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In linguistics, a numeral (or number word) in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of
grammar In linguistics, the grammar of a natural language is its set of structural constraints on speakers' or writers' composition of clauses, phrases, and words. The term can also refer to the study of such constraints, a field that includes doma ...
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. ...
s that act as a determiner that specify the quantity of a noun, 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. Some theories consider "numeral" to be a synonym for "number" and assign all numbers (including ordinal numbers like the compound word "seventy-fifth") to a part of speech called "numerals". Numerals in the broad sense can also be analyzed as a noun ("three is a small number"), as a pronoun ("the two went to town"), or for a small number of words as an adverb ("I rode the slide twice"). Numerals can express relationships like quantity (cardinal numbers), sequence (ordinal numbers), frequency (once, twice), and part ( fraction).


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, 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 noun, "first" serves the function of an adjective, and "twice" serves the function of an adverb. In Old Church Slavonic, 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 part of speech) 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: ...
: ''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. 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 numbers (''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 numbers (''singly'', ''doubly'', and ''triply''). Georgian, Latin, and Romanian (see Romanian distributive numbers) have regular distributive numbers, 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, Polish numerals). 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, which uses either native or Chinese-derived numerals depending on what is being counted. In many languages, such as Chinese, numerals require the use of
numeral classifier A classifier (abbreviated or ) is a word or affix that accompanies nouns and can be considered to "classify" a noun depending on the type of its referent. It is also sometimes called a measure word or counter word. Classifiers play an importan ...
s. Many sign languages, such as ASL, incorporate 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 part of speech, while the words for powers of the base belong to one of the other word classes. In English, these higher words are hundred 102, thousand 103, million 106, and higher powers of a thousand (
short 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 ...
) 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 t ...
—see names of large numbers). 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 104, and powers of myriad. In the Indian subcontinent, they are hundred, thousand, lakh 105, crore 107, and so on. The Mesoamerican system, still used to some extent in Mayan languages, was based on powers of 20: ''bak’'' 400 (202), ''pik'' 8000 (203), ''kalab'' 160,000 (204), etc.


Numerals of cardinal numbers

The cardinal numbers have numerals. In the following tables, ndindicates that the word ''and'' is used in some dialects (such as
British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, "English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in England, or, more broadl ...
), and omitted in other dialects (such as American English). 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. ...
or names of large numbers for more information on naming numbers. There is no consistent and widely accepted way to extend cardinals beyond centillion (
centilliard 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 ...
).


Myriad, Octad, and -yllion 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 numbers 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 many ways, e.g. ''zero-point-one-two'' (0.12), ''twelve percent'' (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: 1 * Pair: 2 (the base of the binary numeral system) * Leash: 3 (the base of the
trinary numeral system A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' ...
) * Dozen: 12 (the base of the duodecimal numeral system) * Baker's dozen: 13 *
Score Score or scorer may refer to: *Test score, the result of an exam or test Business * Score Digital, now part of Bauer Radio * Score Entertainment, a former American trading card design and manufacturing company * Score Media, a former Canadian ...
: 20 (the base of the vigesimal 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 emerge ...
: 60 (the base of the sexagesimal numeral system) * Gross: 144 (= 122) * Great gross: 1728 (= 123)


Basis of counting system

Not all peoples count, 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 B ...
, the Campa languages, Arabela, and Achuar. Some languages of Australia, such as Warlpiri, do not have words for quantities above two, and neither did many Khoisan languages 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 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) 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 ethnic groups, some Sulawesi 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, 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 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 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 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 The Sko or Skou languages are a small language family spoken by about 7000 people, mainly along the Vanimo coast of Sandaun Province in Papua New Guinea, with a few being inland from this area and at least one just across the border in the Indone ...
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 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 of
California California is a state in the Western United States, located along the Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the most populous U.S. state and the 3rd largest by area. It is also the m ...
and in the Pamean languages of
Mexico Mexico (Spanish language, Spanish: México), officially the United Mexican States, is a List of sovereign states, country in the southern portion of North America. It is borders of Mexico, bordered to the north by the United States; to the so ...
, 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, 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 thousands, with variants (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. ...
) * Indian system: crore, lakh (see Indian numbering system. Indian numerals) * East Asian system: based on ten-thousands (see below)


12: duodecimal

Duodecimal systems are based on 12. These include: * Chepang language of Nepal, * 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 ...
in
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
*
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 jo ...
Middle Belt areas such as Janji, Kahugu and the Nimbia dialect of Gwandara. * Melanesia * 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) 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 ...
in market 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 '' dozen'', '' gross'' and '' great gross'', 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 conventions of everyday speech to the internationally ag ...
, such as "two gross six dozen" for 360. Ancient Romans used a decimal system for integers, but switched to duodecimal for fractions, and correspondingly
Latin Latin (, or , ) is a classical language belonging to the Italic languages, 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 ...
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, ...
). A notable fictional duodecimal system was that of J. R. R. Tolkien's Elvish languages, which used duodecimal as well as decimal.


16: hexadecimal

Hexadecimal systems are based on 16. The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) 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 1221) ...
was subdivided into four paisa or twelve pies (thus there were 64 paise or 192 pies in a rupee). The anna was demonetised 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 Mesoamerican cultures, still in use today in the modern indigenous languages of their descendants, namely the Nahuatl and Mayan languages (see Maya numerals). A modern national language which uses a full vigesimal system is Dzongkha in Bhutan. Partial vigesimal systems are found in some European languages: Basque, Celtic languages,
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
(from Celtic), Danish, and Georgian. 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 Score or scorer may refer to: *Test score, the result of an exam or test Business * Score Digital, now part of Bauer Radio * Score Entertainment, a former American trading card design and manufacturing company * Score Media, a former Canadian ...
'' 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 Score or scorer may refer to: *Test score, the result of an exam or test Business * Score Digital, now part of Bauer Radio * Score Entertainment, a former American trading card design and manufacturing company * Score Media, a former Canadian ...
of bob", referring to the 20 shillings in a pound. For Americans the term is most known from the opening of the Gettysburg Address: ''"Four score and seven years ago our fathers..."''.


24: quadrovigesimal

The
Sko languages The Sko or Skou languages are a small language family spoken by about 7000 people, mainly along the Vanimo coast of Sandaun Province in Papua New Guinea, with a few being inland from this area and at least one just across the border in the Indone ...
have a base-24 system with a sub-base of 6.


32: duotrigesimal

Ngiti has base 32.


60: sexagesimal

Ekari has a base-60 system. Sumeria 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) + ��


See also


Numerals in various languages

A databas
Numeral Systems of the World's Languages
compiled by Eugene S.L. Chan of Hong Kong is hosted by the Max Planck Institute for Evolutionary Anthropology in Leipzig, Germany. The database currently contains data for about 4000 languages. * Proto-Indo-European numerals **
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. ...
** Indian numbering system ** Polish numerals ** Hindustani numerals *
Proto-Semitic numerals Proto-Semitic is the hypothetical reconstructed proto-language ancestral to the Semitic languages. There is no consensus regarding the location of the Proto-Semitic '' Urheimat''; scholars hypothesize that it may have originated in the Levan ...
** Hebrew numerals * Chinese numerals ** Japanese numerals ** Korean numerals ** Vietnamese numerals * Australian Aboriginal enumeration *
Balinese numerals The Balinese language has an elaborate decimal numeral system. Basic numerals The numerals 1–10 have basic, combining, and independent forms, many of which are formed through reduplication. The combining forms are used to form higher numbers. ...
*
Dzongkha numerals Dzongkha, the national language of Bhutan, has two numeral systems, one vigesimal (base 20), and a modern decimal system. The vigesimal system remains in robust use. Ten Ten, TEN or 10 may refer to: * 10, an even natural number following 9 and ...
* Finnish numerals *
Javanese numerals The Javanese language has a decimal numeral system with distinct words for the 'tweens' from 21 to 29, called ''likuran''. The basic numerals 1–10 have independent and combining forms, the latter derived via a suffix ''-ng''. The combining for ...
*
Yoruba numerals According to Lounge,Lounge, 2009. the Yoruba language has a rather elaborate vigesimal (base-20) numeral system that involves both addition and subtraction and multiplication. The base of the counting system is ''ogún'' 'twenty' (or 'score'). ...


Related topics

* Long and short scales * Names of large numbers * Numeral system *
Numeral prefix Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-c ...
* Names of small numbers


Notes


Further reading

* {{DEFAULTSORT:Number Names * Names