Non-standard positional numeral systems here designates
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
s 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
base ''b'' is a positive integer, and ''b'' different
numerals are used to represent all
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. The standard set of numerals contains the ''b'' values 0, 1, 2, etc., up to ''b'' − 1, but the value is weighted according to the position of the
digit 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
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
(''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
radix point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
"." and a
minus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...
"−",
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s can 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
sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form†...
Babylonian notation and the Chinese
rod numerals
Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number.
The written fo ...
, 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
glyph
A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...
s 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
Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection (i.e. one-to-one correspondence) that exists in this case betwe ...
.
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 ''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
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
Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalance ...
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 number
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
s, but some
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s 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 imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, the standard set of digits consists of the ''b''
2 numbers from 0 to . It can be generalized to other complex bases, giving rise to the
Complex-base system
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965W. Penney, A "binary" system for co ...
s.
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
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean 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
geometric sequence
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
1, ''b'', ''b''
2, ''b''
3, etc., starting from the least significant position, as given in the polynomial form. In a
mixed-radix system such as the
factorial number system
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digit ...
, 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.
For calendrical use, the
Mayan
Mayan most commonly refers to:
* Maya peoples, various indigenous peoples of Mesoamerica and northern Central America
* Maya civilization, pre-Columbian culture of Mesoamerica and northern Central America
* Mayan languages, language family spoken ...
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,
Fibonacci coding
In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no ...
uses the digits 0 and 1, weighted according to the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
(1, 2, 3, 5, 8, ...); a unique representation of all non-negative integers may be ensured by forbidding consecutive 1s.
Binary-coded decimal
In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for ...
(BCD) are mixed base systems where bits (binary digits) are used to express decimal digits. E.g., in 1001 0011, each group of four bits may represent a decimal digit (in this example 9 and 3, so the eight bits combined represent decimal 93). The weights associated with these 8 positions are 80, 40, 20, 10, 8, 4, 2 and 1. Uniqueness is ensured by requiring that, in each group of four bits, if the first bit is 1, the next two must be 00.
Asymmetric numeral systems
Asymmetric numeral systems are systems used in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
where each digit can have different bases, usually non-integer. In these, not only are the bases of a given digit different, they can be also nonuniform and altered in an asymmetric way to encode information more efficiently. They are optimized for chosen non-uniform probability distributions of symbols, using on average approximately
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum W ...
bits per symbol.
[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.
See also
*
List of numeral systems
There are many different numeral systems, that is, writing systems for expressing numbers.
By culture / time period
By type of notation
Numeral systems are classified here as to whether they use positional notation (also known as place-value ...
*
Komornik–Loreti constant
External links
Expansions in non-integer bases: the top order and the tail
References
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