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 a positional number system is represented as an expansion where The cardinality

is called the level of decomposition.

, and we write the standard set of digits with

digits as Desirable are coding systems with the features: 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[2] is All these coding systems have the mentioned features for

being the imaginary unit): Binary coding systems of complex numbers, i.e. systems with the digits

[9] Listed below are some coding systems

(all are special cases of the systems above) and resp.

codes for the (decimal) numbers −1, 2, −2, i.

The standard binary (which requires a sign, first line) and the "negabinary" systems (second line) are also listed for comparison.

As in all positional number systems with an Archimedean absolute value, there are some numbers with multiple representations.

Examples of such numbers are shown in the right column of the table.

All of them are repeating fractions with the repetend marked by a horizontal line above it.

This is the case with all the mentioned coding systems.

The almost binary quater-imaginary system is listed in the bottom line for comparison purposes.

There, real and imaginary part interleave each other.

Of particular interest are the quater-imaginary base (base 2i) and the base −1 ± i systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign.

Base −1 ± i, using digits 0 and 1, was proposed by S. Khmelnik in 1964[3] and Walter F. Penney in 1965.

[4][6] The rounding region of an integer – i.e., a set

of complex (non-integer) numbers that share the integer part of their representation in this system – has in the complex plane a fractal shape: the twindragon (see figure).

is, by definition, all points that can be written as

can be decomposed into 16 pieces congruent to

is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to

in the center intersects the coordinate axes counterclockwise at the following points:

contains all complex numbers with absolute value ≤ ⁠1/15⁠.

[12] As a consequence, there is an injection of the complex rectangle into the interval

of real numbers by mapping with

[13] Furthermore, there are the two mappings and both surjective, which give rise to a surjective (thus space-filling) mapping which, however, is not continuous and thus not a space-filling curve.

But a very close relative, the Davis-Knuth dragon, is continuous and a space-filling curve.