Zero Padded Basis Representation

Theorem
Let $b \in \Z: b > 1$.

Let $m \in \Z_{> 0}$.

For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that:


 * $(1): \quad \ds n = \sum_{j \mathop = 0}^{m - 1} r_j b^j$
 * $(2): \quad \forall j \in \closedint 0 {m - 1}: r_j \in \N_b$