Exercise 3, § 8

Let $a$ be an integer > 1. A real number $x$ is rational if and only if its expansion to base $a$, say $x = p_0 + \sum_{n=0}^{\infty} u_n a^{-n}$, is periodic, that is to say if and only if there exist two integers $n_0$ and $r > 0$ such that $u_{n+r} = u_n$ for all $n \geq n_0$.