question is then reduced to finding the first of the numbers 70, 700, 7000, 70000, &c., which can be divided by 16 without
Divide these numbers, one after the other, by 16, as follows:
| 16 ) | 70 | (4 | 16 ) | 700 | (43 | 16 ) | 7000 | (437 | 16 ) | 70000 | (4375 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64 | 64 | 64 | 64 | |||||||||||
| 6 | 60 | 60 | 60 | |||||||||||
| 48 | 48 | 48 | ||||||||||||
| 12 | 120 | 120 | ||||||||||||
| 112 | 112 | |||||||||||||
| 8 | 80 | |||||||||||||
| 80 | ||||||||||||||
| 0 |
It appears, then, that 70000 is the first of the numerators which is divisible by 16. But it is not necessary to write down each of these divisions, since it is plain that the last contains all which came before. It will do, then, to proceed at once as if the number of ciphers were without end, to stop when the remainder is nothing, and then count the number of ciphers which have been used. In this case, since 70000 is 16 × 4375,
| 70000 | , which is | 16 × 4375 | , or | 4375 | , |
|---|---|---|---|---|---|
| 160000 | 16 × 10000 | 10000 |
gives the fraction required.
Therefore, to reduce a fraction to a decimal fraction, annex ciphers to the numerator, and divide by the denominator until there is no remainder. The quotient will be the numerator of the required fraction, and the denominator will be unity, followed by as many ciphers as were used in obtaining the