Elements of Algebra

176 ELEMENTS SECT. III. curs, it is easy to find its value. Let there be given, for example, $0.296296$: its value will be $\frac{296}{999} = \frac{8}{27}$, by dividing both its terms by $37$. This fraction ought to give again the decimal fraction proposed; and we may easily be convinced that this is the real result, by dividing $8$ by $9$, and then that quotient by $3$, because $27 = 3 \times 9$: thus, we have 9) $8.000000$ 3) $0.888888$ ------------ $0.296296, \ldots$ which is the decimal fraction that was proposed. 539. Suppose it were required to reduce the fraction $\frac{1}{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10}$ to a decimal. The operation would be as follows: 2) $1.00000000000000$ -------------------- 3) $0.50000000000000$ -------------------- 4) $0.16666666666666$ -------------------- 5) $0.04166666666666$ -------------------- 6) $0.00833333333333$ -------------------- 7) $0.00138888888888$ -------------------- 8) $0.00019841269841$ -------------------- 9) $0.00002480158730$ -------------------- 10) $0.00000275573192$ -------------------- $0.00000027557319$