Complex-base system

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In

).

In general

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

The cardinality ${\displaystyle R:=|Z|}$ is called the level of decomposition.

A positional number system or coding system is a pair

${\displaystyle \left\langle \rho ,Z\right\rangle }$

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

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

Desirable are coding systems with the features:

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

In the real numbers

In this notation our standard decimal coding scheme is denoted by

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

the standard binary system is

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

the negabinary system is

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

and the balanced ternary system^[2] is

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

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):

• ${\displaystyle \left\langle {\sqrt {R}},Z_{R}\right\rangle }$, e.g. ${\displaystyle \left\langle \pm \mathrm {i} {\sqrt {2}},Z_{2}\right\rangle }$ ^[1] and

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

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

${\displaystyle \left\langle {\sqrt {2}}e^{\pm {\tfrac {3\pi }{4}}\mathrm {i} }=-1\pm \mathrm {i} ,Z_{2}\right\rangle }$^[3][5] (see also the section Base −1 ± i below).

• ${\displaystyle \left\langle {\sqrt {R}}e^{\mathrm {i} \varphi },Z_{R}\right\rangle }$, where ${\displaystyle \varphi =\pm \arccos {(-\beta /(2{\sqrt {R}}))}}$, ${\displaystyle \beta <\min(R,2{\sqrt {R}})}$ and ${\displaystyle \beta _{}^{}}$ is a positive integer that can take multiple values at a given ${\displaystyle R}$.^[7] For ${\displaystyle \beta =1}$ and ${\displaystyle R=2}$ this is the system

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

• ${\displaystyle \left\langle 2e^{{\tfrac {\pi }{3}}\mathrm {i} },A_{4}:=\left\{0,1,e^{{\tfrac {2\pi }{3}}\mathrm {i} },e^{-{\tfrac {2\pi }{3}}\mathrm {i} }\right\}\right\rangle }$.^[8]
• ${\displaystyle \left\langle -R,A_{R}^{2}\right\rangle }$, 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.

${\displaystyle \left\langle -2,\{0,1,\mathrm {i} ,1+\mathrm {i} \}\right\rangle .}$^[8]

• ${\displaystyle \left\langle \rho =\rho _{2},Z_{2}\right\rangle }$, where ${\displaystyle \rho _{2}={\begin{cases}(-2)^{\tfrac {\nu }{2}}&{\text{if }}\nu {\text{ even,}}\\(-2)^{\tfrac {\nu -1}{2}}\mathrm {i} &{\text{if }}\nu {\text{ odd.}}\end{cases}}}$ ^[9]

Binary systems

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 ${\displaystyle \langle \rho ,Z_{2}\rangle }$ (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 for i.

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 an

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^[3] and Walter F. Penney in 1965.^[4][6]

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 ${\displaystyle \textstyle \sum _{k\geq 1}x_{k}(\mathrm {i} -1)^{-k}}$ with ${\displaystyle x_{k}\in Z_{2}}$. ${\displaystyle S}$ can be decomposed into 16 pieces congruent to ${\displaystyle {\tfrac {1}{4}}S}$. Notice that if ${\displaystyle S}$ is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to ${\displaystyle {\tfrac {1}{\sqrt {2}}}S}$, 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: ${\displaystyle {\tfrac {2}{15}}\gets 0.{\overline {00001100}}}$, ${\displaystyle {\tfrac {1}{15}}\mathrm {i} \gets 0.{\overline {00000011}}}$, and ${\displaystyle -{\tfrac {8}{15}}\gets 0.{\overline {11000000}}}$, and ${\displaystyle -{\tfrac {4}{15}}\mathrm {i} \gets 0.{\overline {00110000}}}$. Thus, ${\displaystyle S}$ contains all complex numbers with absolute value ≤ 1/15.^[12]

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

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

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

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

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

Furthermore, there are the two mappings

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

and

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

both

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

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

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.

See also

• Dragon curve

References

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
• "Large introduction to complex base numeral systems" with Mathematica sources by Jarek Duda