Numeral (linguistics)

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In linguistics, a numeral 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 that 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 "first") 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").

Contents

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. (*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 (adverbial) 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 (monad, pair/dyad, triad) 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 [5] ), 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 the Indian subcontinent, 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
0 Zero aught, cipher, cypher, donut, dot, duck, goose egg, love, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip, zippo
1Oneace, individual, single, singleton, unary, unit, unity
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
4Fourfoursome, quadruplet, quatern, quaternary, quaternity, quartet, tetrad
5Fivecinque, fin, fivesome, pentad, quint, quintet, quintuplet
6Sixhalf dozen, hexad, sestet, sextet, sextuplet, sise
7Sevenheptad, septet, septuple, walking stick
8Eightoctad, octave, octet, octonary, octuplet, ogdoad
9Nineennead
10Tendeca, decade, das (India)
11Elevenonze, ounze, ounce, banker's dozen
12Twelvedozen
13Thirteen baker's dozen, long dozen [6]
20Twentyscore,
21Twenty-onelong score, [6] blackjack
22Twenty-twoDeuce-deuce
24Twenty-fourtwo dozen
40Fortytwo-score
50Fiftyhalf-century
55Fifty-five double nickel
60Sixtythree-score
70Seventythree-score and ten
80Eightyfour-score
87Eighty-seven four-score and seven
90Ninetyfour-score and ten
100One hundredcentred, century, ton, short hundred
111One hundred [and] eleveneleventy-one [7]
120One hundred [and] twenty long hundred, [6] great hundred, (obsolete) hundred
144One hundred [and] forty-four gross, dozen dozen, small gross
1000One thousandchiliad, grand, G, thou, yard, kilo, k, millennium, Hajaar (India), ten hundred
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
10000Ten thousand myriad, wan (China)
100000One hundred thousand lakh
500000Five hundred thousand crore (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 million crore (Indian)(Pakistan)
100000000One hundred million yi (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 scale Long scale
ValueAmericanBritish
(Nicolas Chuquet)
Continental European
(Jacques Peletier du Mans)
100One
101Ten
102Hundred
103Thousand
106Million
109BillionThousand millionMilliard
1012TrillionBillion
1015QuadrillionThousand billionBilliard
1018QuintillionTrillion
1021SextillionThousand trillionTrilliard
1024SeptillionQuadrillion
1027OctillionThousand quadrillionQuadrilliard
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
10123QuadragintillionThousand vigintillionVigintilliard
10153QuinquagintillionThousand quinvigintillionQuinvigintilliard
10180NovemquinquagintillionTrigintillion
10183SexagintillionThousand trigintillionTrigintilliard
10213SeptuagintillionThousand quintrigintillionQuintrigintilliard
10240NovemseptuagintillionQuadragintillion
10243OctogintillionThousand quadragintillionQuadragintilliard
10273NonagintillionThousand quinquadragintillionQuinquadragintilliard
10300NovemnonagintillionQuinquagintillion
10303 Centillion Thousand quinquagintillionQuinquagintilliard
10360CennovemdecillionSexagintillion
10420CennovemtrigintillionSeptuagintillion
10480CennovemquinquagintillionOctogintillion
10540CennovemseptuagintillionNonagintillion
10600Cennovemnonagintillion Centillion
10603DucentillionThousand centillion Centilliard

There is no consistent and widely accepted way to extend cardinals beyond centillion (centilliard).

Myriad, Octad, and -yllion systems

The following table details the myriad, octad, Ancient Greek Archimedes's notation, 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.

ValueMyriad System NameOctad System NameAncient Greek Myriad Scale Chinese Myriad Scale Chinese Long ScaleKnuth-proposed
System Name
100OneOneεἷς (heîs) One
101TenTenδέκα (déka) Ten
102HundredHundredἑκατόν (hekatón) Hundred
103ThousandThousandχίλιοι (khī́lioi) Ten hundred
104MyriadMyriadμύριοι (mýrioi) ( ) ( )Myriad
105Ten myriadTen myriadδεκάκις μύριοι (dekákis mýrioi)十萬 (十万)十萬 (十万)Ten myriad
106Hundred myriadHundred myriadἑκατοντάκις μύριοι (hekatontákis mýrioi)百萬 (百万)百萬 (百万)Hundred myriad
107Thousand myriadThousand myriadχιλιάκις μύριοι (khiliákis mýrioi)千萬 (千万)千萬 (千万)Ten hundred myriad
108Second myriadOctadμυριάκις μύριοι (muriákis mýrioi) ( 亿 ) ( 亿 )Myllion
109Ten second myriadTen octadδεκάκις μυριάκις μύριοι (dekákis muriákis múrioi)十億 (十亿)十億 (十亿)Ten myllion
1010Hundred second myriadHundred octadἑκατοντάκις μυριάκις μύριοι (hekatontákis muriákis múrioi)百億 (百亿)百億 (百亿)Hundred myllion
1011Thousand second myriadThousand octadχῑλῐάκῐς μυριάκις μύριοι (khīliákis muriákis múrioi)千億 (千亿)千億 (千亿)Ten hundred myllion
1012Third myriadMyriad octadμυριάκις μυριάκις μύριοι (muriákis muriákis mýrioi) 萬億 (万亿)Myriad myllion
1013Ten third myriadTen myriad octadδεκάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis mýrioi)十兆十萬億 (十万亿)Ten myriad myllion
1014Hundred third myriadHundred myriad octadἑκατοντάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis mýrioi)百兆百萬億 (百万亿)Hundred myriad myllion
1015Thousand third myriadThousand myriad octadχιλιάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis mýrioi)千兆千萬億 (千万亿)Ten hundred myriad myllion
1016Fourth myriadSecond octadμυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis mýrioi) Byllion
1017Ten fourth myriadTen second octadδεκάκις μυριάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis muriákis mýrioi)十京十兆Ten byllion
1018Hundred fourth myriadHundred second octadἑκατοντάκις μυριάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis muriákis mýrioi)百京百兆Hundred byllion
1019Thousand fourth myriadThousand second octadχιλιάκις μυριάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis muriákis mýrioi)千京千兆Ten hundred byllion
1020Fifth myriadMyriad second octadμυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis mýrioi) 萬兆Myriad byllion
1021Ten fifth myriadTen myriad second octadδεκάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis muriákis muriákis mýrioi)十垓十萬兆Ten myriad byllion
1022Hundred fifth myriadHundred myriad second octadἑκατοντάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis muriákis muriákis mýrioi)百垓百萬兆Hundred myriad byllion
1023Thousand fifth myriadThousand myriad second octadχιλιάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis muriákis muriákis mýrioi)千垓千萬兆Ten hundred myriad byllion
1024Sixth myriadThird octadμυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis muriákis mýrioi) (in China); 𥝱 (in Japan)億兆Myllion byllion
1028Seventh myriadMyriad third octad 萬億兆Myriad myllion byllion
1032Eighth myriadFourth octad ( ) Tryllion
1036Ninth myriadMyriad fourth octad ( )萬京Myriad tryllion
1040Tenth myriadFifth octad 億京Myllion tryllion
1044Eleventh myriadMyriad fifth octad ( )萬億京Myriad myllion tryllion
1048Twelfth myriadSixth octad ( ) (in China and in Japan)兆京Byllion tryllion
1052Thirteenth myriadMyriad sixth octad 恆河沙 ( 恒河沙 ) (in China)萬兆京Myriad byllion tryllion
1056Fourteenth myriadSeventh octad 阿僧祇 (in China); 恒河沙 (in Japan)億兆京Myllion byllion tryllion
1060Fifteenth myriadMyriad seventh octad 那由他 , 那由多 (in China)萬億兆京Myriad myllion byllion tryllion
1064Sixteenth myriadEighth octad 不可思議 ( 不可思议 ) (in China), 阿僧祇 (in Japan) Quadyllion
1068Seventeenth myriadMyriad eighth octad 無量大數 ( 无量大数 ) (in China)萬垓Myriad quadyllion
1072Eighteenth myriadNinth octad 那由他 , 那由多 (in Japan)億垓Myllion quadyllion
1080Twentieth myriadTenth octad 不可思議 (in Japan)兆垓Byllion quadyllion
1088Twenty-second myriadEleventh octad 無量大数 (in Japan)億兆垓Myllion byllion quadyllion
10128Thirty-second myriadSixteenth octad Quinyllion
10256Sixty-fourth myriadThirty-second octad Sexyllion
10512128th myriadSixty-fourth octad ( )Septyllion
101,024256th myriad128th octad ( )Octyllion
102,048512th myriad256th octad Nonyllion
104,0961024th myriad512th octad ( )Decyllion
108,1922048th myriad1024th octad ( )Undecyllion
1016,3844096th myriad2048th octad 恆河沙 ( 恒河沙 )Duodecyllion
1032,7688192nd myriad4096th octad 阿僧祇 Tredecyllion
1065,53616384th myriad8192nd octad 那由他 , 那由多 Quattuordecyllion
10131,07232768th myriad16384th octad 不可思議 ( 不可思议 )Quindecyllion
10262,14465536th myriad32768th octad 無量大數 ( 无量大数 )Sexdecyllion
10524,288131072nd myriad65536th octadSeptendecyllion
101,048,576262144th myriad131072nd octadOctodecyllion
102,097,152524288th myriad262144th octadNovemdecyllion
104,194,3041048576th myriad524288th octadVigintyllion
102321073741824th myriad536870912nd octadTrigintyllion
102421099511627776th myriad549755813888th octadQuadragintyllion
10252Quinquagintyllion
10262Sexagintyllion
10272Septuagintyllion
10282Octogintyllion
10292Nonagintyllion
102102Centyllion
1021,002Millyllion
10210,002Myryllion

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
11/1One, Unity, Whole
0.99/10Nine tenths, [zero] point nine
0.833333...5/6Five sixths
0.84/5Four fifths, eight tenths, [zero] point eight
0.753/4three quarters, three fourths, seventy-five hundredths, [zero] point seven five
0.77/10Seven tenths, [zero] point seven
0.666666...2/3Two thirds
0.63/5Three fifths, six tenths, [zero] point six
0.51/2 One half, five tenths, [zero] point five
0.42/5Two fifths, four tenths, [zero] point four
0.333333...1/3One third
0.33/10Three tenths, [zero] point three
0.251/4One quarter, one fourth, twenty-five hundredths, [zero] point two five
0.21/5One fifth, two tenths, [zero] point two
0.166666...1/6One sixth
0.142857142857...1/7One seventh
0.1251/8One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
0.111111...1/9One ninth
0.11/10One tenth, [zero] point one, One perdecime, one perdime
0.090909...1/11One eleventh
0.099/100Nine hundredths, [zero] point zero nine
0.083333...1/12One twelfth
0.082/25Two twenty-fifths, eight hundredths, [zero] point zero eight
0.076923076923...1/13One thirteenth
0.071428571428...1/14One fourteenth
0.066666...1/15One fifteenth
0.06251/16One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
0.055555...1/18One eighteenth
0.051/20One twentieth, five hundredths, [zero] point zero five
0.047619047619...1/21One twenty-first
0.045454545...1/22One twenty-second
0.043478260869565217391304347...1/23One twenty-third
0.041666...1/24One twenty-fourth
0.041/25One twenty-fifth, four hundredths, [zero] point zero four
0.033333...1/30One thirtieth
0.031251/32One thirty-second, thirty one-hundred [and] twenty five hundred-thousandths, [zero] point zero three one two five
0.033/100Three hundredths, [zero] point zero three
0.0251/40One fortieth, twenty-five thousandths, [zero] point zero two five
0.021/50One fiftieth, two hundredths, [zero] point zero two
0.016666...1/60One sixtieth
0.0156251/64One sixty-fourth, ten thousand fifty six-hundred [and] twenty-five millionths, [zero] point zero one five six two five
0.012345679012345679...1/81One eighty-first
0.010101...1/99One ninety-ninth
0.011/100One hundredth, [zero] point zero one, One percent
0.009900990099...1/101One hundred-first
0.008264462809917355371900...1/121One over one hundred twenty-one
0.0011/1000One thousandth, [zero] point zero zero one, One permille
0.000277777...1/3600One thirty-six hundredth
0.00011/10000One ten-thousandth, [zero] point zero zero zero one, One myriadth, one permyria, one permyriad, one basis point
0.000011/100000One hundred-thousandth, [zero] point zero zero zero zero one, One lakhth, one perlakh
0.0000011/1000000One millionth, [zero] point zero zero zero zero zero one, One ppm
0.00000011/10000000One ten-millionth, One crorth, one percrore
0.000000011/100000000One hundred-millionth
0.0000000011/1000000000One billionth (in some dialects), One ppb
0.0000000000011/1000000000000One trillionth, One ppt
00/1 Zero, Nil

Other specific quantity terms

Various terms have arisen to describe commonly used measured quantities.

Basis of counting system

Not all peoples use counting, at least not verbally. Specifically, there is not much need for counting among hunter-gatherers who do not engage in commerce. Many languages around the world have no numerals above two to four (if they are actually numerals at all, and not some other part of speech)—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. [10] Some languages of Australia, such as Warlpiri, do not have words for quantities above two, [11] [12] [13] and neither 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). [14]

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.[ citation needed ]

2: binary

Binary systems are based on the number 2, using zeros and ones. Due to its simplicity, only having two distinct digits, binary is commonly used in computing, with zero and one often corresponding to "off/on" respectively.

3: ternary

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

4: quaternary

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]

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). [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.

6: senary

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.

7: septenary

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]

8: octal

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]

9: nonary

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

10: decimal

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:

12: duodecimal

Duodecimal systems are based on the number 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, [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.

16: hexadecimal

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.

20: vigesimal

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...".

24: quadrovigesimal

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

32: duotrigesimal

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

60: sexagesimal

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.

80: octogesimal

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).

kàmpwóò

four hundred

ŋ̀kwuu

eighty

sicyɛɛré

four

and

béé-tàànre

twenty-three

and

kɛ́

ten

and

báár-ìcyɛ̀ɛ̀rè

five-four

kàmpwóò ŋ̀kwuu sicyɛɛré ná béé-tàànre ná kɛ́ ná báár-ìcyɛ̀ɛ̀rè

{four hundred} eighty four and twenty-three and ten and five-four

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

See also

Numerals in various languages

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.

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. 1 2 "What is a numeral?". Archived from the original on 2016-11-25. Retrieved 2017-03-06.
  4. "Walsinfo.com".[ permanent dead link ]
  5. "Numbers in Guaraní (Papapy Avañe'ême)". omniglot.com. Archived from the original on 2021-06-11. Retrieved 2021-06-11.
  6. 1 2 3 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.
  7. Ezard, John (2 Jan 2003). "Tolkien catches up with his hobbit". The Guardian. Retrieved 6 Apr 2018.
  8. "Large Numbers (page 2) at MROB". mrob.com. Archived from the original on 2012-02-13. Retrieved 2020-12-23.
  9. Cardarelli, François (2012). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (Second ed.). Springer. p. 585. ISBN   978-1447100034.
  10. "Hammarström (2009, page 197) "Rarities in numeral systems"" (PDF). Archived from the original (PDF) on 2012-03-08. Retrieved 2010-06-16.
  11. UCL Media Relations, "Aboriginal kids can count without numbers" Archived 2018-06-20 at the Wayback Machine
  12. Butterworth, Brian; Reeve, Robert; Reynolds, Fiona; Lloyd, Delyth (2 September 2008). "Numerical thought with and without words: Evidence from indigenous Australian children". PNAS. 105 (35): 13179–13184. Bibcode:2008PNAS..10513179B. doi: 10.1073/pnas.0806045105 . PMC   2527348 . PMID   18757729. [Warlpiri] has three generic types of number words: singular, dual plural, and greater than dual plural.
  13. The Science Show, Genetic anomaly could explain severe difficulty with arithmetic Archived 2010-03-01 at the Wayback Machine , Australian Broadcasting Corporation
  14. Bernard Comrie, "The Typology of Numeral Systems Archived 2011-05-14 at the Wayback Machine ", p. 3
  15. 1 2 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.
  16. 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.
  17. 1 2 3 Heath, Thomas, A Manual of Greek Mathematics, Courier Dover: 2003. ISBN   978-0-486-43231-1 page, p:11
  18. 1 2 Parkvall, M. Limits of Language, 1st edn. 2008. p.291. ISBN   978-1-59028-210-6
  19. Ascher, Marcia (1994), Ethnomathematics: A Multicultural View of Mathematical Ideas, Chapman & Hall, ISBN   0-412-98941-7
  20. Scientific American Munn& Co: 1968, vol 219: 219
  21. such as twelve months in a year, the twelve-hour clock, twelve inches to the foot, twelve pence to the shilling
  22. "算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus". totton.idirect.com. Archived from the original on 2019-07-06. Retrieved 2019-06-26.
  23. Georges Ifrah, The Universal History of Numbers: The Modern Number System, Random House, 2000: ISBN   1-86046-791-1. 1262 pages

Further reading

Related Research Articles

<span class="mw-page-title-main">Decimal</span> Number in base-10 numeral system

The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve squared, "1000" means twelve cubed, and "0.1" means a twelfth.

The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals sometime between 200 and 78 BCE, the latter being the date of the earliest archeological evidence.

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English number words include numerals and various words derived from them, as well as a large number of words borrowed from other languages.

<span class="mw-page-title-main">Vigesimal</span> Base-20 numeral system

A vigesimal or base-20 (base-score) numeral system is based on twenty. Vigesimal is derived from the Latin adjective vicesimus, meaning 'twentieth'.

In written languages, an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. In English orthography, this corresponds to the suffixes ‑st, ‑nd, ‑rd, ‑th in written ordinals.

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In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on. They differ from cardinal numerals, which represent quantity and other types of numerals.

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