## Complex-base system |

East Asian |
---|

Alphabetic |

Former |

In **complex-base system** is a ** positional numeral system** whose

- in general
- in the real numbers
- in the complex numbers
- binary systems
- base −1 ±
**i** - see also
- references
- external links

Let be an

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

where

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

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 asign ±. - Every number in the
field of fractions , which possibly iscompleted for the metric given by yielding or , is representable as an infinite series which converges under for , and themeasure 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.