Complex-base system

Inarithmetic,acomplex-base systemis apositional numeral systemwhoseradixis animaginary(proposed byDonald Knuthin 1955) orcomplex number(proposed by S. Khmelnik in 1964and Walter F. Penney in 1965).

In general

Let$D$be anintegral domain$\subset \mathbb {C}$,and$|\cdot |$the(Archimedean) absolute valueon 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

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

Thecardinality$R:=|Z|$is called thelevel of decomposition.

A positional number system orcoding systemis 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$\mathbb {Z}$,theGaussian integers$\mathbb {Z} [\mathrm {i} ]$or the integers$\mathbb {Z} [{\tfrac {-1+\mathrm {i} {\sqrt {7}}}{2}}]$,isuniquelyrepresentable as afinitecode, possibly with asign±.
• Every number in thefield of fractions$K:=\operatorname {Quot} (D)$,which possibly iscompletedfor themetricgiven 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 themeasureof the set of numbers with more than one representation is 0. The latter requires that the set$Z$be minimal, i.e.$R=|\rho |$forreal numbersand$R=|\rho |^{2}$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

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

thenegabinarysystem is

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

and the balanced ternary systemis

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

All these coding systems have the mentioned features for$\mathbb {Z}$and$\mathbb {R}$,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 ($\mathrm {i}$being theimaginary unit):

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

$\left\langle \pm 2\mathrm {i},Z_{4}\right\rangle$,thequater-imaginary base,proposed byDonald Knuthin 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 sectionBase −1 ±ibelow).

• $\left\langle {\sqrt {R}}e^{\mathrm {i} \varphi },Z_{R}\right\rangle$,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

Binarycoding 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 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

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
$\mathrm {i} {\sqrt {2}}$	101	10100	100	10.101010100...	${\frac {1}{3}}+{\frac {1}{3}}\mathrm {i} {\sqrt {2}}$←	0.0011= 11.1100
${\frac {-1+\mathrm {i} {\sqrt {7}}}{2}}$	111	1010	110	11.110001100...	${\frac {3+\mathrm {i} {\sqrt {7}}}{4}}$←	1.011= 11.101= 11100.110
$\rho _{2}$	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 anArchimedeanabsolute value,there are some numbers withmultiple representations.Examples of such numbers are shown in the right column of the table. All of them arerepeating fractionswith the repetend marked by a horizontal line above it.

If the set of digits is minimal, the set of such numbers has ameasureof 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

The complex numbers with integer part all zeroes in the basei– 1system

Of particular interest are thequater-imaginary base(base2i) and the base−1 ±isystems discussed below, both of which can be used to finitely represent theGaussian integerswithout sign.

Base−1 ±i,using digits0and1,was proposed by S. Khmelnik in 1964and Walter F. Penney in 1965.

Connection to the twindragon

The rounding region of an integer – i.e., a set$S$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$S$is, by definition, all points that can be written as$\textstyle \sum _{k\geq 1}x_{k}(\mathrm {i} -1)^{-k}$with$x_{k}\in Z_{2}$.$S$can be decomposed into 16 pieces congruent to${\tfrac {1}{4}}S$.Notice that if$S$is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to${\tfrac {1}{\sqrt {2}}}S$,because$(\mathrm {i} -1)S=S\cup (S+1)$.The rectangle$R\subset S$in the center intersects the coordinate axes counterclockwise at the following points:${\tfrac {2}{15}}\gets 0.{\overline {00001100}}$,${\tfrac {1}{15}}\mathrm {i} \gets 0.{\overline {00000011}}$,and$-{\tfrac {8}{15}}\gets 0.{\overline {11000000}}$,and$-{\tfrac {4}{15}}\mathrm {i} \gets 0.{\overline {00110000}}$.Thus,$S$contains all complex numbers with absolute value ≤ ⁠1/15⁠.

As a consequence, there is aninjectionof the complex rectangle

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

into theinterval$[0,1)$of real numbers by mapping

$\textstyle \sum _{k\geq 1}x_{k}(\mathrm {i} -1)^{-k}\mapsto \sum _{k\geq 1}x_{k}b^{-k}$

with$b>2$.

Furthermore, there are the two mappings

${\begin{array}{lll}Z_{2}^{\mathbb {N} }&\to &S\\\left(x_{k}\right)_{k\in \mathbb {N} }&\mapsto &\sum _{k\geq 1}x_{k}(\mathrm {i} -1)^{-k}\end{array}}$

and

${\begin{array}{lll}Z_{2}^{\mathbb {N} }&\to &[0,1)\\\left(x_{k}\right)_{k\in \mathbb {N} }&\mapsto &\sum _{k\geq 1}x_{k}2^{-k}\end{array}}$

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

$[0,1)\qquad \to \qquad S$

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

See also

• Dragon curve

External links

• "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