Numeral (linguistics)

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 (centilliard).

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.[8] In this system, a new word is invented for every 2ⁿ-th power of ten.

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.

Binary systems are based on the number 2, using zeros and ones. With only two symbols binary is used for things with coding like computers.

Ternary systems are based on the number 3, having practical usage in some analog logic, in baseball scoring and in self–similar mathematical structures.

Quaternary systems are based on the number 4. Some Austronesian, Melanesian, Sulawesi, and Papua New Guinea ethnic groups, count with the base number four, using the term asu or aso, the word for dog, as the ubiquitous village dog has four legs.[15] 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.[15][16]

Quinary systems are based on the number 5. It is almost certain the quinary system developed from counting by fingers (five fingers per hand).[17] 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.

Senary systems are based on the number 6. 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.

Septenary systems are based on the number 7. 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 Palikúr language has a base-seven system, but this is dubious.[18]

Octal 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.[19]

Nonary systems are based on the number 9. It has been suggested that Nenets has a base-nine system.[18]

Decimal systems are based on the number 10. 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.[17][20] There are many regional variations including:

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

Duodecimal systems are based on the number 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
• 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.

Because of several measurements based on twelve,[21] 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.

Hexadecimal systems are based on the number 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.[22]

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 (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.

Vigesimal systems are based on the number 20. 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.[17][23] 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 (great score).

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...".

Quadrovigesimal systems are based on the number 24. The Sko languages have a base-24 system with a sub-base of 6.

Duotrigesimal systems are based on the number 32. The Ngiti ethnolinguistic group uses a base 32 numeral system.

Sexagesimal systems are based on the number 60. 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.

Octogesimal systems are based on the number 80. 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 [i.e. 400 + (4 x 80) + (3 x 20) + {10 + (5 + 4)}]’

A database Numeral Systems of the World's Languages Archived 2016-12-21 at the Wayback Machine 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
  • Indian numbering system
  • Polish numerals
  • Hindustani numerals
• Proto-Semitic numerals
  • Hebrew numerals
• Chinese numerals
  • Japanese numerals
  • Korean numerals
  • Vietnamese numerals
• Australian Aboriginal enumeration
• Balinese numerals
• Dzongkha numerals
• Finnish numerals
• Javanese numerals
• Yoruba numerals

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