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) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965).

In general

Let $D$ be an integral domain $\subset \mathbb {C}$, and $|\cdot |$ the (Archimedean) absolute value on it.

A number $X\in D$ in a positional number system is represented as an expansion

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

where

$\rho$		is the radix (or base) $\in D$ with $|\rho |>1$,
$\nu$		is the exponent (position or place) $\in \mathbb {Z}$,
$x_{\nu }$		are digits from the finite set of digits $Z\subset D$, usually with $|x_{\nu }|<|\rho |.$

The cardinality $R:=|Z|$ is called the level of decomposition.

A positional number system or coding system is a pair

$\left\langle \rho ,Z\right\rangle$

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

$Z_{R}:=\{0,1,2,\dotsc ,{R-1}\}.$

Desirable are coding systems with the features:

• Every number in $D$, e. g. the integers $\mathbb {Z}$, the Gaussian integers $\mathbb {Z} [\mathrm {i} ]$ or the integers $\mathbb {Z} [{\tfrac {-1+\mathrm {i} {\sqrt {7}}}{2}}]$, is uniquely representable as a finite code, possibly with a sign ±.
• Every number in the field of fractions $K:={\mathsf {Quot}}(D)$, which possibly is completed for the metric given by $|\cdot |$ yielding $K:=\mathbb {R}$ or $K:=\mathbb {C}$, is representable as an infinite series $X$ which converges under $|\cdot |$ for $\nu \to -\infty$, and the measure of the set of numbers with more than one representation is 0. The latter requires that the set $Z$ be minimal, i. e. $R=|\rho |$ for real resp. $R=|\rho |^{2}$ for complex numbers.

In the real numbers

In this notation our standard decimal coding scheme is denoted by

$\left\langle 10,Z_{10}\right\rangle ,$

the standard binary system is

$\left\langle 2,Z_{2}\right\rangle ,$

the negabinary system is

$\left\langle -2,Z_{2}\right\rangle ,$

and the balanced ternary system is

$\left\langle 3,\{-1,0,1\}\right\rangle .$

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

In the complex numbers

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

• $\left\langle {\sqrt {R}},Z_{R}\right\rangle$, e. g. $\left\langle \pm \mathrm {i} {\sqrt {2}},Z_{2}\right\rangle$ and

$\left\langle \pm 2\mathrm {i} ,Z_{4}\right\rangle$, the quater-imaginary base, proposed by Donald Knuth in 1955.

• $\left\langle {\sqrt {2}}e^{\pm {\tfrac {\pi }{2}}\mathrm {i} }=\pm \mathrm {i} {\sqrt {2}},Z_{2}\right\rangle$ and

$\left\langle {\sqrt {2}}e^{\pm {\tfrac {3\pi }{4}}\mathrm {i} }=-1\pm \mathrm {i} ,Z_{2}\right\rangle$ (see also the section Base −1±i below).

• $\left\langle {\sqrt {R}}e^{\mathrm {i} }\varphi ,Z_{R}\right\rangle$, where $\varphi =\pm \arccos {(-\beta /(2{\sqrt {R}}))}$, $\beta <\min(R,2{\sqrt {R}})$ and $\beta _{}^{}$ is a positive integer that can take multiple values at a given $R$. For $\beta =1$ and $R=2$ this is the system

$\left\langle {\tfrac {-1+\mathrm {i} {\sqrt {7}}}{2}},Z_{2}\right\rangle .$

• $\left\langle 2e^{{\tfrac {\pi }{3}}\mathrm {i} },A_{4}:=\left\{0,1,e^{{\tfrac {2\pi }{3}}\mathrm {i} },e^{-{\tfrac {2\pi }{3}}\mathrm {i} }\right\}\right\rangle$.

• $\left\langle -R,A_{R}^{2}\right\rangle$, where the set $A_{R}^{2}$ consists of complex numbers $r_{\nu }=\alpha _{\nu }^{1}+\alpha _{\nu }^{2}\mathrm {i}$, and numbers $\alpha _{\nu }^{}\in Z_{R}$, e. g.

$\left\langle -2,\{0,1,\mathrm {i} ,1+\mathrm {i} \}\right\rangle .$

• $\left\langle \rho =\rho _{2},Z_{2}\right\rangle$, where $\rho _{2}={\begin{cases}(-2)^{\tfrac {\nu }{2}}&{\text{if }}\nu {\text{ even,}}\\(-2)^{\tfrac {\nu -1}{2}}\mathrm {i} &{\text{if }}\nu {\text{ odd.}}\end{cases}}$ 

Binary systems

Binary coding systems of complex numbers, i. e. systems with the digits $Z_{2}=\{0,1\}$, are of practical interest. Listed below are some coding systems $\langle \rho ,Z_{2}\rangle$ (all are special cases of the systems above) and codes for the numbers −1, 2, −2, $\mathrm {i}$. The standard binary (which requires a sign) and the "negabinary" systems are also listed for comparison. They do not have a genuine expansion for $\mathrm {i}$.

Some bases and some representations

Radix	$-1$	$2$	$-2$	$\mathrm {i}$	Twins and triplets
$2$	$-1$	$10$	$-10$	$\mathrm {i}$	$0.{\overline {1}}=1.{\overline {0}}=1$
$-2$	$11$	$110$	$10$	$\mathrm {i}$	$0.{\overline {01}}=1.{\overline {10}}={\tfrac {1}{3}}$
$\textstyle \mathrm {i} {\sqrt {2}}$	$101$	$10100$	$100$	$10.101010100010...$	$0.{\overline {0011}}=11.{\overline {1100}}={\tfrac {1}{3}}+{\tfrac {1}{3}}\mathrm {i} {\sqrt {2}}$
$-1+\mathrm {i}$	$11101$	$1100$	$11100$	$11$	$0.{\overline {010}}=11.{\overline {001}}=1110.{\overline {100}}={\tfrac {1}{5}}+{\tfrac {3}{5}}\mathrm {i}$
${\tfrac {-1+\mathrm {i} {\sqrt {7}}}{2}}$	$111$	$1010$	$110$	$11.110001100111...$	$1.{\overline {011}}=11.{\overline {101}}=11100.{\overline {110}}={\tfrac {3+\mathrm {i} {\sqrt {7}}}{4}}$
$\rho _{2}$	$101$	$10100$	$100$	$10$	$0.{\overline {0011}}=11.{\overline {1100}}={\tfrac {1}{3}}+{\tfrac {1}{3}}\mathrm {i}$

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.

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.

Base −1 ± i

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

The construction of complex numbers we can get using 6 lowest bits in system with base i + 1 (left) or i − 1 (right)

Base −1±i, using digits 0 and 1, was proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965. 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 a fractal shape, the twindragon.

Example: 3 = 11₂; 11_−1+i = i; the position of 3 on the graph (x, y·i) is (0, 1).

See also

• Dragon curve

External links

• "Number Systems Using a Complex Base" by Jarek Duda, the Wolfram Demonstrations Project
• "The Boundary of Periodic Iterated Function Systems" by Jarek Duda, the Wolfram Demonstrations Project
• "Number Systems in 3D" by Jarek Duda, the Wolfram Demonstrations Project