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

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

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$$\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 $\frac{1}{998}$.

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