# Proof:The base b representation is unique

As usual, in order to justify calling the base b representation of "the" base b representation, we need to prove that it is unique. In fact, it is not necessarily unique if we allow leading 0's (for example, ), but if we rule them out, then we have the following fact:

For all , if and if and if then .

The proof is left as an exercise. Use induction on and the fact that The quotient and remainder are unique.