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{{numeral systems}}

'''Non-standard positional numeral systems''' here designates [[numeral system]]s that may loosely be described as [[Positional notation|positional systems]], but that do not comply with the following description of standard positional systems:

:In a standard positional numeral system, the [[Radix|base]] ''b'' is a positive integer, and ''b'' different [[Numerical digit|numeral]]s are used to represent all [[non-negative]] [[integer]]s.  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 [[numerical digit|digit]] in a number. The value of a digit string like <math>d_3d_2d_1d_0</math> in base ''b'' is given by the '''polynomial form'''

::<math>d_3\times b^3+d_2\times b^2+d_1\times b+d_0</math>.

:The numbers written in superscript represent the [[Exponentiation|powers]] of the base used.
:For instance, in [[hexadecimal]] (''b''=16), using A=10, B=11 etc., the digit string 1F3A means

::<math>1\times16^3+15\times16^2+3\times16+10</math>.

:Upon introducing a [[radix point]] "." and a [[minus sign]] "&ndash;", all [[real number]]s 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 [[Babylonia]]n (standard) [[sexagesimal]] notation or the [[China|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 [[glyph]]s making up the numerals are considered.

==Bijective numeration systems==
A [[Bijective numeration|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#The bijective base-10 system|decimal without a zero]].

===Base one (unary numeral system)===
{{main|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 <math>d_3d_2d_1d_0</math> given by the polynomial form can be simplified into <math>d_3+d_2+d_1+d_0</math> since <math>b^n=1</math> 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==
{{main|Signed-digit representation}}
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 &minus;1, 0 and +1 (rather than 0, 1 and 2 as in the standard [[Ternary numeral system|ternary system]], or 1, 2 and 3 as in the bijective ternary system).

==Bases that are not positive integers==
A few positional systems have been suggested in which the base ''b'' is not a positive integer.
===Negative base===
{{main|negative base}}
Negative-base systems include ''negabinary'', ''negaternary'' and ''negadecimal''; in base &minus;''b'' the number of different numerals used is ''b''.  All integers, positive and negative, can be represented without a sign.

===Complex base===
{{main|Quater-imaginary base}}
In purely imaginary base ''bi'' the ''b''<sup>2</sup> numbers from 0 to {{nowrap|''b''<sup>2</sup> − 1}} are used as digits.<br />
It can be generalized on other complex bases: [[Complex base systems]].

===Non-integer base===
{{main|Non-integer representation}}
In these systems, the number of different numerals used clearly cannot be ''b''.  Example: [[Golden ratio base]] (''phinary'').

==Mixed bases==
{{main|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'', ''b''<sup>2</sup>, ''b''<sup>3</sup>, 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.  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 numbers|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 [[Maya numerals|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.

==External links==
*[http://arxiv.org/abs/math.NT/0608263 Expansions in non-integer bases: the top order and the tail]

[[Category:Non-standard positional numeral systems| ]]