There are many different

numeral system A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent differe ...

s, that is,

writing system A writing system comprises a set of symbols, called a ''script'', as well as the rules by which the script represents a particular language. The earliest writing appeared during the late 4th millennium BC. Throughout history, each independen ...

s for expressing

numbers A number is a mathematical object used to count, measure, and label. The most basic 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

"A ''base'' is a natural number B whose ''powers'' (B multiplied by itself some number of times) are specially designated within a numerical system." The term is not equivalent to

radix In a positional numeral system, the radix (radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, becaus ...

, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base).

By type of notation

Numeral systems are classified here as to whether they use

positional notation Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a posit ...

(also known as place-value notation), and further categorized by

or base.

Standard positional numeral systems

The common names are derived somewhat arbitrarily from a mix of

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

and

Greek Greek may refer to: 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 of all kno ...

, in some cases including roots from both languages within a single name. There have been some proposals for standardisation.Multiplication Tables of Various Bases
p. 45, Michael Thomas de Vlieger,

Dozenal Society of America The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is in ...

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 becau ...

Complex bases

Non-integer bases

''n''-adic number

Mixed radix Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a m ...

Factorial number system In combinatorics, the factorial number system (also known as 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 ...

* 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 * Redundant binary representation * 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 are non-positional, 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.

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 is exp ...

Numeral