Elements of Algebra

172 ELEMENTS SECT. III. The decimal fraction which expresses the value of $2/3$, is also continued ad infinitum; for we have $3)2.0000000$ $0.6666666 = 2/3$. Which is also evident from what we have just said, because $2/3$ is the double of $1/3$. 528. If $1/4$ be the fraction proposed, we have $4)1.0000000$ $0.2500000 = 1/4$. So that $1/4$ is equal to $0.2500000$, or to $0.25$: which is evidently true, since $2/10 + 5/100 = 25/100 = 1/4$. In like manner, we should have for the fraction $3/4$, $4)3.0000000$ $0.7500000 = 3/4$. So that $3/4 = 0.75$: and in fact $7/10 + 5/100 = 75/100 = 3/4$. The fraction $5/4$ is changed into a decimal fraction, by making $4)5.0000000$ $1.2500000 = 5/4$. Now, $1 + 25/100 = 5/4$. 529. In the same manner, $1/5$ will be found equal to $0.2$; $2/5 = 0.4$; $3/5 = 0.6$; $4/5 = 0.8$; $5/5 = 1$; $6/5 = 1.2$, \ldots When the denominator is $6$, we find $1/6 = 0.1666666, \ldots$ which is equal to $0.666666 - 0.5$: but $0.666666 = 2/3$, and $0.5 = 1/2$, wherefore $0.1666666 = 2/3 - 1/2$; or $4/6 - 3/6 = 1/6$. We find, also, $2/6 = 0.333333, \ldots = 1/3$; but $3/6$ becomes $0.5000000 = 1/2$; also, $5/6 = 0.833333 = 0.333333 + 0.5$, that is to say, $1/3 + 1/2$; or $2/6 + 3/6 = 5/6$. 530. When the denominator is $7$, the decimal fractions become more complicated. For example, we find $1/7 = 0.142857$; however it must be observed that these six figures are continually repeated. To be convinced, therefore, that this decimal fraction precisely expresses the value of $1/7$, we may transform it into a geometrical progression, whose first term is $142857/1000000$, the ratio being $1/1000000$; and consequently, the sum $= \frac{\frac{142857}{1000000}}{1 - \frac{1}{1000000}} = \frac{142857}{999999}$ (by multiplying both terms by $1000000$) $= 1/7$. [See Art. 520.] 531. We may prove, in a manner still more easy, that the decimal fraction, which we have found, is exactly equal to $1/7$; for, by substituting for its value the letter $s$, we have