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 be an integral domain , and the (Archimedean) absolute value on it.

A number in a positional number system is represented as an expansion


is the radix (or base) with ,
is the exponent (position or place) ,
are digits from the finite set of digits , usually with

The cardinality is called the level of decomposition.

A positional number system or coding system is a pair

with radix and set of digits , and we write the standard set of digits with digits as

Desirable are coding systems with the features:

  • Every number in , e. g. the integers , the Gaussian integers or the integers , is uniquely representable as a finite code, possibly with a sign ±.
  • Every number in the field of fractions , which possibly is completed for the metric given by yielding or , is representable as an infinite series which converges under for , and the measure of the set of numbers with more than one representation is 0. The latter requires that the set be minimal, i. e. for real numbers and for complex numbers.
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