0.0204081632... as a repeating fraction

Asked Oct 19 '21 at 18:18

Active Oct 19 '21 at 18:26

Viewed 66 times

I calculated $\left(\frac17\right)^2$ and the calculation returned a decimal where a series of numbers going up by an exponent of 2 were all concatenated together at the end of the decimal. the number looked like this:

0.02040816326530612...

It started with 02, 04, 08, 16, 32, and kept going until the hundredth digit inevitably started interfering with previous parts of the pattern. Is there a way to somehow encode all of this into a repeating decimal? because I'm certain that it repeats eventually.

edited Oct 19 '21 at 18:26

ryang

38,879
14
81
179

asked Oct 19 '21 at 18:18

person the human

4

It repeats after $42$ digits – J. W. Tanner Oct 19 '21 at 18:21
2

Note that $49$ is a factor of $10^{42} - 1$. This indicates that the decimal must repeat every 42 places (though it could repeat even more often - it could repeat every 2 places, for instance). – Mark Saving Oct 19 '21 at 18:24
Thanks, but how did you get 42? – person the human Oct 19 '21 at 18:33
3

Cf. Euler's theorem – J. W. Tanner Oct 19 '21 at 18:47
2

Besides the repeating decimal form, we might also write it as $\sum_{n=1}^\infty\frac{2^n}{100^n}=\sum_{n=1}^\infty \frac1{50^n}$. – Berci Oct 19 '21 at 20:00

0.0204081632... as a repeating fraction

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