Nice example. This can be generalized to a theorem saying that a number is rational iff it has a repeating decimal (or any other base) pattern. It'd be kinda interesting to see if this process can be reversed, so constructing a decimal pattern out of a given fraction.

I mean, if you start writing up decimals, you may never know when the pattern is starting to repeat itself (like you may calculate 0.123123 and think that 123 is repeating pattern, but the next digit could be 4).