Help with proving that all ζ ∈ ℚ have a periodic decimal expansion

Question

I've been trying to prove that all ζ ∈ ℚ have periodic decimal expansion. I assume that a terminating decimal expansion counts as a periodic decimal expansion as well since the zeros are periodic as well.

I managed to prove that for all ζ ∈ ℝ with a periodic decimal expansion we have that ζ ∈ ℚ. However, I don't know how to prove that all ζ ∈ ℚ have a periodic decimal expansion. Any tips on how I can prove this?

I've seen answers to my question involving terms like "mod" and "ring" but am not familiar with these terms so my question still stands.

Answer 1

The decimal expansion of a rational number is the result of the operation of long division. Look at the long division algorithm for a specific rational number that results in a repeating decimal. Notice that three are only so many remainders that come up. Once you see the same one again, you're in repetition.

Can you get a proof out of that observation?