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) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965).
A number in a positional number system is represented as an expansion
is the radix (or base) with is the exponent (position or place) are digits from the finite set of digits usually with
The cardinality is called the level of decomposition.
A positional number system or coding system is a pair
with radix and set of digits , and we write the standard set of digits with digits as
Desirable are coding systems with the features
- Every number in , e. g. the integers , the Gaussian integers or the integers , is uniquely representable as a finite code, possibly with a sign.
- Every number in the field of fractions , which possibly is completed for the metric given by yielding or , is representable as an infinite code, where the series converges under for , and the measure of the set of numbers with more than one representation is 0. The latter requires that the set be minimal, i. e. .
In the Real Numbers
In this notation our standard decimal coding scheme is denoted by
the standard binary system is
the negabinary system is
and the balanced ternary system is
All these coding systems have the mentioned features for and , and the latter two do not require a sign.
In the Complex Numbers
Well-known positional number systems for the complex numbers include the following ( being the imaginary unit):
- , e. g.  and
- , where , and is a positive integer that can take multiple values at a given . For and this is the system
- , where the set consists of complex numbers , and numbers , e. g.
- , where 
Binary coding systems of complex numbers, i. e. systems with the digits , are of practical interest. Listed below are some coding systems (all are special cases of the systems above) and codes for the numbers –1, 2, -2, . The standard binary (which requires a sign) and the negabinary systems are also listed for comparison. They do not have a genuine expansion for .
|radix||twins and triplets|
|Some Bases and some Representations|
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.
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.
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.
Example: 3 = 11 base(2); 11 base(-1+i) = i; the position of 3 on the graph (x,y*i) is (0,1).
- Knuth, D.E. (1960). "An Imaginary Number System". Communication of the ACM-3 (4).
- Knuth, Donald (1998). "Positional Number Systems". The art of computer programming. Volume 2 (3rd ed.). Boston: Addison-Wesley. p. 205. ISBN 0-201-89684-2. OCLC 48246681.
- Khmelnik, S.I. (1964). "Specialized digital computer for operations with complex numbers". Questions of Radio Electronics (in Russian) XII (2).
- W. Penney, A "binary" system for complex numbers, JACM 12 (1965) 247-248.
- Jamil, T. (2002). "The complex binary number system". IEEE Potentials 20 (5): 39–41. doi:10.1109/45.983342.
- Duda, Jarek (2008-02-24). "Complex base numeral systems". arXiv:0712.1309.
- Khmelnik, S.I. (1966). "Positional coding of complex numbers". Questions of Radio Electronics (in Russian) XII (9).
- 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.
- Khmelnik, S.I. (2001). Method and system for processing complex numbers. Patent USA, US2003154226 (A1).
- William J. Gilbert, “Arithmetic in Complex Bases” Mathematics Magazine Vol. 57, No. 2, March 1984
- infinite non-repeating sequence