There are many different
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 is,
writing system
A writing system is a method of visually representing verbal communication, based on a script and a set of rules regulating its use. While both writing and speech are useful in conveying messages, writing differs in also being a reliable form ...
s for expressing
numbers
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
.
By culture / time period
By type of notation
Numeral systems are classified here as to whether they use
positional notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
(also known as place-value notation), and further categorized by
radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
or base.
Standard positional numeral systems
![Binary clock](https://upload.wikimedia.org/wikipedia/commons/2/27/Binary_clock.svg)
The common names are derived
somewhat arbitrarily from a mix of
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
and
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
, in some cases including roots from both languages within a single name. There have been some proposals for standardisation.
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:
:In a standard positional ...
Bijective numeration
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 betw ...
Signed-digit representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers.
Signed-digit representation can be used to accomplish fast addition of integers because ...
Negative bases
A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is ...
The common names of the negative base numeral systems are formed using the prefix ''nega-'', giving names such as:
Complex bases
Non-integer bases
''n''-adic number
Mixed radix
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical radix, base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are eac ...
*
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 ...
* Even double factorial number system
* Odd double factorial number system
*
Primorial number system
*
Fibonorial number system
* in timekeeping
* in timekeeping
* (12, 20) traditional English monetary system (£sd)
* (20, 18, 13) Maya timekeeping
Other
*
Quote notation
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
*
Redundant binary representation A redundant binary representation (RBR) is a numeral system that uses more bits than needed to represent a single binary Numerical digit, digit so that most numbers have several representations. An RBR is unlike usual binary numeral systems, includi ...
*
Hereditary base-n notation
*
Asymmetric numeral systems
Asymmetric numeral systems (ANS)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.J. Duda''Asymmetric numeral systems: entropy coding ...
optimized for non-uniform probability distribution of symbols
*
Combinatorial number system In mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natura ...
Non-positional notation
All known numeral systems developed before the
Babylonian numerals
Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform (script), cuneiform, using a wedge-tipped Phragmites, reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record ...
are non-positional,
[Chrisomalis calls the Babylonian system "the first positional system ever" in .] as are many developed later, such as the
Roman numerals
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
. The French Cistercian monks created
their own numeral system.
See also
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Table of bases
This table of bases gives the values of 0 to 256 in bases 2 to 36, using A−Z for 10−35.
"Base" (or "radix") is a term used in discussions of numeral systems which use place-value notation for representing numbers.
Base 10 is in bold.
...
– 0 to 74 in base 2 to 36
*
References
{{Reflist
Systems
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...