Complex base systems

In arithmetic, a complex base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955^[1][2]) or complex number (proposed by S. Khmelnik in 1964^[3] and Walter F. Penney in 1965^[4][5]).

In general

In more general cases the number of Z (real positive, real with any sign, complex) in this positional number system represented as an expansion

$Z = \sum_{m}^{ } r_m \rho^m,$ where

m - number of discharge, a positive or a negative number (including zero),
ρ - the radix, the number (real or complex),
R - level of decomposition, the number taking values from a finite set

$A_R = (a_0, a_1, \dots, a_i, \dots, a_{R-1})$, containing R different numbers $a_i^{ }$, including complexes,

Next, we write the positional number system as follows $<\rho,A_R^{}>$. In particular, the set $A_R^{}$ might look like:

$B_R = (0, 1, 2,\dots, {R-1})$,

$D_R = (-r_1,-r_1+1,\dots, -1,0,1,\dots,r_2-1,r_2)$,

and $R =r_1+r_2+1,r_1\ne0,r_2\ne0)$ (if $r_1^{}=0$ then set $D_R^{}$ is transformed into set $B_R^{}$).

Well-known positional number systems of complex numbers include the following. (i represents the imaginary unit.)

• Quater-imaginary base, proposed by Donald Knuth in 1955: $<\rho=\sqrt{R},B_R>$, example, $<\rho=\pm i\sqrt{2},[0,1]>$^[1] and $<\rho=\pm 2i,[0,1,2,3]>$.^[2]

• $<\rho=\sqrt{2}e^{\pm i \pi / 2},B_2>$, example, $<\rho=[-1\pm i],[0,1]>$ ^[3][4] (see also section "Base −1±i" below).

• $<\rho=\sqrt{R}e^i\varphi,B_R>$, 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.^[6]

• $<\rho=2e^{i \pi / 3},A_4^{}=[0,1,e^{2i \pi / 3},e^{-2i \pi / 3}]>$;^[7]

• $<\rho=-R,A_R^2>$, where the set $A_R^2$ is composed of complex numbers $r_m=\alpha_m^1+i\alpha_m^2$, and numbers $\alpha_m^{ } \in B_R$, example, <-2, [0,1,j,1+i]>.^[7]

• $<\rho=\rho_2^{ },[0,1]>$, where $\rho_2=\left \{ \frac{(-2)^{1/2}~if~m~even}{i(-2)^{(m-1)/2m}~if~m~odd} \right \}$.^[8]

Binary system

Binary systems of complex numbers, with the digits 0 and 1, are of practical interest.^[8] Listed below are those of the system (as a special case shown above systems) and shows code numbers 2, -2, -1. The binary system of real numbers is also listed for comparison.

• $\rho=2: 2=(10)_{\rho}^{ }$ ;
• $\rho=-2: 2=(110)_{\rho}^{ }, -2=(10)_{\rho}, -1=11_{\rho}$ ;
• $\rho=-\rho_2: 2=(10100)_{\rho}^{ }, -2=(100)_{\rho}, -1=101_{\rho}$ ;
• $\rho=i\sqrt{2}: 2=(10100)_{\rho}, -2=(100)_{\rho}, -1=(101)_{\rho}$ ;
• $\rho=-1+i: 2=(1100)_{\rho}^{ }, -2=(11100)_{\rho}, -1=(11101)_{\rho}$ ;
• $\rho=\frac{-1+i\sqrt{2}}2: 2=(1010)_{\rho}, -2=(110)_{\rho}, -1=(111)_{\rho}$ .

Base −1±i

Of particular interest, the quater-imaginary system, 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 i + 1 (left) or i − 1 (right) base system.

Base −1±i, using digits 0 and 1, was proposed by S. Khmelnik in 1964^[3] and Walter F. Penney in 1965.^[5] 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.

See also

• Dragon curve

References

1. ^ ^{a b} Knuth, D.E. (1960). "An Imaginary Number System". Communication of the ACM-3 (4). 
2. ^ ^{a b} Knuth, Donald (1998). "Positional Number Systems". The art of computer programming. Volume 2 (3rd ed.). Boston: Addison-Wesley. pp. 205. ISBN 0-201-89684-2. OCLC 48246681. 
3. ^ ^{a b c} Khmelnik, S.I. (1964 (see also here)). "Specialized digital computer for operations with complex numbers". Questions of Radio Electronics (in Russian) XII (2). 
4. ^ ^{a b} Jamil, T. (2002). "The complex binary number system". IEEE Potentials 20 (5): 39–41. doi:10.1109/45.983342. 
5. ^ ^{a b} Duda, Jarek (2008-02-24). "Complex base numeral systems". arXiv:0712.1309. 
6. ^ Khmelnik, S.I. (1966 (see also here)). "Positional coding of complex numbers". Questions of Radio Electronics (in Russian) XII (9). 
7. ^ ^{a b} Khmelnik, S.I. (2004 (see also here)). Coding of Complex Numbers and Vectors (in Russian). «Mathematics in Computers», Israel, ISBN 978-0-557-74692-7. 
8. ^ ^{a b} Khmelnik, S.I. (2001). Method and system for processing complex numbers. Patent USA, US2003154226 (A1). http://worldwide.espacenet.com/publicationDetails/biblio?DB=EPODOC&adjacent=true&locale=en_EP&FT=D&date=20030814&CC=US&NR=2003154226A1&KC=A1. 

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