Definition:Number Base/Integers

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Let $n \in \Z$ be an integer.

Let $b \in \Z$ be an integer such that $b > 1$.

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

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


$m$ 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 $\ds n = \sum_{j \mathop = 0}^m {r_j b^j}$ as:

$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$

or, if the context is clear:

${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$

Also see

The most common number base is of course base $10$.

So common is it, that numbers written in base $10$ are written merely by concatenating the digits:

$r_m r_{m - 1} \ldots r_2 r_1 r_0$

$2$ is a fundamentally important number base in computer science, as is $16$:

  • Results about number bases can be found here.