On $0$'s in repeating decimals.

Question

Given $a\in\Bbb N$ then $\frac1a$ is of form $0.0\dots0r_1\dots r_k0\dots0r_1\dots r_k\dots$.

What determines the length of $0$ runs?

Answer 1

One way to think about it would be the following:

The zero runs tell you when you approximate this real number using positive decimal steps, how small each subsequent approximation step gets.

Answer 2

If your number starts with precisely $m$ zeroes, it is between $0.\underbrace{0\ldots0}_m1=10^{-m-1}$ (inclusive) and $0.\underbrace{0\ldots0}_{m-1}1=10^{-m}$ (exclusive). In other words, $10^{m}<a\le10^{m+1}$.