Repeating decimals are x/99...9

[Apoapsis]

Today was a review day in my math class, which translates to: Ask questions entirely unrelated to the subject.

Anyway, I stumped the teacher today.

An integer i with n digits can be written as:

d_1d_2d_3...d_n

A repeating decimal with i repeating itself (0.iii...) can be written as

\frac {i}{999...9}

with n nines in the denominator.

The question is simple. Why is that?

Thanks!

[OverBored]

Suppose x = 0.{d_1}{d_2}...{d_n}...

Then :

x*(10^n) - x ={d_1}{d_2}...{d_n}.{d_1}{d_2}...{d_n}... - {d_1}{d_2}...{d_n} = {d_1}{d_2}...{d_n}

so x = \frac{{d_1}{d_2}...{d_n}}{{10^n} - 1} = \frac{{d_1}{d_2}...{d_n}}{99.....99}