Elements of Algebra

CHAP. XII. OF ALGEBRA. 171 CHAP. XII. Of Infinite Decimal Fractions. 525. We have already seen, in logarithmic calculations, that Decimal Fractions are employed instead of Vulgar Fractions: the same are also advantageously employed in other calculations. It will therefore be very necessary to shew how a vulgar fraction may be transformed into a decimal fraction; and, conversely, how we may express the value of a decimal, by a vulgar fraction. 526. Let it be required, in general, to change the fraction $\frac{a}{b}$, into a decimal. As this fraction expresses the quotient of the division of the numerator $a$ by the denominator $b$, let us write, instead of $a$, the quantity $a.0000000$, whose value does not at all differ from that of $a$, since it contains neither tenth parts, hundredth parts, nor any other parts whatever. If we now divide the quantity by the number $b$, according to the common rules of division, observing to put the point in the proper place, which separates the decimal and the integers, we shall obtain the decimal sought. This is the whole of the operation, which we shall illustrate by some examples. Let there be given first the fraction $\frac{1}{2}$, and the division in decimals will assume this form: $\begin{array}{l} 2)1.0000000 \\ \hline 0.5000000 \end{array} = \frac{1}{2}$. Hence it appears, that $\frac{1}{2}$ is equal to $0.5000000$ or to $0.5$; which is sufficiently evident, since this decimal fraction represents $\frac{5}{10}$, which is equivalent to $\frac{1}{2}$. 527. Let now $\frac{1}{3}$ be the given fraction, and we shall have, $\begin{array}{l} 3)1.0000000 \\ \hline 0.3333333 \end{array} = \frac{1}{3}$. This shews, that the decimal fraction, whose value is $\frac{1}{3}$, cannot, strictly, ever be discontinued, but that it goes on, ad infinitum, repeating always the number $3$; which agrees with what has been already shewn, Art. 523; namely, that the fractions $\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \frac{3}{10000} + \ldots$ ad infinitum, $= \frac{1}{3}$.