Are there any bases which represent all rationals in a finite number of digits? In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in which every rational number can be represented in a finite number of digits? My intuition is that the answer is no. If so, what is a proof of this?

Your intuition is correct for instance for all $b > 2$, $\frac{1}{b-1}$ is not going to have a finite representation, and will have the representation
$\frac{1}{b-1} = 0.1111111...._b.$ Eg, $\frac{1}{9} = .11111111$ in base 10.
