Definition:Number Base/Integers

Definition
Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $b$ be any integer such that $b > 1$.

By the Basis Representation Theorem, $n$ can be expressed uniquely in the form:


 * $\displaystyle n = \sum_{j \mathop = 0}^m r_j b^j$

where:
 * $m$ is such that $b^m \le n < b^{m + 1}$
 * all the $r_j$ are such that $0 \le r_j < b$.

The number $b$ is known as the number base to which $n$ is represented.

$n$ is thus described as being (written) in base $b$.

Thus we can write $\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ as:
 * $\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$

or, if the context is clear:
 * ${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$