Decimal expansions and rationality

Forward. For $x=p/q$, long division: remainders mod $q$ are among $0,\ldots,q-1$, so by pigeonhole some remainder repeats within $q$ steps. From that point the decimal is periodic. (If the remainder $0$ is hit, the decimal is finite, considered periodic with period $0$.)

Converse. Up to adding an integer to $x$, we can assume $x=0.\overline{a_1\cdots a_k}$. Multiplying $x$ by $10^k$ and subtracting $x$ gives $x(10^k-1)=\overline{a_1\cdots a_k} \in \mathbb{N}$, so $x=\frac{\overline{a_1\cdots a_k}}{10^k-1}\in\mathbb{Q}$.

Conclusion. Rationals are exactly the eventually periodic decimals.