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For example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying. Also, when I divided $1$ by $998$, I got an amazing result of $0.0010020040080160320641282...$. I have a longer result from my cool memory, which is $0.001002004008016032064128256513026052104208517034068136272545090180360721442885...$What explains why this doubling thing is happening? I love it when I see it! Also I don't know if it repeats (I'm talking about 1/998). I think it does because I looked it up and it said 498 numbers repeat every time, so it's rational. I mean seriously, how does this doubling happen? I hope I can receive good information from you!

VladInTheTaylor
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    $$\frac{1}{1-z} = \sum_{k=0}^\infty z^k,$$ hence $$\frac{1}{10^d - a} = \frac{1}{10^d}\frac{1}{1-\frac{a}{10^d}} = \frac{1}{10^d} \sum_{k=0}^\infty \left(\frac{a}{10^d}\right)^k = \sum_{k=0}^\infty \frac{a^k}{10^{(k+1)d}}.$$ – Daniel Fischer Nov 19 '14 at 15:03
  • Nice pattern-spotting. Similar $\frac1{97}=0.01030927...$ and $\frac1{997}=0.001003009027081..$ etc. :) – Hypergeometricx Nov 19 '14 at 15:09
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    Also look at $1/997$ and, oddly, $1/9899$ – Empy2 Nov 19 '14 at 15:09
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    @Michael - Amazing! A hidden Fibonacci series! – Hypergeometricx Nov 19 '14 at 15:11
  • @DanielFischer Your comment essentially answers OP's question; perhaps it merits being an answer in its own right? – Travis Willse Nov 19 '14 at 15:16
  • Interestingly, taking a step back from the OQ, $$\frac18=0.125=0.1+0.02+0.004+0.0008+0.00016_....$$ and from the example above, $$\frac17=0.142857...=0.1+0.03+0.009+0.0027+0.00081+...$$ – Hypergeometricx Nov 19 '14 at 15:22

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Nice observation. We can write

$$\frac{1}{98} = \frac{1}{100 - 2} = \frac{1}{100}\frac{1}{1 - \frac{2}{100}}$$

Now for $|x| < 1$, $\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$

Apply that result to the expression above and we have

$$\frac{1}{98} = \frac{1}{100}\left( 1 + \left( \frac{2}{100} \right) + \left( \frac{2}{100} \right)^2 + \left( \frac{2}{100} \right)^3 + \cdots \right)$$

That gives the pattern you observe.

You can make a similar analysis for $\displaystyle \frac{1}{998}$.

Simon S
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