# 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[1][2]) or complex number (proposed by S. Khmelnik in 1964[3] and Walter F. Penney in 1965[4][5][6]).

## In general

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

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

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

where

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

The cardinality ${\displaystyle R:=|Z|}$ is called the level of decomposition.

A positional number system or coding system is a pair

${\displaystyle \left\langle \rho ,Z\right\rangle }$

with radix ${\displaystyle \rho }$ and set of digits ${\displaystyle Z}$, and we write the standard set of digits with ${\displaystyle R}$ digits as

${\displaystyle Z_{R}:=\{0,1,2,\dotsc ,{R-1}\}.}$

Desirable are coding systems with the features:

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