TITLE: Are there any bases which represent all rationals in a finite number of digits?
QUESTION [19 upvotes]: 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?

REPLY [24 votes]: A representation, though not positional, which yields a finite string for rational numbers is the Stern-Brocot representation which derives from the Stern-Brocot tree (also here). 
The Stern-Brocot tree is a binary tree which can be used to enumerate all the rationals and which has useful applications in the rational approximation of real numbers. Here is a portion of this curious tree (the horizontal axis has a logarithmic scale):

Since the nodes of this special tree are in one-to-one correspondence with the rationals, one can represent each rational by a string which specifies how to walk down the tree, starting from the root 1/1, to reach the given rational. Since the tree is binary, a rational can be specified by a string of L(eft) and R(ight) letters. The representation is finite because all fractions can be reached with a finite number of turns. 
For example, the fraction 8/5 can be specified by RLRL.
Real numbers have, instead, an infinite representation. For example, the golden ratio has the nice representation RLRLRLRLRL...