129. It will happen in most cases that the annexing of ciphers to the numerator will never make it divisible by the denominator without remainder. For example, try to reduce ¹/₇ to a decimal fraction. 7)1000000000000000000, &c. ------------------- 142857142857142857, &c. The quotient here is a continual repetition of the figures 1, 4, 2, 8, 5, 7, in the same order; therefore ¹/₇ cannot be reduced to a decimal fraction. But, nevertheless, if we take as a numerator any number of figures from the quotient 142857142857, &c., and as a denominator 1 followed by as many ciphers as were used in making that part of the quotient, we shall get a fraction which differs very little from ¹/₇, and which will differ still less from it if we put more figures in the numerator and more ciphers in the denominator. Thus, 1 {is less} 1 3 {which is not} 1 --- { } --- by --- { } --- 10 { than } 7 70 { so much as } 10 14 1 2 1 --- --- --- --- 100 7 700 100 142 1 6 1 ---- --- ---- ---- 1000 7 7000 1000 1428 1 4 1 ----- --- ----- ----- 10000 7 70000 10000 14285 1 5 1 ------ --- ------ ------ 100000 7 700000 100000 142857 1 1 1 ------- --- ------- ------- 1000000 7 7000000 1000000 &c. &c. &c. &c. In the first column is a series of decimal fractions, which come nearer and nearer to ¹/₇, as the third column shews. Therefore, though we cannot find a decimal fraction which is exactly ¹/₇, we can find one which differs from it as little as we please. This may also be illustrated thus: It is required to reduce ¹/₇ to a decimal fraction without the error of say a millionth of a unit; multiply the numerator and denominator of ¹/₇ by a million, and then divide both by 7; we have then 1 1000000 1428571¹/₇ --- = ------- = ----------- 7 7000000 1000000 If we reject the fraction ¹/₇ in the numerator, what we reject is really the 7th part of the millionth part of a unit; or less than the millionth part of a unit. Therefore ¹⁴²⁸⁵⁷/₁₀₀₀₀₀₀ is the fraction required. EXERCISES. Make similar tables} 3 17 1 with } ---, ---, and ---. these fractions } 91 143 247 } 3 The recurring} --- is 329670,329670, &c. quotient of} 91 17 --- 118881,118881, &c. 143