In arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, a complex base system is a positional numeral system whose radix

Radix

In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

 is an imaginary

Imaginary number

An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

 (proposed by Donald Knuth

Donald Knuth

Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

 in 1955) or complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

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

In general

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

where

m - number of discharge, a positive or a negative number (including zero),

- the radix, the number (real or complex),

R - level of decomposition, the number taking values from a finite set

, containing R different numbers , including complexes,

Next, we write the positional number system as follows . In particular, the set might look like:

,

,

and (if then set is transformed into set ).

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

• Quater-imaginary base
  Quater-imaginary base
  The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to a high-school science talent search. It is a non-standard positional numeral system which uses the imaginary number 2i as its base. It is able to uniquely represent every complex number using only...
  
  , proposed by Donald Knuth
  Donald Knuth
  Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
  
   in 1955: , example, and .

• , example, (see also section "Base −1±i" below).

• , where and is a positive integer that can take multiple values at a given R.

• ;

• , where the set is composed of complex numbers , and numbers , example, <-2, [0,1,j,1+i]>.

• , where .

Binary system

Binary systems of complex numbers, with the digits 0 and 1, are of practical interest. 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.

• ;
• ;
• ;
• ;
• ;
• .

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 integer

Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...

s without sign.

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.

External links

• "Number Systems Using a Complex Base" by Jarek Duda, the Wolfram Demonstrations Project
  Wolfram Demonstrations Project
  The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
  
  
• "The Boundary of Periodic Iterated Function Systems" by Jarek Duda, the Wolfram Demonstrations Project
  Wolfram Demonstrations Project
  The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
  
  
• "Number Systems in 3D" by Jarek Duda, the Wolfram Demonstrations Project
  Wolfram Demonstrations Project
  The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...