Complex-base system

In the real numbers

In this notation our standard decimal coding scheme is denoted by

${\displaystyle ⟨10,Z_{10}⟩,}$

the standard binary system is

${\displaystyle ⟨2,Z_2⟩,}$

the negabinary system is

${\displaystyle ⟨-2,Z_2⟩,}$

and the balanced ternary system 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.

In the complex numbers

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

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

${\displaystyle ⟨\pm 2\mathrm{i},Z_4⟩}$, the quater-imaginary base, proposed by Donald Knuth in 1955.

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

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

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

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

• ${\displaystyle ⟨2e^{\frac{\pi }{3}\mathrm{i}},A_4:=\left\{0,1,e^{\frac{2\pi }{3}\mathrm{i}},e^{-\frac{2\pi }{3}\mathrm{i}}\right\}⟩}$.
• ${\displaystyle ⟨-R,A_R^2⟩}$, 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.

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

• ${\displaystyle ⟨\rho ={\rho }_2,Z_2⟩}$, where ${\displaystyle {\rho }_2=\{\begin{array}{ll}(-2{)}^{\frac{\nu }{2}}& \text{if }\nu \text{ even,}\\ (-2{)}^{\frac{\nu -1}{2}}\mathrm{i}& \text{if }\nu \text{ odd.}\end{array}}$

Binary systems

Binary coding systems of complex numbers, i.e. systems with the digits ${\displaystyle Z_2=\{0,1\}}$, are of practical interest. 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 for i.

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.

If the set of digits is minimal, the set of such numbers has a measure 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.

Base −1 ± i

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 and Walter F. Penney in 1965.

Connection to the twindragon

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: the twindragon (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.

As a consequence, there is an injection of the complex rectangle

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

into the interval ${\displaystyle [0,1)}$ 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 ${\displaystyle b>2}$.

Furthermore, there are the two mappings

${\displaystyle \begin{array}{lll}{Z}_2^{\mathbb{N}}& \to & S\\ {\left(x_k\right)}_{k\in \mathbb{N}}& \mapsto & \sum _{k\ge 1}{x}_k(\mathrm{i}-1{)}^{-k}\end{array}}$

and

${\displaystyle \begin{array}{lll}{Z}_2^{\mathbb{N}}& \to & [0,1)\\ {\left(x_k\right)}_{k\in \mathbb{N}}& \mapsto & \sum _{k\ge 1}{x}_k{2}^{-k}\end{array}}$

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

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

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.