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I'm looking to discover more fractions that have interesting* decimal expansions. (I'm asking out of curiosity, there is no particular academic reason as far as I'm concerned).

Here are a few examples:

$\dfrac{1}{99}=0.010101\dots$

$\dfrac{1}{999}=0.001001001\dots$

(and so on...)

This post talks about:

$\dfrac{1}{243}=0.\overline{004115226337448559670781893}$

This Numberphile video talks about how:

$\frac{1}{999^2}=0.00000100200300400500600...$

'generates' the 3-digit integers (except $998$). Similar patterns arise with $1/99$, $1/9999$, and so on.

*I realize that 'interesting, fun or noteworthy' might make this question a bit open-ended or subjective, hence the 'soft-question' tag. Then again, I find it difficult to be more specific about what I'm looking for.

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    $\begingroup$ I’m not sure what you find interesting, but I’ve always been fascinated by how $\frac{1}{7}=0.\overline{142857}$, and the other sevenths have the same 6 numbers repeating, in the same order, but just starting at a different number. For example, $\frac{2}{7}=0.\overline{285714}$ $\endgroup$
    – Joe
    Sep 6, 2019 at 21:03
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    $\begingroup$ You also have $1/9^2=0.0123456\ldots$ (all the single digits except $8$) and $1/99^2 =0.00010203040506\ldots$ (can you guess the missing double digit?) $\endgroup$
    – Henry
    Sep 6, 2019 at 21:05

3 Answers 3

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Given any repeating decimal expansion, you can find the rational number that generates it. For example, let's say I think that $.\overline{02040608}$ is interesting, then there is a process of calculating the corresponding fraction: $$x = .\overline{02040608}$$ $$10^8x = 2040608.\overline{02040608}$$ $$10^8x - 2040608 = x$$ $$x = \frac{2040608}{10^8-1}.$$ You can generate any periodic decimal expansion you want via this method.

Edit: This method also illuminates why we see stuff like 99, 999, 9999 so much in these `interesting' decimal expansions.

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I guess 1/7 is probably the canonical example here. Taking multiples cycles the digits in the decimal expansion.

Here's something I wrote previously that goes into considerably more detail on the topic. To excerpt part of it:

1/7 = 0.142857142857142857...

Multiplying by ten, we see that

10 * 1/7 = 1.42857 142857 142857...

On the other hand, 10 * 1/7 = 10/7 = 1 + 3/7, so

1 + 3/7 = 1.42857142857142857...

Subtracting 1 from each side, we have

3/7 = 0.42857142857142857...

If we multiply by 100 instead of ten, 14 + 2/7 = 100 * 1/7 = 14.2857142857142857... so 2/7 = .2857142857142857...

Continuing with this game, we have

142 + 6/7 = 1000/7 = 142.857142857142857... so 6/7 = 0.857142857142857...

1428 + 4/7 = 10000/7 = 1428.57142857142857... so 4/7 = 0.57142857142857...

14285 + 5/7 = 100000/7 = 14285.7142857142857... so 5/7 = 0.7142857142857...

so we see that the decimal expansions of each of 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 are cycles of the same digits.

So what were the ingredients here?

  • 1/7 is a rational number with lowest-common-terms denominator relatively prime to ten, so we get a repeating decimal that repeats right off the bat.
  • By successively multiplying 1/7 by powers of ten, we get all numbers whose fractional parts cover every multiple of 1/7 between 0 and 1.

The rest of the discussion goes on to characterize such numbers.

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The number $$1/89 = 0.0112359550561798\ldots$$ is interesting as the digits form the Fibonacci sequence $1,1,2,3,5, 8, 13, 21,$... at least for a while. If you want more terms of the sequence you can use, e.g.,

$$1/9899 = 0.000101020305081321\ldots$$

which gives you the two-digit Fibonacci numbers or

$$1 / 998999 = 0.000001001002003005\ldots$$

which gives you the three-digit Fibonacci numbers.

Why does this happen? I cannot tell you; it is a secret you must discover yourself...

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