Complex-base system

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).
In general.
Let formula_1 be an integral domain formula_2, and formula_3 the (Archimedean) absolute value on it.
A number formula_4 in a positional number system is represented as an expansion
where
The cardinality formula_6 is called the "level of decomposition".
A positional number system or coding system is a pair
with radix formula_8 and set of digits formula_9, and we write the standard set of digits with formula_10 digits as
Desirable are coding systems with the features:
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 formula_13 and formula_31, and the last two do not require a sign.
In the complex numbers.
Well-known positional number systems for the complex numbers include the following (formula_32 being the imaginary unit):
Binary systems.
"Binary" coding systems of complex numbers, i.e. systems with the digits formula_54, are of practical interest.
Listed below are some coding systems formula_55 (all are special cases of the systems above) and resp. codes for the (decimal) numbers .
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 .
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.
Base.
Of particular interest are the quater-imaginary base (base ) and the base systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign.
Base , using digits and , was proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965. The rounding region of an integer – i.e., a set formula_56 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 formula_56 is, by definition, all points that can be written as formula_58 with formula_59. formula_56 can be decomposed into 16 pieces congruent to formula_61. Notice that if formula_56 is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to formula_63, because formula_64. The rectangle formula_65 in the center intersects the coordinate axes counterclockwise at the following points: formula_66, formula_67, and formula_68, and formula_69. Thus, formula_56 contains all complex numbers with absolute value ≤ .
As a consequence, there is an injection of the complex rectangle
into the interval formula_72 of real numbers by mapping
with formula_74.
Furthermore, there are the two mappings
and
both surjective, which give rise to a surjective (thus space-filling) mapping
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.