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 $0 \le r_j < b$.
The number $b$ is known as the number base to which $n$ is represented.
$n$ is thus described as being (written) in base $b$.
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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 known as
A number base is also known as a radix.
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