Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

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!

Solution 1:

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 $\displaystyle \frac{1}{998}$.