128)300000000000(234375 256 ---- 440 384 ---- 560 512 ---- 480 384 ---- 960 896 ---- 640 640 --- 0 Since 7 ciphers are used, it appears that 30000000 is the first of the series 30, 300, &c., which is divisible by 128; and therefore ³/₁₂₈ or, which is the same thing (108), ³⁰⁰⁰⁰⁰⁰⁰/₁₂₈₀₀₀₀₀₀₀ is equal to ²³⁴³⁷⁵/₁₀₀₀₀₀₀₀ or ·0234375 (135). From these examples the rule for reducing a fraction to a decimal is: Annex ciphers to the numerator; divide by the denominator, and annex a cipher to each remainder after the figures of the numerator are all used, proceeding exactly as if the numerator had an unlimited number of ciphers annexed to it, and was to be divided by the denominator. Continue this process until there is no remainder, and observe how many ciphers have been used. Place the decimal point in the quotient so as to cut off as many figures as you have used ciphers; and if there be not figures enough for this, annex ciphers to the beginning until there are places enough. 146. From what was shewn in (129), it appears that it is not every fraction which can be reduced to a decimal fraction. It was there shewn, however, that there is no fraction to which we may not find a decimal fraction as near as we please. Thus, ¹/₁₀, ¹⁴/₁₀₀, ¹⁴²/₁₀₀₀, ¹⁴²⁸/₁₀₀₀₀, ¹⁴²⁸⁵/₁₀₀₀₀₀, &c., or ·1, ·14, ·142, ·1428, ·14285, were shewn to be fractions which approach nearer and nearer to ¹/₇. To find either of these fractions, the rule is the same as that in the last article, with this exception, that, I. instead of stopping when there is no remainder, which never happens, stop at any part of the process, and make as many decimal places in the quotient as are equal in number to the number of ciphers which have been used, annexing ciphers to the beginning when this cannot be done, as before. II. Instead of obtaining a fraction which is exactly equal to the fraction from which we set out, we get a fraction which is very near to it, and may get one still nearer, by using more of the quotient. Thus, ·1428 is very near to ¹/₇, but not so near as ·142857; nor is this last, in its turn, so near as ·142857142857, &c. 147. If there should be ciphers in the numerator of a fraction, these must not be reckoned with the number of ciphers which are necessary in order to follow the rule for changing it into a decimal fraction. Take, for example, ¹⁰⁰/₁₂₅; annex ciphers to the numerator, and divide by the denominator. It appears that 1000 is divisible by 125, and that the quotient is 8. One cipher only has been annexed to the numerator, and therefore 100 divided by 125 is ·8. Had the fraction been ¹/₁₂₅, since 1000 divided by 125 gives 8, and three ciphers would have been annexed to the numerator, the fraction would have been ·008.