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]).
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Let be an integral domain and the (Archimedean) absolute value on it.
A number in this positional number system is represented as an expansion
– the radix (or base) with ,
– exponent (position or place)
– 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
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[2] is
All these coding systems have the mentioned features for and , and the latter two do not require a sign.
Well-known positional number systems for the complex numbers include the following ( being the imaginary unit):
Binary coding systems of complex numbers, i. e. systems with the digits , are of practical interest.[8] 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 1 | ||||
0.1 = 1.0 = 1 | |||||
0.01 = 1.10 = | |||||
0.0011 = 11.1100 = | |||||
0.010 = 11.001 = 1110.100 = | |||||
1.011 = 11.101 = 11100.110 = | |||||
0.0011 = 11.1100 = | |||||
1 the underline marks the period |
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.
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.
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.