# Definition:Number Base/Integers

Jump to navigation
Jump to search

## Definition

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 $0 \le r_j < b$.

This article needs to be linked to other articles.In particular: The bounds on $n$ are not stated as part of the Basis Representation Theorem. Is there some other link to these bounds?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The definition is incomplete as the Basis Representation Theorem is only stated for strictly positive integersYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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 $16$:

## 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