Non-standard positional numeral systems

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Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not comply with the following description of standard positional systems:

In a standard positional numeral system, the base b is a positive integer, and b different numerals are used to represent all non-negative integers. Each numeral represents one of the values 0, 1, 2, etc., up to b-1, but the value also depends on the position of the digit in a number. The value of a digit string like d_3d_2d_1d_0 in base b is given by the polynomial form
d_3\times b^3+d_2\times b^2+d_1\times b+d_0.
The numbers written in superscript represent the powers of the base used.
For instance, in hexadecimal (b=16), using A=10, B=11 etc., the digit string 1F3A means
1\times16^3+15\times16^2+3\times16+10.
Upon introducing a radix point "." and a minus sign "–", all real numbers can be represented.

This article summarizes facts on some non-standard positional numeral systems. In most cases, the polynomial form in the description of standard systems still applies.

Certain historical numeral systems like the sexagesimal Babylonian notation or the Chinese rod numerals could be classified as standard systems of base 60 and 10, respectively (unconventionally counting the space representing zero as a numeral). However, they could also be classified as non-standard systems (more specifically, mixed-base systems with unary components), if the primitive repeated glyphs making up the numerals are considered.

Bijective numeration systems[edit]

A bijective numeral system with base b uses b different numerals to represent all non-negative integers. However, the numerals have values 1, 2, 3, etc. up to and including b, whereas zero is represented by an empty digit string. For example it is possible to have decimal without a zero.

Base one (unary numeral system)[edit]

Main article: Unary numeral system

Unary is the bijective numeral system with base b=1. In unary, one numeral is used to represent all positive integers. The value of the digit string d_3d_2d_1d_0 given by the polynomial form can be simplified into d_3+d_2+d_1+d_0 since b^n=1 for all n. Non-standard features of this system include:

  • The value of a digit does not depend on its position. Thus, one can easily argue that unary is not a positional system at all.
  • Introducing a radix point in this system will not enable representation of non-integer values.
  • The single numeral represents the value 1, not the value 0=b-1.
  • The value 0 cannot be represented (or is implicitly represented by an empty digit string).

Signed-digit representation[edit]

In some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a particular system where the base is b=2. In the balanced ternary system, the base is b=3, and the numerals have the values −1, 0 and +1 (rather than 0, 1 and 2 as in the standard ternary system, or 1, 2 and 3 as in the bijective ternary system).

Gray code[edit]

Main article: Gray code

The reflected binary code, also known as the Gray code, is closely related to binary numbers, but some bits are inverted, depending on the values of the higher order bits.

Bases that are not positive integers[edit]

A few positional systems have been suggested in which the base b is not a positive integer.

Negative base[edit]

Main article: negative base

Negative-base systems include negabinary, negaternary and negadecimal; in base −b the number of different numerals used is b. All integers, positive and negative, can be represented without a sign.

Complex base[edit]

Main article: Quater-imaginary base

In purely imaginary base bi the b2 numbers from 0 to b2 − 1 are used as digits.
It can be generalized on other complex bases: Complex base systems.

Non-integer base[edit]

In these systems, the number of different numerals used clearly cannot be b. Example: Golden ratio base (phinary).

Mixed bases[edit]

Main article: mixed radix

It is sometimes convenient to consider positional numeral systems where the weights associated with the positions do not form a geometric sequence 1, b, b2, b3, etc., starting from the least significant position, as given in the polynomial form. In a mixed radix system such as the factorial number system, the weights form a sequence where each weight is an integral multiple of the previous one, and the number of permitted digit values varies accordingly from position to position. Other sequences can be used, but then every integer may not have a unique representation. For example, Fibonacci coding uses the digits 0 and 1, weighted according to the Fibonacci sequence (1, 2, 3, 5, 8, ...); a unique representation of all non-negative integers may be ensured by forbidding consecutive 1s.

For calendrical use, the Mayan numeral system was a mixed radix system, since one of its positions represents a multiplication by 18 rather than 20, in order to fit a 360-day calendar. Also, giving an angle in degrees, minutes and seconds (with decimals), or a time in days, hours, minutes and seconds, can be interpreted as mixed radix systems.

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