Numeral (linguistics)

This article is about the linguistic concept of words that represent numbers. For the mathematical notation for representing numbers of a given set, see Numeral system.

In linguistics, number names (or numerals) are specific words in a natural language that represent numbers.

In writing, numerals are symbols also representing numbers. In mathematics (including computing) there are other meanings and definitions of numbers, over the different stages of the history of science.

Types of numerals

In linguistics, the terms representing numbers can be classified according to their use:^[1]

• Cardinal numerals: these describe quantity – one, two, three, etc.
• Ordinal numerals: describe position in a sequential order – first, second, third, etc. (the terms next and last may also be considered a kind of ordinals)
• Ranking numerals: describe order, based on relevance or importance – primary, secondary, tertiary, etc.^{[citation needed]}
• Partitive numerals: describe division into fractions – whole, half, third, etc.
• Composite numerals: describe composition – unary, binary, ternary, etc.
• Multiplicative numerals: describe repetition - once, twice, and thrice
• Reproductive numerals: describe replication – single, double, triple, etc. (multiple serves as a generic plural)
• Collective numerals: describe sets – pair, triad, dozen, etc. (musical terms: solo, duo/duet, trio, etc. / kinship terms: twin, triplet, etc.) The term is also used for a specialized type of numeral in the Polish number system.
• Distributive numerals: describe an alternating pattern – each person (one), every other week (entailing two weeks), every third person, etc. Note that the English language does not have distinct distributive numerals (though it does have distributive adjectives/pronouns, such as each, either and every), but some other languages such as Georgian^[2] and Latin do have them, e.g. Latin singuli ("one by one"), bini ("in pairs", "by twos"), terni ("three each"), etc.
• Morphological numerals: are mainly used only metalinguistically to describe grammatical number in language – singular, dual, trial, and quadral (to include plural)

Terms such as most, least, some, and others like them are not technically numerals, but quantifiers. Quantifiers do not enumerate, or designate a specific number, but give another, often less specific, indication of amount.

Basis of counting system

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic (Hindu numerals) Eastern Arabic Indian family Tamil	Burmese Khmer Lao Mongolian Thai
East Asian numerals
Chinese Japanese Suzhou	Korean Vietnamese Counting rods
Alphabetic numerals
Abjad Armenian Āryabhaṭa Cyrillic	Ge'ez Greek (Ionian) Hebrew
Other systems
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit	Kharosthi Mayan Quipu Roman Sumerian Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64
List of numeral systems
v · d · e

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 numerals 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.^[3] Some languages of Australia, such as Warlpiri, do not have words for quantities above two,^[4][5] 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).^[6]

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 numerical words, but rather words 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, including the Māori, some Sulawesi and some Papua New Guineans count with the base number four, using the term asu and aso (derived from Javanese asu: dog), as the ubiquitous village dog has four legs.^[7] 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.^[7][8]

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).^[9] An example is Api, a language 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: sexal

Kanum is a rare example of a language with base 6. The Sko languages, however, and base-24 with a subbase of 6.

8: octal

Octal is a counting system 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 Pamean keep count by using the four spaces between their fingers rather than the fingers themselves.^[10]

10: decimal

A majority of traditional number systems are based on the decimal numeral system. Anthropologists hypothesize this may be due to humans having five digits per hand, ten in total.^[9][11][12] There are many regional variations including:

• Western system: based on thousands, with variants (see English-language numerals)
• Indian system: crore, lakh (see Indian numbering system. Indian numerals)
• East Asian system: based on ten-thousands (see below)

Historically, its use was first employed by the ancient Egyptians, who invented a wholly decimal system, and later extended by the Babylonians,^[9] and also a system of pictorial representation, substituting letters and other reminders with symbols. An English farmer coined the term notch, defined as ten, from the tally sticks of the livestock, a full deep score for every twenty, a half score or notch for ten.^[13]

12: duodecimal

Duodecimal systems are based on 12.

These include:

• Chepang language of Nepal,
• Mahl language of Minicoy Island in India
• Nigerian Middle Belt areas such as Janji, Kahugu and the Nimbia dialect of Gwandara.
• Melanesia^{[citation needed]}
• Polynesia^{[citation needed]}
• 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 market and trade settings, such as the numbers 2, 3, 4 and 6. It is still common usage and is found in idiom. For example, "A dime a dozen" refers to something so common or numerous as to be of little worth or noteworthiness.

The system of basing counting on the number 12, is widespread, across many cultures. Examples include:

• time divisions (twelve months in a year, the twelve-hour clock)
• measurement imperial system of units (twelve inches to the foot, twelve troy ounces to the troy pound)
• traditional British monetary system (twelve pence to the shilling)
• the ancient Babylonian concept of 360 days in the year (12 months x 30 days) from which we get 360 degrees to a circle

Consequently, languages evolved or loaned terms such dozen, gross and great gross, which allow for rudimentary and arguably immediately comprehensible duodecimal nomenclature (e.g., stating: "two gross and six dozen" instead of "three hundred and sixty"). Ancient Romans used decimal 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

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^[9][11] 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, where taxmen and farmers would groove a notch for every ten, and a full score for every twenty. The English term score, now rarely used, is a remnant of vigesimal numeration in the word score. 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 forefathers...".

24

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

32

Ngiti has base 32.

60: 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 number of our 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

In many languages, numerals up to the base are a distinct part of speech, while the words used to form higher numbers belong to one of the other classes, such as nouns or adjectives. In English, these higher words are hundred 10², thousand 10³, million 10⁶, and hiɡher powers of thousand (short scale) or of million (long scale). (See names of large numbers.) 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.

Languages may also have numerals for numbers between the base and its powers. Balinese, for example, currently has a decimal system, with numerals for 10, 100, and 1000, but has additional 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.

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.

Main article: List of numbers in various languages

• Proto-Indo-European numerals
  • English-language numerals
  • Indian numbering system
    • Hindustani numerals
• Proto-Semitic numerals
  • Hebrew numerals
• Japanese numerals
• Korean numerals
• Dzongkha numerals
• Javanese numerals
• Balinese numerals
• Finnish numerals
• Polish numerals
• Yoruba numerals
• Australian Aboriginal enumeration

Related topics

• Long and short scales
• Names of large numbers
• Numeral system
• Names of small numbers

Notes

1. ^ LinguaLinks Library
2. ^ Walsinfo.com
3. ^ Hammarström (2009, page 197) "Rarities in numeral systems"
4. ^ UCL Media Relations, "Aboriginal kids can count without numbers"
5. ^ The Science Show, Genetic anomaly could explain severe difficulty with arithmetic, Australian Broadcasting Corporation
6. ^ Bernard Comrie, "The Typology of Numeral Systems", p. 3
7. ^ ^{a b} Ryan, Peter. Encyclopaedia of Papua and New Guinea. Melbourne University Press & University of Papua and New Guinea,:1972 ISBN 0522840256.: 3 pages pp219.
8. ^ Aleksandr Romanovich Luriicac, Lev Semenovich Vygotskiĭ, Evelyn Rossiter. Ape, primitive man, and child: essays in the history of behavior . CRC Press: 1992: ISBN 1878205439: 171 pages
9. ^ ^{a b c d} Heath, Thomas, A Manual of Greek Mathematics, Courier Dover: 2003. ISBN 0486432319 576 page, p:11
10. ^ Ascher, Marcia (1994), Ethnomathematics: A Multicultural View of Mathematical Ideas, Chapman & Hall, ISBN 0412989417 
11. ^ ^{a b} Georges Ifrah, The Universal History of Numbers: The Modern Number System, Random House, 2000: ISBN 1860467911: 1262 pages
12. ^ Scientific American Munn& Co: 1968, vol 219: 219
13. ^ Karl Menninger, Paul Broneer, Number Words and Number Symbols Courier Dover Publications: 1992: ISBN 0486270963: 480 pages