Numeral (linguistics)

This article is about number words. For the mathematical notation of numbers, see Numeral system.

In linguistics, a numeral is a member of a word class (or a subclass of determiners) designating numbers, such as the English word 'two' and the compound 'seventy-seven'.

Identifying numerals

Numerals may be attributive, as in two dogs, or pronominal, as in I saw two (of them).

Many words of different parts of speech indicate number or quantity. Quantifiers do not enumerate, or designate a specific number, but give another, often less specific, indication of amount. Examples are words such as every, most, least, some, etc. There are also number words which enumerate but are not a distinct part of speech, such as 'dozen', which is a noun, 'first', which is an adjective, or 'twice', which is an adverb. Numerals enumerate, but in addition 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. (Note that *dozen dogs played in the park is not grammatical, so 'dozen' is not a numeral.)

Numerals may be simple, such as 'eleven', or compound, such as 'twenty-three'. They 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. 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").

Various other number words are derived from numerals, but are not themselves numerals. Examples are ordinal numbers (first, second, third, etc.; from 'third' up, these are also used for fractions) and multiplicative adverbs (once, twice, and thrice).

In other languages, there may be other kinds of words derived from numerals. For example, in Slavic languages there are collective numbers which describe sets, such as pair or dozen in English (see Polish numerals). Georgian[1] and Latin have distributive numbers, such as Latin singuli "one-by-one", bini "in pairs, two-by-two", terni "three each", etc.

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 or number words, 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 classifiers. Many sign languages, such as ASL, incorporate numerals.

Basis of counting system

Not all languages have numeral systems. Specifically, there is not much need for numeral systems 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 (Krenák), Chiquitano, the Campa languages, Arabela, and Achuar.[2] Some languages of Australia, such as Warlpiri, do not have words for quantities above two,[3][4] 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 (attested in California), and base 12 from counting the knuckles (3 each for the four fingers).[5]

For very large (and very small) numbers, traditional systems have been superseded by the use of scientific notation and the system of SI prefixes. Traditional systems continue to be used in everyday life.

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 pinkie 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.

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.[6] 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.[6][7]

5: quinary

Main article: 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).[8] 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

Main article: Senary

The Morehead-Maro languages of Southern New Guinea are examples of the rare base 6 system. Kanum is one these languages. The Sko languages on the North Coast of New Guinea follow a base-24 system with a subbase of 6.

8: octal

Main article: Octal

Octal counting systems are based on the number 8. It is used 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]

10: decimal

Main article: 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.[8][10] There are many regional variations including:

12: duodecimal

Main article: Duodecimal

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 (which is a highly composite number) 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,[11] 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 (see Roman numerals). A notable fictional duodecimal system was that of J. R. R. Tolkien's Elvish languages, which used duodecimal as well as decimal.

20: vigesimal

Main article: 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[8][12] 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 (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.

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 subbase of 6.

32: duotrigesimal

Ngiti has base 32.

60: sexagesimal

Main article: Sexagesimal

Ekari has a base-60 system. Sumeria had a base-60 system with a decimal subbase (perhaps a conflation of the decimal and a duodecimal systems of its constituent peoples), 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 subbases) up to 80, then by eighties up to 400, and then by 400s (great scores).

kàmpwóò ŋ̀kwuu sicyɛɛré ná béé-tàànre ná kɛ́ ná báár-ìcyɛ̀ɛ̀rè
fourhundred eighty four and twenty-three and ten and five-four

799 [i.e. 400 + (4 x 80) + (3 x 20) + {10 + (5 + 4)}]’

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 become conflated 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) or of a million (long scale—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 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 (202), pik 8000 (203), kalab 160,000 (204), etc.

See also

Numerals in various languages

A database 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.

Related topics

Notes

  1. ↑ Walsinfo.com
  2. ↑ Hammarström (2009, page 197) "Rarities in numeral systems"
  3. ↑ UCL Media Relations, "Aboriginal kids can count without numbers"
  4. ↑ The Science Show, Genetic anomaly could explain severe difficulty with arithmetic, Australian Broadcasting Corporation
  5. ↑ Bernard Comrie, "The Typology of Numeral Systems", p. 3
  6. ↑ 6.0 6.1 Ryan, Peter. Encyclopaedia of Papua and New Guinea. Melbourne University Press & University of Papua and New Guinea,:1972 ISBN 0-522-84025-6.: 3 pages pp219.
  7. ↑ Aleksandr Romanovich Luriicac, Lev Semenovich Vygotskiĭ, Evelyn Rossiter. Ape, primitive man, and child: essays in the history of behavior . CRC Press: 1992: ISBN 1-878205-43-9: 171 pages
  8. ↑ 8.0 8.1 8.2 Heath, Thomas, A Manual of Greek Mathematics, Courier Dover: 2003. ISBN 0486432319576 page, p:11
  9. ↑ Ascher, Marcia (1994), Ethnomathematics: A Multicultural View of Mathematical Ideas, Chapman & Hall, ISBN 0-412-98941-7
  10. ↑ Scientific American Munn& Co: 1968, vol 219: 219
  11. ↑ such as twelve months in a year, the twelve-hour clock, twelve inches to the foot, twelve pence to the shilling
  12. ↑ Georges Ifrah, The Universal History of Numbers: The Modern Number System, Random House, 2000: ISBN 1-86046-791-1: 1262 pages

Further reading