Definition:Basis Representation
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Definition
Let $b \in \Z$ be an integer such that $b > 1$.
Let $n \in \Z$ be an integer such that $n \ne 0$.
The representation of $n$ to the base $b$ is the unique string of digits:
- $\pm \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$
where $\pm$ is:
- the negative sign $-$ if and only if $n < 0$
- the positive sign $+$ (or omitted) if and only if $n > 0$
If $n = 0$, then $n$ is represented simply as $0$.
Also known as
Informally, we can say $n$ is (written) in base $b$ or to base $b$.
Also see
- Basis Representation Theorem which demonstrates that a basis representation exists and is unique for all $n \in \Z$ and $b \in \Z_{> 1}$.
- Definition:Basis Expansion, for real numbers which are not integers
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.2$
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem: Theorem $\text {1-3}$