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).

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

where

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

A positional number system or coding system is a pair

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

Desirable are coding systems with the features: