\begin{definition}[Definition:Number Base/Integers]
Let $n \in \Z$ be an integer.
Let $b$ be any integer such that $b > 1$.
By the Basis Representation Theorem, $n$ can be expressed uniquely in the form:
:$\ds n = \sum_{j \mathop = 0}^m r_j b^j$
where:
:$m$ is such that $b^m \le n < b^{m + 1}$
:all the $r_j$ are such that $0 \le r_j < b$.
<noinclude> {{MissingLinks|The bounds on $n$ are not stated as part of the Basis Representation Theorem. Is there some other link to these bounds?}}</noinclude>
<noinclude>{{Improve|The definition is incomplete as the Basis Representation Theorem is only stated for strictly positive integers}}</noinclude>
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 $\ds n = \sum_{j \mathop = 0}^m {r_j b^j}$ as:
:$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$
or, if the context is clear:
:${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$
\end{definition}