power of 4a^6\/6&. Ans. 16a%V36F'=96a3&2. .#--^- SECT. VI.] EVOLUTION OF RADICALS. 198 3 1 3. Required the third power of ^'6^\/86*x. ^ns, 1288a«6V2i. 4. Required the fourth power of 6\/^. ^ns. 36. 5. Required the third power of b^2a^b, Jlns. 125a^8a^. 6. Required the third power of 3%/? x 2^2. Jlns, 216>/J. CASE IX. EVOLUTION OF RADICALS. 309. From the foregoing operations, the following rule win be sufficiently obvious : RULE. i. Extract the root of the coefficient^ if it is a complete power ; if not, introduce it under the radical sign or index. 2.' If the radical sign is used, multiply the figure over the foot of the radical by the index denoting the root to he taken, 3. If the fractional index is used, divide the index of each let' ter by the index of the required root. EXAMPLES. 1. Extract the square root of IGa'^V^o:". 2. Extract the third root of 27a«^26t/. JJns. 3a^ V^bd. 3. Extract the third root of 8oM. Jlns. 2a^b\ 4. Extract the third root of 6U^VTW. Jlns. i^a^^iW. 5. Extract the square root of 24^3a. Jlns, 2yi08a. 6. Extract the cube root of 54aVVi. Jlns, Sa^b^^Q. CASE X. POLYNOMIALS HAVING RADICAL TERMS. 310. We will give, for the exercise of the learner, some examples of polynomials having one or more of their terms radical quantities. These examples may be solved by an application of the rules laid down in the preceding cases, n Bb ) 194 ELEMENTS OF ALGEBRA. [SECT. VI. EXAMPLES. 1. Required the second power of a-\-Vy* Ans, a^-\-2a>/y+y, 2. Required the third power of a — Vh» Am. (^—^aW~b-{-^ah—y/¥. 3. Required the second power of \/3a4-\/2a7. Ans. 3a+\/6aa;4-2ar. 4. Required the square root of a-\-2^/ab-\-h, Ans. y/a+y/b. 5. Required the square root of 9a+36\/Sax-{-10Sx. Ans. 3Va-\-6V3cc. 6. Required the cube root of (f+3a''^'x-\-3aVoo^-x. Ans, a-\-^x. CASE XI. BINOMIAL AND TRINOMIAL SURDS. 311. Expressions under this form, y/a-\-Vhj or a-\-Vi, are called binomial surds, and may be reduced to rational quan- tities on the principle that the product of the sum and differ- ence of two quantities is equal to the difference of their squares, . Thus the binomial surd \/a-\-\/b Multiplied by - - y/a—s/h uW a-\-y/ab —/ah^l) rives - - - a +^j a rational quantity. 312. Trinomial surds may be reduced, first, to binomial surds, then to rational quantities. Thus, The trinomial surd - ,Ja-\-^fh —^fc Multiplied by - yfa—s/h +n/c . a-\-y/al — \/ac —y/ab —b-\- Vbc -hVac + Vbc — c Gives a ^b-i-2Vbc—c. SECT. VI.] BINOMIAL AND TRINOMIAL SURDS. 191 A Let x=a — b — c / then we shall have Multiplying by - - x — 2>/bc x*-^2xVTc ^ — 2xy/bc — ifbc x* — 46c Restoring value of x - (a — b — cf — ibc. 3. Find a factor which will make 1 + v/2 rational. l + %/2 1— v/2 l + v^2 — v/2— 2 1 — 2=-l. Hence the factor is 1 — \/2. 4. Find a factor which will make n/10 — V2 — \/3 rational. VT6—V2 -V3 Multiplying by - x/'T0-hv'2 4-v/3 10— v/20— v/30 — 2+N/20 — /6 —3 +'/30— v^6 Multiplying by — 2n/6 -f 2v'6 25 — lOv/6 4-10v/6-24 25 —24=1. Hence the factors are v/'io+v/24-v/3, and 5 + 2v/6. 5. Find a factor which will make 3— 2n/2 rational. 6. Find a factor which will make \/6 + 3v/2 — v/5 rational. 313. By the above process, fractions may be cleared from 196 ELEMENTS OF ALGEBRA. [SECT. VI. radical numerators or denominators without altering the value of the fraction, and thus the process of extracting the root be facilitated by confining it either to the numerator or denominator. 1. Let it be required to extract the square root of the fraction ^. a \/a y/a x y/a> Vb \fh X \/a \/ab 2. Extract the square root of the fraction ^ . ooy a-{-b_\/a-\-b s/a-\-bXy/a-\-b a-\-b 4- ^y Vxy Vxy-{-Va-^b ^/axy-\-bxy 3. Extract the square root of -. V- 5_n/5_ n/5xn/5 ^V25_ 5 8 v/8 n/8x\/5~x/40~2n/10' \/2 4. Reduce the fraction = to an equivalent fraction 3— v/2 ^ having a rational denominator. Ans^J^^. 3 5. Reduce the fraction — = =to an equivalent fraction having a rational denominator. .. . Ans.^^^1. 1 6 6. Reduce the fraction ~t to an equivalent fraction hav- 5* ing a rational denominator. 5 g 7. Reduce the fraction -^ = to an equivalent frac- v/3+v^2+l tion having a rational denominator. */??i5. 4— 2V6 + 2v'2. SECT. VI.] BINOMIAL AND TRINOMIAL SURDS. 1^ 8. Reduce the fraction - — v ^-hv ^^ ^^ equivalent fraction having a rational numerator. 84v/10-20>/64-72v/34-20W5 I CASE XII. ROOTS OP BINOMIAL SURDS OF THE FORM a±^'b. 314. It is proposed to obtain a formula for extracting the square roots of expressions in the form of a± y/b. Let - ya-\-Jl=x+y/~y (1). Then - y a-^/b=x-'^/y (2). Squaring both equations, a+>/^=«2+2a:v/y4-y (3). a— %/^=x*— 2xv/y-hy (4). Adding - 2a =2a:* +2y (5). And - - a=oi^-\-y (6). Multiplying the first equation by the second, ^~^^~b=3?-y. (7). Adding the sixth and seventh equations, a^shr^h^lx" (8). Reducing - a'=y/°+^^'~^ (9). Subtracting the seventh from the sixth equations^ a-^ir^z=:'Xy (10). Reducing - v^y^-y/^LZ^J^ (l^)' Substituting these values of x and >/y in the first and sec- ond equations, \/a+V*=v/^^^*V^^ -y/a^^b Va-x/6=V 2 V 2 198 ELEMENTS OF ALGEBRA. [sECT. TI. Or, letting c=y/a^ — b. 1- - ^^^=^/^+\/^ (A). EXAMPLES. 1. Extract the square root of 34-2'/2. Here a=3, andv7z>z=2v/~2=rv/8, or h—^, and c^-ZO— 8=1. 3+2n/2=34-v/8 \/3+2v/2=n/2+1. ^715. 2. Extract the square root of 14— 6V5. Ans. 3 — \/5. 3. Extract the square root of ll + 6\/2. Ans. 3+-/2. 4. Extract the square root of 7 — 2\/10. Ans. V5 — v/2. 5. Extract the square root of 94H-42n/5. ^7^5. 7+3\/5. 6. Extract the square root of 1+W — 3. Ans, 2+1/^. 7. Extract the square root of 28+1073. Ans. 5 + n/3. 8. Extract the square root of ab-\- 4)6^— d^-\-2\/^ahc^ — abd. Ans. V'ab+V^c'—d^' 9. Find the sum of y 16 + 30v/=^+\/l6— 30%/=!. Ans. 10. 10. Find the sum of \/bc-\-2bVbc—b'-\- ybc—2b^bc^b\ Ans. 26 ^ SECT. YII.] EQUATIONS EXCEEDING THE FIRST DECREE. 199 SECTION VII. Equations of the Second Degree, EaUATIONS EXCEEDING THE FIRST DEGREE. 315. The questions heretofore discussed involved only the first power of the unknown quantity, or, if a higher power ever appeared, it was cancelled in the process of the reduction. The enunciation of other questions, however, frequently requires a power or root of the unknown quantity, and for the solution of such cases we must seek for meth- ods different from any heretofore discussed. 316. Equations of this nature are divided into two class- es, viz., pure or incomplete equations, and affected or complete equations. 317. A pure equation is one which, when reduced to its simplest form, involves only one power or root of the unknown quantity. Thus, X =q is a pure equation of the first degree j a^=q^ is a pure equation of the second degree, or a pure quadratic equation; s?=:(f is a pure equation of the third degree, or a pure cubic equation ; x*=q* is a pure equation of the fourth degree, or a pure biquadratic equation, &c. y/Xy or x^=q^y is a sub-quadratic equation ; — L 1 . ^Xf or x^=q^, is a sub-cubic equation ; V«, or X* =q*, is a sub-biquadratic equation, &c. 318. An affected equation is one which, when reduced to its simplest form, involves different powers or roots of the un- known quantity. Thus, a^-{-px=q is an afl!ected quadratic equation ; a'-f-JP*-fpJC=fl is an afi^ected cubic equation, &c. ' 200 ELEMENTS OP ALGEBRA. [SECT. VII. PURE EaUATIONS. 319. Pure equations may be readily solved on the princi- ples, 1. If the same root of loth mernbers of an equation be eX' traded, the results mil be equal. 2. If both members be involv- ed to the same power, the results will be equal (Art. 158). Let us take the equation - - - x^^^q^ ; Extracting the nth root - - - - x =q. Again, the equation - - - - i^x — V q ; Involving to the ?2th power - - - a? =:j. Hence, for the reduction of pure equations, we have the following general RULE. 1. Reduce the equation to such a form that the power or root of the unknown quantity may stand by itself in the first, and the known quantity by itself in the second member of the equation. 2. If the expression containing the unknown quantity is a power, extract the corresponding root of both members. 3. If the expression containing the unknown quantity is a root, involve both members to a power of the same name. .^^ J^ote 1. — If the even root, i. e., the second, fourth, or sixth root of an equation, is to be taken, the resulting second member should be affected by the double sign ± . Thus^ the square root o{ q^—:kq; for -\-qx-\-q—+q^, and — qx —q = +^' : the fourth root of q'^= it g ; (or qxqxqxq=q'^i and — 5'X — qX—qX — q=-{-q'^: and the sixth root of q^=dzq ; for q-q-q'q'q'q=q\ and — qX—qX — qx—qX—qX—q=q\ &c. JSTote 2. — Since, if the even root be taken on both sides of the equation, it would be very natural to suppose that the first member, or x, should be affected with the double sign ± , as well as the second member of the equation. Affect- ing it thus, and arranging the signs in the equation, ±x= ± q, in every possible manner, we shall have the four equa- tions. (1). +0?=+?. (3). -x=+q. (2). -\-oo=—q. (4). •^x=-~q. ^ ••".^\^ SECT. VII.] PURE EQUATIONS. 201 But Still we have in reality no more than the first two equations, as the third equation expresses the same rela- tions with the second ; for, changing the signs, it becomes (2). -\-x=—q : And the fourth expresses the same relations with the first ; for, changing the signs, it becomes (1). -^x=-\-q. EXAMPLES. 1. Find the value of x in the equation — — 10=2» 4 2^—10= 2; Clearing of fractions - - 3a?* — 40= 8 ; Transposing and reducing - 3x* =48 ; Dividing a?* =16; Evolving - - - - X =±4. Verifying on the supposition that x=4-4 ; 2:1^—10= 12—10=2. 4 Verifying on the supposition that a:=— 4 ; ?2ir±'~10=2ll^=12— 10=2. 4 4 2. Find the value of ar in the equation >/x — 16=8 — %/a?. -y/x — 16=8 — Vx ; Involving both members - x — 16=64 — 16v^a:-l-ap; Cancelling and transposing 16\/a:=80 ; Dividing - - - - y/x= 5 ; Involving - - - - x=25. Verification - - V25^6=8--n/25 ; Or v/9=8— 5 = 3. X — ax y/x 3, Find the value of x in the equation — ;=— = — . VX X Clearing the equation of fractions - x' — ax^=x ,* Dividing by X x — ax=l;^ Besolving into factors ... (1 — a).x=l ; Cc 202 ELEMENTS OF ALGEBRA. [sECT. VII. Dividing by the coefficient of a? - a?= 4. Find the value of a? in the equation Vx-j-a-. 1—a a+b Voc — a Clearing of fractions - Vx^ — a^= a+b ; Involving ... oc^ — a^= a^-\-2ab-\-bf^ ; Transposing and reducing - x^='2,a^-\-2ab-\-b^ ; Evolving .... X =±V2a'—2ab-{-b^. . \/^+28 v/a:+38 5. Find the value of x in the equation — -= — - =—rz — — , \/x-{-4f \/a?+6 Clearing of fractions - a?4-34%/^H-168=a?+42\/a?+152; Transposing . - - - 8y/x= 16 ; Dividing ----- Vx— 2j Involving - - - - - x= 4f. . . Vax — b 3s/ax — 25 6. Find the value of x in the equation —= — -:=—-== — — . Vax-\-b 3\/aa?+56 Clearing of fractions, 3ax-{-2bV'ax—6b^=3ax-{-bVax^^b^ ; Transposing - - - bs/ax=z3b^; Dividing by & - - - ^/ax=z3b; Involving - - - - ax=9b^ ; Dividing - - . - x= — . 7. Find the value of x in the equation \/ x+Vco— V ^ 2 V x-\-Vx _ / z _ 3v/f. Multiplying by \/ a:+ v/a? - x+Vx-^s/a^ — a?=— 2~ ' \/a? Transposing and reducing - - x — ---=y/a^ — x; At Dividing by Va; - - - - Va? — \z=L\fx — 1 ; Involving x — s/x-^\—x — 1 ; Transposing and reducing - - - '>Jx=i\ j Involving X — SECT. VII.] PURE EQUATIONS. S08 !* 20" 8. Find the value of x in the equation x-\->/a*-\-3^:=z—rT==z Clearing of fractions - Xy/c^-\-3^-\- (^-\-a^z=2a?-y Transposing and reducing, Xy/a^-\-a^=2(^ — a* — x*=a^ — «• j Involving - - 2c'(a'-{-a^)= a*— 2aV4-a?x Ans, x=\. 12. Find the value of x in the equation \/x+v/3+a?= 6 Ans. a:=l. 13. Find the value of x in the equation x-\-\/W^\^— -^. An.. x=y3. 14. Find the value of x in the equation ap+2= \/4H-«x/64+^. Ans. x^^. 15. Find the value of x in the equation \/2ar'+9x*-f 27x =a:+3. w^TW. x=3. 16. Find the value of a; in the equation ^/x — 32=16 — s/x. Ans, x=%\. v/6x-2 4n/6x~9 17. Find the value of x in the equation >/6a:-f-2 4N/6ar+6 Ans. x=6. 18. Find the value of x in the equation ^x^— a'= 4^3ax« — 3a'x-f86. Ans.x=a^-Wb. 204 ELEMENTS OF ALGEBRA. [sECT. VII, 19. Find the value of a? in the equation c? — 2aa7+a?2=:Z>. Ans, x=a^^>/b. a — s/d^ — a^ 20. Find the value of x in the equation — &. ^ a+v/a^— a?2 Multiplying both numerator and denominator of the first member by a — \/a^ — x^. Reducing and clearing of fractions - - - (a— vV— ^_, Evolving - Transposing Involving - - - Cancelling and transposing Dividing by a? - Re solving- into factors Dividing - a—Va^ — a^=±xy/b; a q= X\/l— y/a^ — x^ ; a^ qp 2aa?x/5"+ bx^ = a^—x? ; bx'^-x^—±1axsfb; bx-\-x^±1a^fb', X—. ^'+1 21. Find the value of x in the equation \/x-\-\/x — a Vx — \/a? — a X — a Multiplying the numerator and denominator of the first fraction by Vx — Vx — a. Whence Evolving Clearing of fractions X— ix-a) n'a (Vx- Wx—af 1 X — a (Vx- Wx-af 1 ~x — a ±n Vx — Vx — a Vx — a Multiplying by Voc — a Or . - - Transposing - (x — a)-\-n{x — a)=:±n\/!X^ — ax; s/x — a= ± »(\/aj — \/x — a) ; X — a= ± niVx^ — ax — x-\-a), X — a= ± nVoc^ — ax — n{x — a) y SECT. VII.] PURE EQUATIONS. 206 Resolvinpf into ) *■» , \ , \ , /Ta ° [ (1-l-n). (x — a)=±n>/a^ — ax; factors - > InTolving - (1-fn)'. (x — af=n\3p^ — ex) =ft'x(x— a) ; Dividing by X — a, (1-fw)^ (x — o)=«'x, Or -. (l-\-nyx—{\—nfa=n^x; Transposing - (l+n)'x — n*x=(l-|-n)^a, Or - (l + 2;i-|-n')x— »"x=(14-»)*a; Addingcoeffi.; ^^2^^^,_^,^^^ ^^^^, . cients of X > Reducing - - (l + 2;i)x=(l+n)^a ; Dividing by l+2;i - x^llt^. ^;w. _ \/a-|-x4-\/a — X 22. Find the value of x in the equation -^ =6. 23. Find the value of x in the equation 1 n/3 24. Find the value of x in the equation \/x'+8=v/125 — 6x*— 12x. ^»5. x=3. 25. Find the value of x in the equation* /.^il5-}- 2* / ^ V X V a-i-a: -V. .^TW. X= — .. Examples of Pure Equations containing two or more unknown, Quantities. 320. In equations of this kind, unknown quantities may be eliminated by the same principles that were applied in equations of the first degree. EXAMPLES. 1. Find the values of x and y in the equations x'+y=28, and ^—^=19. ^ns, x=5, and y=3» 5 3 18 206 ELEMENTS OP ALGEBRA. [sECl^. VII 5 3 Transposing y, and extract- ) ing the square root of the } a?= ^/28 — y first equation - - - J Clearing the second equa- } ^ „ ^ tion of fractions - J 12a:^-5y=285 Transposing, &c. . - - x^=?^^±^ Evolving x=a/ 285 -f5y 12 Forming a new equation of the two values of x, 28-y=5?^±^ ^ 12 336— 122/=285+5y — 12y— 52/=285— 336 172/=51 y=3, and a:=5. 2. Find the values of x and y in the equations 3y=aj+y, and xy= 18. ^?i.9. 3?= db 6, and y= dr 3. * 3. Find the values of x, y, and z in the equations {x-\-y -\-zf =S000, y''-\-2yzS6 = 64>—z\ and 2a?y+202:=200. jins. a?=10, y=8, and z=2, 4. Find the values of x and y in the equations 5a? — 5y=: ^y, and a:^+4y^=:181. .^ns. x=9, and y-=5. 5. Find the values of a?, y, and z in the equations j?'^y=54, y2;=8, and xz—1% Ans. x—Z, y=2, and z—^. 6. Find the values of x and y in the equations a?^4-y^= , and xy= Ans, 07=3 or —2, and y=2 or — 3 x—y x—y x—y 6 xy= ; 0? — V SECT. VII.] PURE EQUATIONS. 207 Multiplying the second equa- > o* — ^^ . n^ tion by 2 - - -) ^~x^ ' ^ '^ Subtracting the third from ) « \ the first - - - - > X — y Contracting the first member - (x — y)*= ; (5.) X — y Clearing of fractions - - - (x — yy=lj (6.) Evolving X — y=lj (7.) Substituting for x — y its val- > a:* -+-«»— 13 • ^8 \ ue in the first equation - i Substituting for x—y its val- > ^ Oj.y_i2 ; (9.) ue in the third equation - > Adding the eighth and ninth > ^^cixy-^f=25 ; equations - - - > Evolving x-\-y=±5. But x^y=l. Hence - • -x=3or — 2, and y=2 or — 3. 7. Find the values of x and y in the equations x'^-{-y^= 13, and a?^+y^=5. ^ns. a?=27 or 8, and y = 8 or 27. 8. Find the values of x and y in the equations 3c^-\-sl*y*-{- y»=273, and x'+xy+y'=21. ^ns.x=±2 or ±2v/— l,and y=±l or ± >/~l. 9. Find the values of x and y in the equations (x' — y*). (a?— y) = 3xy, and (a?*— y«).(x-— y2)=45a:*y'. ^ns. x=z4f or 2, and y=2 or 4. 10. Find the values of x and y in the equations 3^y-\-xy'^= 6, and a^y*+a:'y'=12. Ans. x=2 or 1, and y=l or 2. 11. Find the values of a and y in the equations Vx — Vy = 3, and Vx-\-Vy=^7. Am. x=625, and y=16. r? 5 3 12. Find the values of x and y in the equations x*+a:*y* 4-y'=1009, and r'-ha?V4-y*= 582193. ^n«. ar=8l ar 16, and y=16 or 81. 208 ELEMENTS OF ALGEBRA. [sECT. VII. PROBLEMS PRODUCING PURE EaUATIONS. 321. — 1. What two numbers are those whose difference is to the greater as 2 to 9, and the difference of whose squares is 1281 ^^ns. 18 and 14. 2. A fisherman bfeing asked how many fish he had caught, replied, "If you add 14 to the number, the square root of the sum, diminished by 8, will equal nothing." How many had he caught % Ans. 50. 3. A merchant gains in trade a sum, to which $320 bears the same proportion as five times the sum does to $2500. What is the sum \ Ans. $400. 4. What number is that, the fourth part of whose square being subtracted from 8, leaves a remainder equal to 4 \ Ans. 4, 5. It is required to divide the number 18 into two such parts, that the squares of these parts may be in the propor^ tion of 25 to 16. Ans. 10 and 8. 6. It is required to divide the number 14 into two such parts, that the quotient of the greater part, divided by the less, may be to the quotient of the less, divided by the great- er, as 16 to 9. Ans. 8 and 6. 7. Two persons, A and B, lay out some rnoney on specu- lation. A disposes of his bargain for $11, and gains as much per cent, as B lays out; B gains $36, and it appears that A gains four times as much per cent, as B. Required the capital of each. Ans. A's $5, and B's $120. 8. A gentleman bought two pieces of silk, which together measured 36 yards. Each of them cost as many shillings by the yard as there were yards in the piece, and their whole prices were as 4 to 1. What were the lengths of the pieces \ Ans. 24 and 12 yards. 9. A number of boys set 'out to rob an orchard, each hav- ing as many bags as there were boys in all, and each bag capable of containing as many apples as there were boys. They filled their bags, and found the whole number of ap- ples was 1000. Wto was the number of boys \ Ans. 10., SECT. VII.] PURE EQUATIO^JS. 209 10. Several gentlemen made an excursion, each taking the same sum of money. Each had as many servants as there were gentlemen ; the number of dollars which each had was double the number of all the servants; and the whole sum of money taken out was {^14'58. What was the number of gentlemen 1 *dfu. 9 11. There is a rectangular field, whose length is to the breadth as 6 to 5. After planting one sixth of the whole, there remained 625 square yards. What are the dimen- sions of the field 1 *dns. The sides are 30 and 25 yards. 12. There are two numbers, which are to each other as 3 to 2, and the difference of their fourth powers is to the sum of their cubes as 2G to 7, What are the numbers % An&. 6 and 4-. 13. What two numbers are as 5 ta 4, and the sum of whose cubes is 5103 \ Jlns, 15 and 12. 14. There is a rectangular field containing 360 square rods, and whose length is to its breadth as 8 to 5. What is the length and breadth. Ans, Length 24, breadth 15. 15. There are two square fields, the larger of which con* tains 13941 square rods more than the smaller, and the pro- portion of their sides is as 15 to 8. What is the length of the sides *{ An$. 16. Two travellers, A and B, set out to meet each other. They started at the same time, and travelled on the direct road between the two places \ and on meeting, it appeared that A had travelled 18 miles more than B,and that A could have gone B's distance in 15^ days, while B would have been 28 days in going A'^s distance. What was the distance travelled by each 1 Jlns. A*s 72, B's 54. 17. There are two men whose ages are to each other as 5 to 4, and the sum of the third power of their ages is 137781. What are their ages 1 Am. 45 and 36 years. 18. Find two numbers, such that the second power o( the greater, multiplied by the less, may be equal to 448 j and Dd 210 ELEMENTS OF ALGEBRA. [nECT. VII. the second power of the less, multiplied by the greater, may- be 392. 19. A man wishes to make a cellar that shall contain Sll04f cubic feet, and in such a form that the breadth shall be twice the depth, and the length li the breadth. What must be the length, breadth, and depth 1 jins. Length 48, breadth 36, depth 18. 20. A man wishes to make a cistern that shall contain 500 gallons of wine, in such a form that the length shall be to the breadth as 5 to 4, and the depth to the length as 2 to 5. Now, allowing 231 cubic inches for one wine gallon, what will be the length, breadth, and depth 1 AFFECTED EaUATIONS OF THE SECOND DEGREE. 322. Let 2p nnd q be two variable numbers, 2/? represent- ing the coefficient of the unknown quantity, and q the known quantity ; then, however complicated may be the equations which involve the first and second powers of the unknown quantity, they may be reduced to one of the four following forms : (1.) x'-\-2px=q. (3.) x^-\-2px=-q. (2.) x'' — 2px=q. (4>.) x^ — 2px=z—q. Let us then determine the process by which equations of these forms may be solved. 323. We have already seen that a binomial cannot be a perfect square, and also that the root of a trinomial, which is a perfect square, may be formed by taking the root of the two terms that are complete powers, and connecting them by the sign of the other term (Art. 292). Thus, \/x^-\-2px -{-p^=x-\-p, and \/x^—2px-{-p^=:x—p. 324. We have also seen that the square of a binomial is equal to the square of the first term, plus twice the product of the two terms, plus the square of the last term. Thus, (x-{-pf—x^-\-2px-{-p^ ; And the square of the residual, x—p^ gives (x-py=x^—2px+p\ SECT. VII.] AFFECTED EQUATfONS OF SECOND DEGREE. 211 Hence, if/?' be added to both members of each of the pre- ceding four forms of the affected quadratic equation, the first member of each will be a perfect square. Thus, (5.) :3^+2px+f=q-{.f', (6.) x«— 2pa:+;>*=9-hp'; (7.) s^+lpx^p'^f—q; (8.) j?-^px-\-p^=f^^, 325. If we compare p^ with the coefficient of x, it will be found equal to the square of half of it. Thus, p^= IJl\, Hence, when the quadratic equation is reduced to the first, second, third, or fourth power, the first member may be rendered a perfect square by adding the square of half the coefficient of the first power of the unknown quantity to both members of the equation. This is called completing the square, 326. Each of the above equations may be reduced by ex- tracting the square root of both members, and making the necessary transformations. Extracting the square root of the (5), x-\-p= ± Vq-{-p^ ; Transposing - . . . . x=z—p±y/q-^p^. Extracting th6 square root of the (6), x—p= ± y/q-\-p* ; Transposing x=;)± v/^-f-p\ Extracting the square root of the (7), x-\-p= ± y/p^ — q ; Transposing x=—p±>/p^—q- Extracting the square root of the (8), x — p= ± ^/p^—q ; Transposing x=p± y/p^ — q. 327. Hence, for the solution of affected quadratic equa- tions, we have the following general RULE. 1. Reduce the equation to one of the above four forms. 2. Complete the square by adding to both members of the equa- tion the square of half the coefficient of the first power of the unknown quantity. 212 ELEMENTS OF ALGEBRA. [sECT. VII. 3. Extract the square root of both members^ observing to affect the second member with the double sign ± , and complete the re-' duction by the preceding principles. JVote. — Equations of this nature also give two values of the unknown quantity. Thus, x^-\-4a^x^-{-8apx-\-4fp'^=4>aq-\-4>p' ] (6.) 4fa^a^—Sapx-\-4>p*=4>aq+4a?+4>'=— 4^0^+ V ; (8.) 4a-x* — S(ipx-\-4tp'^ = — 4faq-\-4= ± v/4a9-f-4/ ; (10.) 2ax—2p= ± v/4a9-f 4/ ; (11.) 2ax-\-2p= ± V —4faq-\-4p* (12.) 2ax—2p=± V—^aq-\-4^, Or, reducing, (13.) (14.) (15.) ,_ — 2p± y/^aq-\-^p^ ^ 2a ' _ +2p±>/^aq-\-^p\ 2a — 2p± y/—4>aq-\-^p* 2a (16) ^_ -f2;)dbv/— 4a^+V 2a ' 330. Hence the square of an affected quadratic equation may be completed by the following general RULE. 1. Multiply both members of the equation by four times the 214 ELEMENTS OF ALGEBRA. [sECT, VII. coefficient of the second power of the unknown quantity^ and add the square of the coefficient of the first power to loth members ; the first member will then be a perfect square. 2. Extract the root^ and reduce as before. EXAMPLES. 1. Find the values of a? in the equation 2a;^+3ir=65. 2a?2+3a:z=65. Completing the > 16^2+24^+9^520 + 9 =529; square - > Evolving - - - 4a?+3=±v/'529=±23,- 4a?=— 3±23=20,or— 26; x=b^ or — 6|. 2. Find the values of a: in the equation 3a?^ — ^x — 4—80. Ans. a?=7, or — 4. 35 ^ 3. Find the values of a; in the equations 4a: — =46. X Ans. x—\% or — f. 2ii? X 4f. Find the values of x in the equation x^-\-— — .^=8+ 12 6 * J 11 Sep 5. Find the values of x in the equation 2a?^+8a?+7z= — 4 —^+197. Ans. a;=8, or — lly^. 8 6. Find the values of x in the equation - — _+7-|=8. Ans. a?=:l^, or — |. c 7. Find the values of x in the equation 8— a; 2a?— 11 2 a?— 3 ^ . Ans. a;=6, or \. 6 8. Find the values of x in the equation 5a?-+4a?=273. Ans. x—1, or — 7|. PARTICULAR CASES OF AFFECTED aUADRATICS. 331. It is evident that every equation of the form a?'"+2^a?"=9 r SECT. VII.] AFFECTED EQUATIONS. 215 may be solved by the preceding rules ; for, let y=x", and y^=x^j and the above equation will become f-\-2py=q; Completing the square, i/^-{-2py-^p^=:q-^p'^ ', Evolving and transposing - y= — p±: \/g-\-p*i Substituting the value of y - x'*= — p±Vq-j-p'; Evolving - - - . x=\/^ — p±Vq+p'' 332. Equations also occur in the form > I ar-\-2px^=q. 1 s Let y=ir", then y'=af* ; and substituting these values, y'-\-2py=q; Reducing - - - - y=—p±Vq-\-p^, Or ar=-p±Vq-\-p^j Involving to the nth. power - xz=(—p±y/q-\-p^y. These equations may be readily solved without the for- mality of substitution. Resume the equation a^-\-2px''=q ; Completing the square, af''-\-2px^-{-p^=q-{-p'^ ; Evolving - - - . af'-\-p=±'^q-\-p^ ' Transposing- - - - JL'^=—p±\/q-\-p'; Evolving - - - - x= \/ —p ± ^q-\-f* Resume, also, the equation s t a^-\-2po^z^q ; 3 1 Completing the square, tj^-^2px''-\-p*=.q-\-p^ \ 1 Evolving - - - - a?^-(-^=dr\/^ -!-/>'; Transposing- - - - x'^zzL—p^s/q-^-p^'^ Involving - - - - x—{^—p^s/q-\-p^y, 333. The same principles will apply also to all equations in which there are two terms, simple or compound, and the 216 ELEMENTS OF ALGEBRA. [sECT. VII. exponent of one is double that of the other. Thus, in the equation Letting yz=zx'^-{-2px-\-q^ and f=^{x^-\-'Zpx-\-qy, we shall have then - - y^-{-y=:q', And . - - - 'yz=-^±^//-fiJ Whence - ^-^';ipx-\-q=-^±Vq'+}, And ... - x=-p±\/p^q-i±Vq'+ly Or, in the equation 'W (ax + 2by—2p(ax-{-2b) = q, Letting y='ax-\-2b^ and y'^=(ax-^2bf^ we shall have y^-2py=^q, And - - . . y—p±y/q-^p2- Whence - - ax-{-2b=p±Vq+p'^, And ... - x=P±^l±Pl:i^. a These equations may also be solved without the formality of substitution. 334. If the indeterminate quantity y=0, the affected qua- dratic will assume the form x''±2px=0. This equation may be readily solved ; for, dividing both members by x^ we have x±2p = j Transposing - - x—:p2p. 335. Equations involving more than one unknown quan- tity, as xY-\-^pxy=q, or (a?"+y")^+2p(a?"4-y")=5', if the ex- ponent of one term is double that of the other, may also be reduced to simpler forms by completing the square and per- forming the necessary transformations. 336. When there are two unknown quantities similarly involved in the equations, the work may be simplified by the introduction of two additional symbols which shall rep- SECT. VII.] AFFECTED EQUATIONS. 217 resent known functions of the unknown quantities. Thus, in the equations Let x-{-y=2s, and x — y=2z ; then x=s+z, and y=s — z. And - x'=(s-{-zy=s'-^2sz-^z' ; And - y^z=:(s—zy=s^—2sz + 2^'y Adding - - x'-\-f=2s^-\-2z' j Consequently, bz=2s^-{-22^ j Transposing and > ^^^2^^ ^^^ z=±./^^; reducing - j 2 V 2 Hence - - j=^+-v/-Z_, and y=s—y/ ~~ ; ^ , /b—4>, or ]a^ j , /^— 4-, or ia* Or T=ia-y^ V"^' and y=ia— y/ V~^* Similar operations will reduce any equations of the same form ; As, x + y=a, and a:' + y*=J; or x-\-y=ay and x*-{-y*=b ; or X'{-y=ay and j^-f y^=Z>, &c. 337. Again, let us take the equations X -\-y =ai y ^ Clearing the second equa- > , ,_, tion of fractions - > y — y' Let x-\-y=2sy and x — y=2z, as before; then x=s + z, and y=s—z. Whence - a^={8+zY=s^-^2sz-\-:^; And - - f=(s—zYz=zs''—2sz-\-z'. Adding, x'-\-f=2s^-\-22^=b(s-\-z).(8—z)z=b{s'^2^) = bs'-bs^. Whence - 2r= __ : or 2:=i * 7^-; — —J 2-\-b ' V *+2 Or, substituting and reducing, ,=,± v/ip2, and y=,T ,/]pE. V *+2 * ^V i+2 19 Ee 218 ELEMENTS OF ALGEBRA. [SECT. VII. Restoring the value of s and 5^, Similar operations will reduce any equations of the same form, As, x-\-y—a^ and —+^=5, &c. y X There are a variety of expedients by which complicate equations may be simplified. The above cases will indicate some of the most general. Others must be left to exercise the skill and ingenuity of the learner. AFFECTED EaUATIONS INVOLVING ONLY ONE UN- KNOWN aUANTITY. 35 3^ 1. Find the values of x in the equation 6a7+ =4)4<. X Clearing of fractions - - 6a?^4-35 — 3a?=44a?; Transposing Qx^ — 47a?= — 35 ; Completing the square and reducing - x=l, or |. 3^, 3 2. Find the values of x in the equation 5a? — =2a?4- X — 3 ^^—^, Ans, a?z=4, or —1. 2 3j, 2 2a? 2 3. Find the values of x in the equation — +__=a?4- — _ — . 2 da? 3 Ans. a?n:2±2v/2. 3^ 10 4. Find the values of a? in the equation 3a? — _ — ^^_ = 2-f- \) — ZiX 5^i5. ^;i5. a?=lli, or4.. 2a?— 1 5. Find the values of a? in the equation a?^ — 4a:'' ==621. Ans. a?n:3, or ^—23. 6. Find the values of x in the equation —=—__. 2 4 32 Ans. x=:^~\^^'\^^^ll. 2. i 7. Find the values of a? in the equation 2a?3 + 3a?3z=2. Ans. a;=|, or — 8. SECT. VII.] AFFECTED EQUATIONS. 219' 8. Find the values of x in the equation a:*-|-a:*=756. Jlns. j:=243, or (—28)^ 9. Find the values of x in the equation (lO-hx)' — (10-f-x)* =2. jJns. x=6. 10. Find the values of x in the equation 2(1 -fa? — a?*) — (l+i— x»)^=— |. J^ns. x=i-\-lV'U, 11. Find the values of x in the equation v^ar* — a*=a? — b. Jins. x=-± / . 2 / 126 12. Find the values of ar in the equation 2Vx—a-\-3y/2x 7a-f-5x = — =. *d7is, x=9a. y/x — a Ajr 5 2x 7 13. Find the values of x in the equation — i 1= X 3a:+7 ?^-. jJns. x=2. 13x q /I 14. Find the values of a? in the equation -+— = —, ' J3ns. x=z3. 5x AFFECTED EaUATIONS INVOLVING TWO OR MORE UNKNOWN aUANTITlES. 1. Given And - ^t^^^ + 2^r ''Jtofindxandy. x'-f. xy + f=133S Dividintr the second by the first / , — . _ ._ . ° } x—Vxy-\-y= 7; (3.) equation - - . -^ ^ ^^^ » ^ ' Adding the first and third equa- / Oj:4-2v= 26 (4> ) tions \ ^ > V •; Or x-{-y= 13; (5.) Substituting, in the first equation, Vary 4-13= 19; (6.) Whence ^1^= 6, (7.) And xy= 36; (8.) Multiplying by 3 - - - - 3a?y=108; (9.) Subtractinsf the ninth from the / « « a ok /ia\ ,^. > a^'-'Xxy^f= 25; (10.) second equation - - •\ 220 ELEMENTS OF ALGEBRA. [sECT. VII. Evolving ------ X — y=d=5. (11.) Adding the eleventh and fifth; 2a:::.13±5 = 18, or 8; equations - - - -\ Whence a?=9, or 4. Subtracting the eleventh from; 2^=13^5 =8, or 18; the fifth equation - -* - ) Whence 2/=^> ^^ ^- 2. Given - x^^x^y^ 18-/ > ^^^^^ ^ ^^^ ^_ And - - - xy— 6 \ Transposing in the first; ^2^3,2^^^^^ jg . (3.) equation - - ) Multiplying the second) _ _ 2a^y.= 12; (4.) equation by 2 - ) Adding third and; ^j^^^j^fj^^j^y^^^^ (5.) fourth equations \ Or- - - - (a? + 2/)^+(a7+2/)=30; (6.) Completing the / (^^y)2^(^+2^) + i = 3o+i=.A^5 (7.) square - ) ^ Evolving - - - a7+2/-hi=±Y' ^'^ And - ^+y=±n_|=.:^or-^=5,or-6; (9.) Whence, from the first } x^^f=\% or 24 ; (10.) equation - - ) Subtracting the 4th from ) ^._2^^_^y2^i^ ^^ 12 ; (11.) the 10th equation ) Evolving - ^— y = ±l, or±x/12=:±273; (12.) Adding 13th and ; 2a^=5±l, or — 6±2v/3 ; 9th equations > Whence - - - a^=3 or 2, or — 3± n/3 ; Subtract'g 13thfrom) 2y^4 or 6, or — 6:f 2>/3 ; 9th equation \ Whence - - - 2/=^ ^^ ^' ^^ — 3:F V3. 3. Given - ■ ^xy^m-x^f \ ^^ ^^^ ^ ^^^ ^^ And - - x-\-y— 6 \ ^^ Ans, a:=4 or 2, or 3±v/21 j and 2(=2 or 4, or 3=f\/21. SECT. VII.] 4. Given Aud 5. Given And 6. Given And AFFECTED EQUATIONS. 22! 7. Given And > to find X and y. x-fy = S jJns. x=5 or 3, and y=3 or 5. ..""«., > to find X and y. Jlns, x—b or 2, and y=2 or 5. c .""-^-o > to find X and v. x'+y'=1056\ ^ *^ns. x=^ or 2, and y=2 or 4. xV = =2y^j to find X and 8x3- ^ns. x=2744. or 8, and y=4 or 9604. PROBLEMS PRODUCING AFFECTED EaUATIONS. 1. It is required to divide the number 40 into two such parts that the sum of their squares shall be 818. Jlns. 23 and 17. 2. What two numbers are those whose difference is 9, and their sum, multiplied by the greater, produces 266 1 Ans. 14 and 5. 3. An officer would arrange 1200 men in a solid body, so that each rank may exceed each file by 59 men. How many must be placed in rank, and how many in filel ^ns. Rank 75, file 16 men. 4. Some bees had alighted upon a tree ; at one flight the square root of half of them went away; at another eight ninths of them ; two bees then remained. How many alight- ed on the tree 1 ^ns. 72. 5. A mercer bought a piece of silk for j£16 4*., and the number of shillings he paid per yard was to the number of yards as 4 to 9. How many yards did he buy, and what was the price per yard 1 »dns. 27 yards, at 12* per yard. 6. There is a field in the form of a rectangular parallelo- gram, whose length exceeds the breadth by 16 yard?, and it contains 960 square yards. Required the length and breadth. ^ns. Length 40, breadth 24 yards. 7. A person being asked his age, answered, " If you add 222 ELEMENTS OF ALGEBRA. [sECT. VII. the square root of it to half of it, and subtract 12 from the sum, there will remain nothing." What was his age 1 ^ns. 16. 8. What number is that which, if divided by the product of its digits, the quotient will be 2; but, if 27 be added to the number, the digits will be inverted "i Ans. 36. 9. Find two numbers such that their sum, their product, and the difference of their squares may all be equal to one another. ^?J5. ^±|v/5, and |±^-v/5. 10. A and B hired a pasture, into which A put 4 horses, and B as many as cost him I85. a week. Afterward B put in two additional horses, and found that he must pay 20*. a week. How many horses had B at first, and at what rate was the pasture hired \ Ans. B had 6 horses, and the pasture was hired at 5O5. per week. 11. A labourer dug two trenches, one of which was 6 yards longer than the other, for £>Yi I65., and the digging of each of them cost as many shillings per yard as there were yards in its length. What was the length of each \ Ans, 10 and 16 yards. 12. A and B set out from two towns which were distant from each other 247 miles, and travelled the direct road till they met. A went 9 miles a day, and the number of days at the end of which they met was greater by 3 than the number of miles which B went in a day. How many miles did each go ] Ans. A 117, and B 130 miles. 13. Two merchants each sold the same kind of stuff; the second sold 3 yards more of it than the first, and together they receive 35 crowns. The first said to the second, " I would have received 24 crowns for your stuff;" the other replied, "I would have received 12g crowns for yours." How many yards did each of them sell \ Ans. The first sold 15 or 5, the second 18 or 8. 14. A widow possessed $13,000, which she divided into two parts, and placed them at interest in such a manner that SECT. VII.] AFFECTED EQUATIONS. 223 the incomes from them were equal. If she had put out the* first portion at the same rate as the second, she would have drawn for this part $360 interest ; and if she had placed the second out at the same rate as the first, she would have drawn $490. What were the two rates of interest I ^ns. 7 and 6 per cent. 15. The sum of two numbers is 9, and the sum of their cubes 24-3. What are the numbers 1 j9ns. 3 and 6. 16. The sum of two numbers is 10, and the sum of their fourth powers is 1552. What are the numbers 1 jJns, 4 and 6. 17. The sum of two numbers is 7, and the sum of their fifth powers 3157. What are the numbers 1 ^ns. 5 and 2. 18. There are two square buildings that are paved with stones a foot square each. The side of one building ex- ceeds that of the other by 12 feet, and both their pavements together contain 2120 stones. What are the lengths of them separately ] J^ns, 26 and 38 feet. 19. A regiment of soldiers, consisting of 1066, formed into two squares, one of which has four men more in a side than the other. What number of men are in a side of each of the squares'? j^ns. 21 and 25. 20. The plate of a looking-glass is 18 inches by 12, and is to be framed with a frame of equal width, whose area is to be equal to that of the glass. Required the width of the frame. Ans. 3 inches. 21. A square courtyard has a rectangular gravel-walk round it. The side of the court wants two yards of being six times the width of the gravel walk, and the number of square yards in the walk exceeds the number of yards in the periphery of the court by 164. Required the area of the court. ^ns. 256 yards. 22. There are four towns in the order of the letters A, B, C, and D. The difference between the distances from A to B and from B to C is greater by four miles than the dis- 224 ELEMENTS OF ALGEBRA. [sECT. VII. tance from B to D. Also, the number of miles between B and D is equal to two thirds of the number between A and C ; and the number between A and B is to the number be- tween C and D as seven times the number between A and C is to 208. Required the respective distances. Ans. A B 42, B C 6, C D 26 miles. DISCUSSION OF THE GENERAL EaUATION OF THE SECOND DEGREE.* CASE I. 338. It has already been remarked, and we will now pro- ceed to demonstrate, that every affected equation of the sec- ond degree necessarily admits of two values for the unknown quantity, and only two. 339. Let us resume the first of the four forms of the af- fected quadratic (Art. 322). X'^-1J)X^q', (1.) Adding ;?^ to both members, Q^-\-^])x-\-f-—q-\-f-^ (2.) Or - - - - ■ -^ - {x^^y^q^f-, (3.) Let - - m^=q^f', (4.) Then {x-^-ff^m^-, (5.) Transposing - - - (ar+jp)^— ^'=0 ; (6.) Resolving into factors, (a? H-^+7w).(a?4-p — m)=:0 j (7.) Dividing by a?+pH-?w - - a>+^ — m — ^\ Transpasing - - - a?= — 'p-\-m^ox x— — 'p-^ \fq-\-'p^ '^. Dividincr the 7th equation / , . ^ by x-^-p — m - - ) Transposing - - - x= — p — m, or a?= — j) — \/ q-{-p^. Either of these values of x will answer the conditions of the equation. The same course of demonstration might be applied to the remaining three forms of the quadratic equation. Hence every affected equation of the second degree necessarily admits of two values of the unknown quantity^ and only two. CASE II. 340. We will now resume the results obtained in the four ^ « This discussion is substantially that of ]M. Bourdon. SECT. VII.] AFFECTED EQUATIONS. 225 preceding formulas, and enter into such an analysis of them with reference to the relative values of q and p as will de- termine the particular values of x. These results are (Art. 326), __ (1.) x=-^±Vqj^', I (3.) x=—p±V^ (2.) x=+p±Vq-\-p'', I (4.) X=+p±^p'-^. 1. Since the value of x in each equation is expressed by a rational term, with which a radical is connected by the sign db, in order that this value may be found, the quantity un- der the radical sign must be positive. As p^ is necessarily positive, the value of x may always be found in the first and second equations. If q-\-p^ is a perfect square, the exact value of x will be obtained ; if it is a ^urd, its approximate value. In the third and fourth equations, if qp\ the value of x will be imaginary, since it will involve the ex- traction of the square root of a negative quantity. 2. In the first and second equations, since P', the value of x will be negative ; but if g'^p^, the value of X will be imaginary. 4. In the fourth equation, since p> y/p^—q, if yf^ the value of a: will be imaginary. 5. If y=;>', the radical expression in the third and fourth equations will be reduced to 0, and the values of x will be, Ff 226 ELEMENTS OF ALGEBRA. [sECT. VII. In the third - - - - xz= — p^ In the fourth - - - - x=-\-p. 6. If q—0, the equation will assume the form x^±2px= ± ,• and, consequently, x= ± 2/?, or ± 0. 7. If jo=:0, the equation will assume the form x'^—ztq; and, consequently, x=±V±qi and the value of x will be imaginary in the third and fourth forms of the quadratic. 8. In the equation ax'^±2px=±q, if a—0, the equation will assume the form ±2px—dzq, or be reduced to a simple equation. CASE III. 341. In order to show why we obtain the imaginary re- sults in the third and fourth equations when q^p^, we will demonstrate that these equations, when q^p^, express con- ditions that are incompatible with each other. Kesume the equation X^ — 2pX=: q ; Reducing - - - x=p± y/p^ — q. Designating the first value of x by x', and the second value by x"j we shall have x'=p+Vp^—q; x"=p — Vp —q S Adding - x'-{-x"=zp+ ^ f—qj^p—>j f—q—^p. Hence the sum of the two values of x is equal to the coeffi- cient of the first power of the unknown quantity , taken with the contrary sign. Multiplying the above two equations, Hence the product of the two values of x is equal to the sec- ond member of the equation, taken with the contrary sign. Therefore, in the general equation, x^ — 2px=: — q, 2p is the sum of two numbers, of which q is the product. Now it has already been demonstrated, that if a quantity be re- solved into two factors, their product will be the greatest possible when the factors are equal (Art. 206). SECT. VII.] AFFECTED EQUATIONS. 227 Hence the conditions of the equation limit the value of q ; it may vary between the limits and p*, but can never be- come greater than (-^J =P^' If, then, we assign to y a value greater than the square of half 2p, the equation will express conditions which are in- compatible with each other, and, consequently, the value of X will be imaginary or impossible. Thus, Let it be required to divide 16 into two such parts that their product shall be 72. Let - - x= one of the parts, Then - 16— a?= the other; And, by the conditions of the problem, x.(16— x)z=72; Multiplying ... 16a:— a?^=72; Changing the signs - - o^ — 16x= — 72 j Completing the square, a^ — 16x+64=64 — 72= — 8; Evolving X — 8=db\/ — 8; Transposing - - - - a7=8± ■/— 8. Thus we obtain an imaginary result, which should be the case, as 16 can be divided into no two factors whose prod- uct shall be equal to 72; for, since 2/)=16, ;) will equal 8, and /?*:=64, which is the greatest possible product that can be formed of two numbers whose sum is 16. CASE IV. 344. We will now apply the principles exhibited in this discussion to a few problems, which will give rise to nearly all the circumstances that usually occur in equations of the second degree. First Problem. Find upon a line which joins two luminous bodies, A and Bjthe point where these bodies shine with equal intensity. J^ote. — The solution of this problem depends upon the following principle in physics, viz. : The intensity of light from the same luminous body will be, at different distances^ in the inverse ratio of the squares of the distances. 228 ELEMENTS OP ALGEBRA. [sECT. VII. This being" premised, in the indefinite line (1, 2) let A and B represent the respective position of the two lights, and C the point required. C A C B C 1 1 1 1 1 1 2 Let (z=:A B, the distance between the two lights, And b^=z the intensity of the light A at the unity distance, And c= the intensity of the light B at the unity distance. Let a?=r:A C, the distance from A to the point of equal intensity, Then a — a;=B C, the distance from B to the point of equal intensity. Then, by the above principle in physics, the intensity of A at the distance 1 being Z?, its intensity at the different distances 2, 3, 4 .... a?, will be -, -, — ...._, which last 4 9' 16 s^' term represents its intensity at C. In the same manner, it may be shown that the intensity of B at the distance a — a?, or at C, is equal to -. But the conditions of the ques- {a — xy tion require that their intensities be equal at C ; hence we have the equation b _ x^ {a — xj- Reducing - x=: ^\/- a^h Or, simplifying, X— ^ - — I ; — c B ut - l± 's/U= Vb(V'b ± v/c). And - - b—c=(V'bf—{V'^f={Vb-{-V~c).(V'b—y/~cy, 'whence. - .^-^^M^^±^, (Vb-j-VcXVb-Vc) Taking \/c in the numerator, minus, and dividing the equation by Vb — \/c, we have a\/b * (!)• V4+Vc' SECT. VIl] AFFECTED EQUATIONS. 289 But, taking y/c in the numerator, plus, and dividing the equation by v/5-fVc, we have Hence we also obtain a\/c (2). -"^'^ Discussion, I. Let i>c. 343. — 1. The first value of x, — :r- =, is positive, and less _ Vb-{-Vc _ Vb , , s/b than a ; for, — = ;= being a proper fraction, a . — ^= =>c, we have y/b-\-\/b=i2>/b^y/b-{-^c^ whence —= -=- Vb-\-\/c 1 "^b ay/b a _, >i, and, consequently, a . -^-^=^^^>-. Th.s, .n- deed, should be the case, since we have supposed the inten- sity of A greater than that of B, or J>c. 2. The corresponding value of a — a?, — = =, is also pos- Vb-\-y/C itive, and less than ^; for, since j:>-, aVb a Vb-^s^c 2 ay/b 3. The second value of «, —^- :r, is also positive, but >/b greater than a ; for, — = being an improper fraction, a . vb — v'c 20 230 ELEMENTS OF ALGEBRA. [sECT. VII. -^= --^=— = =:>a. This value of a?, therefore, gives the \/b — -v/c y/b — \/c point of equal intensity to the right of B, at C This should evidently be the case, since the light from A and B radiates in all directions. This point will, moreover, be nearer the body, the light of which is least intense. 4. The second value of a — a?, — = -^ is negative, as it -vb — \/c should be, since x^a ; and the point C is in the direction opposite from A. II. Let &\/Z>, the numerator and denominator will be af- fected with like signs, and, consequently, the value of the fraction will be plus. This result also indicates that C" should be to the left of A. III. Let b—c. aVb aVb aVb a 345.— L The firt value of x, ^/^»+Vc Vb+Vb 2Vb 2' 8BCT. VII.] AFFECTED EQUATIONS. 231 ay/c 2. The corresponding value of a — x, — = =^, also equals -. These two results give the middle point A B for the first required point, and this result conforms to the hypothesis. as/b 3. The second value of x, — = =, since v/6=\/c, willbe y/h \/c reduced to , which indicates that no finite value can be assigned to x, — as/c 4. The corresponding value of a — a:, — =; -^ will also be- come y/b—s/C — ay/c These results also agree with the hypothesis ; for, as the diflference of their intensity decreases, the second values of X and a~x increase, and, when that difference becomes in- finitely small, these values must become infinitely large. IV. Let i=c, and a=0. 346. The first values of x and a — x become 0, and the second — This last character is the symbol of indetermi- nation ; for, on returning to the equation of the problem, Q^c)x^—'labx=—a\ this equation becomes O.a:^— O.x==0, an equation which may be satisfied by any number whatever taken for x. And this agrees with the hypothesis ; for, if the bodies have the same intensity, and are placed at the same point, they will shine with equal light upon any point whatever in the line 1 — 2. V. Let a=0, and b and c unequal. 347. Each of the two values becomes in this case, which proves that on this hypothesis there can be but one point equally illuminated, and that is the point in which the two lights are placed. , 232 ELEMENTS OF ALGEBRA. [SECT. VII. Second Problem. 348. Find two such numbers that the difTerence of their products by the numbers a and h respectively may be equal to a given number \y, and the difference of their squares equal to another given number q. Let X and y represent the numbers sought ; Then - - - ax — h/=Sj , And - - - a^ — y^ —q. Reducing these two equations, we have for the two values of a?, w- of (2). - The two values a'-b' y are, (1). . bs+aVs'-q{a'-b') (2). - bs-aVs'-q(a'-b') Discussion. I. Let a>5. 349. In this case a^ — b^ will be positive ; therefore, in or- der that the values of x and y may be real, we must have q{a'-b')aV— a'9(a'-Z»') ; Or, transposing - bh^-\-a\{a^—b^)>a^s^ j, SECT. VII.] AFFECTED EQUATIONS. 233 Or, subtracting W from both sides of the inequation, Or, dividing - q>-. Thus, if a>-, and g< — — , the question is susceptible a* a* — b^ of a real and direct solution, and will give positive values of y. But, if ?<-j, and y< ^ , ^, the value of y will be nega- tive ; and we shall not obtain a solution of the problem in the sense in which it was enunciated, but of an analogous problem, the equations of which are ax-\-byz=:s, and which differ from the proposed equations in this respect only, that s will express the arithmetical sum instead of the difference of their products. II. Let a<6. 350. In this case a^ — b^ will be negative, and the values of X and y may be put under the form. (!)• ■bVs'+q(b'—a') . To 5 ~~ ' m ^_ ^as-\-bVs'+q{b^^a^) (2). . X ^ ^-^ bs-a^^+q{b^-^) . (i;. y jr^^s y (2). . y= ^ ^, _—hs + ax/s^-\-q{b^—a!') b'—a' The values of x and y, it is evident, will be real, since the quantity placed under the radical is essentially positive. Their first values will be negative. The second value of y may be either positive or negative. s^ In order that it may be positive, we must have q^-^* III. Let a=b, 3.t1. In this case a' — 6*=0, and the first values of x and y will be Go 234 ELEMENTS OF ALGEBRA. [sECT. VIII, x=^, and y=l^-^. The second values of a? and y will be X—-. and y=-. But, if we solve the given equations on the hypothesis that a=^, we shall have a:=— iJ — , and v=:— ^— — The preceding discussions show the precision with which the algebraic results correspond to all the circumstances of the enunciation of a problem. SECTION VIIL Ratioy PropQrtion, and Progression RATIO. 352. By Ratio is meant the relation which one quantity- bears to another with respect to magnitude. The quantities compared must be of the same nature, so as to admit of a common measuring unit. Thus, we compare dollars with dollars, length with length, weight with weight, time with time, &c. 353. The magnitudes of quantities may be compared in two ways. 1. With regard to their difference. This is called jUrith- metical ratio, or ratio by difference. 2. With regard to the number of times one quantity is contained in the other. This is called Geometrical ratioj or ratio by quotient. The Arithmetical ratio of two numbers, as a and bj is ex- pressed, a — hy or a . .b. SECT. VIII.] RATIO. 235 The Geometrical ratio of two numbers, us a and i, is ex- pressed, aiby or -. When the ratio is thus expressed, the first term is called the antecedent^ the last term the consequent, and the two terms, takin together, are called a couplet, 354-. The term arithmetical ratio is only a substitute for the word difference, and involves no principle that is not es- sentially involved in algebraic subtraction. 355. In a geometrical ratio three things are involved, viz., the antecedent, the consequent, and the ratio ; and any two of these being given, the other may be found. Let a= antecedent, c= consequent, and r= ratio : Then, from the geometrical ratio a: c=r, we have r=-, i, e., ratio = antecedent -r- by consequent ; And, a=c . r, i. e., antecedent = consequent x by ratio j And, c=-, i. c, consequent = antecedent -i- by ratio. T 356. When the antecedent is equal to the consequent, the ratio is a unit, or a ratio of equality. When the antecedent is greater than the consequent, the ratio is greater than a unit, or a ratio of greater inequality. When the antecedent is less than the consequent, the ratio is less than a unit, or a ratio of less inequality. A compound ratio is the ratio formed by multiplying the corresponding terms of two or more ratios. A duplicate ratio is the ratio of the squares of the corre- sponding terms of a ratio ; the triplicate ratio, of the cubes of the corresponding terms j the sub-duplicate, of the square roots of the corresponding terms, &c. 357. The ratio, it will be observed, is expressed by a frac- tion, the antecedent becoming the numerator and the conse- quent the denominator. Now it has been demonstrated that, if both terms of the fraction be multiplied or divided by the 236 ELEMENTS OF ALGEBRA. [SECT. VIII. same number, the value of the fraction will not be affected. Hence we infer, 1. If the terms of a ratio he multiplied or divided hy the same number, it does not alter the value of the ratio. 2. j1 ratio may be reduced to its lowest terms by dividing its antecedent and consequent by their greatest common measure. 3. Ratios may be compared with each other by reducing the fractions which represent their values to equivalent fractions having a common denominator. 358. The following are some of the more important theo- rems relating to ratios : 1. A ratio o{ greater inequality is diminished, and a ratio of lesser inequality is increased, by adding the same quantity to both members. First. Let a+bia, or ^^^—-, represent a ratio of greater a inequality : Adding x to both terms - a-\-b-i-x:a-^x, or —^ j a-\-x Then - - - - a-\-b:aya^b-\-x: a-{-Xy Or - - - - - a+b a-\-b -{-x ^ a a-^x For, reducing to a ? a'^-\-ab-\-ax-^bx^a^-{-ab-{-ax common denom. > a{a-\-x) a(a-\-x) Second. Let a — b : a, or , represent a ratio of lesser a inequality : fi h I /P Adding x to bath terms - a — b-\-x : a-\-x, or J— ; a-\-x Then . - - > a — b:a a^-j-ax — ab — bx^a^ — ah-\-ax \ a( common denom. ) a(a-f ^) a{a-\-x^ 2. A ratio of greater inequality is increased:; and a ratio of lesser inequality is diminished, by subtracting the same quan- tity from both terms. SECT. VIII.] RATIO. 237 First. Let a+h:ay or y^, represent a ratio of greater a inequality : Subtracting X from both terms, a+h—X'.a-^x^ox—l- ; Then - - - - a-\-hia a{a — x) a{a — x) 3. A ratio of greater inequality compounded with another ratio increases it ; but a ratio of lesser inequality compound- ed with another ratio diminishes it. First. Let a+bia^ or ^— -, represent a ratio of greater inequality, And - - - - TO : », or -, be any other ratio : n Compounding - - am-{-bm : an^ or ^ "^ _ j an Then ... m:n j n an ■ry J • amn^ amn — hmn For, reducing - - — - > avr ainr 4. If to the terms of any couplet there he added two other terms having the same ratio, the sums will have the same ratio. Let the ratio a : b equal the ratio c:d: Then - W' • ' r-^,=T=-, 5 r b+d b d For, since - - - - a : 6=c : c?. We have ----- -=-. ; b d Clearing of fractions - - - adz=zbc ; Adding cd to both members, ad-\-cd—hc-\-cd; Resolving into factors - d{a-\-c)=:c{b-{-d)y Dividing - - - - -^_= = . b+d d b 5. If from the terms of any couplet two other quantities hav-^ ing the same ratio be subtracted, the remainders will have the same ratio. Let the ratio a : b= the ratio c: d: Then . - - - - For, since - - - - - "We have a- a- ~c -d" a c '-b—d' a : 5= -.c:d, a_ b~ c ad. .Ic; Clearing of fractions Subtracting cd from both members, ad — cd=bc — cd; Resolving into factors - - c?(a — c)=c(b — d); TV' !• a — c c a SECT. VIII.] PROPORTION. , 239 EXAMPLES. 1. Reduce the ratios 360 : 315, and 1595 : 667, to their low- est terms. 2. Which is the greater of the two ratios, 11:9, and 44:35] 3. Which is the least of the three ratios, 20 : 17, 22 : 18, and 25 : 23 1 4. If the consequent be 35, and the antecedent 985, what is the ratio 1 5. If the antecedent be 1512, and the ratio 12, what is the consequent 1 6. If the consequent be 320, and the ratio i, what is the antecedent 1 7. What is the compound ratio of 12 : 21, 18 : 6, and 24 : 5 1 8. What kind of a ratio will be produced by compounding 5x4-7 : 2«— 3, and x-\-2 : ix-{-3 1 9. What kind of a ratio will be produced by compounding c^—3^ : a', a-\-x : h, and b : a—x ? 10. What kind of a ratio will be produced by compound- ing x-\-y : a, x—y : ft, and b : — H^ % 11. What is the ratio produced by compounding 3 : 7, the duplicate ratio of 3 : 5, and the triplicate ratio of 4 : 3 1 12. What is the ratio produced by compounding the sub- duplicate ratio of 49 : 4, and the sub-triplicate ratio of 64 : 1251 PROPORTION. 359. Ratio is a comparison of two quantities to ascertain their difference, or how often one is contained in the other. Proportion is a comparison of two equal ratios. If the ratios are arithmetical, the proportion is called arithmetical proportion^ or proportion by difference. If the ratios are geometrical, the proportion is called geometrical proportion^ or proportion by quotient. 360. There are always two couplets, or four terms, in a proportion. The first and fourth terms are called extremes ; 240 ELEMENTS OF ALGEBRA. [sECT. VIII. the second and third, means. The two antecedents, or the two consequents taken together, are called homologous terms. The terms of the same couplet are called, with reference to the proportion, analogous terms. Three terms are said to he proportional when the ratio formed by the first and second is equal to the ratio formed by the second and third. 361. As an arithmetical proportion, or a proportion by dif- ference^ is nothing more than a simple form of equation, it is unnecessary to give the subject a separate consideration. It is expressed a— 6— c— c/, or a . . &=c . . d. 362. Geometrical proportion is expressed by * a:b=zc:d, Or - a:b::c:dj which expressions are read, " a to b equals c to c?," or, " a is to & as c to d" THEOREMS RELATING TO PROPORTION. 363. — (1.) If four numbers be proportional, the product of the extremes will be equal to the product of the means. Let - - - - a:b'.',c:d;f Then, by equality of ratios, _=- j (i •Clearing of fractions - • ad=:zbc. CoR. 1. Any factor may be transferred from one mean or extreme to the other without destroying the proportion. Thus, \{ a:b:: cmidn, then an:bm::c:d. CoR. 2. If any three terms of a proportion be given, the fourth can always be ascertained ; for if a : & : ; c : cZ, Then - - ad=cb ; cb " Dividing by d, a=— , i. c, the 1st termr=2dx3d-^4th; d Dividing by c, b=—, i. e., the 2d term = 1st x 4th -^ 3d ; c Dividing by b, c=—, i. e., the 3d term=:lstx4th-f-2d ; b Dividing by a, d=z^—, i. e., the 4th term=2dx 3d-T-lst. a 8«CT. VIII.] PROPORTION. 241 364. — (2.) If tkt product of any two numbers he equal to the product of two others^ these four numbers will constitute a pro- portion when so arranged that th^f actors of one product be made the meanSy and the factors of the other product the extremes. Let ad=bc: Dividing by db, and reducing, -=- j b d Hence - - - - a:b::c:d, 365. — (3.) If three numbers be propoitional, the product of the two extremes is equal to the square of the mean. Let - - - - a:b::b:c : Then ?=*, b c Or acz^lr". Cor. The mean proportional, or geometrical mean, be- tween two numbers is equal to the square root of their product. Thus, if ac^b"^, then bz=y/ac. 366. — (4.) If four numbers be proportional^ 1. the order of the extremes^ 2. of the means, 3. of the terms of each couplet, 4. of the couplets, 5. of all the terms, may be inverted withoitt cfe- stroying the proportion. Let . - - - a'.b'.'.c'.d: Then .... t='-. b a Dividing by a, and multiply ) d^c^ ,-. d:b::c:a ; (1.) ing hy d - - - S b a Dividing by c, and multiply- ) a_b^ .-.aiciibid; (2.) ing by 6 ---)crf* Inverting the fractions - -=-» •'• h:a::d:c ; (3.) a c Inverting the order of the ) c_a . - . ^ . . . i . /a\ members • - - }d V Inverting the fractions and j , , changing the order of the > ~~~, .'• d:c::b:a, (5.) terms - - - - ) 367. — (5.) If four numbers be proportional, the ^dnalogous 21 Hh 242 ELEMENTS OP ALGEBRA. [sECT. VIII. or the Homologous terms may be multiplied or divided hy the same number without destroying the proportion. Let - - - - a:b::c',d: Then "L=t b d Multiplying the terms of the } am c , , n r • X { -,— = -,» ''' am : bm : : c : d : nrst traction by m - - S bm d Multiplying the terms of the } a cm , , ^•^ ° [ -=—,.'.a:b::cm:dm; second traction oy m - ) b dm am cm Multiplying the equation hy wz, ==: , ,-, am : b : : cm: d ; b d Dividing the equation by m, J^=_^, .'.a : bm : : c :dm ; bm dm Dividing the terms of the first 1 -^ c a b fraction hy m - - - \ ^~d^ ' ' m' m c : d ; Dividing the terms of the sec- ) ^ ^ c d i 6 1' • mm J m end fraction by m a , c Dividing the numerators hy m - 21= ul, »'* — i b : : — : dj b d m ' m Dividing the denominators by w, x^T") •*.«:— : : c : _. mm ^ m 368. — (6.) If there be two sets of proportions having an ante- cedent and consequent in the one equal to an antecedent and con- sequent of the other, the remaining terms will be proportional. Let - - - - a : :b::c:d, And - a : b : : m: n: By the first proportion - - a c ■b-r By the second - a m Therefore - - f =^, and c : rf d n : n. 369.— (7.) If two homologous or two analogous terms be added to or subtracted from the two others, the proportion will be pre- served. SECT. VIII.] PROPORTION. 243 FirBt, Let - - - aib i\ ci d^ r\ O C ^' • • ■ - i=r Then (by Art. 358, th. 5), i±^=?=^ j bztd b d Hence - - - a-\-c : b-\-d :: a : b^ or as c : d, And - - - a — c : b — d :: a : b^ or as c : d. Second, Inverting the order of the means, a : c : : b : d^ Then - a±b a b . c±d c d* Hence - a-\-b : c-^d :: a : c, or as b : d^ And - a—b : c—d : : a : c^ or as b : d. Cor. 1. Since a±i:_a , a-f c a — c . b±d b b-\-d b—d' Hence we have a + c : a—c : : b-\-d : b — d. Cor. 2. Since a±b_a , a-\-b_a — A. c±d b c-\-d c—d' Hence we have a-{-b : a — b : : c-\-d : c — d. 370. — (8.) If two sets of proportional numbers be multiplied, the products of the corresponding terms will be proportional. Let - - - a : b : : c : dy or -=-=. b d And - - - m : n : : p lOyOT — =?■ : n q Multiplying the two equations - ^=^ ; bn dq Hence - - - - am : bn : : cp : dq. 37 L — (9.) If one set of proportional numbers be divided by the corresponding terms of another set, the quotients will be pro- portional. Let - - a : ^ : : c : (/, or ^=-p b d And - mm: : p : q, or - =?■ : n q Dividing the first equation by ) a . m c^p the second - - - } b n d q* T ^' m n p q 244 ELEMENTS OF ALGEBRA. [sECT. VIII. 372. — (10.) If four numlexs he proportional^ like powers or like roots of them will be proportional. Let - - a : h : : c : d: Then a_c , b~d' Involving - - — =-t » •*• a"" : Jf" : : c^ : d"" / ^ (1 v/c — — Evolving - - -==-=, .-. ^a: :yb:: V'c: ^d. Vb \/a PROBLEMS TO BE SOLVED BY PROPORTION. 1. There are two numbers whose difference is to the l«ss as 100 is to the greater, and the same difference is to the greater as 4 to the less.- What are the two numbers 1 Let x= the greater, and y= the less : Then - - - - - x — y : y ' And . - - - . cc — y : x Multiplying the proportions, (x — yY : xy Dividing consequents - (x — yY : 1 Evolving ... - X — y : 1 Converting into an equation - x — y=20 ; Whence - - - - x—^0-\-y; Substituting this value of x ) ^O+y-y :y:: 100 : 20-f y, in the first proportion ) Or - - - - 20 : y : : 100 : 20 + y; Dividing antecedents - 1 : y : : 5 : 20+y ; Convertiag into an equation - 5y=20+2/ ; Whence . - - - y=5, And a?=20 + 5 = 25. 2. The product of two numbers is 15, and the sum of their squares is to the difference of their squares as 17 to 8. What are the numbers 1 Let x= the greater, and y= the less : 100 X, 4 y; 400 xy; 400 1; 20 ij Then . xy—16, And . x^-\-f : x'-y' : r 17: 8; Adding y and subtracting - 2x' : 2f : : 25 : 9 ;. 4 SECT. VI [1.1 PROPORTION. 246 Dividing first couplet by 2 Evolving - - - Whence - - . Reducing - - - - x' : y« : : 25 : 9 ; - X : y : : 5:3; ar=5y ; a:=5, and y=3. 3. What two numbers are those whose product is 320, and the difference of their cubes is to the cube of their dif- ference as 61 to 11 Let xz=i the greater, and y= the less : Then a?y=z32a, And ----- x^—y^ : (x—yY : Expanding 2d term, a?*— y* : x^—3j^y-\-3xy^—y^ : Subtracting consequents ^02 from antecedents - ) ^ Dividing first ratio by x— y - Dividing antecedents by 3 Substituting value of Jry - Dividing antecedents by 20 - Evolving - - - - Converting into an equation - Whence - - - - 4. It is required to prove that a : x : : V2a — y : v/y^ on the supposition that (a-fx)* : (a—xf : : x-f-y : x— y. Expanding first and 2d terms, •Adding and subtracting, 2o'4-2x* : 4ax : Dividing terms - - a^-\- x^ : 2ax : Transferring the factor x, o'-f- x* : 2a : Inverting means - - a*-f- x* : x* ; -3xy» : (x-yf : 3xy : (x-yf : xy : (x—yY : 320 : (x-yY : 16 : {x-yf : 4 : X — y : X— y=:4 ; - x=20, and y :61. ■• J : 61 • ^ 5 :60 ■'■ 5 :60 ■^ > : 20 1 y :20 1; : 1 1 5 : 1 : * J 16. '' i a2+2ax+x« : a*— 2ax-|-x* : : x+y : x— y; 2x : 2y ; a':x» a : X X :y; x^ry; 2a:y; 2a— y:y; Subtracting terms - Evolving - - - 5. It is required to prove that dx—cy^ on the supposition that X : y : : a^ : b\ and a : 1 V2a—y:y/y, Vc+x ^d+y. Inverting the order of the ratios > _, ,, m the first proportion - 3 Involving second proportion - a' : d' : : c-\-x : d+y ; By equality of ratios - - x :y : : c-\-x : d-\-y; 246 ELEMENTS OF ALGEBRA. [&ECT. VIII. Inverting means -* - - x : c-\-x : : y : (i-\-y ; Subtracting terms - - - x : c : : y : d ^ Converting into an equation - dx=:cy. 6. There are two numbers whose product is 24, and the difference of their cubes : the cube of their difference as 1& : 1. What are the numbers 1 j^ns. 6 and 4. 7. The sum of two numbers is to their difference as 3 : 1, and the difference of their third powers is 56. What are the numbers % Ans. 4 and 2. 8. There are two numbers whose product is 135, and the difference of their squares is to the square of their differ- ence as 4 to 1. What are the numbers'? Ans. 15 and 9. 9. There are two numbers which are to each other in the duplicate ratio of 4 to 3, and 24 is a mean proportional be- tween them. What are the numbers % Ans. 32 and 18. 10. Tl^ere are two numbers which are to each other as 3 to 2. If 6 be added to the greater and subtracted from the less, the sum will be to the remainder as 3 to 1. What are the numbers'? Ans. 24 and 16. 11. What number is that to which if 3, 8, and 17 be sev- erally added, the first sum will be to the second as the sec- ond to the third 1 Ans. 3i. 12. The sum of the third powers of two numbers is to the difference of the third powers as 559 to 127, and the square of the first, multiplied by the second, is equal to 294. What are the numbers % Ans. 7 and 6. ARITHMETICAL PROGRESSION. 373. A series of numbers increasing or decreasing by a constant difference, is called an arithmetical progression^ or progression by difference. 374. When the numbers increase by a common differ- ence, they form an ascending series / when they decrease, a descending series. Thus, the natural numbers, 1, 2, 3, 4, 5, 6, 7, 8, &c., ' form an ascending series. 2. Transposing, &c., a=l — {n — 1)(/, i BECT. VIII.] ARITHMETICAL PROGRESSION. 247 Inverted, they form a descending series j as, 8, 7, 6, 5, 4, 3, 2, 1. 375. From the definition of arithmetical progression, it is evident that in an ascending series each term is found hy adding the common difference to the preceding term. Let a= first term, (f= common difference, and n= the number of terms : Then the terms of the series will be 12 3 4 5 n c, a+d, a4-2J, a-f 3(f, a-\-4>d a-\-(n^l)d. Hence, letting /= the last term, we shall have, - I I / i\ ; ? the formula for the 1. - - - lz=za-{-(n — l}a, \ ^ ' > last term. the formula for the first term. 3. Transposing and > ,_/ — a > the formula for the dividing - - ) n — l' ) common diff. . rr . . / — a . - > the formula for the 4. Transposing, &c., n = — r-+l, J , ^ d 3 number of terms. These four formulas may be enunciated as follows ; 1. The last term is equal to the first term^ plus the common difference multiplied by the number of terms less one. 2. The first term is equal to the last term, minus the common difference multiplied by the number of terms less one. 3. The common difference is equal to the difference between the extremes divided by the number of terms less one. 4. The number of terms is equal to the difference between the extremes divided by the common difference, the quotient increased hy one. 376. If the series is descending, the above formulas wijl evidently become, 1. - l=a—(n—l)d; 2. - a=t+(n—l)d; n — 1 a •248 ELEMENTS OF ALGEBRA. [sECT. VIII. 377. If the common difference and first term are equal, l=za-{-(n — l)d=a-{-(n — l)a=a-\-an — a=any Or, l=a — (n — l)c?=a — (n — l)a=a — an-\-a — 2a—an. 378. From the third formula, d= ^, we may obtain a n — 1 general method for finding any number of arithmetical means between two given numbers. To do this, it is only necessary to obtain, in addition to the given data, the com- mon difl^erence. Let m~ the number of means. Then, since the whole number of terms consists of two extremes, plus the means, we shall have m-\-'2.=n. Hence, substituting for n its value in the above formula,, 1 / — a / — 'a ~m^-2—l~m-^l PROBLEMS FOR SOLUTION. 1. The first term of an arithmetical progression is 50, and the common difference 10. What is the 100th term 1 Ans, /=a-h(7i—l)c/3=:50+(100—l). 10=1040. 2. The first term of an arithmetical series is 120, and the common difference 2. What is the 325th term ] Jlns. 3. The first term of an arithmetical series is 2, the last term 1828, and the number of terms 42. What is the com- mon difference 'X Ans. 4. The last term of an arithrhetical series is 2680, the common difference 5, and the number of terms 30. What is the first term 1 Ans, 5. The first term of an arithmetical series is 8, the last term 1728, and the common difference 2. What is the number of terms 1 ' Ans. 6. The first term of a decreasing arithmetical series is 800, the number of terms 21, and the common difference 2. What is the last term % Ans, 7. Find 4 arithmetical means between 2 and 52. Ans, ci=10 J and the series, 2, 12, 22, 32, 42, 52. SECT. VIII.] ARITHMETICAL PROGRESSION. 349 8. The first term of a descending arithmetical series is 480, the last term 12, and the number of terms 42. What is the common difference \ Ans. 9. Find 8 arithmetical means between 12 and 52920. Ans. d= , and the series 10. The first term m a descending arithmetical series is 46450, the last term 10, and the common difference 2. What is the number of terms ] Ans. SUM OP THE SERIES. 379. The sum of the series may evidently be obtained by the addition of all the terms, nor will this sum be affected if the order of the terms be inverted. Thus, S=[a ]-h[a + (iJ-h[a+2c/] + [a4-3c/]-f ■ - - +[«+ ^n^4,)d] + [a-\-{n^3)d)-\-[a-^{n—2)d']-\-[a-\-(n—l)d]i S=[a-f.(n— l)rf] + [a-f(»-2)(i]-h[a+(«— 3yj + [a+(/i— 4)(/]-f . - . 4- [a-h3(f] + [a-f2(i] + [a-}-rf ]-f [a ]. Adding the two equations, 2S=[2a-|-(»— l)(/]-h[2a+(»— l)(i]4-[2a-h(«— iy] + [2a + {n—l)d]+ . . - -f [.2a+(;i-l)c/] + [2a+(n— l)c/}+[2 . a-^(^ri—l)d] + [2a + {n—l)d]. But, since there are n terms, and all the terms are equa>, 2S=[2a+{n-i)d]n=. Hence, by performing the necessary reductions, 1 o_2a4-(» — l)<^w ) ti^e formula for the sum of the 2 ) terms. 2. a= , the formula for the first term. 2/1 3 d=.^~l , the formula for the common difference. y/{2a —d)^-\-Hds—2a-\-d ^ the formula for the num- 'A 2d ) ber of terms. These four formulas may be enunciated in tho form of general propositions or rules. PROBLEMS TO BE SOLVED BY THE PRECEDING FORMULAS. 1. The first term of an arithmetical series is 5, the num. Ii *^0 ELEMENTS OF ALGEBRA. [sECT. VIII. ber of terms 30, and the common difference 3. What is the sum of all the terms 1 Ans. 1455, 2. The sum of the terms of an arithmetical series is 280, the first term 1, and the number of terms 32. What is the common difference 1 Ans. |. 3. The sum of the terms of an arithmetical series is 950, the common difference 3, and the number of terms 25. What is the first term % Ans. 2. 4. Suppose 100 balls be placed in a straight line, at the distance of a yard from each other ; how far must a person, starting from the box, travel to bring them one by one to a box placed at the distance of a yard from the first ball \ Ans. 5 miles and 1300 yards. 5. In gathering up a certain number of balls, placed on the ground in a straight line, at the distance of 2 yards from each other, the first being placed 2 yards from the box in which they were deposited, a man, starting from the box, travelled 11 miles and 840 yards. How many balls were there 1 Arts. 100. 6. How many strokes do the clocks of Venice, which go on to 24 o'clock, strike in a day \ Ans. 300. 7. In a descending arithmetical series the first term is 730, the common difference 2, and the last term 2. What is the number of terms 1 Ans. 365. 8. A speculator bought 47 house lots in a certain village, giving $10 for the first, $30 for the second, $50 for the third, and so on. What did he pay for the whole 47 1 Ans. $22,090. 9. A man bought a certain number of acres of land, pay- ing for the first $i, for the second $|, and so on. When he came to settle, he had to pay $3775. How many acres did he purchase, and how much did he give per acre 1 Ans. 150 acres, at $25^- per acre. 10. A wealthy gentleman offered to his daughter, on the evening of her marriage, $50,000 as her dowry ; or he would give her on that evening $1, on the next $2, and so on to SECT. VIII.] GEOMETRICAL PROGRESSION. 251 the end of the year, 365 days, and also the balance of inter- est that might be found in her favour if she accepted the lat- ter offer. The lady, being unskilled in mathematit;s, chose the first ofier. Did she gain or lose by this choice 1 Jlns. She lost $16,795. GEOMETRICAL PROGRESSION. / 380. If a series of numbers increase or decrease by the continued multiplication or division by the same number, they are said to be in Geometrical Progression,. 381. When the numbers increase by a common multiplier, they form an ascending geometrical series; and when they decrease by a common divisor, they form a descending geo- metrical series. The common multiplier or divisor is called the ratio. 382. In an ascending geometrical series, each succeeding term is found by multiplying the preceding term by the ratio. 383. The following symbols are used in geometrical pro- gression, viz. : a= first term, /= last term, n= number of terms, r= ratio, and S= sum of the terms. Using the above symbols, we have the series, 1 2 3 4. 5 71—4.. n— 3. n— 2. n—\. n. a, ar^ ar^^ ar^^ ar^ . - - . ar"^, af*, ar"^, ar^-\ ar'^'K Hence we shall have,* the formula for 3. Dividing, evolving, &c., r= (-)**") ( 1 l=L , , ^ , > the last term. 2. Dividing, &c. . . fl=_L, i th« f^^'""^^ f^' ^ r*-' > the first term. the formula for the ratio. These three formulas may be enunciated in the form of general propositions or rules. 384. If the series is descending, the above formulas may still be applied by taking r= to the reciprocal of the com- • The formula for the number of terms is solved by the aid of logarithms, and is, consequently, omitted in this place. 252 ELEMENTS OF ALGEBRA. [SECT. VIII. mon divisor ; for, multiplying by the reciproca,! of a number is the same as dividing by the number itself. In fact, when- ever r To'o J I o> 1 o> 1 OJ 1 o» '*''^*> Or, 1, 10-', 10-^ 10-^ 10-^, 10-^ 10-^, 10-', &c. Hence the corresponding arithmetical series, or the loga- rithms, are 0, —1, —2, —3, —4, —5, —6, —7, &c. 395. It is evident that the logarithms of 1, 10, 100, &c., being 0, 1, 2, &c., respectively, the logarithm of any number between 1 and 10 will be 0+ some decirtial parts ; that of a number between 10 and 100, 1+ some decinial parts j that of a number between 100 and 1000, 2+ some decimal parts, and so on for all the numbers falling between the successive terms of this progression.- It is also evident that the logarithms of y\, y^g^, j^q-q^ SECT. IX.] LOGARITHMS. 259 jjshz^ &c., or .1, .01, .001, .0001, &c., being —1, ~2, —3, — 4-, &c.,the logarithm of any number between 1 and ,1 will be — 1-h some positive decimal parts ; that of a number be- tween .1 and .01, — 2-f- some decimal parts; that of a num- ber between .01 and .001, — 3-|- some decimal parts, &c. 396. Between each two adjoining terms of both series in Art. 392 a term may be interpolated, and a new series of numbers, and logarithms will be produced, each consisting of double the number of terms. T4iis interpolation nruiy be effected by finding the geometrical mean (Art. 365, Cor.), or taking the square root of the product of the two terms in the geometrical series, and the arithmetical mean^ or half their sum, in the arithmetical series. The term interpola- ted between 1 and iO in the geometrical series would be >/lX 10=3,1622777; between 10 ©nd 100 would be v/lOx" 100=31,62280 ; the corresponding terms interpolated in the arithmetical series would be (0+l)-r2=,5, and (l-h2)-r2 = 1,5. ' The two series, then, would be 1, 3.162277, 10, 31.62280, 100, &c. 0, .5, 1, 1.5, 2, &c. These two series may again be interpolated as before, and so on continiKilly. The nulhber of terms in the two series will continually increase, and the differences between them continually decrease, with each succeeding interpolation* 397. To construct a table of logarithms, however, it is unnecessary to interpolate systematically throughout the series ; for, if the logarithm of some few of the prime num- bers be calculated, those of the composite numbers may be obtained by the process indicated in Art. 393. Indeed, these interpolations may be limited to any two adjoining terms in the series. Hence, the logarithm of any number, whele or fractional, between any two terms of the series in Art. 392, may be cal- culated by the following general 260 ELEMENTS OF ALGEBRA. [sECT. IX. RULE* 1. Find a geometrical mean between 1 and 10, 10 and 100, or any other two adjacent terms of the series between which the number proposed lies. Also^ between the mean thus found and the nearest extreme^ observing that the proposed number shall fall between the mean found and that extreme^ find another geometri- cal mean^ as before ; and so on, till you have arrived sufficiently near the number whose logarithm is sought. 2. Find as many arithmetical means between the correspond, ing terms of the arithmetical series 0, 1, 2, 3, <^c., in the same order as the geometrical means were found, and the last of these will be the logarithm of the proposed number EXAMPLES. • 1. Calculate the logarithm of 5. Here the proposed number lies between 1 and 10. Firsty then, the logarithm of 10 is 1, and the logarithm of 1 is 0. Then, (10 X 1)^ = 3.162277, which is the geometrical mean, And, (l + 0)-r-2— 1 = .5, which is the arithmetical mean. Hence, the logarithm of 3.162277 is .5. Secondly, the logarithm of 10 is 1, anr5^ 3x0.002316237 5x0.000055590 7x5' 7x0. 000001588 ''^ 9x5' 9x0.000000050 =0.301029995 =0.173717793 =0.002316237 =0.000055590 =0.000001588 =0.000000050 =0.000000002 (A.) (B.) (C.) (D.) (E.) (F.) 11x5* Whence the log. (2-hl)= log. 3 =0.477121255. The above logarithm is correct as far as to the ninth place of decimals. 2. Required the logarithm of 11. ^ns. 1.04139269. 400. The only numbers whose logarithms it will be found necessary to compute by the preceding formula, of by in- terpolating the series, are the prime numbers 3, 7, 11, 13, 264 ELEMENTS OF ALGEBRA. [sECT. IX. 17, 19, 23, 29, &;c. The logarithms of composite numbers may be computed by the propositions verified in Art. 393. COMPUTATION OF LOGARITHMIC TABLES. 401. The following table will exhibit the manner in which the logarithms of the natural series of numbers 1, 2, 3, 4, &c., to 30, may bccomputed: ' 4 Nos. Method of Computation. Logarithmi. 1. log. Izz: 0.00000000. 2. Since 10H-5 = 2,log. 10- log. 5= log. 2=0.30103000. 3. Computed by formula in Art. 399, log. 3 = 0.47712126. 4. Since 2x2=4, log. 2+ log. 2 = log. 4 = 0.60206000. 5. Computed by interpolating the > j 5=0 69897000. series in Art. 397 - 5 ' ^' 6. Since 2x3 = 6, log. 27f log. 3 = log. 6=0.77815125. 7. Computed by formula in Art. 399, log. 7 = 0.84509804. 8. Since 2X4=8, log. 2+ log. 4 = log! 8 = 0.90308999. 9. Since 3x3=9, log. 3+ log. 3 = log. 9 = 0.95424251. 10. :-..-. log. 10=1.00000000. 11. Computed by formula in Art. 399, log. 11 = 1.04139269. 12. Since 3x4=12, log. 3+ log. 4 = log. 12=1.07918125. 13. Computed by formula in Art. 396, log. 13=1.11394335. 14. Since2x7=14, log. 2+log.7 = log. 14=1.14612804. 15. Since3x5 = 15,log. 3+log.5 = log. 15 = 1.17609126. 16. Since 4x4=16, log. 4+ log. 4 = log. 16=1.20411998. 17. Computed by formula in Art. 399 log 17= 1.23044892. 18. Since3x6 = 18, log. 3+log.6 = log. 18=1.25527251. 19. Computed by formula in Art. 399, log. 19 = 1.^7875360. 20. Since 2 x 10=20, log. 2+ log. 10= log. 20= 1.30103000. 21. Since 3x 7=21, log. 3+ log. 7= log.^l = 1.32221929. 22. Since 2x 11 = 22, log. 2+ log. lI=log. 22= L34242268. 23. Computed by formula in Art. 399, log. 23=1.36172784. 24. Since4x 6 = 24, log.4+ log. 6= log. 24= 1.38021124. 25. Since5x 5=25, log. 5.+ log. 5= log. 25 = 1.39794001. 26. Since 2 x 13 = 26, log. 2+ log. 13= log. 26 = 1.41497335. 27.- Since ^X 9 = 27, log. 3 -f- log. 9= log. 27= 1.43136376. 28. Since 4 X 7=28, log. 4+ log. 7= log. 28= 1.44715803. SECT. IX.] LOGARITHMS. 265 29. Computed by formula in Art. 399, log. 29 = 1.46239800. 30. Since 3 X 10=30, log. 3-Mog.4. = log. 30= 1.47712125. The logarithm usually consists of two parts, the integral part, usually called the index or characteristic^ and a decimal. 402. It will also be perceived that the multiplying or di- viding of any number by 10, 100, 1000, &c , is performed by increasing or diminishing the integral part of its logarithm by 1, 2, 3, &c. J hence, all numbers which consist of the same figures, whether, they be integers, decimals, or mixed numbers, will have for the decimal part of their logarithms the same positive number. Thus, according to the tables now in common use, the logarithm of 3854 is 3.58591171. Log. 3854 =3.58591171 J Log. 38540= log. (3854 x 10)= log. 3854+ 1=4.58591171 j Log. 385,4 = log. ??^ = log. 3854-1=2.58591171 ; Log. 38,54 = log.^^ = log. 3854-2=1.58591171 J Log. 3,854 = log. ??5^ = log. 3854-3=0.58591 171 ; Log. ,3854 = log. ^?5i = log. 3854-4= 1.58591171 ; ^ ' ^ 10000 ^ Log. ,03854= log. J?5i. = log. 3854-5=2.58591171. ^ ^ 100000 ^ The number of units in the characteristic of a logarithm is one less than the number of digits in the natural number j and for decimals, the negative characteristic denotes how far the first significant figure is removed from the place of units. The decimal part of the logarithm is always positive. APPLICATIONS OF LOGARITHMS. 403. The tables of logarithms in common use contain the logarithms of numbers from 1 to 10000. An explanation of these tables, and also of the methods of finding from them the logarithm of any number, or the number of any logarithm 23 L L 266 ELEMENTS OF ALGEBRA. [sECT. IX. whatever, usually accompanies them, so that such explana- tions are unnecessary here. The numbers and logarithms used in the following applications of logarithms are taken from these tables. I. MULTIPLICATION AND DIVISION. 404. Since logarithms are a series of exponents denoting different powers of the common number 10, it is evident that the sum of the logarithms of any two numbers will be the logarithm of their product, and the difference of their logarithms will be the logarithm of the quotient produced by dividing the greater by the less. Hence, I. To multiply by logarithms, take the logarithms of the fac- tors from the table^ add them together^ and then find the natural number corresponding to their sum ; this will he the product re- quired. 1. Multiply 16 by 5, by logarithms. Logarithm 16 =1.20411998; Logarithm "5 =z0.69897000 ; Logarithm 80 =1.90308998. Ans.QO, 2. Multiply 37153 by 4086, by logarithms. Logarithm 37153 =4.5699939 ; Logarithm 408,6 =2.6112984; Product, 15180715.8 . . 6.1812923. 3. Muhiply 4675,12 by .03275, by logarithms. Logarithm 4675.12 =3.6697928; Logarithm 03275 =2.5152113; Product, 153,1102, &c. . 2.3850041. II. To divide by logarithms, subtract the logarithm of the divisor from the logarithm of the dividend, and the remainder will be the logarithm of the quotient. EXAMPLES. 1. Divide 72 by 24, by logarithms. Logarithm 72 =1.85733250; Logarithm 24 =1.38021124; Quotient, 3 0.47712126. •BCT. IX.] LOGARFTHMS. 26T 2. Divide 4768,2 by 36,954, by logarithms. Logarithm 4768,2 = 3 6783545 ; Logarithm 36,954 =1 567 6615; Quotient, 129,032 .... 2.1106930. 3. Divide 46257 by ,17608, by logarithms. Logarithm 46257 =4.6651725; Logarithm ,17608 =1.24 57100 ; Quotient, 2^2741 . . . . 5.4194625.- II. INVOLUTION AND EVOLUTION. 405. Involution is performed by multiplying the exponent of the number to be involved by the exponent denoting the power (Art. 157) ; and Evolution by dividing the exponent of the number by the exponent denoting the root to be ta- ken (Art. 289). Hence, I. To involve by logarithms, multiply the logarithm of the number to be involved by the number denoting the power ; the product will be the logarithm of the power. EXAMPLES. 1. Involve 9 to the second power, by logarithms. Logarithm 9 =0.95424251; Multiplying by 2, 2 ; Square, 81 1.90848502. 2. Involve 7.0851 to the third power, by logarithms. Logarithm 7.0851 =0.8503399 ; Multiplying by 3, 3 ; Cube, 355,6475 .... 2.5510197. 3. Involve 0.9061 to the seventh power, by logarithms. Logarithm 0.9061 =1.9571761304 ; Multiplyihg by 7, 7 ; Power, 0.5015 .... 1.7002329128. 4. Involve 1.0045 to the 365th power, by logarithms Logarithm 1.0045 =0.0019499 ; Multiplying by 365, 365 ; 97495" 116994 58497 Power, 5.148888 . . . 0.7117135. 268 ELEMENTS OF ALGEBRA. [sECT. IX. II. To evolve by logarithms, divide the logarithm of the given number by the number denoting the root to be taken ; the quotient will be the logarithm of the root. EXAMPLES. 1. Evolve 81 to the fourth root, by logarithms. Logarithm 81 =r 1.90848502 ; Dividing by 4, -r-4 ; Root, 3 0.47712120. 2. Evolve 7.0825 to the fifth root, by logarithms. Logarithm 7.0825 =0.8501866 j Dividing by 5, 4-5 j Root, 1.479235 .... 0.1700373. 3. Evolve 1.045 to the 365th root, by logarithms. Logarithm 1.045 =0.0019116; Dividing by 365, -i-365j Root, 1.000121 .... 0.0000524. 4. What is the 8th power of the 9th root of 654 % Logarithm 654 r=2.8 155777483 ; Multiplying by 8, 8 ; * 22.5246219864 ; Dividing by 9, -r-9 j Root, 318.3 2.507357762. III. EXPONENTIAL EQUATIONS. 406. Equations into which the unknown quantity enters in the form of an index are called exponential equations. Such equations may be most readily solved by logarithms. Thus, d^=b^ but a=^=(log. a)xx ; therefore, (log. a)xx= log. 5, or, dividing by log. a, x— ^^' . log. a EXAMPLES. 1. Reduce the equation 5''i=100. As the two members are equal, their logarithms must also be equal y therefore, SECT. IX.] LOGARITHMS. 269 (Log. 5)xa?= log. 100; Ti' .' log. 100 2 00000000 OQC1 Dividing - - x= _-° = — — =2.861. log. 5 0.69897000 2. Reduce the equation 3'=24.3. (Log. 3)xx= log. 24.3; Tx- ... log. 243 2.38561 ^ Dividing - - x= —P- = =5. ^ log. 3 0.47712 407. Another and a more difficult form of exponential equation is a"*=b. Here the exponent x is the exponent of the exponent m. In this equation assume rrfzuy^ then a?=by And - (log. a)xy= log. h ; Dividing - - y= ^^' ; log. a Hence - - »t*= —°^ (which let) =c; log. a Then - (log. »i)xa?= log. c; Dividing • - ir= ^^' ^ . log. m EXAMPLE. X 1. Reduce the equation 9'= 1000. (Log. 9) X 3'= log. 1000 J Dividing . . 3x^ log. 1000 ^ 3.00000000 ^3^ ^ log. 9 0.95424251 ' Then - - - 3*= 3.14.-. (log. 3) Xic= log. 3.14, And - . . x=l^^Jll!=2^9^5-1.04. log. 3 0.47712126 2. Reduce the equation 4^=4096. ^n«.a:= 1:^=1.6309 + . log. 3 408. A third and a still more difficult form of the expo- nential equation is af=b. Taking the logarithms of both sides, we have (Log. x)xx= log. b. This equation naay be solved by " Trial and Errors Thus, 270 ELEMENTS OF ALGEBRA. [sECT. IX. make two suppositions of the value of the unknown quanti- ty, and find their errors j then institute the following pro- portion : / Diff. of the errors : diff. of the assumed numbers : : least error : to the correction required in the corresponding assumed number. EXAMPLES. 1. Reduce the equation a;*=i256. Then - (log. x)xcc= log. 256 ; Suppose - - 07=3.5, or 3.6. ' By first Supposition. By second Supposition. Log. x=[og, 3.5=0.54406804 Multiplying by 3 5 (Log. a?)xa7= 1.90423814 But, log. 256 =2 40823997 Error* - 0.50400183 Log. X- log. 3.6=0.55630250 Muhiplying by 3.6 (Log. x)xx =2.00263900 Log. 256 =2.40823997 Error* - 0.39555097 Difference of the errors, 0.10844086. Then, 0.10844086 : 0.1 : ; 0.39555097 : 0.365 j hence x 3.965 + . To correct this still farther, suppose a?=3.96, or 4.01. By first Supposition. By second Supposition. Lo^.a;=loa. 3.96=0.59769519 Multiplying by 3 96 (Log. a;)xa: =2 36687295 Loa.x= log. 4,01=0.60314437 Multiplying by 4.01 {Log. x) XX =2.41860892 256 =2.40823997,Lo^. 256 =2.40823997 First errorf =0.04136702! Second errort=0.01036895 Difference of the errors =0 05173597. Then, 05173597 : 0.05 : : 0.01036895 : 0.01. Hence, a:=4.01 — 0.01=4, which value for x satisfies the conditions of the equation ; for 4^=256. 2. Reduce the equation 4a;''=100a?^ * Both these suppositions are discovered to be less than the true number; hence the errors are like with reference to their signs. t One of these suppositions is less, the other greater than the true value of a;; hence the errors are unlike with reference to their signs. •ECT. IX.] LOGARITHMS. 271 Dividing by 4 - - - - af=25a^; Dividing by a:* .... jf-«=r25j Taking the logarithm, (log. af)x(x— 3)= log. 25; Suppose x=4fy or 6. Bjr wcoDd Suppokition. Log. x= log. 6 =0.77815125 By fitrt SappMition. Log. x= log. 4 =0.60205999 Multiplying } _ , ultiplying ) by X— 3 S (Log. x) X (ar— 3) =p 0.60205999 Log. 25 =1.39794001 First error =0.79588002 Difference of the errors =1.73239376. Then, 1.73239376 : 2 : : 0.79588002 : 0.092 Hence, x=4-|-0.92=4.92. Again, suppose 07=4.92, or 4.93. By Am Supptwitioo. Log.x=log.4.92=0.69196510 Multiplying ^ ^ J ^2 :— 3 S Multiplying^ by X — 3 (Log. a) X (a:— 3) =2.33445375 Log. 25 =1.39794001 Second error =0.93651374 by X- (Log. x) X (x--3)= 1 .32857299 By tecood Sappotilion. Log. x= log. 4.93 = 0.69284692 Multiplying by X— 3 ! = 1.93 (Log. x) X (X— 3) = 1 .337 1 9456 1.39794001'Log. 25 = 1.39794001 Log. 25 First error =0.06936702,' Second error =0.06074545 Difference of errors =0.00851157. Then, 0.00851157 : 0.01 : : 0.06074545 : 0.07. Hence, i= 4. 93 -1-0.07 — 5.00, which value for x satisfies the conditions of the equation ; for 4X5^=100X5'. IV. GEOMETRICAL SERIES. 409. Logarithms are also very convenient in finding the last term, and also the sum of the series in Geometrical Progression, when n is not a very small number. The num- ber of terms may also be obtained by the aid of logarithms. I. The formula for the last term is (Art. 383), Taking the log., log. /= log. a4-(log. r)x(n — \). Hence, to find the last term in n geometrical series by 272 ELEMENTS OF ALGEBRA. [sECT. IX. logarithms, add the logarithm of the first term to the logarithm of the ratio multiplied by the number of terms less one ; the sum will be the logarithm of the last term, EXAMPLE. 1. The first term of a geometrical series is 4, the ratio 5, and the number of terms 61. Required the last term. Or, log. l=z log. 4+(log. 5) X 60=0.60205999+41.9382000 = 42.54025999. ' Hence, finding the natural number corresponding with 42.54025999, 7=3469479392577934009746744427570344331708876. 2. The formula for the sum of the terms in a geometrical series is (Art. 386) g_ar"— + a +a» H-a* -f» + 0' x+o* 426. By comparing these results with the products from which they have been derived, we perceive, 1. That the multiplier of x in the second term has been converted into the first power of a, repeated as many times as there are units in the number of binomial factors used, or, which is the same thing, as there are units in the expo- nent denoting the power to which x-\-a was to be involved, 2. That the multiplier of x in the third term has been converted into a^ repeated as many times as there can be formed different products from a number of letters equal to the number of binomials employed, taken two in a set. 3< That the multiplier of the fourth term has been con- verted into a', repeated as many times as there can be form- ed different products from a number of letters, equal to the number of binomials employed, taken three in a set, and so on. 427. It is therefore evident that, whatever may be the power to which the binomial a: + a is to be raised, the for- mation of its power will be subject to the following laws, viz. : 1. The exponent of x in the first term will be equal to the ex- ponent of the power , and in the succeeding terms will decrease regularly by 1 to the last teivi^ in which it will le 0, 2. The exponent of a in the first term will be 0, in the second 1, and that it will go on increasing by 1 until it becomes equal to the exponent of the power to which the binomial was to be in* volved. 3. That the numerical coefficient of x in the first term will b€ 1 ; in the second it will be equal to the exponent denoting th% S8^ > ELEMENTS OF ALGEBRA. [sECT. X. power to which the binomial was to le involved ; in the third term it will be equal to the number of products^ which may be formed from a number of letters^ equal to the exponent denoting the power of the binomial, taken two in a set ; in the fourth term it will be equal to the numbev of products which may be formed from the same number of letters taken three in a set, Sfc. ♦ 428. The above theorem, with reference to the coeffi- cients, is too cohiplicated for general use. In order to sini- plify it, let it be required to expand (oj+a)"*. The first few. and the last few terms, without the numerical coefficients, will be +a^ (A). The numerical coefficient of the first term is 1 ; that of the second is m /that of the third is equal to the number of products which may be formed of m letters taken two in a set ; this is expressed by the formula —A 1 : the coeffi- cient of the fourth term is — — ' ' — — ^ , &;c. 1x2x3 By inspecting the above formulas for the numerical coef- ficients of X, it will be perceived that the coefficient of the third term is equal to the coefficient of the second term (m) multiplied by the exponent of x {m — 1) in that term, the product divided by the number (2) which marks the place of this term, counting from the left. And, also, the coefficient of the fourth tferm is equal to the coefficient of the third term / ^(^— ; \ ^ multiplied by the exponent of x (m — 2) in that term, the product divided by the number (3) denoting the place of that term &c. 429. Again, since in the expression [x-{-a) , a may be sub- stituted for X, and x for a, without altering its value,^ it fol- lows that the same thing may be done in the development of it. Hence, if this development contains a term of the form Ka"a?"'~" (K representing the numerical coefficient), it SECT. X.] BINOMIAL THEOREM. 287 must have another equal to Kx^aT-''^ or Ka'^'^x". These two terms are evidently at equal distances from the two extremes, for the number of terms whioh precede any term being indicated by the exponent of a in that term, it follows that the term Ka"x"*~" has n terms before it, and that the term Ka^^x" has m — n terms before it, and, consequently, n terms after it, since the whole number of terms is denoted by m-fl. Therefore, in the development of any power of a binomial, the coefficients at equal distances from the extremes are equal to each other. Hence, the numerical coefficients of the series A will be l-LffT-L ^^— -j- K^— 1) (^— -) ■ m{m—l) (ct— 2) 1x2 1x2x3 ^ ' ' *' ' 1x2x3 +!fcL)+m+l (B). Compounding the two series A and B, we have (x+ar=x--\-max-' + ^lil^^^a'x"-' + m(m-l) (m-2)^ ^ ^ ^ 1x2 1x2x3 x-'-f . . . 4. ^(^-1) (^ -2)^^.^^^(m-l)^^ , 1x2x3 1x2 4-mxa"'~' + a"'. 430. The preceding operations give rise to the following simple theorem for obtaining the coefficients : 1. The coefficient of the first term is 1 j that of the second is equal to the number of units in the exponent, which denotes the power to which the binomial is to be raised. 2. .^nd univp-sally^ if we multiply the numerical coefficient by the exponent of x in that term, and then divide the product ly the number which marks the place of that term from the left, the quotient will be the coefficient of the succeeding term. 3. The terms in the last half of the series of coefficients will be found to correspond with those in the first half placed in the inverse order. 431. These results of the Binomial Formula are substanr tially the same as those obtained by a different process, and practically applied in Articles 259-272. 288 ELEMENTS OF ALGEBRA. [SECT. X. It should also be remarked that the same formula will apply whether m represent a positive or negative whole number or a fraction. CONTINUED FRACTIONS. 432. A continued fraction is one which has 1 for its nu- merator, and for its denominator an entire number plus a fraction \ which fraction also has 1 for its numerator, and for its denominator an entire number plus a fraction, and so on. Thus, 1 «+l c+l <3+0 : '2x3+1 _ 3 7' 3 The fourth is ?+i_ 3x2+1 7x2+2 7 16" 3+1 2 The fifth is 2+L^ 7x2+3 16x2+7 17 39' 3+1 2+1. 2 The sixth is 5+L_ _ 17xl+7_ 39x1+16 24 ■ 55 3+1 2+1 2+k The seventh is ^+^ i _24x2+17_ 55x2+39 .65 149 3+1 .2+1 - 2+1 1+1 2 ^ote 2. — The successive reductions, it will be perceived by inspecting the above results, are alternately less and greater than the whole continued fraction, and they ap- proximate this fraction nearer find nearer. The first re- duction is always less than the whole continued fraction. Hence the reductions of an odd rank are always less than the whole continued fraction^ and those of an even rank are greater. In the above reductions. The second difi^ers from the true value of the continued 1 65 19 fraction by - The third differs by 2 149 ~ 298 66 __ 3 __ 8 149 7 1043' .LGEI 5RA. [sect. X, 7 le- 65 149 3 2384 es 17 2 149 39 5811 24. 55 e5 149 _ 1 8195 e5 149 es 149 = 0. 292 ELEMENT The fourth diifers by - The fifth differs by The sixth differs by The seventh differs by - J^Tote 3. — If a vulgar fraction which is not expressed in its lowest terms be converted into a continued fraction, and all the reductions be formed to the last inclusive, the last re- daction will not be the proposed fraction, but this fraction reduced to its lowest terms. 348 For example, let the fraction be converted into a ^ 954 continued fraction. Thug, 348^0 1 954 12+1 1+1 1+1 1+1 The reductions of this continued fraction are, 1 1 2 3 , 29 1' 2' 3' 5' 8' W 29 . 348 The last reduction, — , is the same as reduced to its 77 954 lowest terms. EXAMPLES. 1. Required the vulgar fraction which is equivalent to the continued fraction 1 3+T 2+1 5+1 6 992 SECT. X.] CONTINUED FRACTIONS. 2^3 2. Required the approximate values of the continued fraction 1 IH-l 2+1 3+1 4+1 5+1 6+1 7+1 8+1 3. Required the approximative values of the continued fraction 1 9+1 8+1 7+1 6+1 5+1 4+1 3+1 2- 4. The ratio of the circumference of a circle to its diam- eter may he expressed hy the fraction ^^7^;^.^-^ > required some of the approximative values of this ratio. Converting the given fraction into a continued fraction^ we have 314159^3^1 100000 7+1 15+1 1 + 1 25+j 1+1 7+1 4* 994l elements of algebra. [sect. x. The successive reductions are, 3 22 333 355 9208 9563 76149 ^^^ 314159' r 7' 106' 113' 2931' 3044' 24239' ^" lOOOOO' INFINITE SERIES. 433. An Infinite Series is a progression of numbers con- nected together by the signs + or — , proceeding onward without termination, but usually according to some regular law, which may be discovered by tracing a few of the lead- ing terms. A Converging Series is one whose successive terms de- crease. Thus, -+-2 + 3+4 +-5 +r^^'> X x^ a^ x^ x^ and - - 1+A +1+2+^4-, &c., 2 4 8 16 32 ' ' are converging series, when a?>l in the first series. A Diverging Series is one whose successive terms increase. Thus, x-\-x^-{-x'-{-x* + x'-\-, &c., and - 2-1-4, 4-8-1- 16+32+, &c., are diverging series, when a7< 1 in the first series. I. EXPANSION OF INFINITE SERIES.. 434. There are four general methods of converting alge- braic expressions into an infinite series of equivalent value. First. We have already seen that the division of alge- braic quantities (Art. 110) will sometimes produce an infi- nite series. Also, a fraction may sometimes be expanded into an infinite series by dividing the numerator by the de- nominator. examples.. 1. Divide 1+a by 1 — a. SECT. X.] EXPANSION OF INFINITE SERIES. 899 1— a l4-2a4-2a'+2a»+2a*-f , &c., ad infin. 2a— 2a« 2a'— 2a« 2a» 2a*— 2a* 2^ 2. Reduce the fraction to an infinite series. 1 — a Since the value of a fraction is the quotient resulting from the division of the numerator by the denominator (Art. 125), the vahie of the above fraction will be obtained by dividing 1 by 1— a. 1 |1— fl 1 — a l-|-o+a'4-a*+fl^-f > &c., ad inftn, a a^—d? a' KoH, — By observing that the value of a fraction is equal to the terms of the quotient -|- the fraction formed by pla- cing the remainder over the denominator^ we shall have --i_ = l+a+a»+a»-fa'+a*+ a"+:^. 1— a \—a 3. Reduce the fraction to an in£aite series. 296 ELEMENTS OF ALGEBRA. [sECT. X. 4. Reduce the fraction to an infinite series. a—-b a or a^ 5. Reduce the fraction to an infinite series. 1 + a Ans. \—a-\-d!' — o^-\-a^ — a^, &c., adinjin, 6. Reduce the fraction^ to an infinite series. 1-2 Ans. 1 + 2+4+8+ 16 + 32+64+, &c., adinfin. J^ote. — The above result might, at first sight, seem ab- surd ; but it should be remarked that, if we wish to stop at any term of the above series, we must add the fraction that remains to the terms taken. Thus, if we stop after taking seven terms of the quotient, we shall have 1 1 + 2+4+8+ 16 + 32+64+-!??-=: 127+1^=— 1. 1_2~ \ 1-2 —1 7. Reduce the fraction -^ — to an infinite series. a—x 8. Reduce the fraction . to an infinite series. 435. Secondly. An infinite series may be formed by ex- tracting the root of a compound surd. EXAMPLES. 1. Reduce -s/ c^-\-h^ to an infinite series. Extracting the square root, according to the rule given in Art. 291, a2+^,^(a+^__|l+JL, &c., adinfin, 2a 8a' 16a' 2aY SECT. X.] EXPANSION OF INFINITE SERIES. 297 2. Reduce V<^—t^ to an infinite series. 3. Reduce ^/l^x to an infinite series. Thirdly. We have already seen (Art. 295) that if a bino- mial which has a negative or fractional index be expanded by the Binomial Theorem, it will produce an infinite series. This case has already been sufficiently explained and il- lustrated in the article referred to above. 436. Foitrthly, An algebraic expression may also be ex- panded by assuming a series with indeterminate coefficients^- and afterward finding the value of these coefficients. To give some idea of this method of development, we will suppose it is required to expand — ^ into a series arran- ged according to the ascending powers of x. This expres- sion may evidently be expanded so as to answer these con- ditions ; for _=a(c-f ia?)"*. Expanding this last expres- sion by the binomial theorem, and representing the known terms and coefficients successively by A, B, C, &c., we shall have =A4-Bx-fCa:2-fDx*+Ex*-f , &c., adinjin. c-\-bx The above coefficients A, B, C, &c., being functions of a, by and c, that is, dependant on them for their values, but independent of x, are called indeterminate coefficients. It is now required to determine thp value of these coeffi- cients. Multiplying both members of the equation by the denom- inator c-j-^^j and transposing a, we obtain 0={Ac-a)-\-Ab x-{-Bb I x^+Cb r'+j &c., adinjin, +Bc +Cc I -f-Dc Here it is evident that if Ac — a, A6+Bc, Bi+Cc, &c., be made each equal to 0, the several terms of the second mem- ber will be reduced to 0, and, consequently, the mendber will Pp 298 ELEMENTS OF ALGEBRA. [SECT. X. equal 0, and thus the conditions of the equation may be sat- isfied. From the above assumption we derive the following values of the successive coefficients : 1st - Ac— a =0; hence A — -. c ♦ 2d - Ah+Bc=0',henceB=-^=-^X ": =-^. c c c c^ 3d - BJ+Cc=:Oj henceC=-i^=-^X-^=:. + -. c c c^ a? 4th - CZ.+Dc=0ihenceD = -2^=-^X^' =-^'. Hence we have — ^-f-__ar— __a; , &c., ad mjin. c-\-bx c c c^ c'* 437. By inspecting the preceding operations, we shall per- ceive that each succeeding coefficient is equal to the prece- ding multiplied by — - ; consequently, — - is the ratio of the c c progression of the coefficients, and — — is the ratio of the c progression of the series. EXAMPLES, 1. Expand into an infinite series. b—ax Assume =A-\-Bx-\-Cc(^-{-'Dx'^-{-Ex'^-{-, &c., ad infin. b — ax Multiplying both members of the equation by b — ax, and transposing c?, we have 0=z(Ab—d)—Aa -{-Bb X — Ba + Cb x^—Ca +Db oc^—Da +Eb x\ &c. Whence, making the several coefficients equal to 0, we have, 1st ' Ab — d =0; hence A=_. b id .' Bb—Aa=.0', hence B=^= ^ X^=~. b b b' 3d - C^-Ba:=0 ', hence C=^=^ X?^^. b b^ b b SECT. X.] EXPANSION OF INFINITE SERIES. 299 4th - DJ-Ca=Oi hence D=^=^x?=^. 5th - Ei-Da=0; hence E=°?=t''x?=^. tr br Hence we have 2. Expand — "^ into an infinite series. Ans. l+3x-h4x'+7x*+llat*+18a:', &c. 3. Expand . = into an infinite series. Am. l+a?+2a?'+2x»-H3a!:*-|-3x»+4a?«4-4a?', &c. 1— X 1— 2x— 3x- Ans, l+x4-5x'+13ar'4-41x*+121i»4-365x«, &c. 4«. Expand - — ^— , into an infinite series. 5. Expand ^"'"^ into an infinite series. Remark. — The method of indeterminate coefficients re- quires that we should know the form of the development with reference to the exponents of x. The terms are gen- erally supposed to be arranged according to the ascending powers of x, commencing with x°. Sometimes, however, this form is not exact \ in this case the calculus detects the error in the supposition. For example, let it be required to expand the fraction 1 Si-x"' Suppose — L_-A+Ba:-hCx»+Da:*, &c. 3x — :r Multiplying both members by Zx — a?*, and transposing 1, we have 0=— l + 3Ax— A Ix*, + 3B -I-3C +3D [ x«-C [x*,&c. + 3D x«— B -I-3C Whence the conditions of the equation require that — 1 = 0, which is absurd j hence the above form will not apply to the development of the expression — . OX— x^ 300 ELEMENTS OF ALGEBRA. [SECT. X. II. SUMMATION OF INFINITE SERIES. 438. The summation of a series is the finding a finite ex- pression equivalent to the series. But as different series are often governed by very differ- ent lavi^s, the methods of finding the sum which are appli- cable to one class of series, will not apply universally. Hence result different methods of summation. I. First Method. — If the series is a regular descending geometrical series, that is, if its terms decrease by a com- mon divisor, the sum of the series may be obtained by the following formula : (Art. 389.) 1—2' As this formula has been explained and applied (see Arti- cles 380 to 39p) in Geometrical Progression, we need add nothing more concerning it in this place. II. Second Method. — The summation of certain classes of infinite series may be effected by subtraction. EXAMPLES. 1. Let it be required to find the sum of the infinite series ' ;+.4^+:r^. + rL+^.'&c. (1.) 1.2.3 2.3.4 3.4.5 4.5.6 5.6.7 By removing the last two factors from each of the de- nominators in the preceding series, let us form a new series whose value may be expressed by S j thus, S=i-fl-f l+-+i+i &c., ad injin, (2.) 1-23456 '' ^ ' By transposition, ^-\=l+\+\+l+\^''-^''^'''fi'>- (3-) By subtracting the last equation (3) from the second (2), 1= — + — + — -j- — + — . &cc.t ad infin. (4.) 1.2 2.3 3.4 4.5 5.6' ' *^ ^ ^ By transposition, 1 — _= — + — + — + — , &c., ad iniin, (5.) 2 2.3 3.4 4.5 5.6' ' '^ ^ ^ SECT. X.] SUMMATION OP INFINITE SERIES. 801 Whence, by subtracting this last equation (5) from the fourth (4), we have !=_!_ +_^4— ^4— i5_, &c., ad infin, 2 1.4.3 2.9.4 3.16.5 4.25.6' Or, i=_?_-f_^-f -^ +-^, &c., ad infin. '2 1.2.3^2.3.4 3.4.5 4.5.6' -^ Whence, dividing by 2, l=_L_+_J_-f-— 1- + -?— , &c., ad infin. 4 1.2.3 2.3.4 3.4.5 4.5.6' -^ Hence the sum of the given series is -. 4 2. Required the sum of the infinite series 1.3^2.4^3.5^4.6^5.7'^ Let . Sr^l+l+l+l-^l Sec, ad infin. (1.) J> O 't Or - S= ?4.1+l-fl, &c., a^ x_:^_x^_^_;^ '23456 ^ /y> nn^ /)o3 /y^4 ^S Eed„ci„g,S(.-l)=-l + ^^+£-3+iL+?^+^^,&c. By making a;=:l, the equation becomes* 0=— ! + —+— +— + — +—, &c. 1.2 2.3 3.4 4.5 5.6' III. RECURRING SERIES. 439. A recurring series is one which is so constituted that a certain number of contiguous terms, taken in any part of the series, have a given relation to the term immediately succeeding. Thus, in the series l-^3x-\-4^x'-\-lx^-\-nx'+lSx\ &c., the sum of the coefficients of any two contiguous terms is equal to the coefficient of the following term. If the series be expressed by A+B-i-C+D+E+F, &c., then The 1st term - - A=l ; The 2d term - - B = 3a?; The 3d term - - Cz=zBx-{-Ax^=4 = Cx-^Bx'=lx' i SECT. X.] RECURRING SERIES. 303 The 5th term - - E=Dx+Ca:«=lla?* j The 6th term - - F=Ex-^Dj^=1Sx\ 6cc. That is, each of the terms after the second is equal to the one immediately preceding multiplied by x, plus the one next preceding multiplied by x^. Hence all the terms after the first two are subject to a definite law. 440. The particular expression from which any term of the series may be found when the preceding terms are known is called the scale of the series^ and that from which the coefficients may be formed the scale of the coefficients. Recurring series are divided into orders, and the order is estimated by the number of terms contained in the scale. In the expansion of — — in Art. 436, we have a recurring series of the first order. Thus, a a abx , ah^oc^ aPx^ . c-^bx c cr* The scale of the coefficients here is — _ j that of the terms c bx is — — . This is the simplest form of the recurring series. 441, In a recurring series of the second order the law of progression depends upon two terms, and, consequently, the scale consists of two parts. Let m-{-n represent the scale of the series, and A + B-f C + D-f E+F, &c., represent the recurring series. Then The 3d term - - C-Bmx-\-knx^ \ The 4th term - - 'D = Qmx-\-Bnx^ ] The 5th term - - E = Dotx + C»x^ dec. Taking the last two terms in the above expression, we have the two equations D = Cmx+Bnx' ) ^^ g^^ ^^^ ^^^^^^ ^^ ^ ^^^ ^ Ez^D^nx+Cwx' S Since the scale of the series is the same, whatever be the value of J?, the reduction may be rendered more simple by making x=l. The equations then become 304 ELEMENTS OF ALGEBRA. [sECT. X. E=:I>m+Cn, These, reduced, give DC-BE CE-DD CC-BD' CC— BD In the series l + 3a7+5j?'+7a?='+9j?*+lla7^ &c., A=:l, B=:3x, C^So^VD^Tar', E=9a?^ Then, making x=. 1, we have 7x5— 3x9_^2^ I ^^ 5x9— 7x7 ^_.^ 5x5—3x7 I 5x5-3x7 44*2. In a recurring series of the third order the law of progression depends upon three contiguous terms. Letting m-{-n-\-r. represent the scale of series, and A+B+C+D+E+F, &c., the series, then The 4th term - - J) — Q>mx-\.Bnx^-\-krT^-, The 5th term - - '£,:=zDmx-\.Cnx'-\-Brx'' -, The 6th term - - Y-Emx-^J)nx'-\-Qrx\ &c. In a similar manner, we may obtain the succeeding terms in the higher orders of the recurring series. 443. To ascertain whether th§ law of progression depends on two, or three, or more terms, we may first make trial of two terms ; and if the scale of the series thus found does not correspond with the series, we may try three or more terms. If we begin with a number of terms greater than is necessary, one or more of the values found will be 0, and the others will constitute the true scale of valuation. 444. When the scale of a decreasing series is known, the sum of the terms may be found. Let - a-^hx-\-cx'''-\-dx^-\-ex^-\-fx^^ &c., be a recurring series, whose scale of relation is m-\-n. Then The 1st term - - =:A ; The 2d term - - =B ; The 3d term - - C = 'Bxmx-{-Axnx^ ; The 4th term - - 'D = Cxmx-\-Bxnx'^ ; The 5th term - - E=:Dxwia?-f-Cx»a?^, &c. SECT. X.] RECURRING SERIES. 20Sk If the series be infinitely extended, the la^t term may be neglected as of no comparative value j and if S= the sum of the terms, we shall have S=rA-f B + mxx(B + C + D, &c.) + »ir*x(A-|-B-f C, &c.). But B+C-f D, &c.» =S— A, and A+B+C, &c., =S. Hence, by substitution, Or - S=A+B+Smx— Amx+Sna?*. Transposing, S — Sotx — S«x^=A-f B— Amx, Or - S{l—mx—nx')=A+E-Amx, T\. .J. o A-fB — Amx Dividmsr, o= — ! -. •&» 1 — mx — nx^ EXAMPLES. 1. Required the sum of the infinite series l + 6x+12xH48jc'+120a?*, &c. A = l, B=6j:, C=12a:', D=48ar', E = 120a;*, &;c. Then, making x=l, we have ^^ 12x48-6x12 0^.^ 12x12—6x4:8 ^_ 12x120-48x48 ^ 12x12—6x48 Substituting the values of A, B, wi, and n in the formula, o_A + B — Amx 1—vix—nx^^ We shall have - S=J"^^^""^. 1 — X — 6ar Or - - - S=^±^. 1 — X — 6a:' 2. Required the sum of the infiMte series l+2x-f 8x'+28a:'+100x*, &c. Substituting, as before, ^ = a ^^8x100—28x28 8x8—2x28 ""' "~ 8—8—2x28 q_ l+2j— 3j __ 1- -X l-^Sx—^x" 1— 3x— 2x» 3. Required the sum of the infinite series l + 3a:-}-4x'+7x'+llx*+18x'+29j?«, &c. Qq 306 ELEMENTS OF ALGEBRA. [sECT. X. 4>. Required the sum of the infinite series l-\-2x-\-3x^-}-^3c^+5x'+6x\ &c. 5. Required the sum of the infinite series l + 3x-\-bx'-\-lx'-{-9x'-\-llx''-{-13x% &c. IV. METHOD OF DIFFERENCES. ^ ' 445. We will now proceed to point out another process by which the summation of various kinds of series to a lim- ited number of terms may be obtained. This is termed Method of Differences^ as it depends on finding the several orders of differences belonging- to the series. 1. Orders of Differences. 1. If we take the first term from the second, the second from the third, the third from the fourth, &c., in the given series, the remainders will form a new series, which is called the first order of differences. 2. If we proceed with this new series in the same manner as with the given series, we shall obtain the second order of differences. 3. In the same manner we may obtain the third^ fourth, fifths &c., orders of differences. 446. It should be observed, however, that when the sev- eral terms of the series increase, the differences will all be positive ; but when they decrease, the differences will be negative and positive alternately. EXAMPLES. 1. Required the several orders of differences in the series 1^ 2^ 3^ 4^ b\ 6^ &c. The proposed series - 1, 4, 9, 16, 25, 36, &c. 1st order of difference - - 3, 5, 7, 9, 11, &c. 2d order of difference - - 2, 2, 2, 2, &c. 3d order of difference - - - 0, 0, 0, &c. 2. Required the several orders of differences in the series 1, 6, 20, 50, 105, 196, &c. 1st order of difference - 5, 14, 30, 55, 91, &c. 2d order of difference - - 9, 16, 25, 36, &c. SECT. X.] METHOD OF DIFFERENCES. 80f 3d order of difference - - -.7, 9, 11, A:c. 4th order of difference - - - - 2, 2, ^c* 2. Law of the Coefficients. 44-7. Letting rr, ft, c, rf, &c., represent a series, and pro- ceeding with this series in the same manner as with the preceding, we shall likewise obtain the several orders of differences. Proposed series, a, ft, c, c/, e, /, &c. 1st order of differ., ft — a, c — ft, d — c, c — (/, /— e, &c. 2d differ., c— 2ft-|-a, rf— 2c+ft, A2rf+c, /— 26+b-^a, &c. Transposing^ and reducing these several equations, we ob- tain the following expressions for the terms of the original series : 2d term - hz=a-{-d' ; 3d term - c=a-\-'id'-\-d" p 4th term - d=a'-\-W-\-W^d"' ; ' 5th term - e=za^^d'f6d"-\-4>d'" + d"'\ &c. 450. By inspecting the above, we shall discover that the coefficients observe the same law as in the powers of a bi- nomial, with this difference, that the coefficients of the nth term of the series are the coefficients of the {n—l)th. power of a binomial. Substituting, then, 7i—l for n in the formula for the coefficients of an involved binomial (Art. 448), and applying the coefficients thus obtained to d\ d'\ d'"^ d"", «fec., as in the preceding equations, we have the following gen- eral expression for the nth term of the series, a, b, c, d, &c. : T^th term^a+^izl . c/'+!^ . !^ . J"+^ZZ_1 . !Ll^ . ^11? . 1 12 12 3 d'^+^'szl .'^^.'Lrl.'LrA. d'"' &c. 12 3 4 JN'ote. — When the differences, after a few of the first or- ders, become 0, any term of the series is easily found. EXAMPLES. 1. Eequired the 12th term of the series 2, 6, 12, 20, 30, &c. Proposed series - . . 2, 6,, 12, 20, 30,. &c. SECT. X.] METHOD OF DIFFERENCES. 309 Ist or-der of difference - - - 4, 6, 8, 10, &c. 2d order of difference - - - - 2, 2, 2, &c. 3d order of difference - - - - 0, 0, &c. Here , e— 2£;+c,/-2e+c/, &c. 4thdiff.,(^— 3c+36— a,e— 3c/+3c— Z*,/— 3e+3ri— c, &c. Here it will be observed that the first order of differences in the new series is the same as the original series, and the second order of differences is the same as the first order in the original series a, 6, c, d^ &c. j and, generally, that the (n-{-\)i\i order in the new series i^ the same as the nih. order in the original series. In this case, = 1st term; a = 1st order of difference ; d' =:2d order of difference ; d" =:3d order of difference ; d"'z=iA>\\i order of difference ; 6?""=: 5th order of difference. Resuming now the formula (Art. 450) , n — 1 1, , n — 1 n — 2 ,,- , n — 1 \ — 2 n — 3 ,,,, „ ^1 1 2 1 2 3 ' ' which is the general expression for the n\\v term of a series whose first term is a ; applying it to the new series, in which the first term is 0, and substituting n-\-l for n,we have n , , n n — 1 1, , n n — 1 n — 2 j., , n n — 1 n — 2 ^12 12 3 12 3 __ . d , &c., 4 ^ which is a general expression for the (w + l)th term of the series 0, a, a+J, a+Z>+c, a+o+c+t^, &:c. ; Or the Tith term of the series a, a-\-h^ a+^+c, a-\-h-\-c-\-d^ &c. But the n\\v term of the latter series is evidently the sum of n terms of the series a, Z>, c, d^ &;c. Hence, the general formula for the sum of n terms, a series of which a is the first term, is , n n — 1 7, , n n — 1 n — 2 »,, , n n — 1 n — 2 n — 3 na-\-- . d +- . . . a +- • • • • • ^12 1, ,2 3 ^12 3 4 d"', &c. SECT.!.] METHOD OF DIFFERENCES. 311 EXAMPLES. 1. Required the sum of n terms of the series 1, 2, 3, 4, 5, 6, &c. Proposed series - - - 1, 2, 3, 4, 5, 6, &c. 1st order of difference - - - 1, 1, 1, 1, 1, &c. 2d order of difference - - - 0, 0, 0, 0, &c. Here a=l, (/'=1, and d=zO j therefore, na+ .(f=n-\-- . =n+ = — I— = sum of n 12 12 2 2 terms. In the above example let n=20 j then n^^400-20^210. ^ns. 2 2 2. Required the nth term of the series of odd numbers 1, 3, 5, 7, 9, 4:c. Proposed series - - - - 1, 3, 5, 7, 9, &c. Ist order of difference - - - 2, 2, 2, 2, «fec. 2d order of diflJerence - - - - 0, 0, 0, &c. Here a=l, t/'=2, and d'=0 j therefore, na-\-- . . a = -\- . 2=n*. 12 2 Hence, Me sum of the terms is equal to the square of the number of terms. ' 3. Required the sum of n terms of the series P, 2', 3*, 4*, 5^ 6', r, &c., or 1, 4, 9, 16, 25, 36, 49, &c. Also, the sum of 20 terms. Proposed series - - 1, 4, 9, 16, 25, 36, 49, &c. 1st order of diflference - 3, 5, 7, 9, 11, 13, ' ■ac xy*- 56'-5ffic-f-a*c/ x/4-,&c. (1) a cr a- If the series be of the form y= _Q _^---fta«-»>_ca"-*— —ta S Or, by uniting the corresponding terms, the equation be- comes (x"— a'')-hi(a:^'— a'-')+c(a:*-«— a''-»)-h . . +t(x—a)=zO. In this equation X n— 1 „n—l a>n-2_^n_«^ X — a, are each divisible by a?— a (Art. 203, th. 7) j therefore, the first member of the original equation is also divisible by X — a. EXAMPLES. Suppose 2 to be a root of the equation a:*— 16x^4-56 = 0. By the theorem just demonstrated, the first member of this equation must be divisible by x— 2. Thus, olG ELEMENTS OF ALGEBRA. [^ECT. XI. gg— I6ar^4-56 |a?— 2 a?3_ 2x^ x^—Ux-2a — 14a;24-28a? —28074-56 —280? +56 Corollary 1. If we divide the general equation a7" + 5a?"-' + ca;"-'+ . . . -^tx-\-u=0, (1.) by a; — a, there will result a new equation one degree less than the given equation, which may be put under the gen- eral form a?"-'+Z»V-2+c'a7"-^+, &c.=iO. , (2.) Hence, the original equation may be transformed into the following equivalent expression : (a?— a)(a?'^-' + 6V-2+c'a?"-''+, &c.)=0. The conditions of this equation are satisfied on the sup-'' position that x — a. Cor. 2. The result (2) obtained in the preceding corollary may evidently be divided by x — a', if a^ represent a root of that equation 5 then, x^-'-\-b"x^+c"x^-^-{-, 6lc.,=0. (3.) Hence we shall have a:n-i4.2,'a?"-'+cV-^+, &c. = (x—a) (a?"-^ + fx""^ + c"a?"-*+, &C.)=::0, and the original equation becomes (x—a) (x—a') (a:"-2+^"ar-«+c"ic^+, &c.)=0. The conditions of this equation are satisfied on either of the following suppositions, viz. : x=a, Or - - - - x=za\ Proceeding in the same way to find the remaining roots, a", a'", &c., the original equation will eventually assume the form (x-a) (x—a) (x—a") (x—a'") (x—a""), &c.,=0. (4.) SECT. XI.] GENERAL THEORY OF EQUATIONS. 317 Cor. 3. In the above (4) equation there are evidently n factors. Hence, the number of roots of an equation is denoted by the degree of the equation. Thus, An equation of the second degree has tioo roots ; An equation of the third degree has three roots ; An equation of the fourth degree has four roots^ &c. Scholium. If any of the factors into which the first mem- . ber of the equation may be resolved are equal, the number of unequal roots will evidently be less than the number of units in the exponent expressing their degree. EXAMPLE. The equation (x— a)* {x—a'f (x — a")' (x — a")=0 has but four different roots, although it is an equation of the 10th degree. Cor. 4. If one root of a cubic equation be found, and the equation be divided by the simple equation containing that root, the quotient will be an equation of the second degree containing the other roots. EXAMPLES. 1 If one root of the cubic equation a^ — 7x'+36=0 is 3, what are the other two roots 1 By the conditions of the problem, x=3 .*. x— 3=0. Dividing the given equation by this, we have X*— Tx'-h 36 |x— 3 x*— 3x« x'— 4x— 12 —4x^4-36 — 4ir'-|-12x — 12XH-36 — 12X+36 Hence we have the quadratic equation x"— 4x— 12:r0, which, reduced, gives x=6, or — 2, the other two roots of the cubic equation. Hence, the three roots of the proposed equation are 3, 6, and — 2. 318 ELEMENTS OF ALGEBRA. [sECT. XI. 2. If one root of the equation x^-{-ixr — 16a;-f 20 = 0, is — 5, what are the other two roots ] ^ns. 2 and 2. 3. If one root of the equation x^-\-Zx'' — 10y=0, is 2, what are the other two roots ] Ans. — 5 and 0. CoR. 5. If two roots of an equation of the fourth degree be given, the remaining two njay also be found ,• and so of the higher equations. EXAMPLE. 1. Two roots of the equation a;*--3a;^— 1437^4-4807—32 = 0, are 1 and 2 ; what are the other roots 1 *Mns. 4 and — 4. CoR. 6. Equations of the form a;"=a would appear to have but one root j but, from the preceding reasoning, it must have n roots. EXAMPLES. 1. What are the two foots of the equation a?^=4 1 Ans, 2 and —2. 2. What are the roots of the equation x^—l % Extracting the cube root, we obtain a:=l. Consequently, 1 is one of the roots j then, to ascertain if it has any more roots, we may put the equation under the form 0?^— 1=0. This equation must be divisible by x — 1 ; therefore, 3?='— l = (a?— 1) (a?2+a?+l)=0, Or - - - - x'+x+l =0. The roots of this last equation are |( — 1-\-\/ — 3), and |( — 1 — ^ — 3). Hence, the three roots of the equation a?'=l are 1, ^(— I+n/"^), and |(— 1— x/^^3). 3. What are the roots of the equation 37" = ! 1 Ans. 1,-1, s/~—i, and —^'—i, 4. What are the roots of the equation aj^=l 1 Ans. 1, andi(^— 1— v/5± y/— 10±2v^5J. COMPOSITION OF EQUATIONS. 454. From what has been said, it will be readily inferred SECT. XI.] GENERAL THEORY OF EQUATIONS. 319 that equations of any degree higher than the first may be produced by the successive multiplication of equations of the first degree. Let - - - x— 2=0, And - - - X— 3=0: X— 4 =0, ar*— 9x^+26a?- X — 5 -24=0. = 0, Multiplying Again, let - Multiplying Again, let - Multiplying - a:*— 14x»+71a:«--154a:-f-120=0, &c. Hence, the product of two equations of the first degree is an equation of the second degree ; the product of three equations of the first degree is an equation of the third* degree, &c. The above equation of the fourth degree has evidently four roots, viz., 2, 3, 4, and 5. 455. The law by which the coefficients are governed may be seen by inspecting the results obtained by the actual multiplication of the factors. Let a, a\ a", a'", a"'\ &c., represent the roots of the gen- eral equation a?'*+6x'^'+ca:''-2+, &c., = 0. Then we shall have (by Art. 453, Cor. 2) x^-f- Jj:'^' + cjr-'+, &c.,=(x— a) {x—d) (x—a") (x— a"0,&c., =0. Or, multiplying the factors and writing the coefficients of the same power of x under each other, we have 1. [x — o) (* — a') - - =x' — a la: +aa'=0. :0. T-ffla'a"tt"'=0 2.(jr-a)(x-a')(x-a") 3. [x—a) (r— «') (r— a") (x— a'")=a —a' —a xHoa' x—aa'a": —a' +aa" — a" +a'a" *-a T^-aa' x^aa'a'' — a' -\-aa" —aa'a"' —a" +aa'" -aa"a" -a'" +a'a" —a'a"a' +aV" •^a"a"' 320 ELEMENTS OF ALGEBRA. [SECT. XI. 456. By attending carefully to the above results, we shall discover the following properties : 1. The coefficient of x in the first term is always 1. 2. The coefficient of x in the second term is the sum of all the roots of the equation taken with contrary signs. Thus, the roots of the equation of the second degree are a and a'; the coefficients of x in the second term are — a, and — a\ In the cubic equation the roots are a, a\ and a"; the coefficients are — a, — a\ and — a". In the equation of the fourth degree the roots are a, a\ a'\ and a'"; the coef- ficients are — a, — a\ — a", and — a"', 3. The coefficient of x in the third term is the sum of all the products of the roots taken two and two, and so on. Thus, in the equation of the fourth degree, the-roots are a, a J a", and a"; and the coefficients in the third term are aa\ aa"y aa"\ a'a[\ a'a"\ a" a'". 4i The last term, which is independent of x, is the product formed from all the roots of the equation after the signs are changed. Thus, in the cubic equation, the last term — aaa'= — ax — a' X — a" ; and in the biquadratic equation, the last term -\-aaa"a"'—-^a X — a' X — a" x — a"\ CoR. 1. If the roots are all negative, the terms of the equa- tion to which they belong will all be positive. For, letting - - x——a, x^—a" xz= — a'\ &c. By transposition, we have a;+a = 0, a;+a'=0, a?4-a"=r0, &c. Consequently, {x-^a) (x-\-a) {x-\-a")', &c., = 0. CoR. 2. If part of the roots are positive and part negative, part of the terms of the equation to which they belong will be positive and part negative. TRANSFORMATION OF EQUATIONS. 457. The transformation of an equation consists in chan- ging its form without destroying the equality of its members. SECT. XI.] GENERAL THEORY OF EQUATIONS. 321 458. TuEOREM. — *dny proposed equation may be transformed into another, the roots of which shall be any multiples or sub' multiples of those of the former. First, In order to demonstrate the above, let us resume the general equation x"+^'-'+cx"-»+ +^x+tt=0. Let y represent the unknown quantity of a new equation, of which the roots are a times greater than those of the pro- posed equation ; then y=ax, and x=±. ^ ' a Substituting this value for x in the general equation, y-A^b^ ^c^ U _l/Llw=0. Multiplying by a", y"4-%'-'+ca'y''--+ +/a'*-'y+a''tt=0. (1.) This last equation will evidently fulfil the conditions re- quired, since y=ax Secondly, Lety=:-; then a: = ay. a Substituting and reducing, as before, we shall obtain V'+ly'-'+~y-'+ +^y+^=o- (2-) Corollary. Since the coefficients in the preceding equa- tion (1) are multiples of the coefficients in the general equa- tion, it is evident that any equation having fractional coeffi- cients may be transformed into another, in which all the terms shall be entire numbers, and the coefficient of whose first term shall be unity. EXAMPLE. Transform the equation ar'-fir^-|-|a?+J=0. Multiplying this equation by 12, the least common multi- ple of the denominators, l2x''\-Gx'-\'Sx-\-9=0, In this equation all the terms are entire numbers, but the coefficient of the first terra is greater than unity. Ss 322 ELEMENTS OF ALGEBRA. [sect. XI. Then, let y=12a;, and a?=J^. ^ ' 12 Whence, by substitution, 122^12^^12 ^ Multiplying by 12^, 2/='4-6/+96y+1296 = 0. If the value of y in this equation be found, that of x can be readily obtained, since x=^. 459. Theorem. — »dn equation may ie transformed into an- other^ the roots of which shall be greater or less than those of the former by a given number. Let us resume the general equation a:"+Z'a:"~'+ca;"~2+ .... ■^tx-\-u=0^ and suppose it were required to transform it into another, whose roots (y) shall be less or greater than those of the given equation by e. First. Let x=y-^e. By substituting y+e for x in the general equation, we shall obtain (y+e)"+i(y+e)'-'+c(y+e)'-2+ +^(y+e)+i*=0. Or, expanding, -Vb ^ 2 -\-{n-\)be + c y"-2+ +e" 0. (L) ■\te This equation will evidently fulfil the conditions required, since y—X—e. Secondly. Let x^=:y—e ; then. Substituting, as before, we have {y—ey-\-b{y—e)"-'-\-c(y—ef-''-\- -\-t[y—e)-\-u = 0. Or, expanding. 8BCT. XI.] GENERAL THEORY OF EQUATIONS. 323 y* — n^ {n I)?.' 3^-'+, »— ^) d'y'^-'e' (n— 2). r(y+e)'=ry'H-3ry'e+3rye'+re* (n— 1). *(y4-e)'=^*+2^« +s^ (»). ^(y+c) =/y 4-^e - - - (n-4-1). w =M By inspecting the above results, it will be perceived that the exponents of e form an ascending series, 0, 1, 2 .n—% n— 1, 71. Then, putting V= to the given equation, W= to the sum of the coefficients of c°, X= to the sum^of the coefficients of Y Z e. — = to the sum of the coefficients of e*. = to the sum 2 '2.3 of the coefficients of c*, &c., we shall have Y =x"+6x'^'H-cx''-'4-(2/+e)'*-^+, (fee, =(e+y— a) (e-{-y—a) .... (e-\-y—ar). The first member, by Cor. 4, equals Xe+L^ e\ With respect to the second member, it follows, from the preceding theorem, 1. The 'part involving e , or the last term, is equal to the prod- uct (y — a) (y — a') .... (y — a"") of the factors of the proposed equation; hence, W=(y—a) (y—a) . (y— «")• 2. The coefficient of e is equal to the sum of the products of these n factors, taken n — 1 and n — 1, or equal to the sum of all the quotients that can be obtained by dividing W by each of the n factors of the first degree in the given equation j hence, V w , vv , w w y — a y—a y — a y — a ' 3. The coefficient of e^ is equal to the sum of the prod- ucts of these n factors, taken n — 2 and n — 2, or equal to the sum of the quotients that can be obtained bydividing W by each of the factors of the second degree > hence, Y W W W 2 {y—a) [y—a) (y—a) {y—a') {y—a"'-') {y-a"% CoR. 6. If two or more of the roots of the given equation are equal to each other ; that is, « = «'=:«", &c., the derived polynomial, which is the sum of the products n factors^ taken n — 1 and n — 1, contains a factor in its difTer- SECT. XI.] GENERAL THEORY OF EQUATIONS. 329 ent parts, which is two or more times a factor of the pro- posed equation. Hence, if the equation contain equal roots^ there must be a common divisor between the first member of the proposed equation and its first derived polynomial. 460. Problem. — Having given an equation^ it is required to discover whether it has equal roots^ and to discover the method ef determining these roots. Resume the general equation j or, since the polynomial W differs from V only by the substitution of y for x, y^Jf-by''-^-\-cy'^^ ty-\-u-0. Then, supposing the equation to contain m factors equa] to y — a\ &c., and also to contain the simple factors y— />, y—qy &;c., then will W^Cy— a)'" {y—a'y {y—a'y — iy—p) (y—q) iy—r) — Whence, by the preceding corollary, j^^mW m'W m'^W WWW y-a y—a y—a" y—p y—q y—r Now (y— a)"~^ (y— a')'"'~^&c., are factors common to all the terms of the above polynomial j hence their product (y—a)'^-^x(y — a')*"'-^ x(y— a")'"'~* is the greatest common divisor of the polynomials W and X; or, D=:(y — a)'»'-^x(y — a')'"'-* x (y— a")'""~* > that is, the greatest common divisor is composed of the product of those factors which enter two or more times in the given equation, each being raised to a power less by unity than in the given equation. Hence, to discover whether an equation W=0 contains any equal roots^ form X, or the derived polynomial of W ; then seek for the greatest common divisor between W and X j if one can- not be obtained, the equation has no equal factors, and, const' quently, no equal roots. 461. Again, if the greatest common divisor (D) is of the first degree, or of the form y—a, make y—a — 0, whence y=a ; and we may conclude that the equation has two roots Tt 330 ELEMENTS OF ALGEBRA. [sECT. XI. equal to a ; if it is of the form [y — a)'*, we may conclude that the equation has n-{-l roots, each equal to a. If the greatest common measure (D) is of the form we must find the two values of y. Let a and a represent those values, then the equation will have two roots each equal to a, and two each equal to a. Hence, the equal roots of an equation may be obtained by finding the greatest common divisor of its first member and its derived polynomial, and solving the equation obtained by putting this common divisor equal to 0. EXAMPLE, Has the equation aP — '7x'^-\-lQx — 12=0 equal roots'? if so, how many, and what are they ] The derived polynomial of this equation is Performing upon this and the first member of the given equation the operations indicated in Art. 137 to find the greatest common divisor, we obtain x—% Then - - - a?— 2=0, And - ' - X *=2. Therefore, we conclude the equation has two roots equal to 2. Now, since the equation has two roots equal to 2, it must (Art. 460) be divisible by (x—2y=x^—4>x-{-4>. Whence - a;^— 737^+ 16a7— 12=(a;— 2)'(a?-3)=0, And - - - - 'x — 3=0, or 07=3, which is the other root of the equation. 462. To show the application of the preceding principles, we will subjoin a few equations with equal roots. EXAMPLES. 1. Reduce the equation 2x'—12x^-\-19x''—6x+9z^0j which has eoual roots. SECT. XI.] GENERAL THEORY OF EQUATIONS. 331 The derived polynomial is 8i»— 36aH-38a:— 6. Whence - - D=«— 3=0, And - - - - X . = 3. Therefore, thfe equation has two roots equal to 3. Dividing its first member by (x — 3y=x^ — 6a7+9, we obtain 2x^+1=0, OTX=±V'^h Hence, the four roots of the equation are 3, 3, v^"^, and — v/^. 2. Reduce the equation x^— 2T*+ac'— 7x»+8x— 3 = 0, which has equal roots. The first derived polynomial is 5a:«_8ar'H-9x»— 14H-8. Whence - D=x'—2x+l, or (x—lf ; And the given equation has three roots equal to 1. Dividing the first member by (x — 1)', or x' — 3x'+3x — 1, we have a:*+x+3=0, or x=—i±is/—n. The five roots are, 1, 1, 1, — i+iv/— 11, and _^-iv/— n. " 3. Reduce the equation xH5x«-f6x^— 6x^— ISr*— 3a:»+8x+4=0, which has equal roots. W= x'4-5x« + 6x'— 6x*— ISr*— 3a^+8x+4.,- X =7x«+30x'+30x*— 24f'— 45x'— 6x+8; D = x*-\- 3jr'+ a^— 3x — 2. Since D surpasses the second degree, we must apply to it the same process we have to W. Its first derived polynomial is 4r'+9x*+2x— 3, And the greatest common divisor ; or, D'=a:+1. Hence, D has two equal roots equal to — 1 ; and, dividing it by (x+l)», or x*+2a;+l, we have • x'-|-x — 2=0 5 or x=l, or — 2. 33!^ ELEMENTS OF ALGEBRA. [sECT. XII. Therefore, D, or x'+3x'+x''—3x—2={x+lf{x—l) (.r+2), And - - - W = {x+lf{x—lY(x+2f. The roots of the equation, then, are 1, 1, —1, —1, —1, —2, and —2. 4. Required the equal roots of the equation x'—Sx'-]-26x'—4>ox'-\'4^bx'—21x'—10x'^20x—^z^0. Ans. 1 and 2. SECTION XIL RESOLUTION OF THE HIGHER EaUATIONS. Resolution of the Cubic Equations ly Cardan's Rule. — Young's Method. — Des Cartes'* Method of Resolving Biquadratic Equations. — JVewton^s Method of ^Approximation. — Resolu- tion of Higher Equations by Trial and Error, 463. We will now proceed to investigate the methods by which affected equations of the third degree may be solved. Equations of this nature may all be exhibited under the ttiree following forms, in which p, p\ and g may be either + or — : (1.) x^-\-px =q; (2.) x'+px' =.q; (3.) x^-\-p'x'^-\-px = q. JSTote. — The known quantities p, p\ and q are here used iii their most general sense, and may be entire or fractional, positive or negative quantities. First Form. 464. In order to deduce a general formula for the reduc- tion of cubic equations of the first form, let us take x^-\-px = q. Let - y-\-z—x^ and Syz= — p; Then - - ^=(y+zY=f+3fz+Syz'+z'i SECT.XIl] AFFECTED EQUATIONS OF THIRD DEGREE. 333 Resolving into factors - ar'=y'-|-3y2(y-f «)+2'; Substituting X for y+2 - a?*=y'4-3y2X+2^} Substituting this value of x^^ y^^z'-}-3yzx-\-px=q ; in the 1st equation • S JResolving into factors - y*-f 2*H-(3y2-f^)x=g, Or - - - ^ - r y^-{-z^+{—p-\-p)x=q; Whence - - * - - y^+z* =q* To determine the values of y and 2, we have the two equations, y»H-2»=y; (L) 3yz=^p; (2.) Dividing the 2d 3 - - yz =—lp ; (3.) Cubing . . - - f2^=-^\jp»; (4.) Squaring the lst7 yS^2fz^+z'=f j (5.) equation - 5 Multiplying the 4.th by 4 - 4^2*=: — ^\p^; (6.) Subtracting the 6th ) ^.o^a^+^e^ 2^ , 3 /^ x from the 5th - i ^ ^ ^ ___ Extracting the square root, y'' — z^=± Vq^+-sSp\ (^O Or y^— r»^±2v/ ig^+^S/.( 9.) Adding the 9th to the 1st - ^f=q±^\^iq'-\-^\p^y 3/ ===r Dividing and evolving - y=\/ i9^ '^i9^'^iSP^i Subtracting: the 9th from the } , ^ — 3 r- 1st ! . . . 1 ^2^=q^^2V}q'+^\p'; Dividing and evolving - ^=\/h9^ >/W'^j\P^9 3/ , Consequently, we take - y=\/i?+ v^4?^+lV?^ And «=\/k— v/T?+SP- Adding the last two equations, and observing that y-[-2z=x, ^=\/iq+ ^W+i^'+\/k-^iq'-^i\p* (A.) Or, since 2= — 2E and x=y+2, y Y/ij+v/ie^+^P'-j^ \^k+^^W-^i\p'- (A^) 834 irLEMENTS OF ALGEBRA. [sECT. XII. Again, taking the equation x^—px—q^ and letting y-{-z—x, and 3yz=+j9, and proceeding as before, we shall obtain* x=^hq+ Viq'-j\f-\-\/-hq-Vh'-^\p\ (B.) Or, since z—'^, and x=:y-\'Z, ^=\/h+Vif-^\f+- ip By the above formulas we may obtain the exact or ap- proximate roots of cubic equations of the first form.f EXABIPLES. 1. Find the value of a: in the equation x^+6x—2. Substituting 6 for jo, and 2 for g, in formula A, we have Whence a?z.r ^/4+^^= 1,587401— 1,259921=,32748 + . jlns. 2. Find the value of x in the equation x^ — 2a:— — 4. By formula B, we have x= \ Ai+ v/-L«.— 8-4- \/z^— n/JJ.— _8_ : V 2 4 27 \ 2 4 27 = y_2+ioy 3+y^-2-J-0 V3 J = ^— 2+l',9245 + e^=^2^1";9245 j =z^^=;o755—e/3,9245z=— ,41226- 1,5773} =—1,9999 + , or —2. ^ns, * In formirias B and B', it is evident that, if-^^p^'^^q^,ihe equation can- not be reduced, since it involves the extraction of the square root of a neg- ative quantity; hence, the value of x can only be obtained by imaginary quantities, and the conditions of the question are incompatible with each other. t These formulas are substantially what is known under the cognomen " Garden's Rule for Cubic Equations.'? The invention of the rule, however, is due to Nicholas Tartalea and to Scipio Ferreus, who found it independ- ently of each other; but Garden first published it to the world. — See Ed. Encycl., Art. Alg. SECT. XII.] AFFECTED EQUATIONS OF THIRD DEGREE. 335 3. Find the value of xin^he equation x* — 6x=12. By formula B', we have 3 / 2 a?=\/ll-f v/JJ.'+z±«-h — =3,1392. Ans, V a ' 4 2 7 4. Find the value of x in the equation o? — 15x=4*. Ans, x=:4. 5. Find the value of a: in the equation x'-|-9x=584'. Ans. x=8. Second Form, 465. If the second term be made to disappear from a cubic equation of this form, there will result a cubic equation of the first form (Art. 459, Cor. 1). Hence, to reduce equations of the second form, we have the following general EULE. Transform the given equation into one of the first form^ and then reduce as before. EXAMPLES. 1. Find the value of x in the equation x'+3x*=54. Let x=z — l = z — Ij Then af'=z'—32^-{-3z—l, And 3x»= +.32;'— 62+3; Adding the two equations, x*+3x'=2r' — 3z +2; Hence - - - 2'— 32+2=54 ; Transposing - - - 2* — 32=52. Applying formula B, we have V 2^ 4 n^y 2 4 ^T?* 3/ =izzr 3/ zzzzzz = \/26+\/2_l±4_-?_I+\/26 — v'iUL* u ; V 4 aT y 4*7 = y/26+v/676^+ y/26— v/676— 1 j = ^26+25,980761921+^26-25,980761921 j 336 ELEMENTS OF ALGEBRA. [sECT. XII. 3. 1 8^ _3^ 2 7 =:^51,980761921+>2/,019238079 ; =:3,732+,267=i 3,999. Hence jt zn 3,999 -l=z 2,999+, or 3. Ans. 2. Find the value of x in the equation x^ — 3a?^=: 16 Let - - x—z-\-%—z-\-\ ; Then - ^—fz^ IG+^V X ^\ or ^ — 32^= 18. Applying formula B' to this last equation, 3/ ZZIZZI. 3 Z—\/uLa_s/11} 1L\ 2 V 2 4 Hence a?=2:+ 1 = 34- 1=4. •^'i*- 3. Find the value of a? in the equation,ir^+6a;^=1600. Ans. 10. 4. Find the value of x in the equation x^ — ,3a7^=^-,004. Ans. \2. Third Form. x^-\-p'x^-\-px=zq. 466. Making the second term disappear, we shall have, as before, an equation of the first form ; hence, the method of reducing an equation of the second form will be the same as that for.the second form. EXAMPLES. 1. Find the value of x in the equation x^ — 6a?^+ 18a? ==22. Let x=z-{-^—z-\-2f then we shall have (Art. 459) 2^-\-6z=2. Applying formula A, we shall find z= v^4 — v^2. Whence a?=z+2=^4— ^2+2=1,5874— 1,2599+2=2,3274. Ans. 2. Find the value of x in the equation x^-\-Sx'^ — 4a? =32. Ans. a? =2. 3. Find the value of x in the eq'^iation a^ — 10a?^+10a?=100. Ans. x=10. SECT. XII. ] RESOLUTION OP HIGHER EQUATIONS. 337 young's method OF RESOLVING CUBIC EQUATIONS. 467. Every cubic equation may be transformed so as to appear under the form r'-f6a^+ca:=N. (A.) 468. Now, suppose that two consecutive numbers in either of the series 1, 2, 3, &c., or 10, 20, 30, frc, or . 1, . 2, . 3, &c., are found such, that, substituting the first for x in the above equation, the result shall be less than N, and, by substituting the second, the result shall be greater than N ; then the first of these numbers will be the first figure of one of the roots of the equation. Let this figure be represented by r, and the other succeeding figures of the same root by $, t, «, &c. ; then, substituting for x the first figure (r) of its root in the equation (A), we shall have r»+^,7-»-fcr=N; *(B.) Whence - - - r=—^--, (C.) 469. Let the remaining figures of the root equal y, then x=r-\-y: substituting this value for x in the first equation (A), we have cy-f-cr =cx bf^2bry-^br': f+S rf-^-Sr'y Adding, y*+(3r+W+(3'^+2ir-|-c)y-h(r'+ir»+cr)=N. (D.) But, if - 6' =3r +A, (1.) c' =3r»+2ir+c, (2.) N'=N-r»-ir'-cr, (3.) the above equation becomes y34.fty+cy=N'. (E.) 470. This equation is in all respects similar to the first (A) ; and, since s is the first figure of the root y of this equation, substituting as before. Whence - - s=-^-^ — ;• (^0 29 U ^ f-^cr =cx \ r4-Jr»=ii» ) =:N. + r'=x' ) 338 ELEMENTS OF ALGEBRA. [sECT. XII. Supposing the value of s found, and putting t, u^ &c , equal to 2r, or y—z-rs, we have c'z-\-c s=c'y ; Adding, !_' z'+{3s-\-¥)z'+(3s'-\-2Vs-]-c')z+(s' + b's'-\-c's)^'N\ (G.) But, if - b''=Ss+b\ (4.) c'' =^Ss'-\-Ws+c\ (5.) W=W—s'—b's'—c's,{6.) the above equation becomes z'-\-b''z' + c''z=W\ (H.) an equation which is in all respects similar to the first. Hence we may proceed in the same way to find the first figure f, in the root z, and so on till we have found all the figures in the root x of the proposed equation. 471. Now, by observing the formation of the coefficients b\ c' in the equation marked (F), and recollecting that r, being the first figure of the root, must be greater than s, it will appear obvious that c' must form a part of the divisor s^-\-b's+c\ and if r be already known, the value of c' will become known (2), which may, therefore, be used as a trial divisor for finding s ; the same may be observed of the next and the succeeding divisors j but these trial divisors, c'', c"', &c., will continually approach nearer the true divisors. 472. Now, if the first figure of the root r be found by trial, and r-\-b be multiplied by it, and the product added to c, the sum will be the first divisor ; thus, r{r+b)= r'-\-br 7'^-\-b7'-\-c=: 1st divisor. (7.) u N N Hence - r= —— , ,, . - r-1roj'-\-c r{r-\-b)-\-c If under these two expressions we write r', and add up the three, we shall obtain c'; thus, SECT. XII.] RESOLUTION OF HIGHER EQUATIONS. 339 r* 3r«+2Z»r+c=c'. (8.) Having obtained c', we have a trial divisor of N' that will enable us to determine more easily the next figure s of the root. When s is found, the second divisor may be computed ; thus, s*-^-b's-\-c'= 2d divisor. Hence - - s= ^ = ^1 s'-{-b's-\-c' s{s-\-3r+b)-{-3r'-\'2br^t By a similar process we shall obtain /'+*"/+c"= 3d divisor. Hence, /= Also, Hence, tt= Also, tt'+6'"tt+c'"= 4th divisor. &c., Scc\, &c. The above formulas may be readily applied to the reduc- tion of cubic equations. By a careful inspection of them, we may obtain the following general RULE. 1. Put down c, the coefficient of a?, and a little to the right place the absolute number^ which is to be considered as a divi' dend, the figures of the root forming the quotient. 2. Place the first figure of the root^ found by trials in the quo- tient^ above which u^ite the coefficient of a?-, observing that its unites place he over the unit's place of the quotient. 3. Multiply the value of the quotient figure, taking in those 340 ELEMENTS OF ALGEBRA. [sECT. XII. above by that value ; add the product to c, and the sum is the first divisor, 4. Write the square of the quotient figure just found under the first divisor^ add it to the two sums immediately above^ and the result will be the trial divisor for finding the next figure of the root. 5. Find now the next figure of the root^ and to its value (in- cluding those above it) prefix three times the preceding^ taking the value of the figure above it ; multiply the result by the last found figure ; add the product to the trial divisor ^ and we shall have the true divisor ; and in the same manner are the succeed- ing divisors to be obtained. EXAMPLES 1. Reduce the equation a:^-f8a:^-f 6a7=75 .9. SECT. XII.] RESOLUTION OP HIGHER EQUATIONS. 3^1 § H- Id CI r-t 00 II i 3 + 3 CO 1 II •E '« CO v** kt 5 «+ ^ IS "» Z II 11 II II II II II CO 00 :> ^ s CO O CO ^ lO >r> c< c* 3 5S d CO in 05 CO kO SP CO CO 00 o 00 o oo t- CO t- t- .^ CO CO Oi O (N O 00 o i« Id r-^ r^ O c< o CM »o \Ci CO CO CO CO II II If II II il II II II 00 + c< ' • II l^ C< 00 + 4- II z Cd + + C •? •a C4 CO ^ CO [I 4- G\ 00 + + il » ^ o •* oo II > s ^ CO CO 342 ELEMENTS OF ALGEBRA. [sect. XII. J^ote. — By inspecting the preceding example, we shall ob- serve that if, after obtaining three places of decimals in the right-hand column, we had continued to reject the remain- ing decimals, we should have had the root equally correct to three places of decimals. Now, in order that the num- ber of decimals in the last column may not exceed three, it is obvious that the divisor corresponding to the first deci- mal in the root must contain but two decimals, that corre- sponding to the next decimals of the root but one, and that for every succeeding decimal in the root the right-hand di- git of the corresponding divisor must be cut off. It should, however, be observed, that whatever would have been car- ried had the complete multiplication been performed, is still to be carried for the mcrease of the next figure 5 and, indeed, if the figure cut off exceed 5, one is to be carried to the next figure. Hence the work of the above example may be rendered more concise, and will stand as follows, the figures cut off being placed a little to the right : 6 75.9 12.4257 + 20 52 26 4 "50 5.76 55.76 .16 ./ 61.68 .3a 44 61.9 844 004 62.3 1 892 76325 23.9 22 . 304 1.596 1.240 688 .356 .312 312 827625 .044 484375 62.4 SECT. XII.] RESOLUTION OF HIGHER EQUATIONS. 343 2. Reduce the equation a:'-f.x'=500. This equation is the same as x'-f x'-}-0x=500 ; hence i=l, c=0, and N=500. The first figure of the root is 7. 56 56 49 500 392 108 104. |7. 61727975, &c.,=x. 736 161 13.56 3 1, .264 .887181 174.56 36 1. 1, .376819 . 323862 188.48 .2381 52957 37859 188.7181 1 15098 13251 188 . 9563 1669 1847 1704 189 . 123|2 189 . 290 5 143 133 10 . iiQio loioi;;, 9 3. Reduce the equation x* — 17x'-|-54a:=350. An8. 1=14 . 954, &c* 4. Reduce the equation x'-f2x'+3x= 13089030. An8, a?=235. 5. Reduce the equation j^-|-2x^— 23x=70. Ans. a:=5 . 1345, &c. 6. Reduce the equation a:* — 2x=5. ^iM.a;=2» 09455 14815423265917, &c. 344 ELEMENTS OF ALGEBRA. [sECT. XII. DES cartes' method OF RESOLVING EQUATIONS OF THE FOURTH DEGREE. 473. Every equation of the fourth degree may be reduced to the form x*-\-bx^-\-cx''-\-dx+e=zO. (A.) This equation may also be transformed into another which shall want the second term (Art. 459) j thus, x'-]-c'x^-{-d'x+e'=zO. (B.) 474. Now if we can arrive to a solution of the equation in this form, in which the roots sustain a given relation to the original equation (A), the complete solution of that equation may be effected. Now, suppose B to be formed by the product of x'+px+q^O, (1.) x^-i-rx-{-s=0, (2.) two equations in which p, q, r, and s are unknown quanti- ties, and we shall obtain by the actual multiplication of the factors (1) (2), and, taking the sum of the coefficients of the equal powers of a:, co'+{p+r)af'-^{s-hq+pr)x'+{ps-[-qr)x-{-qsz=zO. (C.) Whence - p+r =0, or r= — p; (3.) s+q-{-pr=c'; (4.) ps+qr =d'} (5.) qs =e'. (6.) Or, substituting — p for r in (4) and (5), and transposing, they will become S+q^c'+p'-, (7.) S-q=t (8.) P And, by subtracting the square of (8) from the square of (7), we have c'2+2c>2_j_^4_» ^4^5^ or 4e'., P Or, clearing of fractions, and arranging the terms with reference to the highest power of ^, we have /+2cy+(c'2_4eV-(/'2:=0. (D.) SECT. XH.] RESOLUTION OP HIGHER EQUATIONS. 346 li p*z=z, this equation will become , z^+ac'z^-f-Cc''— 4«>— ^"=0. (E.) Now, if we add and subtract equations (7) and (8), and divide the result by 2, we shall have ,= ic'+ii^+±; (9.) 9=ic'+iF'-~ (10-) 4-75. From these two formulas (9) and ( 10), p being known from equation (E), s and q can be obtained. Hence, substituting —p for r in equation (2), and reducing the two equations (1) and (2), we shall have X=-i,p±V j^^q ; (11.) X= + ip±Vip'-s. (12.) These equations (11) and (12) give the four roots of the biquadratic equation (B). Cor. 1. The cubic equation (E) gives three roots; but the same values of x will be obtained, whichever of the roots be used. Cor. 2. If the roots of the cubic equation (E) are all real, the roots of the biquadratic equation (B) will be real also. If only one root of the cubic equation (E) be real, then the proposed biquadratic (B) will have two real and two imaginary roots. 476. The above formulas may be readily applied to the reduction of equations of the fourth degree. EXAMPLES. 1. Reduce the equation x*— 3a:*H-6x+8=0. Comparing this equation with formula (B), we shall have c'= — 3, (/'=r6, and e'=:S; and substituting these values (oi c', d\ and e' in formula (E), it becomes 2^-62*-- 23z-36=0. Reducing this equation, 2=9 j hence p=y/z=V^=~-:t3» Substituting -|-3 for;? in formulas (9) and (10), we have «=ic'H-i;>«+^=~^+^4-f=4; 346 ELEMENTS OF ALGEBRA. [sECT. XII. Substituting these 3, 4, and 2 for 7?, s, and q in formulas (11) and (12), we have X::=z—ip± Vif^9-=^-^± v/f=:2 = -|±i=:— 1, Or —2 ,' Hence the four roots of the biquadratic equation are — 1, -% l+lV^T, and i-v/^=7. 2. Reduce the equation x*—4,x^—8x-{-32—0. Ans. 4, 2, — 1 + v^^^, and —\—^~—3, 3. Reduce the equation o:^— 9a:^+30a;2— 46a?+24 = 0. *> ^7i5. 1, 4, 2+ y^^, and 2— v/^^. 4. Reduce the equation a:' + 1 6a;' + 9 9a;^4- 228a; +144=0. Arts. —1, —3, — 6 + 2v^^^, and — 6— 2v/"^. 477. This is an expeditious method of finding the approx- imate root of an equation, when its near root is given or has been ascertained by trial, and is equally applicable, whatever be the degree of the equation. 478. Let us resume the general equation a.n_^j^n-i_^^^«-2_|_ *a;'4-^a;4-^~.0. (A.) Then let a represent the near root of the equation which is known, and z represent the part to be added to make the root complete \ then a?=a + S. (1.) Substituting this value for x in the first equation (A), we have («+2:)"+Z>(a4-z)"-' + c(a+2r)"-2+ 5(a+2f)2+jf(a+2r) + M = 0. (B.) Then, transforming, as in Art. 459, W+X^+I^^+^^ z-=.^, (C.) 479. Now, since r, by hypothesis, is a proper fraction, the terms that involve z^^ z^^ &;c., being less than 2, may be re- SECT. XII.] RESOLUTION OF HIGHER EQUATIONS. 347 jected from the equation without departing far from rigid exactness. The equation (C) will then become W + X2=0. (2.) W Whence - - ^=~^' ^^'^ But, comparing with the transformations in Art. 459, we shall find W= or +bar-' -{-car-* «a'-f/a+w; (4.) X =na'^' + {n—l)bar-^+(n—2)ca'*~^. . . 2sa +e. (5.) Substituting these values for W and X in equation (3), we have aT-^-ba^'^ + car-* sa^-^ta-\-u ^jv v ^~ na''-'-{-{n-i)ba''-'-\-{n-2)car^ 2sa+t' ^ '' The numeral value of this expression should be calculated to within one or two places of decimals, and added to the root (a) found by trial. Let the resulting approximate root be represented by a', then a'=a-{-z ; and if z' represent the part still to be added, we shall have a"'+ba'^'-\ -c a'^^ sa'^+ta'-{-u ^ ~ na'^'+(»— l)6a"*^H (n—2)ca"^ 2sa'+i ^ Letting a" represent the third approximate root, we shall have a"=a'+z', 480. Proceeding in this manner, the approximation may be carried to any assigned degree of exactness. EXA3IPLES. 1. Reduce the equation r'-f-2x^—8x=24. By making trial of 1, 2, 3, and 4, we shall find that the root of the equation is between 3 and 4, and very nearly equal to 3. Then n = 3, a=3, b or 5=2, c or ^=—8, u= — 24, and Xz= a+z. By substituting these values in formula (D), we have z=— ?!±^ • ^'~:?- ^~"--=: A = 0.09 j hencex=3.09,nearly. 3.3^+4.3—8 ^r > Again, if 3 . 09 be substituted for a' in formula (D'), we shall have 348 ELEMENTS OF ALGEBRA. [sECT. XII. 3.(3.09/+4.(3.09)— 8 ' a?=3. 09364. 2. Reduce the equation x^-\-x^-{-x = 90. Here 7^=3, b or s=zl, c or ^=1, w= — 90, and a will be found =4< ; hence x=4f-\-Zj and we shall have ^=— i!±iji±ri?=:/^ = 0. 1 J hence a:=4. 1, nearly. 3.4^ + 2.4+1 '' ' » ^ Again, ^.^ _(4.ir+(4.ir+(4.1)- 9_0^Q QQ,33 hence .=.4. 3.(4.1)+2.(4.1)+1 10283. 3. Reduce the equation a;*— 38a:'+210j:=^+538a:+289=:0. ^7X5.07=30.535653. 4. Reduce the equation x'-\-6x^—10x'—n2x''—201x-\-110 =0. ^ns. 07=4.46410161. RESOLUTION OF HIGHER EaUATIONS BY TRIAL AND ERROR. 481. The roots of cubic equations may also be found to a sufficient degree of exactness by successive approxima- tions. From the laws of the coefficients, as stated in Art. 455, it is evident that the roots must be such that, when their signs are changed, their product shall be equal to the last term of the equation, and their sum equal to the coeffi- cient of the second term. By considering this law, some estimate may be formed of the values of the roots, and a trial may then be made, by substituting in the place of the unknown letter its supposed value. If this proves too small or too great, it may be increased or diminished, and the tri- als repeated till one is found which will nearly satisfy the conditions of the equations. 482. Now, since the errors in the results will be very nearly proportioned to the errors in the assumed numbers, after we have assumed two approximate values, and calcu- lated the errors which result from them, we may obtain a more exact correction of the root by the following propor- tion : SECT. XII.] RESOLUTION OP HIGHER EQUATIONS. 349 The difference of the errors : to the difference of the assumed numbers : : the least error : to the correction required, E'er, letting N and n= the assumed numbers, S and s= the errors of these numbers, and R and r= the errors of the results, we shall have R : r : : S : 5 very nearly. Hence, by Art. 369, R-r : S-5 i:r:s, 483. If the value which is first found is not sufficiently correct, this may be taken as one of the numbers for a sec- ond trial J and the process may be repeated till the error is diminished as much as is required. There will generally be an advantage in assuming two numbers whose difference is . 1, or .01, or .001, &c. EXAMPLES. 1. Reduce the equation ar'—8x^-f 1707—10=0. Here the signs are alternately positive and negative, therefore (Art. 455) the roots must all be positive ; their product =10, and their sum =8. Suppose J7=5 . 1 or 5 . 2 ; then, By 1st supposition, :! (5 . l)'-8 . (5 . 1)'+ 17 . (5 . 1)- 10 = 1 . 271 . , ^ ' ^ ' ^ ' ^ errors. By 2d, (5 . 2)='-8 . (5 . 2)*+ 17 . (5 . 2)- 10 =2 . 688 Difference of errors - - - - 1.417 Then - - 1 . 4 : . 1 : : 1 . 27 : . 09. Hence - - a:=5 . 1-0 . 09=5 . 01, nearly. To correct this farther, suppose a?=5 . 01, or 5 . 02; then, By 1st supposition, (5.01)»-8.(5.01)»+17.(5.01)-10=;0.121> By2d,(5.02)'-8.(5.02y+n.(5.02)-10=0.246 S Difference of errors - - - - 0.125 Then - - . 125 : .01 : : . 121 : . 01. Hence - - a:=5 . 01— . 01=5. This value of x satisfies the conditions of the equation; for, 5'-8x 5^-1- 17x5—10=0. Therefore, one of the roots of the equation is 5. 30 350 ELEMENTS OF ALGEBRA. [sECT. XII. To find the other two roots, let the first member be divi- ded by X — 5 (Art. 453), and the quotient put equal to 0. a?3_8x2+17a?— 10 |a?— 5 x^ — 5a?^ x^ — 'Sx-\-2 — 3a?2— 170-— 10 — 3a?2_15a? 2a?— 10 2a?— 10 Hence - - a?^— 3a? + 2=0. Reducing - - x r=;2 or 1. The three roots of the given equation, then, are 5, 2, and 1. 2. Reduce the equation x^ — 8a:^+4a?+48=0. Let x=4f . 1, or 4 . 2. Substituting successively these values for x in the equa- tion, we have, 1st, (4 . 1)^-8 . (4 . 1)^+4 . (4 . l)+48:=-l . 159 ) 2d, (4.2)'— 8.(4.2)2+4.(4.1) + 48=— 2.282i Difference of errors - - - — 1.123 Then - - — 1 . 1 : . 1 : : — 1 . 1 : . 1. ^ Hence - - a?=4 . 1 — 0.1=4. This value of a? satisfies the conditions of the equation j for, 4'— 8.42+4.4+48=0. Therefore, one of the roots is 4. Dividing the first member of the given equation by x — 4, the quotient is a?2_4a?— 12=0. Reducing - a?=6, or— 2. The roots of the equation are — 2, 4, and 6 3. Reduce the equation x^~\~16x'^-^65x — 50=0. ^ns. 1, 5, and 10. 4. Reduce the equation 2a:^— 16a?='+40a?2— 30a7=— 1. ^ns. a? =1.2847. 5. Reduce the equation a:5+2a?^+3a?'+4a?'+5a?=5 . 4321. Jlns. x=S .414455. SECT. XII.] RESOLUTION OP HIGHER EQUATIONS. 351 young's METHOD OF RESOLVING HIGHER EQUATIONS. ^S4f. The method of solving cubic equations in Art. 4-72 is obviously adapted to equations of any higher degree j and, by carefully inspecting the properties of equations, and the mode of reduction there employed, we shall be able to deduce, for the reduction of equations of the ;ith degree, the following general RULE. 1. Arrange the coefficients of the given equation in a row^ commencing with that of the first term then find by trial the first figure of the root. 2. Jldd the product of the first root figure and the first coeffi- cient to the second coefficient ; the product of this sum and the same figure to the third coefficient^ and so on to the last coeffi- cient^ and the last sum will be the divisor. Multiply this by the first figure of the root^ and subtract the product from the tei'm constituting the right-hand member of the equation ; the remain- der will form the first dividend. 3. Repeat this process with the first coefficient and these sums, and the number under the last sum will be the .trial divisor /or the next figure. 4. Perform a similar process with the first coefficient and these second sums, stopping under the n — Ith coefficient, jjgain, per- form a similar process with the same first coefficient and these last sums, stopping under the n — 2/A coefficient, and so on till the last sum falls under the second coefficient. 5. Find now, from the trial divisor and the first dividend^ the next figure of the root, and proceed with the last set of sums and this new figure exactly the same as with the original coefficients and the first figure in finding the preceding divisor, and the sec- ond divisor will be obtained. Then proceed, as before, to find the SECOND DIVIDEND, and so on till the work has been carried to a sufficient degree of exactness, Kote. — The work may be contracted by cutting ofl' deci- mals as before 352 ELEMENTS OF ALGEBRA. [sECT. XII. EXAMPLES. 1. Eeduce the equation a;'— 3x2+7507=10000. Operation. • 9 —3 81 78 162 240 243 483 29 . 44 75 702 777 2160 10000 6993 |9 . 8860027, &c., =x. 9 9 3007 2677 .5616 18 9 2937 409 . 952 329 306 .4384 .1662 27 9 3346 . 952 434 . 016 23 23, . 2722 .2616 36.8 .8 512.44 30.08 3780 . 968 46. 110 106 78 37.6 .8 542 . 52 30.72 3827 . 07|8 46 . 36 28 27 38.4 .8 573 . 24 3. 14 3873 . 44 3.50 1 3|9 . |2 576 ..3.8 3876. 9|4 3.1 3.5 579 . |5 3 3880 . 4 518|3 Kote. — By bringing down one period of decimals, we have found the root to eight places of figures. If another period, or eight decimals, had been brought down, the root might have been found to twelve plav:jes of figures, or x^=: 9 . 88600270094. 2. Eeduce the equation a?'+6x4— 10a?'— 112ac2—207a?= 110. SECT. XII.] RESOLUTION OF HIGHER EQUATIONS. 353 Operation, 6 4 —10 40 30 56 "86 72 158 88 246 10.56 —112 120 ,624 —207 32 110 14.46410161. -700 10 4 8 344 —175 1408 810 667 . 05984 14 4 352 632 1233 434 . 6496 142.94016 133.46395 18 4 984 102. 1667.6496 477.4144 9.47621 9 . 24089 22 4 1086. 106. .624 .912 2145.0640 79 . 3352 23532 23158 26.4 .4 256 . 56 10.72 1193. 111. 536 264 2224 . 399,2 80 . 389 374 232 26.8 267.28 1304.800 2304.788 142 4 10.88 17.453 5.434 139 27.2 278.16 1322.2513 2310.2212 4 11.04 17.56 5.44 27.6 289.20 1339. 8|1 2315.66 4 1.68 17.6 14 218. |0 290 .^8j8 1357.4 2315. 8|0 1.7 1.2 1 292.16 1358.16 2131 1|5. 9 1 1 2|9]4 1|3;5|9 3. Reduce the equation ar'-h2j:*+3a:'+4j:*+5a?=54321. Ans. x=8 . 41445475, &c. 4. Reduce the equation x«-|-2x5+3x*+4ar'-|-5x--f 6x= 654321. ^ns. Xz=% . 95697957, &c. 5. Reduce the equation a:'— 3x<'— 2 . 5x'-|- lOx^+ar*— 9x»-j- 2j=2. Jlns. x=1 , 62599736, &c. 6. Reduce the equation a:«-f 10x'-h21x— 55x'— 100J?*-f- 525x'^-8(>4x«— 630x=216. Ans, x= . 79128785, &c. NOTES. Note A, page 13, Art. 2. The term Quantity seems to be used by writers on Mathematics with a great degree of vagueness, and the definitions of it are liable to many objections. For instance : '• Quantity is a general term, embracing every- thing which admits of increase or diminution."* Now, it is with perfect consistency that the natural philosopher speaks of " increasing or dimin- ishing" heat : so mental power or energy may be increased or diminished ; and so, also, passion, resentment, anger, benevolence, or love may be in- creased or diminished. Hence, by the above definition, they are included under the term Quantity, and are, consequently, objects of mathematical investigation. The incorrectness of this definition needs no farther illus- tration. With regard to Number, we believe it cannot properly be included under Quantity. Dugald Stewartf remarks : '• As to number and propor- tion, it might be easily shown that neither of them fall under the definition of Quantity, in any sense of that word." Believing the term Quantity in- correctly applied in most treatises on Algebra, we have endeavoured to substitute the word number in its place. Dr. Reidt suggests a distinction of Quantitj into proper and improper. Proper Quantity is that which is measured by its own kind, such as ex- tension and- duration. Improper Quantity is that which cannot be meas- ured by its own kind, but to which we assign a measure in some proper quantity that is related to it. Velocity, density, elasticity, '^7(2«+iy^9(2'/j+l)»'^ll(2/i+iy"'^^- } NOTES. 367 The reductioQ of the above formula was not introdaced into the body of the work, on account of its length and complexity. Let a represent the base of the system, t+1 be any number in the com- mon arithmetical scale, and x its logarithm ; then (Art. ) a'=r+l. Again, let a=14-6, then (l+6)'=l+r; and to find the log. of l+u, we must solve this equation, in which x is the unknown quantity. Involving both members to the power m, we have (I4.i)"«=(l+t,)"'; mrfmx— 1) ,, mx(mT— l)(mr— 2) Expanding, l+7iu:i+— ^-^ *X4»+ ^ 'xA»+,&c. fn(m— 1) . m(m— l)(m— 2) _ , =l-t-mp+ ^ g ^ Xi>'-H— ^ ^Xr'+, »-^t.*-f j^r'-^, & C. Or, if we take M=, r^— -, — L_ t^ r— (a_l)-i(a-l)»4-i(a-l)»-,&c., log. (r-f l)=M(r-ir»4-Jr'-U*4-»p»— , &c.) (A.) Since a is a constant quantity, M, which is termed the modulus of sys- tem, must also be a constant quantity. ^=2-;302k509=*^294482. Or, more correctly, M— 434294481903251827651 1289189166. This is the series (A) earliest known for the calculation of logarithms But the difficulty with it is that it will either diverge, or not converge so quickly as to make the summation of a few terms of it a irafficient ap- 358 NOTES. proximation to the value of x or log. {v-\-l), unless tj be a proper fraction sufficiently small. If v be nearly equal to 1, the series converges too slowly to be of any use ; and if v be greater than 1, the series diverges, and is, consequently, useless. We may, however, transform this series (A) into others, so as to ob- tain a series that will apply in every possible case. For if 1 — v instead of l-\-v be used, we shall obtain ■»= v^ log. {l—v)=M{—v—-—-—\v*—^v'^—, &c.) But log. (1 — v)=: — log.- — ; therefore =—M{v+^v''-\-lv^+\v*-\-^v^+, &c.) 1 log.— -=M(?j+ii;='-}-^2)3-|-iv*-|-|«5+, &c.) (B.) By adding together the formulas (A) and (B), and observing that log. {li-v)+ log. j^= log. YZT^, we have log. J-^=M(2r+iv3+fr*+|r^+, &c.) =2M[v-hiv'+^v'+\v-'+, &c.) (C. ) Again, let us put — =-, then «=-— , and substituting in the above formula, it becomes Transposing - log. u^log. ^+2M ^^^H-j ^^-^y +| ^^-=^y^ Now, letting u=n-\-l and t—n, we shall have u—t=l, and u-\-t=2n-{-l ; substituting these values in the preceding formula, it becomes log. (n+l)=log. «+2M ^^_j^g-^^g^^+^.jj,+, &c.^ This series evidently converges very rapidly, even when w=l ; but converges more rapidly as n increases. Hence, having found the log- arithm of any number, we may easily find the logarithm of the next higher in the natural series of numbers by the application of this formula. See Edinburgh and Recs's Encyclopedias. 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 2Apr'503S£- IIBRARY USE Airrtise^ REC D LP MAR 19 1960 9 '53 D Tf REC-n ' n APR 3 5 '3 53 . -^Sfci'eainl l4Jar.'65MsX '^^C'D to ^2 9'64. ilAM- i\ ^ >- ^ ^ ' ^-^ ' I- K i-m3 M, ^:a£3 LD 21A-50w-4,'59 (Al724sl0)476B General Library University of California m: Berkeley 911266 THE UNIVERSITY OF CALIFORNIA UBRARY