Definition:Basis Representation

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

• Definition:Number Base

• 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 $\text {1-3}$