Elements of Algebra

CHAP. XII. OF ALGEBRA. 175 Let us first suppose, that a single figure is constantly repeated, and let us represent it by a, so that $s = 0 \cdot aaaaaaa$. We have $10s = a \cdot aaaaaaa$ and subtracting $s = 0 \cdot aaaaaaa$ we have $9s = a$; wherefore $s = \frac{a}{9}$. 538. When two figures are repeated, as ab, we have $s = 0 \cdot ababab$. Therefore $100s = ab \cdot ababab$; and if we subtract $s$ from it, there remains $99s = ab$; consequently, $s = \frac{ab}{99}$. When three figures, as abc, are found repeated, we have $s = 0 \cdot abcabcabc$; consequently, $1000s = abc \cdot abcabc$; and subtracting $s$ from it, there remains $999s = abc$; wherefore $s = \frac{abc}{999}$, and so on. Whenever, therefore, a decimal fraction of this kind oc- Suppose that when this happens we have added $s$ cyphers, and that $q$ is the integral part of the quotient: then abstracting from the point, we shall have $\frac{n \times 10^s}{d} = q + \frac{n}{d}$; wherefore $q = \frac{n}{d} \times (10^s - 1)$. Now, as $q$ must be an integer number, it is required to determine the least integer number for $s$, such that $\frac{n}{d} \times (10^s - 1)$ or only that $\frac{10^s - 1}{d}$ may be an integer number. This problem requires several cases to be distinguished: the first is that in which $d$ is a divisor of 10, or of 100, or of 1000, &c and it is evident that in this case there can be no circulating fraction. For the second case, we shall take that in which $d$ is an odd number, and not a factor of any power of 10; in this case, the value of $s$ may rise to $d - 1$, but frequently it is less. A third case is that in which $d$ is even, and, consequently, without being a factor of any power of 10, has nevertheless a common divisor with one of those powers: this common divisor can only be a number of the form $2^c$; so that if, $\frac{d}{2^c} = e$, I say, the period will be the same as for the fraction $\frac{n}{d}$, but will not commence before the figure represented by $c$. This case comes to the same therefore with the second case, on which it is evident the theory depends. F. T.