In linguistics, a numeral is a word class (part of speech) of number words, specifically of counting words such as English 'two', 'eleven', and 'seventy-seven'.

In mathematics and writing, the word "numeral" is used for a different concept: the symbols used to write numerals and numbers. (See numeral system.)

Identifying numerals [link]

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. 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 [link]

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 [link]

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 [link]

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.^[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 [link]

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 is Api,^[which?] 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: senary [link]

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 [link]

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 Pame keep count by using the four spaces between their fingers rather than the fingers themselves.^[9]

10: decimal [link]

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.^[8][10][11] 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,^[8] 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.^[12]

12: duodecimal [link]

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 [link]

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][10] 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: quadrivigesimal [link]

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

32: duotrigesimal [link]

Ngiti has base 32.

60: sexagesimal [link]

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 [link]

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).

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

Larger numerals [link]

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 hiɡher 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), as 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 [link]

Numerals in various languages [link]

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.

• 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 [link]

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

Notes [link]