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

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

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.

References

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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 ...