Non-standard positional numeral systems
This article may be too technical for most readers to understand.(November 2023) |
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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 entirely comply with the following description of standard positional systems:
- In a standard positional numeral system, the in a number. The value of a digit string like pqrs in base b is given by the polynomial form
- .
- The numbers written in superscript represent the powers of the base used.
- For instance, in hexadecimal (b = 16), using the numerals A for 10, B for 11 etc., the digit string 7A3F means
- ,
- which written in our normal decimal notation is 31295.
- Upon introducing a minus sign "−", real numberscan be represented up to arbitrary accuracy.
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
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
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 p + q + r + s 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 (or is implicitly represented by an empty digit string).
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 (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
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.
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 b2 − 1. 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 (phinary), 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
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.
Sequences where each weight is not an integer multiple of the previous weight may also be used, but then every integer may not have a unique representation. For example,
Asymmetric numeral systems
Asymmetric numeral systems are systems used in
See also
External links
References
- ^ J. Duda, K. Tahboub, N. J. Gadil, E. J. Delp, The use of asymmetric numeral systems as an accurate replacement for Huffman coding, Picture Coding Symposium, 2015.