Numeral system - Citizendia

This article is about different methods of expressing numbers with symbols. For completely different sets of numbers, see number system. In Mathematics, a number system is a set of Numbers (in the broadest sense of the word together with one or more operations such as Addition

Numeral systems by culture
Hindu-Arabic numerals
Indian Eastern Arabic Khmer	Indian family Brahmi Thai
East Asian numerals
Chinese Suzhou Counting rods	Japanese Korean
Alphabetic numerals
Abjad Armenian Cyrillic Ge'ez	Hebrew Greek (Ionian) Āryabhaṭa
Other systems
Attic Babylonian Egyptian Etruscan	Mayan Roman Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…
v • d • e

A numeral system (or system of numeration) is a mathematical notation for representing numbers of a given set by symbols in a consistent manner. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system The Eastern Arabic numerals (also called Arabic-Indic numerals and Arabic Eastern Numerals) are the symbols used to represent the Hindu-Arabic numeral system Khmer numerals are the numerals used in the Khmer language of Cambodia. Most of the positional Base 10 Numeral systems in the world have originated from India, which first developed the concept of positional numerology The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens Thai numerals (เลขไทย are a set of numerals traditionally used in Thailand, although the Arabic numerals are more common Chinese numerals are characters for writing Numbers in Chinese. The Suzhou numerals or huama is a Numeral system used in China before the introduction of Arabic numerals. Counting rods ( Japanese: 算木 sangi are small bars typically 3-14 cm long used by mathematicians for calculation in China, Japan The Korean language has two regularly used sets of numerals a Sino-Korean system and a native Korean system The Abjad numerals are a decimal Numeral system in which the 28 letters of the Arabic alphabet are assigned numerical values The system of Armenian numerals is a historic Numeral system created using the Majuscules (uppercase letters of the Armenian alphabet. Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. Ge'ez (gez ግዕዝ) also called Ethiopic, is an Abugida script that was originally developed to write Ge'ez, a Semitic language The system of Hebrew numerals is a quasi-decimal alphabetic Numeral system using the letters of the Hebrew alphabet. ʹ the numeral sign redirects here For the accent ´ see Acute accent. The Āryabhaṭa numeration is a system of numerals based on Sanskrit phonemes. Attic numerals were used by the ancient Greeks, possibly from the 7th century BC Babylonian numerals were written in cuneiform, using a wedge-tipped reed Stylus to make a mark on a soft Clay tablet which would be exposed The system of Ancient Egyptian numerals was a Numeral system used in ancient Egypt aka Kemet The Etruscan numerals were used by the ancient Etruscans The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman The Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. Discovery In 1946 a deposit with more than 250 sickles corresponding to the period 1500-1250 BC was discovered in Frankleben (in the region of Merseburg - Querfurt This is a list of Numeral system topics (and "numeric representations" by Wikipedia page A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral The decimal ( base ten or occasionally denary) Numeral system has ten as its base. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. Quaternary is the base - Numeral system. It uses the digits 0 1 2 and 3 to represent any Real number. The octal Numeral system, or oct for short is the base -8 number system and uses the digits 0 to 7 In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a Base 32 or duotrigesimal is a Numeral system with 32 as its base The base - system is a Numeral system with 64 as its base It is the largest power-of-two base that can be represented using single printable ASCII The unary numeral system is the bijective base - 1 Numeral system. Ternary or trinary is the base - Numeral system. Analogous to a " Bit " a ternary digit is known as a trit ( Nonary is a base - Numeral system, typically using the digits 0-8 but not the digit 9 The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten The base - system is a Numeral system with 24 as its base There are 24 hours in a day so our time keeping system includes a base-24 component Base 30 or trigesimal is a positional numeral system using 30 as the Radix. Base 36 is a positional numeral system using 36 as the Radix. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in different bases. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral

Ideally, a numeral system will:

Represent a useful set of numbers (e. g. all whole numbers, integers, or real numbers)
Give every number represented a unique representation (or at least a standard representation)
Reflect the algebraic and arithmetic structure of the numbers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the real numbers may be described informally in several different ways

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics and Computer science, a digit is a symbol (a number symbol e In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 31, and another that recurs, such as 2. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is 309999999. . . . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2. 31 and 2. 310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero.

Numeral systems are sometimes called number systems, but that name is misleading, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc. In Mathematics, a number system is a set of Numbers (in the broadest sense of the word together with one or more operations such as Addition In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 Such systems are not the topic of this article.

1 Types of numeral systems
2 Bases used
3 Positional systems in detail
4 Change of radix
5 Generalized variable-length integers
- 5.1 Properties of numerical systems with integer bases
6 See also
7 References
8 External links

Types of numeral systems

The most commonly used system of numerals is known as Hindu-Arabic numerals, and two great Indian mathematicians could be given credit for developing them. The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Brahmagupta ( (598–668 was an Indian mathematician and astronomer.

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. The unary numeral system is the bijective base - 1 Numeral system. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an If the symbol / is chosen, for example, then the number seven would be represented by ///////. Tally marks represent one such system still in common use. Tally marks are an implementation of the Unary numeral system. In practice, the unary system is normally only useful for small numbers, although it plays an important role in theoretical computer science. Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such Also, Elias gamma coding which is commonly used in data compression expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral. Elias gamma code is a universal code encoding positive integers

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ //// and number 123 as + - - /// without any need for zero. This is called sign-value notation. In Computers Sign-value notation (sign-magnitude notation in computers is the use of the high-order bit (left end of a binary word to represent the numeric sign 0 for + and The ancient Egyptian system is of this type, and the Roman system is a modification of this idea. The system of Ancient Egyptian numerals was a Numeral system used in ancient Egypt aka Kemet Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D/ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them

More elegant is a positional system, also known as place-value notation. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a Again working in base 10, we use ten different digits 0, . . . , 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, borrowed from India, is a positional base 10 system; it is used today throughout the world. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found . . respectively. The sign-value systems use only the geometric numerals and the positional system use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. ʹ the numeral sign redirects here For the accent ´ see Acute accent. However, the spoken language uses both arithmetic and geometric numerals.

In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, . Bijective numeration is any Numeral system that establishes a Bijection between the set of non-negative Integers and the set of finite strings . . , k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Bijective base-k numeration is also called k-adic notation, not to be confused with p-adic numbers. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 Bijective base-1 the same as unary.

Bases used

Computing

Switches, mimicked by their electronic successors built originally of vacuum tubes and in modern technology of transistors, have only two possible states: "open" and "closed". This article is about the electronic device not an evacuated pipe used for experiments in Free-fall. In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. This base-2 system (binary) is the basis for digital computers. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. A digital system uses discrete (discontinuous values usually but not always Symbolized Numerically (hence called "digital" to represent information for It is used to perform integer arithmetic in almost all digital computers; some exotic base-3 (ternary) and base-10 computers have also been built, but those designs were discarded early in the history of computing hardware. Ternary or trinary is the base - Numeral system. Analogous to a " Bit " a ternary digit is known as a trit ( The history of computer hardware encompasses the hardware, its architecture, and its impact on software.

Modern computers use transistors that represent two states with either high or low voltages. A computer is a Machine that manipulates data according to a list of instructions. In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals The smallest unit of memory for this binary state is called a bit. Bits are arranged in groups to aid in processing, and to make the binary numbers shorter and more manageable for humans. More recently these groups of bits, such as bytes and words, are sized in multiples of four. A byte (pronounced "bite" baɪt is the basic unit of measurement of information storage in Computer science. Thus base 16 (hexadecimal) is commonly used as shorthand. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a Base 8 (octal) has also been used for this purpose.

A computer does not treat all of its data as numerical. For instance, some of it may be treated as program instructions or data such as text. However, arithmetic and Boolean logic constitute most internal operations. Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of Whole numbers are represented exactly, as integers. In computer science the term integer is used to refer to a Data type which represents some finite subset of the mathematical Integers These are also known as Real numbers, allowing fractional values, are usually approximated as floating point numbers. In Mathematics, the real numbers may be described informally in several different ways In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. The computer uses different methods to do arithmetic with these two kinds of numbers. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone

Five

A base-5 system (quinary) has been used in many cultures for counting. Quinary ( base -) is a Numeral system with five as the base This originates from the five Fingers on either Hand. Plainly it is based on the number of fingers on a human hand. It may also be regarded as a sub-base of other bases, such as base 10 and base 60.

Eight

A base-8 system (octal) was devised by the Yuki of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. The octal Numeral system, or oct for short is the base -8 number system and uses the digits 0 to 7 The Yuki are a Native American tribe from the zone of Round Valley, in what today is part of the territory of Mendocino County, Northern California There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997).

Ten

The Algorists versus the Abacists by Gregor Reisch: Margarita Philosophica, 1508

The base-10 system (decimal) is the one most commonly used today. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. It is assumed to have originated because humans have ten fingers. Human beings, humans or man (Origin 1590–1600 L homō man OL hemō the earthly one (see Humus A finger is a type of digit, an organ of manipulation and sensation found in the Hands of Humans and other Primates Normally humans have five digits These systems often use a larger superimposed base. See Decimal superbase. Many numeral systems with base 10 use a superimposed larger base of 100 1000 10000 or 1000000

Twelve

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtracting being just as easy. The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. 12 is a useful base because it has many factors. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without It is the smallest multiple of one through four and of six. There is still a special word for "dozen" and just like there is a word for 10², hundred, there is also a word for 12², gross. Base-12 could have originated from the number of knuckles in the four fingers of a hand excluding the thumb, which is used as a pointer in counting.

Twelve is a common British unit of measurement. There are twelve inches to a foot. Prior to 1971, in British currency, there were 12 pennies to a shilling. [1]. English words for numbers are also 'base-12' in that there is a unique word for the numbers one through twelve, with 'thirteen' being the first word that was formed by combining numbers (three and ten).

There are 24 hours per day, usually counted till 12 until noon (p.m.) and once again until midnight (a.m.), often further divided per 6 hours in counting (for instance in Thailand) or as switches between using terms like 'night', 'morning', 'afternoon', and 'evening', whereas other languages use such terms with durations of 3 to 9 hours often according to switches at some of the 3 hour interval marks. The Kingdom of Thailand (ˈtaɪlænd ราชอาณาจักรไทย, râːtɕʰa-ʔaːnaːtɕɑ̀k-tʰɑj

Multiples of 12 have been in common use as English units of resolution in the analog and digital printing world, where 1 point equals 1/72 of an inch and 12 points equal 1 pica, and printer resolutions like 360, 600, 720, 1200 or 1440 dpi (dots per inch) are common. In Typography, a point is the smallest unit of measure being a subdivision of the larger pica. This page is for the unit of measure For the eating disorder see Pica (disorder. These are combinations of base-12 and base-10 factors: (3×12)×10, (5×12)×10, (6×12)×10, (10×12)×10 and (12×12)×10.

Twenty

The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base-20 (vigesimal), possibly originating from the number of a person's fingers and toes. The Maya civilization is a Mesoamerican Civilization, noted for the only known fully developed written language of the Pre-Columbian Americas The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences Mesoamerica or Meso-America (Mesoamérica is a Region extending approximately from central Mexico to Honduras and Nicaragua, defined The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten Evidence of base-20 counting systems is also found in the languages of central and western Africa.

Possible remnants of a base-20 system also exist in French, as seen in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores"). "Twenty" redirects here For the village in England, see Twenty Lincolnshire. For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve).

The Irish language also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. Irish (ga ''Gaeilge'' is a Goidelic language of the Indo-European language family originating in Ireland and historically spoken by the Irish. A remnant of this system may be seen in the modern word for 40, daoichead.

Danish numerals display a similar base-20 structure. Danish ( d̥ænsɡ̊ is one of the North Germanic languages (also called Scandinavian languages a sub-group of the Germanic branch of the

Sixty

Base 60 (sexagesimal) was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. Sumer ( Sumerian: sux-Latn [[Ki (earth ki]]-[[EN (cuneiform en]]-'''ĝir15''', Akkadian: Šumeru; possibly Biblical Shinar Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding The hour (symbol h) is a unit of Time. It is not an SI unit but is accepted for use with the SI A minute is a Unit of measurement of Time or of Angle. The minute is a unit of Time equal to 1/60th of an Hour or 60 The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units This article describes the unit of angle For other meanings see Degree. A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60 of one degree. A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60 of one degree. 60 also has a large number of factors, including the first six counting numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Base-60 systems are believed to have originated through the merging of base-10 and base-12 systems. The Chinese Calendar, for example, uses a base-60 Jia-Zi甲子 system to denote years, with each year within the 60-year cycle being named with two symbols, the first being base-10 (called Tian-Gan天干 or heavenly stems) and the second symbol being base 12 (called Di-Zhi地支 or earthly branches). The Chinese calendar is lunisolar, incorporating elements of a Lunar calendar with those of a Solar calendar. The Chinese sexagenary cycle ( is a cyclic numeral system of 60 combinations of the two basic cycles the ten Heavenly Stems (天干 tiāngān The ten Celestial Stems ( sometimes known as Heavenly Stems, are the elements of an ancient Chinese cyclic character Numeral system: Jia (甲 Yi (乙 The Earthly Branches ( or) provide one Chinese system for reckoning Time. Both symbols are incremented in successive years until the first pattern recurs 60 years later. The second symbol of this system is also related to the 12-animal Chinese zodiac system. The Chinese Zodiac is a 12 year cycle Each year of the 12 year cycle is named after one of the original 12 animals The Jia-zi system can also be applied to counting days, with a year containing roughly six 60-day cycles.

Dual base (five and twenty)

Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). There are an estimated 2000 Languages spoken in Africa. About a hundred of these are widely used for inter-ethnic communication The Republic of Guinea-Bissau (ˈgɪni bɨˈsaʊ República da Guiné-Bissau ʁɛˈpublikɐ dɐ giˈnɛ biˈsau is a country in Western Africa, and one of the smallest Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region. Sudan (officially the Republic of Sudan) ( السودان al-Sūdān is a country in northeastern Africa.

Base names

Number	From Latin			From Greek	Other
	Cardinals	Ordinals	Distributives
1	unary	primal	singulary	henadic
2	dual		binary	dyadic
3		tertial	ternary, trinary	triadic
4		quartal	quaternary	tetradic
5		quintal	quinary	pentadic	quinternary
6		sextal	senary	hexadic	heximal, hexary
7		septimal	septenary	hebdomadic	septuary
8	octal	octaval, octavary	octonary	ogdoadic	octonal
9		nonary	novenary	enneadic	novary, noval
10		decimal	denary	decadic
11		undecimal	undenary	hendecadic	unodecimal
12		duodecimal	duodenary	duodecadic	dozenal
13		tridecimal, tredecimal			triodecimal
14		quattuordecimal, quadrodecimal			tetradecimal
15		quindecimal	quindenary		pentadecimal
16		sedecimal	sedenary		hexadecimal, sexadecimal
17		septendecimal			heptadecimal
18		octodecimal			decennoctal
19		nonadecimal			novodecimal, decennoval
20		vicesimal, vigesimal	vicenary	icosadic	bigesimal, bidecimal
30		tricesimal, trigesimal	tricenary	triacontadic	triogesimal
40		quadragesimal	quadragenary
50		quinquagesimal	quinquagenary		pentagesimal
60		sexagesimal	sexagenary	hexecontadic
70		septuagesimal	septuagenary
80		octogesimal	octogenary
90		nonagesimal	nonagenary
100		centesimal	centenary	hecatontadic
200		ducentesimal	ducenary		bicentesimal, bicentimal
300		trecentesimal	trecenary		tercentimal, tricentesimal
400		quadringentesimal	quadringenary		quadricentesimal, quattrocentimal
500		quingentesimal	quingenary		pentacentesimal, quincentimal
600		sescentesimal			hexacentesimal, hexacentimal
700		septingentesimal	septingenary		heptacentesimal, heptacentimal
800		octingentesimal	octingenary		octacentesimal, octacentimal
900		noningentesimal	nongenary
1000		millesimal	millenary	chiliadic
10000				myriadic	decamillesimal

   21 - unovigesimal / unobigesimal
   22 - duovigesimal
   23 - triovigesimal
   24 - quadrovigesimal / quadriovigesimal
   26 - hexavigesimal / sexavigesimal
   27 - heptovigesimal
   28 - octovigesimal
   29 - novovigesimal
   31 - unotrigesimal
        (. The unary numeral system is the bijective base - 1 Numeral system. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. Ternary or trinary is the base - Numeral system. Analogous to a " Bit " a ternary digit is known as a trit (  Quaternary is the base - Numeral system. It uses the digits 0 1 2 and 3 to represent any Real number. Quinary ( base -) is a Numeral system with five as the base This originates from the five Fingers on either Hand. Quinary ( base -) is a Numeral system with five as the base This originates from the five Fingers on either Hand. In Mathematics, a senary Numeral system is a base - numeral system In Mathematics, a senary Numeral system is a base - numeral system The septenary Numeral system is the base - number system and uses the digits 0-6 The octal Numeral system, or oct for short is the base -8 number system and uses the digits 0 to 7 Nonary is a base - Numeral system, typically using the digits 0-8 but not the digit 9 The decimal ( base ten or occasionally denary) Numeral system has ten as its base. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. The undecimal ( base -) Positional notation system is based on the number eleven rather than ten as in Decimal or eight in  The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. The tetradecimal (base-14 Positional notation system is based on the number fourteen The pentadecimal (base-15 positional notation system is based on the number fifteen In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a  The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten Base 30 or trigesimal is a positional numeral system using 30 as the Radix. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. The base - system is a Numeral system with 24 as its base There are 24 hours in a day so our time keeping system includes a base-24 component A Hexavigesimal Numeral system has a base of Twenty-six. Base 26 is a fairly natural way of representing numbers as text using the 26-letter Latin alphabet . . repeat naming pattern. . . )
   36 - hexatridecimal / sexatrigesimal
        (. Base 36 is a positional numeral system using 36 as the Radix. . . repeat naming pattern. . . )
   41 - unoquadragesimal
        (. . . repeat naming pattern. . . )
   51 - unoquinquagesimal 
        (. . . repeat naming pattern. . . )
   64 - quadrosexagesimal
        (. The base - system is a Numeral system with 64 as its base It is the largest power-of-two base that can be represented using single printable ASCII . . repeat naming pattern. . . )
  110 - decacentimal
  111 - unodecacentimal
        (. . . repeat naming pattern. . . )
  210 - decabicentimal
  211 - unodecabicentimal
        (. . . repeat naming pattern. . . )
  800 - octocentimal / octocentesimal
 2000 - bimillesimal
        (. . . repeat naming pattern. . . )

Positional systems in detail

See also: Positional notation

In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.

For example, in the decimal system (base 10), the numeral 4327 means (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), noting that 10⁰ = 1. The decimal ( base ten or occasionally denary) Numeral system has ten as its base.

In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a_nbⁿ + a_n − 1b^n − 1 + a_n − 2b^n − 2 + . . . + a₀b⁰ and writing the enumerated digits a_na_n − 1a_n − 2 . . . a₀ in descending order. The digits are natural numbers between 0 and b − 1, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10. 11 denotes 1×2¹ + 0×2⁰ + 1×2⁻¹ + 1×2⁻² = 2. 75.

In general, numbers in the base b system are of the form:

$(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b = \sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.$

The numbers b^k and b^−k are the weights of the corresponding digits. A weight function is a mathematical device used when performing a sum integral or average in order to give some elements more of a "weight" than others The position k is the logarithm of the corresponding weight w, that is $k = log b w = log b b k$ . In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The highest used position is close to the order of magnitude of the number. An order of magnitude is the class of scale or magnitude of any amount where each class contains values of a fixed ratio to the class preceding it

The number of tally marks required in the unary numeral system for describing the weight would have been w. Tally marks are an implementation of the Unary numeral system. The unary numeral system is the bijective base - 1 Numeral system. In the positional system the number of digits required to describe it is only $k + 1 =$ $log b w$ $+ 1$ , for $k \ge 0$ . E. g. to describe the weight 1000 then 4 digits are needed since $log 10 1000 + 1 = 3 + 1$ . The number of digits required to describe the position is $log b k + 1 = log b log b w + 1$ (in positions 1, 10, 100. . . only for simplicity in the decimal example).

Position	3	2	1	0	-1	-2	. . .
Weight	$b 3$	$b 2$	$b 1$	$b 0$	$b - 1$	$b - 2$	. . .
Digit	$a 3$	$a 2$	$a 1$	$a 0$	$c 1$	$c 2$	. . .
Decimal example weight	1000	100	10	1	0. 1	0. 01	. . .
Decimal example digit	4	3	2	7	0	0	. . .

Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. ↔ In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions A number that terminates in one base may repeat in another (thus 0. 3₁₀ = 0. 0100110011001. . . ₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 1415926. . . ₁₀ can be written down as the unperiodic 11. 001001000011111. . . ₂.

If b = p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897

Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation E. g. , 1020304 base 10 into base 7:

1020304 / 7 = 145757 r 5
 145757 / 7 =  20822 r 3
  20822 / 7 =   2974 r 4
   2974 / 7 =    424 r 6
    424 / 7 =     60 r 4
     60 / 7 =      8 r 4
      8 / 7 =      1 r 1
      1 / 7 =      0 r 1   => 11446435

E. g. , 10110111 base 2 into base 5:

10110111 / 101 = 100100 r 11  (3)
  100100 / 101 =    111 r  1  (1)
     111 / 101 =      1 r 10  (2)
       1 / 101 =      0 r  1  (1)  => 1213

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E. g. , 0. 1A4C base 16 into base 9:

0. 1A4C × 9 = 0. ECAC
0. ECAC × 9 = 8. 520C
0. 520C × 9 = 2. E26C
0. E26C × 9 = 7. F5CC
0. F5CC × 9 = 8. A42C 
0. A42C × 9 = 5. C58C  => 0. 082785. . .

Generalized variable-length integers

More general is using a notation (here written little-endian) like $a 0 a 1 a 2$ for $a 0 + a 1 b 1 + a 2 b 1 b 2$ , etc.

This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. Punycode is a Computer programming encoding syntax by which a Unicode string of characters can be translated into the more-limited character set permitted A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i. e. 1) then a (i. e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight b₁ is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence:

a (0), ba (1), ca (2), . . , 9a (35), bb (36), cb (37), . . , 9b (70), bca (71), . . , 99a (1260), bcb (1261), etc.

Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed.

The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero. Bijective numeration is any Numeral system that establishes a Bijection between the set of non-negative Integers and the set of finite strings

Properties of numerical systems with integer bases

Numeral systems with base A, where A is a positive integer, possess the following properties:

If A is even and A/2 is odd, all integral powers greater than zero of the number (A/2)+1 will contain (A/2)+1 as their last digit

If both A and A/2 are even, then all integral powers greater than or equal to zero of the number (A/2)+1 will alternate between having (A/2)+1 and 1 as their last digit. (For odd powers it will be (A/2)+1, for even powers it will be 1)

Proof of the first property:

Define ${A \over 2} + 1 = x$ Then x is even, and all $x p$ for p greater than 0 must be even. The property is equivalent to

$\!\ x^p \equiv\ x\ (\mbox{mod}\ A)$

We first check the case for p=1

$\!\ x \equiv\ x\ (\mbox{mod}\ A)$

x is less than A, so the result is trivial. We then check for p=2:

$\!\ x^2 = xx$

$\!\ x^2 = x(x-1) + x$

Since $x-1 = ({A \over 2} + 1) - 1 = {A \over 2}$ , then for all even N:

$\!\ {NA \over 2} = N(x-1) \equiv\ 0\ (\mbox{mod}\ A)\ (1)$

Because x is even, then $x (x - 1)$ is congruent to zero modulo A. Therefore:

$\!\ x^2 \equiv\ x\ (\mbox{mod}\ A)$

Using induction, assuming that the property holds for p-1:

$\!\ x^p = {x^{p-1}}x = {x^{p-1}}(x-1) + x^{p-1}$

Since the case holds for p-1, then ${x^{p-1}} \equiv\ x\ (\mbox{mod}\ A)$ . Since

$\!\ {x^{p-1}}(x-1)$

is a case of Equation 1, then ${x^{p-1}}(x-1) \equiv\ 0\ (\mbox{mod}\ A)$ . This leaves, for all p greater than 0,

$\!\ x^p \equiv\ x\ (\mbox{mod}\ A)$

Q.E.D.

Proof of the second property:

Define ${A \over 2} + 1 = x$ Then x is odd, and all $x p$ for p greater than or equal to 0 must be odd. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" The property is equivalent to

$\!\ x^p \equiv\ 1\ (\mbox{mod}\ A);\ \mbox{if}\ p \equiv\ 0\ (\mbox{mod}\ 2)$

$\!\ x^p \equiv\ x\ (\mbox{mod}\ A);\ \mbox{if}\ p \equiv\ 1\ (\mbox{mod}\ 2)$

Since $x-1 = ({A \over 2} + 1) - 1 = {A \over 2}$ , then for all odd E:

$\!\ {EA \over 2} = E(x-1) \equiv\ {A \over 2}\ (\mbox{mod}\ A)\ (2)$

The case is first checked for p=0:

$\!\ x^0 = 1$

$\!\ 1 \equiv\ 1\ (\mbox{mod}\ A)$

This result is trivial

Next, for p=1:

$\!\ x^1 = x$

$\!\ x \equiv\ x\ (\mbox{mod}\ A)$

This result is also trivial

Next, for p=2:

$\!\ x^2 = xx = x(x-1) + x$

Because x is odd, then x(x-1) is a case of Equation 2,

$x(x-1) + x \equiv\ {{A \over 2} + x}\ (\mbox{mod}\ A)$

$\!\ {A \over 2} + x = {A \over 2} + {A \over 2} + 1 = A+1$

$\!\ A+1 \equiv\ 1\ (\mbox{mod}\ A), (\mbox{so}\ x(x-1) + x = x^2 \equiv\ 1\ (\mbox{mod}\ A)$

Next, for p=3:

$\!\ x^3 = {x^2}x = {x^2}(x-1) + x^2$

Because $x 2$ is odd, $x 2 (x - 1) + x 2$ is a case of Equation 2,

$\!\ {x^2}(x-1) + x^2 \equiv\ {{A \over 2} + x^2}\ (\mbox{mod}\ A)$

Since $x^2 \equiv\ 1\ (\mbox{mod}\ A)$ ,

$\!\ {x^2}(x-1) + x^2 \equiv\ {{A \over 2} + 1}\ (\mbox{mod}\ A)$

${{A \over 2} + 1} = x$ , so $x^3 \equiv\ x\ (\mbox{mod}\ A)$ .

Using induction, assuming that the property holds for p-1:

$\!\ x^p \equiv\ {x^{p-1}}(x-1) + x^{p-1}$

If p is odd:

$\!\ x^{p-1} \equiv\ 1\ (\mbox{mod}\ A)$

Since $x p - 1 (x - 1)$ is a case of Equation (2), ${x^{p-1}}(x-1) + x^{p-1} \equiv\ {{A \over 2} + 1}\ (\mbox{mod}\ A)$ , so

$x^p \equiv\ x\ (\mbox{mod}\ A)$

If p is even:

$\!\ x^{p-1} \equiv\ x\ (\mbox{mod}\ A)$

Since $x p - 1 (x - 1)$ is a case of Equation (2), ${x^{p-1}}(x-1) + x^{p-1} \equiv\ {{A \over 2} + x}\ (\mbox{mod}\ A)$ .

${A \over 2} + x = {A \over 2} + {A \over 2} + 1 = A+1$

$A+1 \equiv\ 1\ (\mbox{mod}\ A)$ , so

$x^p \equiv\ 1\ (\mbox{mod}\ A)$