The decimal expansion of 1/19

Question

Solve in your head, no long division. What is 	frac{1}{19} as a decimal, to 8 decimal places? Aim: under 25 seconds.

Accepted Answer

\frac{1}{19} = \frac{1}{20 - 1} = \frac{1/20}{1 - 1/20} = \frac{1}{20}\left(1 + \frac{1}{20} + \frac{1}{20^2} + \cdotsight). Each term is \left(	frac{1}{20}ight)^k = 	frac{5^k}{10^{2k}}, so successive powers of 5 land two decimal places apart: 0.05 + 0.0025 + 0.000125 + 0.00000625 + \cdots Read off the pairs 05, 25, 125, 625, \dots (the powers of five), stacked two places apart and allowed to carry: \frac{1}{19} = 0.05\,26\,31\,57\ldots Adding carefully: 0.05 + 0.0025 = 0.0525, then +0.000125 = 0.052625, then +0.00000625 = 0.05263125, and so on, settling toward 0.05263157\ldots True val…

The decimal expansion of 1/19

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