Inarithmetic, acomplex-base system is apositional numeral system whoseradix is animaginary (proposed byDonald Knuth in 1955^[1][2]) orcomplex number (proposed by S. Khmelnik in 1964^[3] and Walter F. Penney in 1965^[4][5][6]).

Let${\displaystyle D}$ be anintegral 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

${\displaystyle X=\pm \sum _{\nu }^{}{x}_{\nu }{\rho }^{\nu },}$

where

${\displaystyle \rho \in D}$		is theradix (orbase) with${\displaystyle |\rho |>1}$,
${\displaystyle \nu \in \mathbb{Z}}$		is the exponent (position or place),
${\displaystyle x_{\nu }}$		are digits from thefinite set of digits${\displaystyle Z\subset D}$, usually with

Thecardinality${\displaystyle R:=|Z|}$ is called thelevel of decomposition.

A positional number system orcoding system is a pair

${\displaystyle ⟨\rho ,Z⟩}$

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

${\displaystyle Z_R:=\{0,1,2,\dots ,R-1\}.}$

Desirable are coding systems with the features:

• Every number in${\displaystyle D}$, e. g. the integers${\displaystyle \mathbb{Z}}$, theGaussian integers${\displaystyle \mathbb{Z}[\mathrm{i}]}$ or the integers${\displaystyle \mathbb{Z}[\frac{-1+\mathrm{i}\sqrt{7}}{2}]}$, isuniquely representable as afinite code, possibly with asign ±.
• Every number in thefield of fractions${\displaystyle K:=\mathrm{Quot}(D)}$, which possibly iscompleted for themetric given by${\displaystyle |\cdot |}$ yielding${\displaystyle K:=\mathbb{R}}$ or${\displaystyle K:=\mathbb{C}}$, is representable as an infinite series${\displaystyle X}$ which converges under${\displaystyle |\cdot |}$ for${\displaystyle \nu \to -\mathrm{\infty }}$, and themeasure of the set of numbers with more than one representation is 0. The latter requires that the set${\displaystyle Z}$ be minimal, i.e.${\displaystyle R=|\rho |}$ forreal numbers and${\displaystyle R=|\rho {|}^2}$ for complex numbers.

In this notation our standard decimal coding scheme is denoted by

${\displaystyle ⟨10,Z_{10}⟩,}$

the standard binary system is

${\displaystyle ⟨2,Z_2⟩,}$

thenegabinary system is

${\displaystyle ⟨-2,Z_2⟩,}$

and the balanced ternary system^[2] is

${\displaystyle ⟨3,\{-1,0,1\}⟩.}$

All these coding systems have the mentioned features for${\displaystyle \mathbb{Z}}$ and${\displaystyle \mathbb{R}}$, and the last two do not require a sign.

Well-known positional number systems for the complex numbers include the following (${\displaystyle \mathrm{i}}$ being theimaginary unit):

• ${\displaystyle ⟨\sqrt{R},Z_R⟩}$, e.g.${\displaystyle ⟨\pm \mathrm{i}\sqrt{2},Z_2⟩}$^[1] and

${\displaystyle ⟨\pm 2\mathrm{i},Z_4⟩}$,^[2] thequater-imaginary base, proposed byDonald Knuth in 1955.

• ${\displaystyle ⟨\sqrt{2}{e}^{\pm \frac{\pi }{2}\mathrm{i}}=\pm \mathrm{i}\sqrt{2},Z_2⟩}$ and

${\displaystyle ⟨\sqrt{2}{e}^{\pm \frac{3\pi }{4}\mathrm{i}}=-1\pm \mathrm{i},Z_2⟩}$^[3][5] (see also the sectionBase −1 ±i below).

• ${\displaystyle ⟨\sqrt{R}{e}^{\mathrm{i}\phi },Z_R⟩}$, where${\displaystyle \phi =\pm \arccos (-\beta /(2\sqrt{R}))}$, and${\displaystyle {\beta }_{}^{}}$ is a positive integer that can take multiple values at a given${\displaystyle R}$.^[7] For${\displaystyle \beta =1}$ and${\displaystyle R=2}$ this is the system

${\displaystyle ⟨\frac{-1+\mathrm{i}\sqrt{7}}{2},Z_2⟩.}$

• ${\displaystyle ⟨2e^{\frac{\pi }{3}\mathrm{i}},A_4:=\left\{0,1,e^{\frac{2\pi }{3}\mathrm{i}},e^{-\frac{2\pi }{3}\mathrm{i}}\right\}⟩}$.^[8]
• ${\displaystyle ⟨-R,A_R^2⟩}$, where the set${\displaystyle A_R^2}$ consists of complex numbers${\displaystyle r_{\nu }={\alpha }_{\nu }^1+{\alpha }_{\nu }^2\mathrm{i}}$, and numbers${\displaystyle {\alpha }_{\nu }^{}\in Z_R}$, e.g.

${\displaystyle ⟨-2,\{0,1,\mathrm{i},1+\mathrm{i}\}⟩.}$^[8]

• ${\displaystyle ⟨\rho ={\rho }_2,Z_2⟩}$, where ^[9]

Binary coding systems of complex numbers, i.e. systems with the digits${\displaystyle Z_2=\{0,1\}}$, are of practical interest.^[9]Listed below are some coding systems${\displaystyle ⟨\rho ,Z_2⟩}$ (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. They do not have a genuine expansion fori.

Some bases and some representations^[10]

Radix	–1 ←	2 ←	–2 ←	i ←	Twins and triplets
2	–1	10	–10	i	1 ←	0.1 = 1.0
–2	11	110	10	i	⁠1/3⁠ ←	0.01 = 1.10
${\displaystyle \mathrm{i}\sqrt{2}}$	101	10100	100	10.101010100...[11]	${\displaystyle \frac{1}{3}+\frac{1}{3}\mathrm{i}\sqrt{2}}$ ←	0.0011 = 11.1100
${\displaystyle \frac{-1+\mathrm{i}\sqrt{7}}{2}}$	111	1010	110	11.110001100...[11]	${\displaystyle \frac{3+\mathrm{i}\sqrt{7}}{4}}$ ←	1.011 = 11.101 = 11100.110
${\displaystyle {\rho }_2}$	101	10100	100	10	⁠1/3⁠ + ⁠1/3⁠i ←	0.0011 = 11.1100
–1+i	11101	1100	11100	11	⁠1/5⁠ + ⁠3/5⁠i ←	0.010 = 11.001 = 1110.100
2i	103	2	102	10.2	⁠1/5⁠ + ⁠2/5⁠i ←	0.0033 = 1.3003 = 10.0330 = 11.3300

As in all positional number systems with anArchimedean absolute value, there are some numbers withmultiple representations. Examples of such numbers are shown in the right column of the table. All of them arerepeating fractions with the repetend marked by a horizontal line above it.

If the set of digits is minimal, the set of such numbers has ameasure of 0. 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 thequater-imaginary base (base2i) and the base−1 ±i systems discussed below, both of which can be used to finitely represent theGaussian integers without sign.

Base−1 ±i, using digits0 and1, 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${\displaystyle S}$ of complex (non-integer) numbers that share the integer part of their representation in this system – has in the complex plane a fractal shape: thetwindragon (see figure). This set${\displaystyle S}$ is, by definition, all points that can be written as${\displaystyle \sum _{k\ge 1}{x}_k(\mathrm{i}-1{)}^{-k}}$ with${\displaystyle x_k\in Z_2}$.${\displaystyle S}$ can be decomposed into 16 pieces congruent to${\displaystyle \frac{1}{4}S}$. Notice that if${\displaystyle S}$ is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to${\displaystyle \frac{1}{\sqrt{2}}S}$, because${\displaystyle (\mathrm{i}-1)S=S\cup (S+1)}$. The rectangle${\displaystyle R\subset S}$ in the center intersects the coordinate axes counterclockwise at the following points:${\displaystyle \frac{2}{15}\leftarrow 0.\overline{00001100}}$,${\displaystyle \frac{1}{15}\mathrm{i}\leftarrow 0.\overline{00000011}}$, and${\displaystyle -\frac{8}{15}\leftarrow 0.\overline{11000000}}$, and${\displaystyle -\frac{4}{15}\mathrm{i}\leftarrow 0.\overline{00110000}}$. Thus,${\displaystyle S}$ contains all complex numbers with absolute value ≤ ⁠1/15⁠.^[12]

As a consequence, there is aninjection of the complex rectangle

${\displaystyle [-\frac{8}{15},\frac{2}{15}]\times [-\frac{4}{15},\frac{1}{15}]\mathrm{i}}$

into theinterval${\displaystyle [0,1)}$ of real numbers by mapping

${\displaystyle \sum _{k\ge 1}{x}_k(\mathrm{i}-1{)}^{-k}\mapsto \sum _{k\ge 1}{x}_k{b}^{-k}}$

with${\displaystyle b>2}$.^[13]

Furthermore, there are the two mappings

and

bothsurjective, which give rise to a surjective (thus space-filling) mapping

${\displaystyle [0,1)\to S}$

which, however, is notcontinuous and thusnot aspace-fillingcurve. But a very close relative, theDavis-Knuth dragon, is continuous and a space-filling curve.

• Dragon curve

• "Number Systems Using a Complex Base" by Jarek Duda, theWolfram Demonstrations Project
• "The Boundary of Periodic Iterated Function Systems" by Jarek Duda, theWolfram Demonstrations Project
• "Number Systems in 3D" by Jarek Duda, theWolfram Demonstrations Project
• "Large introduction to complex base numeral systems" with Mathematica sources by Jarek Duda

• Non-standard positional numeral systems
• Fractals
• Complex numbers
• Ring theory

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