In linguistics, a numeral (or number word) in the broadest sense is a

word A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...

phrase In syntax and grammar, a phrase is a group of words or singular word acting as a grammatical unit. For instance, the English expression "the very happy squirrel" is a noun phrase which contains the adjective phrase "very happy". Phrases can consi ...

that describes a numerical

quantity Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms o ...

. Some theories of

grammar In linguistics, the grammar of a natural language is its set of structure, structural constraints on speakers' or writers' composition of clause (linguistics), clauses, phrases, and words. The term can also refer to the study of such constraint ...

use the word "numeral" to refer to

cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...

s that act as a

determiner A determiner, also called determinative (abbreviated ), is a word, phrase, or affix that occurs together with a noun or noun phrase and generally serves to express the reference of that noun or noun phrase in the context. That is, a determiner m ...

that specify the quantity of a

noun A noun () is a word that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for: * Living creatures (including people, alive, d ...

, for example the "two" in "two hats". Some theories of grammar do not include determiners as a part of speech and consider "two" in this example to be an

adjective In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that generally grammatical modifier, modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun. Tra ...

. Some theories consider "numeral" to be a

synonym A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all ...

for "number" and assign all numbers (including

ordinal numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

like the

compound word In linguistics, a compound is a lexeme (less precisely, a word or sign) that consists of more than one stem. Compounding, composition or nominal composition is the process of word formation that creates compound lexemes. Compounding occurs when ...

"seventy-fifth") to a

part of speech In grammar, a part of speech or part-of-speech (abbreviated as POS or PoS, also known as word class or grammatical category) is a category of words (or, more generally, of lexical items) that have similar grammatical properties. Words that are assi ...

called "numerals". Numerals in the broad sense can also be analyzed as a noun ("three is a small number"), as a

pronoun In linguistics and grammar, a pronoun (abbreviated ) is a word or a group of words that one may substitute for a noun or noun phrase. Pronouns have traditionally been regarded as one of the parts of speech, but some modern theorists would not co ...

("the two went to town"), or for a small number of words as an

adverb An adverb is a word or an expression that generally modifies a verb, adjective, another adverb, determiner, clause, preposition, or sentence. Adverbs typically express manner, place, time, frequency, degree, level of certainty, etc., answering ...

("I rode the slide twice"). Numerals can express relationships like quantity (cardinal numbers),

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

(ordinal numbers),

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

(once, twice), and part (

fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

Identifying numerals

Numerals may be

attributive In grammar, an attributive expression is a word or phrase within a noun phrase that modifies the head noun. It may be an: * attributive adjective * attributive noun * attributive verb or other part of speech, such as an attributive numeral. ...

, as in ''two dogs'', or

pronominal In linguistics and grammar, a pronoun (abbreviated ) is a word or a group of words that one may substitute for a noun or noun phrase. Pronouns have traditionally been regarded as one of the parts of speech, but some modern theorists would not c ...

, as in ''I saw two (of them)''. Many words of different parts of speech indicate number or quantity. Such words are called quantifiers. Examples are words such as ''every'', ''most'', ''least'', ''some'', etc. Numerals are distinguished from other quantifiers by the fact that they designate a specific number. Examples are words such as ''five, ten, fifty, one hundred, etc.'' They may or may not be treated as a distinct part of speech; this may vary, not only with the language, but with the choice of word. For example, "dozen" serves the function of a

, "first" serves the function of an

, and "twice" serves the function of an

. In

Old Church Slavonic Old Church Slavonic or Old Slavonic () was the first Slavic languages, Slavic literary language. Historians credit the 9th-century Byzantine Empire, Byzantine missionaries Saints Cyril and Methodius with Standard language, standardizing the lan ...

, the cardinal numbers 5 to 10 were feminine nouns; when quantifying a noun, that noun was declined in the genitive plural like other nouns that followed a noun of quantity (one would say the equivalent of "five of people"). In English grammar, the classification "''numeral''" (viewed as a

) is reserved for those words which have distinct grammatical behavior: when a numeral modifies a noun, it may replace the

article Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: G ...

: ''the/some dogs played in the park'' → ''twelve dogs played in the park''. (Note that *''dozen dogs played in the park'' is not grammatical, so "dozen" is not a numeral in this sense.) English numerals indicate

cardinal numbers In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...

. However, not all words for cardinal numbers are necessarily numerals. For example, ''million'' is grammatically a noun, and must be preceded by an article or numeral itself. Numerals may be simple, such as 'eleven', or compound, such as 'twenty-three'. In linguistics, however, numerals are classified according to purpose: examples are

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

s (''first'', ''second'', ''third'', etc.; from 'third' up, these are also used for fractions), multiplicative (adverbial) numbers (''once'', ''twice'', and ''thrice''), multipliers (''single'', ''double'', and ''triple''), and

distributive number In linguistics, a distributive numeral, or distributive number word, is a word that answers "how many times each?" or "how many at a time?", such as ''singly'' or ''doubly''. They are contrasted with multipliers. In English, this part of speec ...

s (''singly'', ''doubly'', and ''triply'').

Georgian Georgian may refer to: Common meanings * Anything related to, or originating from Georgia (country) ** Georgians, an indigenous Caucasian ethnic group ** Georgian language, a Kartvelian language spoken by Georgians **Georgian scripts, three scrip ...

, Latin, and Romanian (see Romanian distributive numbers) have regular

s, such as Latin ''singuli'' "one-by-one", ''bini'' "in pairs, two-by-two", ''terni'' "three each", etc. In languages other than English, there may be other kinds of number words. For example, in Slavic languages there are collective numbers (monad, pair/dyad, triad) which describe sets, such as ''pair'' or ''dozen'' in English (see

Russian numerals Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

Polish numerals Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, ...

). Some languages have a very limited set of numerals, and in some cases they arguably do not have any numerals at all, but instead use more generic quantifiers, such as 'pair' or 'many'. However, by now most such languages have borrowed the numeral system or part of the numeral system of a national or colonial language, though in a few cases (such as Guarani), a numeral system has been invented internally rather than borrowed. Other languages had an indigenous system but borrowed a second set of numerals anyway. An example is

Japanese Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...

, which uses either native or Chinese-derived numerals depending on what is being counted. In many languages, such as

Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of ...

, numerals require the use of numeral classifiers. Many

sign language Sign languages (also known as signed languages) are languages that use the visual-manual modality to convey meaning, instead of spoken words. Sign languages are expressed through manual articulation in combination with non-manual markers. Sign l ...

s, such as

ASL American Sign Language (ASL) is a natural language that serves as the predominant sign language of Deaf communities in the United States of America and most of Anglophone Canada. ASL is a complete and organized visual language that is express ...

incorporate Incorporation may refer to: * Incorporation (business), the creation of a corporation * Incorporation of a place, creation of municipal corporation such as a city or county * Incorporation (academic), awarding a degree based on the student having ...

numerals.

Larger numerals

English has derived numerals for multiples of its base (''fifty, sixty,'' etc.), and some languages have simplex numerals for these, or even for numbers between the multiples of its base. Balinese, for example, currently has a decimal system, with words for 10, 100, and 1000, but has additional simplex numerals for 25 (with a second word for 25 only found in a compound for 75), 35, 45, 50, 150, 175, 200 (with a second found in a compound for 1200), 400, 900, and 1600. In Hindustani, the numerals between 10 and 100 have developed to the extent that they need to be learned independently. In many languages, numerals up to the base are a distinct

, while the words for powers of the base belong to one of the other word classes. In English, these higher words are

hundred 100 or one hundred (Roman numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to de ...

10²,

thousand 1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000. A group of one thousand th ...

10³,

million One million (1,000,000), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian ''millione'' (''milione'' in modern Italian), from ''mille'', "thousand", plus the au ...

10⁶, and higher powers of a thousand (

short scale The long and short scales are two of several naming systems for integer power of 10, powers of ten which use some of the same terms for different magnitudes. For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one m ...

) or of a million (

long scale The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes. For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the ...

—see

names of large numbers Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...

). These words cannot modify a noun without being preceded by an article or numeral (*''hundred dogs played in the park''), and so are nouns. In East Asia, the higher units are hundred, thousand,

myriad A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinospher ...

10⁴, and powers of myriad. In the

Indian subcontinent The Indian subcontinent is a list of the physiographic regions of the world, physiographical region in United Nations geoscheme for Asia#Southern Asia, Southern Asia. It is situated on the Indian Plate, projecting southwards into the Indian O ...

, they are hundred, thousand,

lakh A lakh (; abbreviated L; sometimes written lac) is a unit in the Indian numbering system equal to one hundred thousand (100,000; scientific notation: 105). In the Indian 2,2,3 convention of digit grouping, it is written as 1,00,000. For ex ...

10⁵,

crore A crore (; abbreviated cr) denotes ten million (10,000,000 or 107 in scientific notation) and is equal to 100 lakh in the Indian numbering system. It is written as 1,00,00,000 with the local 2,2,3 style of digit group separators (one lakh is e ...

10⁷, and so on. The Mesoamerican system, still used to some extent in

Mayan languages The Mayan languagesIn linguistics, it is conventional to use ''Mayan'' when referring to the languages, or an aspect of a language. In other academic fields, ''Maya'' is the preferred usage, serving as both a singular and plural noun, and as ...

, was based on powers of 20: ''bak’'' 400 (20²), ''pik'' 8000 (20³), ''kalab'' 160,000 (20⁴), etc.

Numerals of cardinal numbers

The

have numerals. In the following tables, ndindicates that the word ''and'' is used in some

dialect The term dialect (from Latin , , from the Ancient Greek word , 'discourse', from , 'through' and , 'I speak') can refer to either of two distinctly different types of Linguistics, linguistic phenomena: One usage refers to a variety (linguisti ...

s (such as

British English British English (BrE, en-GB, or BE) is, according to Lexico, Oxford Dictionaries, "English language, English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in ...

), and omitted in other dialects (such as

American English American English, sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. English is the Languages of the United States, most widely spoken lan ...

). This table demonstrates the standard English construction of some cardinal numbers. (See next table for names of larger cardinals.)

English names for powers of 10

This table compares the English names of cardinal numbers according to various American, British, and Continental European conventions. See

English numerals English number words include numerals and various words derived from them, as well as a large number of words borrowed from other languages. Cardinal numbers Cardinal numbers refer to the size of a group. In English, these words are numerals. ...

for more information on naming numbers. There is no consistent and widely accepted way to extend cardinals beyond

centillion Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...

( centilliard).

Myriad, Octad, and
-yllion -yllion (pronounced ) is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger n ...
systems

The following table details the myriad, octad, Chinese myriad, Chinese long and -yllion names for powers of 10. There is also a Knuth-proposed system notation of numbers, named the -yllion system. In this system, a new word is invented for every ''2ⁿ''-th power of ten.

Fractional numerals

This is a table of English names for non-negative

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s less than or equal to 1. It also lists alternative names, but there is no widespread convention for the names of extremely small positive numbers. Keep in mind that rational numbers like 0.12 can be represented in

infinitely Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

many ways, e.g. ''zero-point-one-two'' (0.12), ''twelve

percent In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also use ...

'' (12%), ''three twenty-fifths'' (), ''nine seventy-fifths'' (), ''six fiftieths'' (), ''twelve hundredths'' (), ''twenty-four two-hundredths'' (), etc.

Other specific quantity terms

Various terms have arisen to describe commonly used measured quantities. *

Unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

: 1 *

Pair Pair or PAIR or Pairing may refer to: Government and politics * Pair (parliamentary convention), matching of members unable to attend, so as not to change the voting margin * ''Pair'', a member of the Prussian House of Lords * ''Pair'', the Frenc ...

: 2 (the base of the

binary numeral system 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 ...

) *

Leash A leash (also called a lead, lead line or tether) is a rope or similar material used to control an animal by attaching it to a collar, harness, or halter. In British English, a leash is generally for a larger (possibly dangerous or aggressive ...

: 3 (the base of the trinary numeral system) *

Dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...

: 12 (the base of the

duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...

numeral system) *

Baker's dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...

: 13 * Score: 20 (the base of the

vigesimal vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...

numeral system) *

Shock Shock may refer to: Common uses Collective noun *Shock, a historic commercial term for a group of 60, see English numerals#Special names * Stook, or shock of grain, stacked sheaves Healthcare * Shock (circulatory), circulatory medical emergen ...

: 60 (the base of 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� ...

numeral system) * Gross: 144 (= 12²) *

Great gross 1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross). It is the number of cubic inches in a cubic foot. It is also the number of daily chants of the Hare Krishna mantra by a Har ...

: 1728 (= 12³)

Basis of counting system

Not all peoples

count Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York: ...

, at least not verbally. Specifically, there is not much need for counting among hunter-gatherers who do not engage in commerce. Many languages around the world have no numerals above two to four (if they're actually numerals at all, and not some other part of speech)—or at least did not before contact with the colonial societies—and speakers of these languages may have no tradition of using the numerals they did have for counting. Indeed, several languages from the Amazon have been independently reported to have no specific number words other than 'one'. These include Nadëb, pre-contact

Mocoví The Mocoví ( Mocoví: ''moqoit'') are an indigenous people of the Gran Chaco region of South America. They speak the Mocoví language and are one of the ethnic groups belonging to the Guaycuru peoples. In the 2010 Argentine census, 22,439 peopl ...

and Pilagá, Culina and pre-contact Jarawara, Jabutí, Canela-Krahô, Botocudo (Krenák),

Chiquitano The Chiquitano or Chiquitos are an indigenous people of Bolivia, with a small number also living in Brazil. The Chiquitano primarily live in the Chiquitania tropical savanna of Santa Cruz Department, Bolivia, with a small number also living in Be ...

, the

Campa languages The Campa (Kampa) or Campan (Kampan) languages, a.k.a. Pre-Andine Maipurean / Arawakan, are closely related Arawakan languages of the Peruvian Amazon. Languages The Glottolog uses the term ''Pre-Andine'' for this group of languages and classifi ...

, Arabela, and

Achuar The Achuar are an Amazonian community of some 18,500 individuals along either side of the border in between Ecuador and Peru. As of the early 1970s, the Achuar were one of the last of the Jivaroan groups still generally unaffected by outside c ...

. Some languages of Australia, such as Warlpiri, do not have words for quantities above two, and neither did many

Khoisan languages The Khoisan languages (; also Khoesan or Khoesaan) are a group of African languages originally classified together by Joseph Greenberg. Khoisan languages share click consonants and do not belong to other African language families. For much of th ...

at the time of European contact. Such languages do not have a word class of 'numeral'. Most languages with both numerals and counting use base 8, 10, 12, or 20. Base 10 appears to come from counting one's fingers, base 20 from the fingers and toes, base 8 from counting the spaces between the fingers (attested in California), and base 12 from counting the knuckles (3 each for the four fingers).

No base

Many languages of

Melanesia Melanesia (, ) is a subregion of Oceania in the southwestern Pacific Ocean. It extends from Indonesia's New Guinea in the west to Fiji in the east, and includes the Arafura Sea. The region includes the four independent countries of Fiji, Va ...

have (or once had) counting systems based on parts of the body which do not have a numeric base; there are (or were) no numerals, but rather nouns for relevant parts of the body—or simply pointing to the relevant spots—were used for quantities. For example, 1–4 may be the fingers, 5 'thumb', 6 'wrist', 7 'elbow', 8 'shoulder', etc., across the body and down the other arm, so that the opposite little finger represents a number between 17 (

Torres Islands The Torres Islands are in the Torba Province of the country of Vanuatu, and is that country’s the northernmost island group. The chain of islands that make up this micro-archipelago straddles the broader cultural boundary between Island Melane ...

) to 23 ( Eleman). For numbers beyond this, the torso, legs and toes may be used, or one might count back up the other arm and back down the first, depending on the people.

2: binary

Binary systems are base 2, using zeros and ones. With only two symbols binary is used for things with coding like computers.

3: ternary

Base 3 counting has practical usage in some analog logic, in baseball scoring and in self–similar mathematical structures.

4: quaternary

Some Austronesian and

Melanesian Melanesian is the adjectival form of Melanesia. It may refer to: * Melanesians * Melanesian mythology * Melanesian languages In linguistics, Melanesian is an obsolete term referring to the Austronesian languages of Melanesia: that is, the Oceani ...

ethnic groups, some

Sulawesi Sulawesi (), also known as Celebes (), is an island in Indonesia. One of the four Greater Sunda Islands, and the world's eleventh-largest island, it is situated east of Borneo, west of the Maluku Islands, and south of Mindanao and the Sulu Ar ...

and some

Papua New Guinea Papua New Guinea (abbreviated PNG; , ; tpi, Papua Niugini; ho, Papua Niu Gini), officially the Independent State of Papua New Guinea ( tpi, Independen Stet bilong Papua Niugini; ho, Independen Stet bilong Papua Niu Gini), is a country i ...

ns, count with the base number four, using the term ''asu'' and ''aso'', the word for

dog The dog (''Canis familiaris'' or ''Canis lupus familiaris'') is a domesticated descendant of the wolf. Also called the domestic dog, it is derived from the extinct Pleistocene wolf, and the modern wolf is the dog's nearest living relative. Do ...

, as the ubiquitous village dog has four legs.Ryan, Peter. ''Encyclopaedia of Papua and New Guinea''. Melbourne University Press & University of Papua and New Guinea,:1972 .: 3 pages p 219. This is argued by anthropologists to be also based on early humans noting the human and animal shared body feature of two arms and two legs as well as its ease in simple arithmetic and counting. As an example of the system's ease a realistic scenario could include a farmer returning from the market with fifty ''asu'' heads of pig (200), less 30 ''asu'' (120) of pig bartered for 10 ''asu'' (40) of goats noting his new pig count total as twenty ''asu'': 80 pigs remaining. The system has a correlation to the

dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...

counting system and is still in common use in these areas as a natural and easy method of simple arithmetic.

5: quinary

Quinary systems are based on the number 5. It is almost certain the quinary system developed from counting by fingers (five fingers per hand).Heath, Thomas, ''A Manual of Greek Mathematics'', Courier Dover: 2003. page, p:11 An example are the

Epi languages The half dozen Epi languages are spoken on Epi Island Epi (or Épi, Api; formerly known as Tasiko or Volcano Island) is an island in Shefa Province, Vanuatu, at the north end of the Shepherd Islands. The island is long northwest–southeast ...

of Vanuatu, where 5 is ''luna'' 'hand', 10 ''lua-luna'' 'two hand', 15 ''tolu-luna'' 'three hand', etc. 11 is then ''lua-luna tai'' 'two-hand one', and 17 ''tolu-luna lua'' 'three-hand two'. 5 is a common ''auxiliary base'', or ''sub-base'', where 6 is 'five and one', 7 'five and two', etc.

Aztec The Aztecs () were a Mesoamerican culture that flourished in central Mexico in the post-classic period from 1300 to 1521. The Aztec people included different Indigenous peoples of Mexico, ethnic groups of central Mexico, particularly those g ...

was a vigesimal (base-20) system with sub-base 5.

6: senary

The Morehead-Maro languages of Southern New Guinea are examples of the rare base 6 system with monomorphemic words running up to 6⁶. Examples are Kanum and Kómnzo. The Sko languages on the North Coast of New Guinea follow a base-24 system with a sub-base of 6.

7: septenary

Septenary systems are very rare, as few natural objects consistently have seven distinctive features. Traditionally, it occurs in week-related timing. It has been suggested that the

Palikur The Palikur are an indigenous people located in the riverine areas of the Brazilian state of Amapá and in French Guiana, particularly in the south-eastern border region, on the north bank of the Oyapock River. The Palikur Nation, or ''naoné'', ...

language has a base-seven system, but this is dubious.Parkvall, M. ''Limits of Language'', 1st edn. 2008. p.291.

8: octal

Octal counting systems are based on the number 8. Examples can be found in the

Yuki language Yuki, also known as Ukomno'm, is an extinct language of California, formerly spoken by the Yuki people. The Yuki are the original inhabitants of the Eel River area and the Round Valley Reservation of northern California. Yuki ceased to be used ...

California California is a U.S. state, state in the Western United States, located along the West Coast of the United States, Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the List of states and territori ...

and in the Pamean languages of

Mexico Mexico (Spanish: México), officially the United Mexican States, is a country in the southern portion of North America. It is bordered to the north by the United States; to the south and west by the Pacific Ocean; to the southeast by Guatema ...

, because the Yuki and Pame keep count by using the four spaces between their fingers rather than the fingers themselves.

9: nonary

It has been suggested that Nenets has a base-nine system.

10: decimal

A majority of traditional number systems are decimal. This dates back at least to the ancient

Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian ...

, who used a wholly decimal system. Anthropologists hypothesize this may be due to humans having five digits per hand, ten in total. There are many regional variations including: * Western system: based on

s, with variants (see

) * Indian system:

(see

Indian numbering system The Indian numbering system is used in all South Asian countries (Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan, Sri Lanka and Afghanistan) to express large numbers. The terms ''lakh'' or 1,00,000 (one hundred thousand, written as ''100,00 ...

Indian numerals Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...

) * East Asian system: based on ten-thousands (see below)

12: duodecimal

Duodecimal systems are based on 12. These include: *

Chepang language Chepang may refer to: * Chepang people, a group indigenous to the lands of Nepal * Chepang language, the language of the Chepang people {{dab ...

Nepal Nepal (; ne, नेपाल ), formerly the Federal Democratic Republic of Nepal ( ne, सङ्घीय लोकतान्त्रिक गणतन्त्र नेपाल ), is a landlocked country in South Asia. It is mai ...

, * Mahl language of

Minicoy Island Minicoy, locally known as Maliku (), is an island in Lakshadweep, India. Along with Viringili, it is on ''Maliku atoll'', the southernmost atoll of Lakshadweep archipelago. Administratively, it is a census town in the Indian union territory o ...

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

Nigerian Nigerians or the Nigerian people are citizens of Nigeria or people with ancestry from Nigeria. The name Nigeria was taken from the Niger River running through the country. This name was allegedly coined in the late 19th century by British jour ...

Middle Belt The Middle Belt (also spelt Middle-Belt or Central Nigeria) is a term used in human geography to designate a belt region stretching across central Nigeria longitudinally and forming a transition zone between Northern and Southern Nigeria. It ...

areas such as Janji, Kahugu and the Nimbia dialect of

Gwandara Gwandara is a West Chadic language, and the closest relative of Hausa. Its several dialects are spoken in northern Nigeria, predominantly in the north central region of Nigeria by the Gwandara people and some settlers who are about 30,000 people. ...

. *

* reconstructed proto- Benue–Congo Duodecimal numeric systems have some practical advantages over decimal. It is much easier to divide the base digit twelve (which is a

highly composite number __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...

) by many important

divisors In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

market Market is a term used to describe concepts such as: *Market (economics), system in which parties engage in transactions according to supply and demand *Market economy *Marketplace, a physical marketplace or public market Geography *Märket, an ...

and trade settings, such as the numbers 2, 3, 4 and 6. Because of several measurements based on twelve, many Western languages have words for base-twelve units such as ''

'', '' gross'' and ''

great gross 1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross). It is the number of cubic inches in a cubic foot. It is also the number of daily chants of the Hare Krishna mantra by a Har ...

'', which allow for rudimentary duodecimal

nomenclature Nomenclature (, ) is a system of names or terms, or the rules for forming these terms in a particular field of arts or sciences. The principles of naming vary from the relatively informal naming conventions, conventions of everyday speech to the i ...

, such as "two gross six dozen" for 360.

Ancient Romans In modern historiography, ancient Rome refers to Roman people, Roman civilisation from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD. It encompasses the Roman Kingdom ...

used a decimal system for

integers 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 o ...

, but switched to

for

fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, and correspondingly

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 ...

developed a rich vocabulary for duodecimal-based fractions (see

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 ...

). A notable fictional duodecimal system was that of

J. R. R. Tolkien John Ronald Reuel Tolkien (, ; 3 January 1892 – 2 September 1973) was an English writer and philology, philologist. He was the author of the high fantasy works ''The Hobbit'' and ''The Lord of the Rings''. From 1925 to 1945, Tolkien was ...

Elvish languages Elvish languages are constructed languages used by Elves in a fantasy setting. The philologist and fantasy author J. R. R. Tolkien created the first of these languages, including Quenya and Sindarin. Tolkien's Elvish languages The philologist ...

, which used duodecimal as well as decimal.

16: hexadecimal

Hexadecimal systems are based on 16. The traditional

Chinese units of measurement Chinese units of measurement, known in Chinese as the ''shìzhì'' ("market system"), are the traditional units of measurement of the Han Chinese. Although Chinese numerals have been decimal (base-10) since the Shang dynasty, Shang, several Chine ...

were base-16. For example, one jīn (斤) in the old system equals sixteen

tael Tael (),"Tael" entry
at the

suanpan The suanpan (), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely ''Supplementary Notes on the Art of Figures'' written by Xu Yue. However, the exact design ...

(Chinese

abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...

) can be used to perform hexadecimal calculations such as additions and subtractions. South Asian monetary systems were base-16. One rupee in Pakistan and India was divided into 16 annay. A single

anna Anna may refer to: People Surname and given name * Anna (name) Mononym * Anna the Prophetess, in the Gospel of Luke * Anna (wife of Artabasdos) (fl. 715–773) * Anna (daughter of Boris I) (9th–10th century) * Anna (Anisia) (fl. 1218 to 12 ...

was subdivided into four

paisa Paisa (also transliterated as ''pice'', ''pesa'', ''poysha'', ''poisha'' and ''baisa'') is a monetary unit in several countries. The word is also a generalised idiom for money and wealth. In India, Nepal, and Pakistan, the ''Paisa'' currently equa ...

or twelve

pies A pie is a baked dish which is usually made of a pastry dough casing that contains a filling of various sweet or savoury ingredients. Sweet pies may be filled with fruit (as in an apple pie), nuts ( pecan pie), brown sugar ( sugar pie), sweet ...

(thus there were 64 paise or 192 pies in a rupee). The anna was

demonetised Legal tender is a form of money that courts of law are required to recognize as satisfactory payment for any monetary debt. Each jurisdiction determines what is legal tender, but essentially it is anything which when offered ("tendered") in pa ...

as a currency unit when India decimalised its currency in 1957, followed by Pakistan in 1961.

20: vigesimal

Vigesimal numbers use the number 20 as the base number for counting. Anthropologists are convinced the system originated from digit counting, as did bases five and ten, twenty being the number of human fingers and toes combined.Georges Ifrah, ''The Universal History of Numbers: The Modern Number System'', Random House, 2000: . 1262 pages The system is in widespread use across the world. Some include the classical

Mesoamerica Mesoamerica is a historical region and cultural area in southern North America and most of Central America. It extends from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica. W ...

n cultures, still in use today in the modern indigenous languages of their descendants, namely the

Nahuatl Nahuatl (; ), Aztec, or Mexicano is a language or, by some definitions, a group of languages of the Uto-Aztecan language family. Varieties of Nahuatl are spoken by about Nahua peoples, most of whom live mainly in Central Mexico and have smaller ...

and

(see

Maya numerals The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols; zero (a shell), one (a dot) and fiv ...

). A modern national language which uses a full vigesimal system is

Dzongkha Dzongkha (; ) is a Sino-Tibetan language that is the official and national language of Bhutan. It is written using the Tibetan script. The word means "the language of the fortress", from ' "fortress" and ' "language". , Dzongkha had 171,080 n ...

in Bhutan. Partial vigesimal systems are found in some European languages:

Basque Basque may refer to: * Basques, an ethnic group of Spain and France * Basque language, their language Places * Basque Country (greater region), the homeland of the Basque people with parts in both Spain and France * Basque Country (autonomous co ...

Celtic languages The Celtic languages ( usually , but sometimes ) are a group of related languages descended from Proto-Celtic. They form a branch of the Indo-European language family. The term "Celtic" was first used to describe this language group by Edward ...

, French (from Celtic),

Danish Danish may refer to: * Something of, from, or related to the country of Denmark People * A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark * Culture of Denmark * Danish people or Danes, people with a Danish a ...

, and

. In these languages the systems are vigesimal up to 99, then decimal from 100 up. That is, 140 is 'one hundred two score', not *seven score, and there is no numeral for 400 (great score). The term '' score'' originates from

tally stick A tally stick (or simply tally) was an ancient memory aid device used to record and document numbers, quantities and messages. Tally sticks first appear as animal bones carved with notches during the Upper Palaeolithic; a notable example is the ...

s, and is perhaps a remnant of Celtic vigesimal counting. It was widely used to learn the pre-decimal British currency in this idiom: "a dozen pence and a score of

bob Bob, BOB, or B.O.B. may refer to: Places * Mount Bob, New York, United States *Bob Island, Palmer Archipelago, Antarctica People, fictional characters, and named animals *Bob (given name), a list of people and fictional characters *Bob (surname ...

", referring to the 20

shilling The shilling is a historical coin, and the name of a unit of modern currencies formerly used in the United Kingdom, Australia, New Zealand, other British Commonwealth countries and Ireland, where they were generally equivalent to 12 pence o ...

s in a pound. For Americans the term is most known from the opening of the

Gettysburg Address The Gettysburg Address is a Public speaking, speech that President of the United States, U.S. President Abraham Lincoln delivered during the American Civil War at the dedication of the Gettysburg National Cemetery, Soldiers' National Cemetery, ...

: ''"Four score and seven years ago our fathers..."''.

24: quadrovigesimal

The Sko languages have a base-24 system with a sub-base of 6.

32: duotrigesimal

Ngiti The Ngiti , or South Lendu, is an ethnolinguistic group located in the Ituri Province of the Democratic Republic of the Congo. Ngiti speakers call their language ''Ndruna''. In 1991, the Ngiti numbered 100,000 located in the Irumu territory sout ...

has base 32.

60: sexagesimal

Ekari has a base-60 system.

Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...

ia had a base-60 system with a decimal sub-base (with alternating cycles of 10 and 6), which was the origin of the numbering of modern degrees, minutes, and seconds.

80: octogesimal

Supyire is said to have a base-80 system; it counts in twenties (with 5 and 10 as sub-bases) up to 80, then by eighties up to 400, and then by 400s (great scores). 799 .e. 400 + (4 x 80) + (3 x 20) + ��

Identifying numerals

Larger numerals

Numerals of cardinal numbers

English names for powers of 10

Myriad, Octad, and
-yllion -yllion (pronounced ) is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger n ...
systems

Fractional numerals

Other specific quantity terms

Basis of counting system

No base

2: binary

3: ternary

4: quaternary

5: quinary

6: senary

7: septenary

8: octal

9: nonary

10: decimal

12: duodecimal

16: hexadecimal

20: vigesimal

24: quadrovigesimal

32: duotrigesimal

60: sexagesimal

80: octogesimal

See also

Numerals in various languages

Related topics

Notes

Further reading

Identifying numerals

Larger numerals

Numerals of cardinal numbers

English names for powers of 10

Myriad, Octad, and -yllion -yllion (pronounced ) is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger n ... systems

Fractional numerals

Other specific quantity terms

Basis of counting system

No base

2: binary

3: ternary

4: quaternary

5: quinary

6: senary

7: septenary

8: octal

9: nonary

10: decimal

12: duodecimal

16: hexadecimal

20: vigesimal

24: quadrovigesimal

32: duotrigesimal

60: sexagesimal

80: octogesimal

See also

Numerals in various languages

Related topics

Notes

Further reading

Myriad, Octad, and
-yllion -yllion (pronounced ) is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger n ...
systems