Non-standard positional numeral systems
Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems:
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.
Some historical numeral systems may be described as non-standard positional numeral systems. E.g., the sexagesimal Babylonian notation and the Chinese rod numerals, which can be classified as standard systems of base 60 and 10, respectively, counting the space representing zero as a numeral, can also be classified as non-standard systems, more specifically, mixed-base systems with unary components, considering the primitive repeated glyphs making up the numerals.
However, most of the non-standard systems listed below have never been intended for general use, but were devised by mathematicians or engineers for special academic or technical use.
Bijective numeration systems
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)
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 pqrs given by the polynomial form can be simplified into since bn = 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.
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 −1, 0 and +1.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 parity of the higher order bits.Graphical and physical variants
Cistercian numerals are a decimal positional numeral system, but the positions are not aligned as in common decimal notation; instead, they are attached to the top-right, top-left, bottom-right and bottom-left of a vertical stem, respectively, and thus limited to four in number. The system has close similarities to standard positional numeral systems, but may also be compared to e.g. Greek numerals, where different sets of symbols are used for the ones, tens, hundreds and thousands, likewise giving an upper limit on the numbers that can be represented.Similarly, in computers, e.g. the long integer format is a standard binary system, but it has a limited number of positions, and the physical locations for the representations of the digits may not be aligned. In an analog odometer and in an abacus, the decimal digits are aligned but limited in number.
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
Negative-base systems include negabinary, negaternary and negadecimal, with bases −2, −3, and −10 respectively; in base −b the number of different numerals used is b. Due to the properties of negative numbers raised to powers, all integers, positive and negative, can be represented without a sign.Complex base
In a purely imaginary base bi system, where b is an integer larger than 1 and i the imaginary unit, the standard set of digits consists of the b2 numbers from 0 to. It can be generalized to other complex bases, giving rise to the complex-base systems.Non-integer base
In non-integer bases, the number of different numerals used clearly cannot be b. Instead, the numerals 0 to are used. For example, golden ratio base, uses the 2 different numerals 0 and 1.Mixed bases
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. Examples include:- Measuring time often uses a mix of base 24 for hours, and base 60 for minutes and seconds, with each part often written base 10, as in 20:15:00 representing twenty hours and fifteen minutes.
- Similarly, giving an angle in degrees, minutes and seconds, can be interpreted as a mixed-radix system.
- Non-decimal currencies have been common, e.g. in Commonwealth countries that before decimalization used pounds, shillings and pennies.
- The Mayan numeral system was base 20, but when applied to the calendar it was a mixed-radix system as one of its positions represented a multiplication by 18 rather than 20, in order to fit a 360-day calendar.
- The factorial number system is a mixed-radix system where the weights form a sequence where each weight is an integer multiple of the previous one, and the number of permitted digit values varies accordingly from position to position.