..z-a is the G.C.M. of the first and last of the proposed quantities; and their least com. mult. is (ax3+ a3x — a2x2- a1)(x2— a2) . . . . . . (1). The other quantity is (x2+ a3)(x2—a3) . . . . . . . . . (2). ......... The G.C.M. of (1) and (2) is (x2-a3)x the G.C.M. of ax3+a3x—a2x2—a1 and x2+a3. Rejecting the factor a in the former quantity, x2+ a3) x2+a2x—ax3—a3 ( x—a x2+ a2x -ax3-a3 - ax3- a3 .. the G.C.M. of (1) and (2) is (x2+a3)(x2—a3); .. least com. mult. required is (ax3+a3x—a2x3—a1)(x2— a2), or ax3- a2x2-a3x+a®. 119. A more expeditious method of applying the preceding rule to find the Least Com. Mult., when it can readily be done, is that of resolving each quantity into its component factors, as follows:-taking the last Example, (1) x3-a3x-ax2+ a3= x2(x− a)—a2(x—a), =(x2—a3)(x−a). (2) x1-a1=(x2+a2)(x2—a3)=(x2+a2)(x− a)(x+a). =a(x2+a1)(x-a). Now the G.C.M. of (2) and (3) is (x2+a2)(x−a); .. least com. mult. of (2) and (3) is a(x-a)......(4). Again, the G.C.M. of (1) and (4) is x2-a2; .. least com. mult. required is a(x^—a1)(x−a), or ax3-a31-a3x+a®. 119*. The G.C.M. of two or more quantities is the L.C.M. of all the common measures. For the L.C.M. of all the common measures contains all the factors that appear in them, and therefore contains all the factors common to the proposed quantities: but their G.C.M. contains all these common factors: therefore the L.C.M. in question is either equal to, or a multiple of the G.C.M. But since every common measure of the proposed quantities measures their G.C.M., i.e. their G.C.M. is a common multiple of all their common measures, therefore (Art. 117) this G.C.M. is either equal to, or a multiple of the L.C.M. of all the common measures. But these two conditions cannot be both satisfied unless the above G.C.M. and L.C.M. are equal, therefore they are equal. [Exercises H.] ADDITION AND SUBTRACTION OF FRACTIONS. 120. If the fractions to be added together have a common denominator, their sum is found by adding the numerators together for a new numerator and retaining the common denominator. This follows from the principle laid down in Art. 99, Ex. 1. Or thus; in each of the fractions the unit is divided into the same number, b, of equal parts; and it is plain that a of these parts, or toge ther with c of the same parts, or, a 121. If the fractions have not a common denominator they must be transformed to others of the same value, which have a common denominator (Art. 112...114), and then the addition may take place as before. Here a is considered as a fraction whose denominator is 1. By Art. 116 or Art. 119 the Least Com. Mult. of the denominators is found to be a1-1; therefore the sum required is 122. If two fractions have a common denominator, their difference is found by taking the difference of the numerators for a new numerator and retaining the common denominator. Or, the same reasoning will apply as that in. Art. 120. 123. If they have not a common denominator, they must be transformed to others of the same value, which have a common denominator, and then the subtraction may take place as before. The sign of bd in the numerator is negative, because every part of the latter fraction is to be taken from the former. (See Art. 87.) a2+2ab+b2 a2-2ab+b2 bc + bd ac-ad-bc-bd = 3+2x-2-2x [Exercises I.] 1 MULTIPLICATION AND DIVISION OF FRACTIONS. 124. To multiply a fraction by any quantity multiply the numerator by that quantity and retain the denominator. number of equal parts (since the denominators are the same), and c times as many of these equal parts are taken in the latter as in the former, therefore the latter fraction is c times as great as the former. plied by its denominator, the product is the numerator. 126. COR. 2. The result is the same, whether the numerator be multiplied by a given quantity, or the denominator divided by it. from the division of its denominator by c. 127. To divide a fraction by any quantity multiply the denominator by that quantity, and retain the numerator. a part of this is the quantity to be divided being a cth part of b what it was before, and the divisor the same. (Art. 96.) 128. COR. The result is the same, whether the denominator is multiplied by the quantity, or the numerator divided by it. Let the fraction be ac c, it becomes or a the quantity which arises from the division bdc bd' of the numerator by c. 129. To prove the Rule for the Multiplication of Fractions*. RULE. The product of two fractions is found by multiplying the numerators together for a new numerator, and the denominators for a new denominator. OBS. In the strict sense of the word Multiplication, which supposes a quantity to be added to itself a certain number of times, to multiply by a fraction would be impossible: the operation must therefore be understood in an extended sense. Since to multiply a by b is the same thing as to take b of it, we shall easily perceive that the extension of the sense of multiplication will lead us to conclude that to multiply by will be the same thing as to take с a с ths of it; i. e. to take the dth part of it, and then to take c of such parts. a a But the ath part ofis (Art. 127), and c of such quantities as being ac bd bd taken will produced (Art. 124): this therefore is the result required. 130. To prove the Rule for the Division of Fractions. RULE. To divide one fraction by another, invert the numerator and denominator of the divisor, and use it as a multiplier according to the rule for multiplication. OBS. Since ordinary division is the inverse of ordinary Multiplication, the signification of division here must be extended to mean the inverse of multiplication in all cases. Then the Quotient will always be such a quantity as when multiplied by the Divisor will produce the Dividend; therefore if the Dividend can be put into two factors, one of which is the Divisor, the other must be the Quotient. The proofs of the Rules for Multiplication and Division of Fractions given in former Editions of this work are very objectionable. The fallacy consists in assuming bdx x bx.dy-bd.xy and symbols. ED. bdy = which have only been proved true for integral values of the |