The decimal expansion of 1/998

Question

Solve in your head, no long division. What is 	frac{1}{998} as a decimal, to about 12 places? Aim: under 25 seconds.

Accepted Answer

\frac{1}{998} = \frac{1}{1000 - 2} = \frac{1/1000}{1 - 2/1000} = \frac{1}{1000}\left(1 + \frac{2}{1000} + \frac{2^2}{1000^2} + \cdotsight). Each term is 	frac{2^k}{1000^{k+1}} = 2^k 	imes 10^{-3(k+1)}, so the powers of two land three decimal places apart, in tidy triples: 0.001 + 0.000002 + 0.000000004 + 0.000000000008 + \cdots Read off the groups 001, 002, 004, 008, 016, 032, 064, 128, \dots: \frac{1}{998} = 0.001\,002\,004\,008\,016\,032\ldots True value: 0.001002004008016032064\ldots The pattern keeps doubling (256, then 512) until a group crosses 1000 and carries into the previous tri…

The decimal expansion of 1/998

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