Numeral (linguistics)


In linguistics, a numeral in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word "numeral" to refer to cardinal numbers that act as a determiner to 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 to a part of speech called "numerals" Numerals in the broad sense can also be analyzed as a noun, as a pronoun, or for a small number of words as an adverb.
Numerals can express relationships like quantity, sequence, frequency, and part.

Identifying numerals

Numerals may be attributive, as in two dogs, or pronominal, as in I saw two .
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. In English grammar, the classification "numeral" is reserved for those words which have distinct grammatical behavior: when a numeral modifies a noun, it may replace the article: the/some dogs played in the park → twelve dogs played in the park. 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, multiplicative numbers, multipliers, and distributive numbers. Georgian, Latin, and Romanian 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 which describe sets, such as pair or dozen in English.
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, 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 classifiers. Many sign languages, such as ASL, incorporate numerals.

Larger numerals

English has derived numerals for multiples of its base, 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, 35, 45, 50, 150, 175, 200, 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 or of a million. These words cannot modify a noun without being preceded by an article or numeral, and so are nouns.
In East Asia, the higher units are hundred, thousand, myriad 104, and powers of myriad. In India, 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, pik 8000, kalab 160,000, etc.

Numerals of cardinal numbers

The cardinal numbers have numerals. In the following tables, indicates that the word and is used in some dialects, and omitted in other dialects.
This table demonstrates the standard English construction of some cardinal numbers.
ValueNameAlternate names, and names for sets of the given size
0Zeroaught, cipher, cypher, donut, dot, duck, goose egg, love, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip, zippo, Sunya
1Oneace, individual, single, singleton, unary, unit, unity, Pratham
2Twobinary, brace, couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twin, twosome, yoke
3Threedeuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika, hat-trick
4Fourfoursome, quadruplet, quatern, quaternary, quaternity, quartet, tetrad
5Fivecinque, fin, fivesome, pentad, quint, quintet, quintuplet
6Sixhalf dozen, hexad, sestet, sextet, sextuplet, sise
7Sevenheptad, septet, septuple, walking stick
8Eightoctad, octave, octet, octonary, octuplet, ogdoad
9Nineennead
10Tendeca, decade, das
11Elevenonze, ounze, ounce,
12Twelvedozen
13Thirteenbaker's dozen, long dozen
20Twentyscore,
21Twenty-onelong score, blackjack
22Twenty-twoDeuce-deuce
24Twenty-fourtwo dozen
40Fortytwo-score
50Fiftyhalf-century
55Fifty-fivedouble nickel
60Sixtythree-score
70Seventythree-score and ten
80Eightyfour-score
87Eighty-sevenfour-score and seven
90Ninetyfour-score and ten
100One hundredcentred, century, ton, short hundred
111One hundred eleveneleventy-one
120One hundred twentylong hundred, great hundred, hundred
144One hundred forty-fourgross, dozen dozen, small gross
One thousandchiliad, grand, G, thou, yard, kilo, k, millennium, Hajaar
One thousand twenty-fourkibi or kilo in computing, see binary prefix
One thousand one hundredEleven hundred
One thousand seven hundred twenty-eightgreat gross, long gross, dozen gross
Ten thousandmyriad, wan
One hundred thousandlakh
Five hundred thousandcrore
One millionMega, meg, mil,
One million forty-eight thousand five hundred seventy-sixMibi or Mega in computing, see binary prefix
Ten millioncrore
One hundred millionyi

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 or names of large numbers for more information on naming numbers.
There is no consistent and widely accepted way to extend cardinals beyond centillion.

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. For instance, in this system, 1032 would be represented as 1'0000,0000;0000,0000:0000,0000;0000,0000.
ValueMyriad System NameOctad System NameChinese Myriad ScaleChinese Long ScaleKnuth-proposed
System Name
100OneOneOne
101TenTenTen
102HundredHundredHundred
103ThousandThousandTen hundred
104MyriadMyriad Myriad
105Ten myriadTen myriad十萬 十萬 Ten myriad
106Hundred myriadHundred myriad百萬 百萬 Hundred myriad
107Thousand myriadThousand myriad千萬 千萬 Ten hundred myriad
108Second MyriadOctad Myllion
1012Third myriadMyriad Octad萬億Myriad myllion
1016Fourth myriadSecond octadByllion
1020Fifth myriadMyriad second octad萬兆
1024Sixth myriadThird octad億兆Myllion byllion
1028Seventh myriadMyriad third octad萬億兆
1032Eighth myriadFourth octad Tryllion
1036Ninth myriadMyriad fourth octad 萬京
1040Tenth myriadFifth octad億京
1044Eleventh myriadMyriad fifth octad 萬億京
1048Twelfth myriadSixth octad 兆京
1052Thirteenth myriadMyriad sixth octad 萬兆京
1056Fourteenth myriadSeventh octad ; 億兆京
1060Fifteenth myriadMyriad seventh octad, 萬億兆京
1064Sixteenth myriadEighth octad , Quadyllion
1068Seventeenth myriadMyriad eighth octad 萬垓
1072Eighteenth myriadNinth octad, 億垓
1080Twentieth myriadTenth octad 兆垓
1088Twenty-second myriadEleventh Octad 億兆垓
10128Quinyllion
10256Sexyllion
10512 Septyllion
101,024 Octyllion
102,048Nonyllion
104,096 Decyllion
108,192 Undecyllion
1016,384Duodecyllion
1032,768Tredecyllion
1065,536Quattuordecyllion
10131,072Quindecyllion
10262,144Sexdecyllion
10524,288Septendecyllion
101,048,576Octodecyllion
102,097,152Novemdecyllion
104,194,304Vigintyllion
10232Trigintyllion
10242Quadragintyllion
10252Quinquagintyllion
10262Sexagintyllion
10272Septuagintyllion
10282Octogintyllion
10292Nonagintyllion
102102Centyllion
1021,002Millyllion
10210,002Myryllion

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, twelve percent, three twenty-fifths, nine seventy-fifths, six fiftieths, twelve hundredths, twenty-four two-hundredths, etc.
ValueFractionCommon names
1One, Unity, Whole
0.9Nine tenths, point nine
Five sixths
0.8Four fifths, eight tenths, point eight
0.75three quarters, three fourths, seventy-five hundredths, point seven five
0.7Seven tenths, point seven
Two thirds
0.6Three fifths, six tenths, point six
0.5One half, five tenths, point five
0.4Two fifths, four tenths, point four
One third
0.3Three tenths, point three
0.25One quarter, one fourth, twenty-five hundredths, point two five
0.2One fifth, two tenths, point two
One sixth
One seventh
0.125One eighth, one-hundred-twenty-five thousandths, point one two five
One ninth
0.1One tenth, point one, One perdecime, one perdime
One eleventh
0.09Nine hundredths, point zero nine
One twelfth
0.08Two twenty-fifths, eight hundredths, point zero eight
One thirteenth
One fourteenth
One fifteenth
0.0625One sixteenth, six-hundred-twenty-five ten-thousandths, point zero six two five
One eighteenth
0.05One twentieth, five hundredths, point zero five
One twenty-first
One twenty-second
One twenty-third
One twenty-fourth
0.04One twenty-fifth, four hundredths, point zero four
One thirtieth
0.03125One thirty-second, thirty one-hundred twenty five hundred-thousandths, point zero three one two five
0.03Three hundredths, point zero three
0.025One fortieth, twenty-five thousandths, point zero two five
0.02One fiftieth, two hundredths, point zero two
One sixtieth
0.015625One sixty-fourth, ten thousand fifty six-hundred twenty-five millionths, point zero one five six two five
One eighty-first
One ninety-ninth
0.01One hundredth, point zero one, One percent
One hundred-first
One over one hundred twenty-one
0.001One thousandth, point zero zero one, One permille
One thirty-six hundredth
0.0001One ten-thousandth, point zero zero zero one, One myriadth, one permyria, one permyriad, one basis point
One hundred-thousandth, point zero zero zero zero one, One lakhth, one perlakh
One millionth, point zero zero zero zero zero one, One ppm
One ten-millionth, One crorth, one percrore
One hundred-millionth
One billionth, One ppb
One trillionth, One ppt
0Zero, Nil

Other specific quantity terms

Various terms have arisen to describe commonly used measured quantities.
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 —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 and Pilagá, Culina and pre-contact Jarawara, Jabutí, Canela-Krahô, Botocudo, Chiquitano, the Campa languages, Arabela, and Achuar. Some languages of Australia, such as Warlpiri, do not have words for quantities above two, as 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, and base 12 from counting the knuckles.

No base

Many languages of Melanesia have counting systems based on parts of the body which do not have a numeric base; there are 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 to 23. 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, often using zeros and ones. With only two symbols binary is useful for logical systems 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 Guineans, count with the base number four, using the term asu and aso, the word for dog, as the ubiquitous village dog has four legs. 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, less 30 asu of pig bartered for 10 asu 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. 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 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 language has a base-seven system, but this is dubious.

8: octal

Octal counting systems are based on the number 8. Examples can be found in the Yuki language of California and in the Pamean languages of Mexico, 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:
Duodecimal systems are based on 12.
These include:
Duodecimal numeric systems have some practical advantages over decimal. It is much easier to divide the base digit twelve by many important divisors 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, 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 developed a rich vocabulary for duodecimal-based fractions. 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 in the old system equals sixteen taels. The suanpan 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 was subdivided into four paisa or twelve pies. 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.
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. 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, 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.
The term score originates from tally sticks, 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", 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 have a base-24 system with a sub-base of 6.

32: duotrigesimal

has base 32.

60: sexagesimal

has a base-60 system. Sumeria had a base-60 system with a decimal sub-base, which was the origin of the numbering of modern degrees, minutes, and seconds.

80: octogesimal

is said to have a base-80 system; it counts in twenties up to 80, then by eighties up to 400, and then by 400s.
799 ’

Numerals in various languages

A database 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.