Elements of Algebra

CHAP. VIII. OF ALGEBRA. 25 $\frac{a}{b}, \frac{2a}{2b}, \frac{3a}{3b}, \frac{4a}{4b}, \frac{5a}{5b}, \frac{6a}{6b}, \frac{7a}{7b}$ &c. 87. To be convinced of this, we have only to write for the value of the fraction $\frac{a}{b}$ a certain letter $c$, representing by this letter $c$ the quotient of the division of $a$ by $b$; and to recollect that the multiplication of the quotient $c$ by the divisor $b$ must give the dividend. For since $c$ multiplied by $b$ gives $a$, it is evident that $c$ multiplied by $2b$ will give $2a$, that $c$ multiplied by $3b$ will give $3a$, and that, in general, $c$ multiplied by $mb$ will give $ma$. Now, changing this into an example of division, and dividing the product $ma$ by $mb$, one of the factors, the quotient must be equal to the other factor $c$; but $ma$ divided by $mb$ gives also the fraction $\frac{ma}{mb}$, which is consequently equal to $c$; and this is what was to be proved: for $c$ having been assumed as the value of the fraction $\frac{a}{b}$, it is evident that this fraction is equal to the fraction $\frac{ma}{mb}$, whatever be the value of $m$. 88. We have seen that every fraction may be represented in an infinite number of forms, each of which contains the same value; and it is evident that of all these forms, that which is composed of the least numbers, will be most easily understood. For example, we may substitute, instead of $\frac{2}{3}$, the following fractions, $\frac{4}{6}, \frac{6}{9}, \frac{8}{12}, \frac{10}{15}, \frac{12}{18}$, &c. but of all these expressions $\frac{2}{3}$ is that of which it is easiest to form an idea. Here therefore a problem arises, how a fraction, such as $\frac{8}{12}$, which is not expressed by the least possible numbers, may be reduced to its simplest form, or to its least terms; that is to say, in our present example, to $\frac{2}{3}$. 89. It will be easy to resolve this problem, if we consider that a fraction still preserves its value, when we multiply both its terms, or its numerator and denominator, by the same number. For from this it also follows, that if we divide the numerator and denominator of a fraction by the same number, the fraction will still preserve the same value. This is made more evident by means of the general expression $\frac{ma}{mb}$; for if we divide both the numerator $ma$ and the denominator $mb$ by the number $m$, we obtain the fraction $\frac{a}{b}$, which, as was before proved, is equal to $\frac{ma}{mb}$.