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Non-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from 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 d₃d₂d₁d₀ in base b is

.

For instance, in hexadecimal (b=16), using A=10, B=11 etc., the digit string 1F3A means

.

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 all cases except the last one below on mixed bases, the expression

in the description of standard systems applies.

Certain historical numeral systems like the Babylonian (standard) sexagesimal notation or the Chinese rod numerals could be classified as standard systems, if the 60 resp. 10 distinct numerals are considered as digits, 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.

Contents 1 Bijective numeration systems 1.1 Base one (unary numeral system) 2 Signed-digit representations 3 Bases that are not positive integers 4 Mixed bases 5 External links

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, where as zero is represented by an empty digit string. For example it is possible to have decimal without a zero.

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 d₃d₂d₁d₀ can be simplified into d₃ + d₂ + d₁ + d₀ since bⁿ = 1 for all n. The non-standard features of this system are:

1. 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.
2. Introducing a radix point in this system will not enable representation of non-integer values.
3. The single numeral represents the value 1, not the value 0=b-1.
4. The value 0 cannot be represented (or is implicitly represented by an empty digit string).

In some systems, while the base is a positive integer, negative digits are allowed. The articles Signed-digit representation and Non-adjacent form consider systems 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).

A few positional systems have been suggested, in which the base b is not a positive integer. In these systems, the number of different numerals used clearly cannot be b. For details, see the relevant articles, Golden ratio base, Negabinary, Negaternary, Non-integer representation and Quarter-imaginary base.

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. In a mixed radix system such as the factoradic system, the weights form a sequence where each weight is an integral multiple of the previous one. However, any sequence can be used, but in the more general case, every number does not necessarily have a unique representation, though unique representations may be guaranteed by imposing suitable constraints on the digit sequence. For example, using the Fibonacci sequence (1, 2, 3, 5, 8, ...) and the digits 0 and 1 leads to Fibonacci coding; requiring no consecutive 1's ensures a unique representation of all non-negative integers.

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.

• Expansions in non-integer bases: the top order and the tail