MULTIPLICATION AND DIVISION OF FRACTIONS. RULE.-Multiplication. Multiply the numerators together; the result will be the numerator of the product. Multiply the denominators together; the result will be the denominator of the product. Division. Invert the terms of the divisor, and proceed as if it were multiplication. α If is to be multiplied by c, the meaning is, that a things If is b of the denomination b, are to be multiplied by c: the product will, therefore, be c times those things; that is, b ac But if, in stead of c, the multiplier be c divided by d, that is, mer result will evidently be d times what it ought; that result must, therefore, be divided by d; that is, the true product will ac be bd : hence α с ac : so that the product of two fractions is found by multiplying together their numerators for a new numerator, and their denominators for a new denominator. If α с e ac e there be three fractions, then, since 7=bd they are reducible to two, namely, and of which the product, as ace bd just shown, is and, in a similar manner, may the truth of bdf' the rule be proved for four fractions, and for any greater number. Again, if is to be divided by then that which, multiplied ī с α by the divisor ď tient. But, from what is shown of multiplication, produces the dividend a hence, ad с acd which, multiplied by, produces; because X = bc a bcd d x-; which proves the rule for division. a C ad α Зах (1) Multiply 3x by 4y In this second example, it is seen that x, y, enter as factors into both numerators and denominators; and, if allowed to remain, would, of course, appear in both numerator and denominator of the product: they are, therefore, at once expunged, and the product obtained free of these superfluous factors; and, in general, it is better to remove such superfluous factors thus early, rather than to allow them to find their way into the product, where, from being involved with other quantities, they may sometimes be lost sight of. It is well, therefore, to search the numerators and denominators for common factors before applying the rule; and, if any be found, to suppress them, treating the fractions as if such common factors did not enter. If numerator and denominator of this last result be multiplied in example 7; but the preceding is the better form, as fewer factors enter. a a + x (3) (2+3) ((−1). (4) (1+2) (1–3)—–— y y a 2 x2+-+2x (17) ( x2 + 1 + 2 x − 2 ) + ( x −1 +1). (18) √(x2+2x−2+1/1−1). (19) √(~ +6—2). (20) √(x2+ * a2 b2 2ax 3 ab-bx). (21) Divide x by x+a, as far as five terms in the quotient, and then annex the correction furnished by the remainder. (22) Divide a+x by b+x, as far as four terms in the quotient, and annex the correction. (23) Divide a2 by (a+x)2 as far as four terms, and annex the correction. (24) Required the square of 1—2+ far as four terms. 2x2 3x3 + &c. as 3 4 43. NOTE.-Division of fractions is sometimes indicated by writing the divisor under the dividend with a line between, the sign being dispensed with. In such a case we may change the form into that of multiplication, making the dividend one factor and the divisor inverted the other, and proceed as in the foregoing examples; but, without doing this, the simplified result may often be more speedily obtained by leaving the form unchanged, and multiplying dividend and divisor by the product of their denominators. Thus, in the following instance, if the numerator and denominator, that is, the quantities above and below the black line, be multiplied by 2, we shall have the numerator and denominator of this last being each multiplied by 12, the product of 3 and 4. And it is plain that, instead of the product of the denominators of the proposed quantities, we may use any common multiple of them: thus, in the following example, employing a2-12, the least common multiple of the denominators belonging to the principal numerator and denominator, we have And it may be further remarked of division that, whichever way we indicate the operation, whether as in these latter examples, or by interposing the sign÷between dividend and divisor, we may always expunge factors which are common to the two numerators, and factors which are common to the two denominators, before applying the rule; for instance, we may copy off examples 12 and 14 above, thus:a2-x2 a3x+ax3 h-x result. and 1 ;; or, rather, - which is indeed the final ax And this hint will be of service to the learner in treating some of the above examples: examples 5 and 6, for instance. If, in all cases, the divisor be at once inverted, and superfluous factors expunged only after the form has been changed to that of multiplication, there will frequently be a needless repetition of symbols. It is not, however, always possible to discover, by mere inspection, whether or not two expressions have a factor in common: it is in general easy to do so when the factor is a simple quantity; but not when it is more complicated: to ascertain whether or not two polynomials admit of a compound factor, requires in general a special rule for the purpose: this rule we shall now investigate and apply, not so much with a view of bringing it into operation in the reduction of fractions, since, as already observed, we do not think the advantage gained worth the trouble, but because the general method of finding the common measure of two polynomials has been recently applied to the important problem of the analysis of numerical equations. We shall give the investigation in a smaller type, to imply that the learner may, for the present, pass it over or not, as he feels disposed. 44. To find the greatest common measure of two or more quantities: In speaking of the greatest common measure, that is, of the greatest common divisor of two algebraic polynomials, we do not imply anything as to numerical magnitude: this has already been noticed at page 84: what is meant by the greatest divisor is that divisor which is composed of the greatest number of algebraic factors, or which involves the highest powers of the letters according to which the polynomials are arranged. The greatest divisor is, therefore, the divisor of highest dimensions, as respects these letters. This term, dimension, and the equivalent term, degree, is of frequent occurrence in algebra: thus, such an expression as ax2+bx+c, is said to be an expression of two dimensions, as respects x, or an expression of the second degree; ax3+px+q, is an expression of three dimensions, or an expression of the third degree, and so on, the highest exponent always, pointing out the dimensions or degree of the |