Complex Base Systems - in General

In General

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

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

, where

– the radix (or base) with ,
– exponent (position or place)
– 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 Gaussian integers, is uniquely representable as a finite code, possibly with a sign.
  • Every number in is representable as an infinite code, where the series 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. .

In this notation our standard decimal coding scheme is denoted by

,

the standard binary system is

,

the negabinary system is

,

and the balanced ternary system is

.

All these coding systems have the mentioned features for and, and the latter two do not require a sign.


Well-known positional number systems for the complex numbers include the following ( being the imaginary unit):

  • , e. g. and
, the quater-imaginary base, proposed by Donald Knuth in 1955.
  • and
(see also the section Base −1± below).
  • , where, and is a positive integer that can take multiple values at a given . For and this is the system
.
  • ;
  • , where the set consists of complex numbers, and numbers, e. g.
.
  • , where \rho_2=\begin{cases} (-2)^{\tfrac{\nu}2} & \text{if } \nu \text{ even,}\\ (-2)^{\tfrac{\nu-1}2}\mathrm i & \text{if } \nu \text{ odd.} \end{cases}

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