# Definition:Number Base/Integers

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## Definition

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:

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

where:

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

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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**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem