Repeating decimals are rational

Statement

Let $x \ge 0$ be a real number with a decimal expansion that is eventually periodic. That is, there are $N \ge 0$ and $p \ge 1$ with $d_{k + p} = d_k$ for each $k > N$. Then $x$ is a rational number.

Remarks

Proof: $10^{N+p} x - 10^N x$ is an integer, because the periodic tails cancel. Thus $x$ is a quotient of an integer by $10^N (10^p - 1)$. For example, $x = 0.\overline{36}$ gives $100 x - x = 36$, thus $x = 36/99 = 4/11$. The converse is also true: the decimal expansion of each rational number is eventually periodic.

Explanations

L1 human

A decimal that repeats the same block of digits forever is always a fraction. Shift the decimal point by the length of the block and subtract: the endless tail disappears. For $0.777\dots$, this gives $10 x - x = 7$, thus $x = 7/9$.