complex-base system

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

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

: $X\; =\; \backslash pm\; \backslash sum\_\{\backslash nu\}^\{\; \}\; x\_\backslash nu\; \backslash rho^\backslash nu,$

where

:

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

A positional number system or coding system is a pair

: $\backslash left\backslash langle\; \backslash rho,\; Z\; \backslash right\backslash rangle$

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

: $Z\_R\; :=\; \backslash \{0,\; 1,\; 2,\backslash dotsc,\; \{R-1\}\backslash \}.$

Desirable are coding systems with the features:

• Every number in $D$, e. g. the integers $\backslash Z$, the Gaussian integers $\backslash Z[\backslash mathrm\; i]$ or the integers $\backslash Z[\backslash tfrac\{-1+\backslash mathrm\; i\backslash sqrt7\}2]$, is uniquely representable as a finite code, possibly with a sign ±.
• Every number in the field of fractions $K:=\backslash operatorname\{Quot\}(D)$, which possibly is completed for the metric given by $|\backslash cdot|$ yielding $K:=\backslash R$ or $K:=\backslash C$, is representable as an infinite series $X$ which converges under $|\backslash cdot|$ for $\backslash nu\; \backslash to\; -\backslash infty$, 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. $R=|\backslash rho|$ for real numbers and $R=|\backslash rho|^2$ for complex numbers.

In this notation our standard decimal coding scheme is denoted by

:$\backslash left\backslash langle\; 10,\; Z\_\{10\}\; \backslash right\backslash rangle,$

the standard binary system is

:$\backslash left\backslash langle\; 2,\; Z\_2\; \backslash right\backslash rangle,$

the negabinary system is

:$\backslash left\backslash langle\; -2,\; Z\_2\; \backslash right\backslash rangle,$

and the balanced ternary system is

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

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

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

• $\backslash left\backslash langle\backslash sqrt\{R\},Z\_R\backslash right\backslash rangle$, e.g. $\backslash left\backslash langle\backslash pm\; \backslash mathrm\; i\; \backslash sqrt\{2\},Z\_2\backslash right\backslash rangle$ and

:$\backslash left\backslash langle\backslash pm\; 2\backslash mathrm\; i,Z\_4\backslash right\backslash rangle$, the quater-imaginary base, proposed by Donald Knuth in 1955.

• $\backslash left\backslash langle\backslash sqrt\{2\}e^\{\backslash pm\; \backslash tfrac\{\backslash pi\}2\; \backslash mathrm\; i\}=\backslash pm\; \backslash mathrm\; i\backslash sqrt\{2\},Z\_2\backslash right\backslash rangle$ and

:$\backslash left\backslash langle\backslash sqrt\{2\}e^\{\backslash pm\; \backslash tfrac\{3\; \backslash pi\}4\; \backslash mathrm\; i\}=-1\backslash pm\backslash mathrm\; i,Z\_2\backslash right\backslash rangle$ (see also the section Base −1 ± i below).

• $\backslash left\backslash langle\backslash sqrt\{R\}e^\{\backslash mathrm\; i\backslash varphi\},Z\_R\backslash right\backslash rangle$, where $\backslash varphi=\backslash pm\; \backslash arccos\{(-\backslash beta/(2\backslash sqrt\{R\}))\}$, and $\backslash beta\_\{\; \}^\{\; \}$ is a positive integer that can take multiple values at a given $R$.{{cite journal |last=Khmelnik |first=S.I. |title=Positional coding of complex numbers

|journal=Questions of Radio Electronics (In Russian)|volume=XII |issue=9 |year=1966}} For $\backslash beta=1$ and $R=2$ this is the system

:$\backslash left\backslash langle\backslash tfrac\{-1+\backslash mathrm\; i\backslash sqrt7\}2,Z\_2\backslash right\backslash rangle.$

• $\backslash left\backslash langle\; 2e^\{\backslash tfrac\{\backslash pi\}3\; \backslash mathrm\; i\},A\_4:=\backslash left\backslash \{0,1,e^\{\backslash tfrac\{2\; \backslash pi\}3\; \backslash mathrm\; i\},e^\{-\backslash tfrac\{2\; \backslash pi\}3\; \backslash mathrm\; i\}\backslash right\backslash \}\backslash right\backslash rangle$.{{cite book |last=Khmelnik |first=S.I. |title=Coding of Complex Numbers and Vectors (in Russian)|publisher=Mathematics in Computer |location=Israel |isbn=978-0-557-74692-7 | year=2004 |url=http://mic34.com/Magazine/94846.pdf}}
• $\backslash left\backslash langle-R,A\_R^2\backslash right\backslash rangle$, where the set $A\_R^2$ consists of complex numbers $r\_\backslash nu=\backslash alpha\_\backslash nu^1+\backslash alpha\_\backslash nu^2\backslash mathrm\; i$, and numbers $\backslash alpha\_\backslash nu^\{\; \}\; \backslash in\; Z\_R$, e.g.

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

• $\backslash left\backslash langle\backslash rho=\backslash rho\_2,Z\_2\backslash right\backslash rangle$, where $\backslash rho\_2=\backslash begin\{cases\}$

(-2)^{\tfrac{\nu}2} & \text{if } \nu \text{ even,}\\

(-2)^{\tfrac{\nu-1}2}\mathrm i & \text{if } \nu \text{ odd.}

\end{cases} {{cite book |last=Khmelnik |first=S.I. |title=Method and system for processing complex numbers |url=http://worldwide.espacenet.com/publicationDetails/biblio?DB=EPODOC&adjacent=true&locale=en_EP&FT=D&date=20030814&CC=US&NR=2003154226A1&KC=A1 |publisher=Patent USA, US2003154226 (A1) |year=2001 }}

Binary coding systems of complex numbers, i.e. systems with the digits $Z\_2=\backslash \{0,1\backslash \}$, are of practical interest.

Listed below are some coding systems $\backslash langle\; \backslash rho,\; Z\_2\; \backslash rangle$ (all are special cases of the systems above) and resp. codes for the (decimal) numbers {{math|−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 {{math|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.

Image:ComplexTwindragon.svg

Of particular interest are the quater-imaginary base (base {{math|2i}}) and the base {{math|−1 ± i}} systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign.

Base {{math|−1 ± i}}, using digits {{math|0}} and {{math|1}}, was proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965.

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: the twindragon (see figure). This set $S$ is, by definition, all points that can be written as $\backslash textstyle\; \backslash sum\_\{k\backslash geq\; 1\}x\_k\; (\backslash mathrm\; i-1)^\{-k\}$ with $x\_k\backslash in\; Z\_2$. $S$ can be decomposed into 16 pieces congruent to $\backslash tfrac14\; S$. Notice that if $S$ is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to $\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}S$, because $(\backslash mathrm\; i-1)S=S\backslash cup(S+1)$. The rectangle $R\backslash subset\; S$ in the center intersects the coordinate axes counterclockwise at the following points: $\backslash tfrac2\{15\}\backslash gets\; 0.\backslash overline\{00001100\}$, $\backslash tfrac1\{15\}\; \backslash mathrm\; i\backslash gets\; 0.\backslash overline\{00000011\}$, and $-\backslash tfrac8\{15\}\backslash gets\; 0.\backslash overline\{11000000\}$, and $-\backslash tfrac4\{15\}\; \backslash mathrm\; i\backslash gets\; 0.\backslash overline\{00110000\}$. Thus, $S$ contains all complex numbers with absolute value ≤ {{sfrac|1|15}}.Knuth 1998 p.206

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

: $[-\backslash tfrac8\{15\},\backslash tfrac2\{15\}]\backslash times[-\backslash tfrac4\{15\},\backslash tfrac1\{15\}]\backslash mathrm\; i$

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

: $\backslash textstyle\; \backslash sum\_\{k\backslash geq\; 1\}x\_k\; (\backslash mathrm\; i-1)^\{-k\}\; \backslash mapsto\; \backslash sum\_\{k\backslash geq\; 1\}x\_k\; b^\{-k\}$

with $b\; >\; 2$.Base $b\; =\; 2$ cannot be taken because both, $\backslash textstyle\; 2^\{-1\}\; =\; 0.1\_\{\backslash text\{bin\}\}\; =\; 0.5\_\{\backslash text\{dec\}\}$ and $\backslash textstyle\; \backslash sum\_\{k\backslash geq\; 2\}\; 2^\{-k\}\; =\; 0.0\backslash overline\{1\}\_\{\backslash text\{bin\}\}\; =\; 0.1\_\{\backslash text\{bin\}\}\; =\; 0.5\_\{\backslash text\{dec\}\}$. However, $\backslash textstyle\; (\backslash mathrm\; i-1)^\{-1\}\; =\; -0.1\_\{\backslash text\{bin\}\}\; -0.1\_\{\backslash text\{bin\}\}\; \backslash mathrm\; i\; =\; -0.5\_\{\backslash text\{dec\}\}\; -0.5\_\{\backslash text\{dec\}\}\; \backslash mathrm\; i$   is unequal to   $\backslash textstyle\; \backslash sum\_\{k\backslash geq\; 2\}\; (\backslash mathrm\; i-1)^\{-k\}\; =\; 0.1\_\{\backslash text\{dec\}\}\; +0.3\_\{\backslash text\{dec\}\}\; \backslash mathrm\; i$.

Furthermore, there are the two mappings

:$\backslash begin\{array\}\{lll\}$

Z_2^\N & \to & S \\

\left(x_k\right)_{k\in\N} & \mapsto & \sum_{k\geq 1}x_k (\mathrm i-1)^{-k}

\end{array}

and

:$\backslash begin\{array\}\{lll\}$

Z_2^\N & \to & [0,1) \\

\left(x_k\right)_{k\in\N} & \mapsto & \sum_{k\geq 1}x_k 2^{-k}

\end{array}

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

:$[0,1)\; \backslash qquad\; \backslash to\; \backslash qquad\; S$

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.

• Dragon curve

{{reflist}}

• "[http://demonstrations.wolfram.com/NumberSystemsUsingAComplexBase/ Number Systems Using a Complex Base]" by Jarek Duda, the Wolfram Demonstrations Project
• "[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ The Boundary of Periodic Iterated Function Systems]" by Jarek Duda, the Wolfram Demonstrations Project
• "[http://demonstrations.wolfram.com/NumberSystemsIn3D/ Number Systems in 3D]" by Jarek Duda, the Wolfram Demonstrations Project
• "[https://community.wolfram.com/groups/-/m/t/3035669 Large introduction to complex base numeral systems]" with Mathematica sources by Jarek Duda

Category:Non-standard positional numeral systems

Category:Fractals

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Category:Ring theory