{ tal opera tions. Fundamen- Rule I. The terms of the dividend are to be ranged is still a remainder from which the operation may be Fundamen according to the powers of some one of its letters ; continued without end. This expression of a quotient tal operations. and those of the divisor, according to the powers is called an infinite feries; the nature of which shall be considered afterwards. By comparing a few of the first terms, the law of the series may be discovered, by divisor, they are ranged according to the powers of a. to any number of terms wanted. of the General Rule. main: that, therefore, from the nature of division, must be the true quotient. Note. The fign • is sometimes used to express the quantities between which it is placed: divided by att. $ 2. OF FRACTIONS. Definitions. called a proper fraction. it is called an improper fraction. ther a fraction, it is called a mixt quantity. 5. The reciprocal of a fraction, is a fraction whose nu. merator is the denominator of the other; and whose a+b)2 + 2ab + b(a+b denominator is the numerator of the other. The a't ab reciprocal of an integer is the quotient of divided by that integer. Thus, ab + b* b ab to bi is the reciprocal of g; and is the reciprocal a a m a of th. The distinctions in Def. 2, 3, 4, properly belong to common arithmetic, from which they are borrowed, and are scarcely used in algebra. The operations concerning fractions are founded on the following proposition: If the divisor and dividend be either both multiplied or both divided by the same quantity, the quotient is the same ; or, if both the numerator and denominator of the fraction be either multiplied or divided by the fame quantity, the value of that fraction is the faine. +6ab--26 +6ab-25* Fonda ticos 401 viding both numerator and denominater by any com- tions are also to be divided by their fimple divisors, and the quantities are always to be ranged according to the powers of the same letter. The fimple divisors in the given quantities, or in the remainders, do not affect a compound divisor which is wanted ; and hence also, to make the division succeed, any of the dividends may be multiplied by a simple Cor, 2. A fraction is multiplied by any integer, by quantity, Besides the fimple divisors in the remainders can make no part of the common measure sought; and any compound divisor which does not measure the die multiplying the denominator by that integer. vifor from which it proceeds, it may be taken out. EXAMPLES. ai-b)a'- 2ab +6°(1: a62 - 2ab +26* Remainder, which divided by - 2b is a-ba-b'(a+b a'-62 If the quantities given are 8a2b-10ab3 +- 264, and 4a2-5ab +67, and 3ab out of the second, it is, 3a1-3ab + ab-13. As the latter is to be divided by 45)63(1 187)391(2 the former, it must be multiplied by 4, to make the 45 374 operation succeed, and then it is as follows: 18)45(2 17)187111 402-5ab2) 12a1-12a*b+4ab-4b3(39 1.29-15a*b+ 3ab2 3ab+ab-463 18. This remainder is to be divided by b, and the new dividend multiplied by 3, to make the division pro ceed. Thus, za'+ab-4b4)12a-15ab +362( 4 122°+4ab--1664 - 1gab + 1962 mainder must measure the dividend, 2. A quantity which measures both divisor and divi. and this remainder, divided by -19b, givesa-b, which being made a divisor, divides 3a' to ab-462 without a appears, that the Prob. II. To reduce a Fraction to its lowest Terms. their greateft common measure, which may be found by Prob. 1, which is absurd. 75abc_3a When the greatest common measure of algebraical Thus, 125bex Š x 2gbc being the greateft common quantities is required, if either of them be limple, any 5x' a4-64 a' +1 Also, a'-9362 9a4b-9a3b2 + 3a*b3--3ab4 _9a3+3ab2 greateś compound divisor is wanted, the preceding 8a?be-loab3 +26* the greatest Sab---262 rule is to be applied ; only, common measure being a-bxb, by Prob. 1. PROBA a3 ma m .bt za 2a 2 Fundamen. PROB. III. To reduce an Integer to the Form of a Cor. 2. A fraction, whose numerator is a compound Fundamen- quantity, may be distinguished into parts, by dividing al operations. tions. Rule. Multiply the given integer by any quantity for the numerator into several parts, and setting each over a numerator, and set that quantity under the pro. the original denominator, and uniting the new frac- tions (reduced if necessary) by the figns of their nu merators. a-2ab + b? a 2ab f? + PROB. VI. To multiply Fractions, . Rule. Multiply their numerators into one another, to nators to Fractions of equal Value, that all have minators, multipled into one another, shall give the denominator of the product. atb amb a_a Ex. xarici a od , and the product For, if is to be multiplied by c, the product is on shall give the common denominator. but if it is to be multiplid only by the former pro. duct must be divided by d, and it becomes (Cor. 2. to the preceding problem.) Then a=bm, and cdn,, = AC a ca b b Prob. V. To add and subtract Fraclions. Prob. VII. To divide Fractions. add or subtract the numerators ; and the sum or dif- Rule. Multiply the numerator of the dividend by the the numerator of the quotient. Then multiply the denominator of the dividend by the numerator of the divisor, and their product shall give the denomi. nator. ad bc From fubt.the difference is Or, Multiply the dividend by the reciprocal of the dibd visor; the product will be the quotient wanted. From the nature of division it is evidenty that, when bc b Fax but ais , s; the former quotient must be multiplied by b, and it bc da. Or, let 7 =m, and a =n; then a=bm, and c=dn; bdn be Scholium. 2-6* By these problems, the four fundamental operations may be performed, when any terms of the original. 2a*--a?+62 _a* + b3 quantities, or of those which arise in the course of the operation, Ex, in into the form of a fraction, with the fame denominator also ad=bdm and be=bdn; therefore (odym =) 2% * 2x3 , &c. -ax-** 2** 2x3 2x a a 2x* a A L G E B R 406 L Part I. A. Of Propose 3ax Cor. 1. Of four arithmetical proportionale, ary three of Propor, tion. Example. Mult. tion. 26 being given, the fourth may be found. Thus, let a, b, c, be the ift, 2d, and 4th terms, and let x be the 3d which ia fought. Then by def. atc=b+x, and x=a+c-b. a33 a? бах? Prod. - 20'+7 6x* Cor. 2. If three quantities be arithmetical propor2** tionals, the sum of the extremes is double of the midB**)a' *x(aast dle term; and hence, of three such proportionals, any 4 + ax two being given, the third may be found. 2. Of Geometrical Proportion. -axtx Definition. If of four quantities, the quotient of the firft and second is equal to the quotient of the third and fourth, these quantities are said to be in geometri- cal proportion. They are also called proportionals. :c:d. b ő and the proportion is expressed thus, a : 6:c. + Prop. I. The product of the extremes of four quan. tities geometrically proportional is equal to the product Let a:b::c: d. Then by Def.7= If ad=bc, then dividing by bd, 6 = áo that is, ) a:b::c:d. Cor, 1. The product of the extremes of three quan- tities, geometrically proportional, is equal to the square of the middle term. Cor. 2. Of four quantities geometrically proportional, Ex. Let a, b, c, be the three firit ; to find the 4th. ax=bc der the difference of the two quantities, it is called bc arithmetical proportion; and if we consider their quo- and divining both by a, x= tient, it is called geometrical proportion. This laft be This coincides with the Rule of Three in arithmetic, ing moft generally useful, is commonly called fimply and may be considered as a demonftration of it. In proportionals. applying the rule to any particular case, it is only to be observed, that the quantities must be so connected 3: Of three geometrical proportionals, any two being given, the third may be found. Prop. II. If four quantities be geometrically pro- portional, then if any equimultiples whatever be taken Prop. In four quantities arithmetically proportional, of the firk and third, and also any equimultiples whatthe sum of the extremes is equal to the sum of the first be greater than that of the second, the multiple of of the fourth the third will be greater than that of the fourth; and the a means. =12, 2x Of Equa- the first and third, ma and mc may represent any equi- and is expressed by placing the fign = between Of Equae tions. any equimultiples of the second and fourth. Since 2. When a quantity stands alone upon one side of an . on one side of an equation, and there are only known quantities on the other, that equation is said to be the highelt power of the unknown quantity to be The E. S Simple, the unknown quanti- ad, quation Quadrat, 17, E, of the fame book. ty in any term be the ) 3d, &c. ) is called Gubic, &c. But the exponents of the unknown quantities are sup- posed to be integers, and the equation is supposed to be cleared of fractions, in which the unknown quanti- ty, or any of its powers, enter the denominators. 5 is a simple equation; 3x Comp. Thus, x tau 4+b:b::c+d:d ; Convert. 2:4- -6::6- when cleared of the fraction by multiplying both fides by 2x, becomes 6x-5=24% a quadratic. For fince adzbc, it is obvious, that in each of these x3_-27+=x®-20 is an equation of the sixth order, &c. cases the product of the extremes is equal to the pro As the general relations of quantity which may be duct of the means ; the quantities are therefore pro- treated of in algebra, are almost universally either that portionals. (Prop. 1.) of equality, or such as may be reduced to that of equa- branches of the science. The most common and useful application of algebra from certain given relations to each other, and to such as are known; and hence it has been called the analy. tical art. The equations employed for expressing these be, the deducing equations containing only one un- The solution of the different orders of equations will tionals. be successively explained. The preliminary rules in the the Ś 1. Of fimple Equations, and their Resolution. Rule 1. Any quantity may be transposed from one side of an equation to the other, by changing its figa. CHAP III. Thus, if 3x-10=2x+5 Then, 3x-2x=10+5 or x=15 Thus also, 5x+b=a to 2x By trapfp. 3x=1-b. This rule is obvious from prob. 1. and 2.; for it is equivalent to adding equal quantities to both sides of 1. An Equation may in general be defined to be a the equation, or to subiracting equal quantities from propofition afferting the equality of two quantities; both sides. Cars |