# Numeral (linguistics)

In linguistics, a numeral (or number word) 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 (including ordinal numbers like the compound word "seventy-fifth") to a part of speech called "numerals"[1][2] Numerals in the broad sense can also be analyzed as a noun ("three is a small number"), as a pronoun ("the two went to town"), or for a small number of words as an adverb ("I rode the slide twice").

Numerals can express relationships like quantity (cardinal numbers), sequence (ordinal numbers), frequency (once, twice), and part (fraction).[3]

## 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. 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.[3] 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 (one would say the equivalent of "five of people"). In English grammar, the classification "numeral" (viewed as a part of speech) 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 parktwelve dogs played in the park. (Note that *dozen dogs played in the park is not grammatical, so "dozen" is not a numeral in this sense.) 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 (first, second, third, etc.; from 'third' up, these are also used for fractions), multiplicative numbers (once, twice, and thrice), multipliers (single, double, and triple), and distributive numbers (singly, doubly, and triply). Georgian,[4] Latin, and Romanian (see Romanian distributive numbers) 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 (see Russian numerals, Polish numerals).

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 (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.

## 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 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 (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.

## Numerals of cardinal numbers

The cardinal numbers have numerals. In the following tables, [and] indicates that the word and is used in some dialects (such as British English), and omitted in other dialects (such as American English).

This table demonstrates the standard English construction of some cardinal numbers. (See next table for names of larger cardinals.)

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 (Sanskrit)
1Oneace, individual, single, singleton, unary, unit, unity, Pratham(Sanskrit)
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
5Fivecinque, fin, fivesome, pentad, quint, quintet, quintuplet
6Sixhalf dozen, hexad, sestet, sextet, sextuplet, sise
11Elevenonze, ounze, ounce, banker's dozen
12Twelvedozen
13Thirteenbaker's dozen, long dozen[5]
20Twentyscore,
21Twenty-onelong score,[5] 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 [and] eleveneleventy-one[6]
120One hundred [and] twentylong hundred,[5] great hundred, (obsolete) hundred
144One hundred [and] forty-fourgross, dozen dozen, small gross
1000One thousandchiliad, grand, G, thou, yard, kilo, k, millennium, Hajaar (India)
1024One thousand [and] twenty-fourkibi or kilo in computing, see binary prefix (kilo is shortened to K, Kibi to Ki)
1100One thousand one hundredEleven hundred
1728One thousand seven hundred [and] twenty-eightgreat gross, long gross, dozen gross
100000One hundred thousandlakh
500000Five hundred thousandcrore (Iranian)
1000000One millionMega, meg, mil, (often shortened to M)
1048576One million forty-eight thousand five hundred [and] seventy-sixMibi or Mega in computing, see binary prefix (Mega is shortened to M, Mibi to Mi)
10000000Ten millioncrore (Indian)(Pakistan)
100000000One hundred millionyi (China)

#### 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.

Short scaleLong scale
ValueAmericanBritish
(Nicolas Chuquet)
Continental European
(Jacques Peletier du Mans)
100One
101Ten
102Hundred
103Thousand
106Million
109BillionThousand millionMilliard
1012TrillionBillion
1018QuintillionTrillion
1021SextillionThousand trillionTrilliard
1030NonillionQuintillion
1033DecillionThousand quintillionQuintilliard
1036UndecillionSextillion
1039DuodecillionThousand sextillionSextilliard
1042TredecillionSeptillion
1045QuattuordecillionThousand septillionSeptilliard
1048QuindecillionOctillion
1051SexdecillionThousand octillionOctilliard
1054SeptendecillionNonillion
1057OctodecillionThousand nonillionNonilliard
1060NovemdecillionDecillion
1063VigintillionThousand decillionDecilliard
1066UnvigintillionUndecillion
1069DuovigintillionThousand undecillionUndecilliard
1072TrevigintillionDuodecillion
1075QuattuorvigintillionThousand duodecillionDuodecilliard
1078QuinvigintillionTredecillion
1081SexvigintillionThousand tredecillionTredecilliard
1084SeptenvigintillionQuattuordecillion
1087OctovigintillionThousand quattuordecillionQuattuordecilliard
1090NovemvigintillionQuindecillion
1093TrigintillionThousand quindecillionQuindecilliard
1096UntrigintillionSexdecillion
1099DuotrigintillionThousand sexdecillionSexdecilliard
10120NovemtrigintillionVigintillion
10153QuinquagintillionThousand quinvigintillionQuinvigintilliard
10180NovemquinquagintillionTrigintillion
10183SexagintillionThousand trigintillionTrigintilliard
10213SeptuagintillionThousand quintrigintillionQuintrigintilliard
10300NovemnonagintillionQuinquagintillion
10303CentillionThousand quinquagintillionQuinquagintilliard
10360CennovemdecillionSexagintillion
10420CennovemtrigintillionSeptuagintillion
10480CennovemquinquagintillionOctogintillion
10540CennovemseptuagintillionNonagintillion
10600CennovemnonagintillionCentillion
10603DucentillionThousand centillionCentilliard

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. For instance, in this system, 1032 would be represented as 1'0000,0000;0000,0000:0000,0000;0000,0000.

System Name
100 One One One
101 Ten Ten Ten
102 Hundred Hundred Hundred
103 Thousand Thousand Ten hundred
1048 Twelfth myriad Sixth octad () (in China and in Japan) 兆京
1056 Fourteenth myriad Seventh octad 阿僧祇 (in China); 恆河沙 (恒河沙) (in Japan) 億兆京
10128 Quinyllion
10256 Sexyllion
10512 () Septyllion
101,024 () Octyllion
102,048 Nonyllion
104,096 () Decyllion
108,192 () Undecyllion
1016,384 Duodecyllion
1032,768 Tredecyllion
1065,536 Quattuordecyllion
10131,072 Quindecyllion
10262,144 Sexdecyllion
10524,288 Septendecyllion
101,048,576 Octodecyllion
102,097,152 Novemdecyllion
104,194,304 Vigintyllion
10232 Trigintyllion
10252 Quinquagintyllion
10262 Sexagintyllion
10272 Septuagintyllion
10282 Octogintyllion
10292 Nonagintyllion
102102 Centyllion
1021,002 Millyllion
10210,002 Myryllion

## 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 (0.12), twelve percent (12%), three twenty-fifths (3/25), nine seventy-fifths (9/75), six fiftieths (6/50), twelve hundredths (12/100), twenty-four two-hundredths (24/200), etc.

ValueFractionCommon names
1 1/1 One, Unity, Whole
0.9 9/10 Nine tenths, [zero] point nine
0.833333... 5/6 Five sixths
0.8 4/5 Four fifths, eight tenths, [zero] point eight
0.75 3/4 three quarters, three fourths, seventy-five hundredths, [zero] point seven five
0.7 7/10 Seven tenths, [zero] point seven
0.666666... 2/3 Two thirds
0.6 3/5 Three fifths, six tenths, [zero] point six
0.5 1/2 One half, five tenths, [zero] point five
0.4 2/5 Two fifths, four tenths, [zero] point four
0.333333... 1/3 One third
0.3 3/10 Three tenths, [zero] point three
0.25 1/4 One quarter, one fourth, twenty-five hundredths, [zero] point two five
0.2 1/5 One fifth, two tenths, [zero] point two
0.166666... 1/6 One sixth
0.142857142857... 1/7 One seventh
0.125 1/8 One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
0.111111... 1/9 One ninth
0.1 1/10 One tenth, [zero] point one, One perdecime, one perdime
0.090909... 1/11 One eleventh
0.09 9/100 Nine hundredths, [zero] point zero nine
0.083333... 1/12 One twelfth
0.08 2/25 Two twenty-fifths, eight hundredths, [zero] point zero eight
0.076923076923... 1/13 One thirteenth
0.071428571428... 1/14 One fourteenth
0.066666... 1/15 One fifteenth
0.0625 1/16 One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
0.055555... 1/18 One eighteenth
0.05 1/20 One twentieth, five hundredths, [zero] point zero five
0.047619047619... 1/21 One twenty-first
0.045454545... 1/22 One twenty-second
0.043478260869565217391304347... 1/23 One twenty-third
0.041666... 1/24 One twenty-fourth
0.04 1/25 One twenty-fifth, four hundredths, [zero] point zero four
0.033333... 1/30 One thirtieth
0.03125 1/32 One thirty-second, thirty one-hundred [and] twenty five hundred-thousandths, [zero] point zero three one two five
0.03 3/100 Three hundredths, [zero] point zero three
0.025 1/40 One fortieth, twenty-five thousandths, [zero] point zero two five
0.02 1/50 One fiftieth, two hundredths, [zero] point zero two
0.016666... 1/60 One sixtieth
0.015625 1/64 One sixty-fourth, ten thousand fifty six-hundred [and] twenty-five millionths, [zero] point zero one five six two five
0.012345679012345679... 1/81 One eighty-first
0.010101... 1/99 One ninety-ninth
0.01 1/100 One hundredth, [zero] point zero one, One percent
0.009900990099... 1/101 One hundred-first
0.008264462809917355371900... 1/121 One over one hundred twenty-one
0.001 1/1000 One thousandth, [zero] point zero zero one, One permille
0.000277777... 1/3600 One thirty-six hundredth
0.0001 1/10000 One ten-thousandth, [zero] point zero zero zero one, One myriadth, one permyria, one permyriad, one basis point
0.00001 1/100000 One hundred-thousandth, [zero] point zero zero zero zero one, One lakhth, one perlakh
0.000001 1/1000000 One millionth, [zero] point zero zero zero zero zero one, One ppm
0.0000001 1/10000000 One ten-millionth, One crorth, one percrore
0.00000001 1/100000000 One hundred-millionth
0.000000001 1/1000000000 One billionth (in some dialects), One ppb
0.000000000001 1/1000000000000 One trillionth, One ppt
0 0/1 Zero, Nil

## Other specific quantity terms

Various terms have arisen to describe commonly used measured quantities.

## 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.[7] Some languages of Australia, such as Warlpiri, do not have words for quantities above two,[8][9] 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).[10]

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 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.

### 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.[11] 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.[11][12]

### 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).[13] 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

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.[14]

### 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.[15]

### 9: nonary

It has been suggested that Nenets has a base-nine system.[14]

### 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.[13][16] There are many regional variations including:

### 12: 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,[17] 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 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.[18]

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.

### 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.[13][19] 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...".

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

### 32: duotrigesimal

Ngiti has base 32.

### 60: sexagesimal

Ekari has a base-60 system. Sumeria had a base-60 system with a decimal sub-base (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 sub-bases) 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)}]’

### 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.

## Notes

1. Charles Follen: A Practical Grammar of the German Language. Boston, 1828, p. 9, p. 44 and 48. Quote: "PARTS OF SPEECH. There are ten parts of speech, viz. Article, Substantive or Noun, Adjective, Numeral, Pronoun, Verb, Adverb, Preposition, Conjunction, and Interjection.", "NUMERALS. The numbers are divided into cardinal, ordinal, proportional, distributive, and collective. [...] Numerals of proportion and distribution are [...] &c. Observation. The above numerals, in fach or fäl´tig, are regularly declined, like other adjectives."
2. Horace Dalmolin: The New English Grammar: With Phonetics, Morphology and Syntax, Tate Publishing & Enterprises, 2009, p. 175 & p. 177. Quote: "76. The different types of words used to compose a sentence, in order to relate an idea or to convey a thought, are known as parts of speech. [...] The parts of speech, with a brief definition, will follow. [...] 87. Numeral: Numerals are words that express the idea of number. There are two types of numerals: cardinal and ordinal. The cardinal numbers (one, two, three...) are used for counting people, objects, etc. Ordinal numbers (first, second, third...) can indicate order, placement in rank, etc."
3. "What is a numeral?".
4. Walsinfo.com
5. Blunt, Joseph (1 January 1837). "The Shipmaster's Assistant, and Commercial Digest: Containing Information Useful to Merchants, Owners, and Masters of Ships". E. & G.W. Blunt via Google Books.
6. Ezard, John (2 Jan 2003). "Tolkien catches up with his hobbit". The Guardian. Retrieved 6 Apr 2018.
7. Hammarström (2009, page 197) "Rarities in numeral systems"
8. UCL Media Relations, "Aboriginal kids can count without numbers" Archived 2018-06-20 at the Wayback Machine
9. The Science Show, Genetic anomaly could explain severe difficulty with arithmetic, Australian Broadcasting Corporation
10. Bernard Comrie, "The Typology of Numeral Systems", p. 3
11. 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 p 219.
12. 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.
13. Heath, Thomas, A Manual of Greek Mathematics, Courier Dover: 2003. ISBN 978-0-486-43231-1 page, p:11
14. Parkvall, M. Limits of Language, 1st edn. 2008. p.291. ISBN 978-1-59028-210-6
15. Ascher, Marcia (1994), Ethnomathematics: A Multicultural View of Mathematical Ideas, Chapman & Hall, ISBN 0-412-98941-7
16. Scientific American Munn& Co: 1968, vol 219: 219
17. such as twelve months in a year, the twelve-hour clock, twelve inches to the foot, twelve pence to the shilling
18. "算盤 Hexadecimal Addition & Subtraction on an Chinese Abacus". totton.idirect.com. Retrieved 2019-06-26.
19. Georges Ifrah, The Universal History of Numbers: The Modern Number System, Random House, 2000: ISBN 1-86046-791-1. 1262 pages
• James R. Hurford (2010) [1975]. The Linguistic Theory of Numerals. Cambridge University Press. ISBN 978-0-521-13368-5.