          # Complex-base system

In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965).

## In general

Let $D$ be an integral domain $\subset \mathbb {C}$ , and $|\cdot |$ the (Archimedean) absolute value on it.

A number $X\in D$ in a positional number system is represented as an expansion

$X=\pm \sum _{\nu }^{}x_{\nu }\rho ^{\nu },$ where

 $\rho$ is the radix (or base) $\in D$ with $|\rho |>1$ , $\nu$ is the exponent (position or place) $\in \mathbb {Z}$ , $x_{\nu }$ are digits from the finite set of digits $Z\subset D$ , usually with $|x_{\nu }|<|\rho |.$ The cardinality $R:=|Z|$ is called the level of decomposition.

A positional number system or coding system is a pair

$\left\langle \rho ,Z\right\rangle$ with radix $\rho$ and set of digits $Z$ , and we write the standard set of digits with $R$ digits as

$Z_{R}:=\{0,1,2,\dotsc ,{R-1}\}.$ Desirable are coding systems with the features:

• Every number in $D$ , e. g. the integers $\mathbb {Z}$ , the Gaussian integers $\mathbb {Z} [\mathrm {i} ]$ or the integers $\mathbb {Z} [{\tfrac {-1+\mathrm {i} {\sqrt {7}}}{2}}]$ , is uniquely representable as a finite code, possibly with a sign ±.
• Every number in the field of fractions $K:={\mathsf {Quot}}(D)$ , which possibly is completed for the metric given by $|\cdot |$ yielding $K:=\mathbb {R}$ or $K:=\mathbb {C}$ , is representable as an infinite series $X$ which converges under $|\cdot |$ for $\nu \to -\infty$ , and the measure of the set of numbers with more than one representation is 0. The latter requires that the set $Z$ be minimal, i. e. $R=|\rho |$ for real numbers and $R=|\rho |^{2}$ for complex numbers.
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