Editing Numeral (linguistics)

{{short description|Word or phrase which describes a numerical quantity}}
{{About|number words|the mathematical notation of numbers|numeral system}}

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 number]]s 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 numeral|ordinal numbers]] like "first") to a [[part of speech]] called "numerals".<ref>Charles Follen: ''A Practical Grammar of the German Language''. Boston, 1828, p.&nbsp;9, p.&nbsp;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."</ref><ref>Horace Dalmolin: ''The New English Grammar: With Phonetics, Morphology and Syntax'', Tate Publishing & Enterprises, 2009, p.&nbsp;175 & p.&nbsp;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."</ref> 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]]).<ref name=GloL>{{Cite web|url=http://www-01.sil.org/linguistics/glossaryoflinguisticterms/WhatIsANumeral.htm|title=What is a numeral?|access-date=2017-03-06|archive-date=2016-11-25|archive-url=https://web.archive.org/web/20161125083926/http://www-01.sil.org/Linguistics/GlossaryofLinguisticTerms/WhatIsANumeral.htm|url-status=live}}</ref>

==Identifying numerals==
{{Redirect-distinguish|collective numeral|collective number|collective noun}}
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 [[quantifier (linguistics)|quantifier]]s. 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.<ref name=GloL/> 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 [[Declension|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 (grammar)|article]]: '''''the/some''' dogs played in the park'' → '''''twelve''' 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 number (linguistics)|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 number (linguistics)|ordinal number]]s (''first'', ''second'', ''third'', etc.; from 'third' up, these are also used for fractions), [[Adverbial number|multiplicative (adverbial) numbers]] (''once'', ''twice'', and ''thrice''), [[multiplier (linguistics)|multipliers]] (''single'', ''double'', and ''triple''), and [[distributive number]]s (''singly'', ''doubly'', and ''triply''). [[Georgian language|Georgian]],<ref>{{cite web| url = http://wals.info/feature/description/| title = Walsinfo.com}}{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> Latin, and Romanian (see [[Romanian numbers#Distributive numbers|Romanian distributive numbers]]) have regular [[distributive number]]s, such as Latin {{lang|la|singuli}} "one-by-one", {{lang|la|bini}} "in pairs, two-by-two", {{lang|la|terni}} "three each", etc. In languages other than English, there may be other kinds of number words. For example, in Slavic languages there are [[English numerals#Collective numbers|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 language|Guarani]]<ref>{{Cite web|title=Numbers in Guaraní (Papapy Avañe'ême)|url=https://omniglot.com/language/numbers/guarani.htm|access-date=2021-06-11|website=omniglot.com|archive-date=2021-06-11|archive-url=https://web.archive.org/web/20210611191500/https://omniglot.com/language/numbers/guarani.htm|url-status=live}}</ref>), 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 numerals|Japanese]], which uses either native or Chinese-derived numerals depending on what is being counted.

In many languages, such as [[Chinese numerals|Chinese]], numerals require the use of [[numeral classifier]]s. Many [[sign language]]s, such as [[American Sign Language|ASL]], [[Incorporation (linguistics)|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 language|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 numerals|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]] 10<sup>2</sup>, [[thousand]] 10<sup>3</sup>, [[million]] 10<sup>6</sup>, 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]] 10<sup>4</sup>, and [[Chinese numerals#Large numbers|powers of myriad]]. In the [[Indian subcontinent]], they are hundred, thousand, [[lakh]] 10<sup>5</sup>, [[crore]] 10<sup>7</sup>, and [[Indian numbering system|so on]]. The [[Maya numerals|Mesoamerican system]], still used to some extent in [[Mayan languages]], was based on powers of 20: ''bak’'' 400 (20<sup>2</sup>), ''pik'' 8000 (20<sup>3</sup>), ''kalab'' 160,000 (20<sup>4</sup>), etc.

== Numerals of cardinal numbers ==
{{More citations needed|section|date=April 2019}}

The [[cardinal numbers]] have numerals. In the following tables, [and] indicates that the word ''and'' is used in some [[dialect]]s (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.)

{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Value !! Name !! Alternate names, and names for sets of the given size
|-
| align="right" | 0 ||[[Names for the number 0|Zero]]|| aught, cipher, cypher, donut, dot, duck, goose egg, [[Tennis scoring system#Game score|love]], nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip, zippo
|-
| align="right" | 1 || One || ace, individual, single, singleton, unary, unit, unity
|-
| align="right" | 2 || Two || binary, [[wikt:brace|brace]], couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twin, twosome, yoke
|-
| align="right" | 3 || Three || deuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika, hat-trick
|-
| align="right" | 4 || Four || foursome, quadruplet, quatern, quaternary, quaternity, quartet, tetrad
|-
| align="right" | 5 || Five || cinque, fin, fivesome, pentad, quint, quintet, quintuplet
|-
| align="right" | 6 || Six || half dozen, hexad, sestet, sextet, sextuplet, sise
|-
| align="right" | 7 || Seven || heptad, septet, septuple
|-
| align="right" | 8 || Eight || octad, octave, octet, octonary, octuplet, ogdoad
|-
| align="right" | 9 || Nine || ennead
|-
| align="right" | 10 || Ten || deca, decade, das ([[India]])
|-
| align="right" | 11 || Eleven || onze, ounze, ounce, [[wikt:banker's dozen|banker's dozen]]
|-
| align="right" | 12 || Twelve || dozen
|-
| align="right" | 13 || Thirteen ||[[baker's dozen]], long dozen<ref name="ShipAssistant">{{cite web|url=https://books.google.com/books?id=cDkSAAAAYAAJ&q=%22long%20score%22%2021&pg=PA417|title=The Shipmaster's Assistant, and Commercial Digest: Containing Information Useful to Merchants, Owners, and Masters of Ships|last=Blunt|first=Joseph|date=1 January 1837|publisher=E. & G.W. Blunt|via=Google Books}}</ref>
|-
| align="right" | 20 || Twenty || score,
|-
| align="right" | 21 || Twenty-one || long score,<ref name="ShipAssistant" /> [[blackjack]]
|-
| align="right" | 22 || Twenty-two || Deuce-deuce
|-
| align="right" | 24 || Twenty-four || two dozen
|-
| align="right" | 40 || Forty || two-score
|-
| align="right" | 50 || Fifty || half-century
|-
| align="right" | 55 || Fifty-five ||[[double nickel]]
|-
| align="right" | 60 || Sixty || three-score
|-
| align="right" | 70 || Seventy || three-score and ten
|-
| align="right" | 80 || Eighty || four-score
|-
| align="right" | 87 || Eighty-seven ||[[Gettysburg Address|four-score and seven]]
|-
| align="right" | 90 || Ninety || four-score and ten
|-
| align="right" | 100 || One hundred || centred, century, ton, [[long hundred|short hundred]]
|-
| align="right" | 111 || One hundred [and] eleven || eleventy-one<ref>{{cite news|url=https://www.theguardian.com/uk/2003/jan/02/jrrtolkien.books|title=Tolkien catches up with his hobbit|last=Ezard|first=John|date=2 Jan 2003|work=The Guardian|access-date=6 Apr 2018}}</ref>
|-
| align="right" | 120 || One hundred [and] twenty || [[long hundred]],<ref name="ShipAssistant" /> great hundred, ''(obsolete)'' hundred
|-
| align="right" | 144 || One hundred [and] forty-four ||[[Gross (unit)|gross]], dozen dozen, small gross
|-
| align="right" |{{gaps|1|000}}|| One thousand || chiliad, grand, G, thou, yard, kilo, k, [[millennium]], Hajaar ([[India]]), ten hundred
|-
| align="right" |{{gaps|1|024}}|| One thousand [and] twenty-four || kibi or kilo in [[computing]], see [[binary prefix]] (kilo is shortened to K, Kibi to Ki)
|-
| align="right" |{{gaps|1|100}}|| One thousand one hundred || Eleven hundred
|-
| align="right" |{{gaps|1|728}}|| One thousand seven hundred [and] twenty-eight || great gross, long gross, dozen gross
|-
| align="right" |{{gaps|10|000}}|| Ten thousand ||[[myriad]], [[Myriad#Sinosphere|wan]] (China)
|-
| align="right" |{{gaps|100|000}}|| One hundred thousand ||[[lakh]]
|-
| align="right" |{{gaps|500|000}}|| Five hundred thousand ||[[crore]] (Iranian)
|-
| align="right" |{{gaps|1|000|000}}|| One million || Mega, meg, mil, (often shortened to M)
|-
| align="right" |{{gaps|1|048|576}}|| One million forty-eight thousand five hundred [and] seventy-six || Mibi or Mega in [[computing]], see [[binary prefix]] (Mega is shortened to M, Mibi to Mi)
|-
| align="right" |{{gaps|10|000|000}}|| Ten million ||[[crore]] (Indian)(Pakistan)
|-
| align="right" |{{gaps|100|000|000}}|| One hundred million ||[[Myriad#Sinosphere|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.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|- style="text-align: center"
!  !![[long and short scales|Short scale]]!! colspan="2" |[[long and short scales|Long scale]]
|- style="background: #eeeeff; text-align: center"
! Value !! American!! British<br> ([[Nicolas Chuquet]]) !! Continental European <br> ([[Jacques Peletier du Mans]]) 
|-
| 10<sup>0</sup>|| colspan="3" align="center" | One 
|-
| 10<sup>1</sup>|| colspan="3" align="center" | Ten 
|-
| 10<sup>2</sup>|| colspan="3" align="center" | Hundred 
|-
| 10<sup>3</sup>|| colspan="3" align="center" | Thousand 
|-
| 10<sup>6</sup>|| colspan="3" align="center" | Million 
|-
| 10<sup>9</sup>|| Billion || Thousand million || Milliard 
|-
| 10<sup>12</sup>|| Trillion || colspan="2" align="center" | Billion 
|-
| 10<sup>15</sup>|| Quadrillion || Thousand billion || Billiard 
|-
| 10<sup>18</sup>|| Quintillion || colspan="2" align="center" | Trillion 
|-
| 10<sup>21</sup>|| Sextillion || Thousand trillion || Trilliard 
|-
| 10<sup>24</sup>|| Septillion || colspan="2" align="center" | Quadrillion 
|-
| 10<sup>27</sup>|| Octillion || Thousand quadrillion || Quadrilliard 
|-
| 10<sup>30</sup>|| Nonillion || colspan="2" align="center" | Quintillion 
|-
| 10<sup>33</sup>|| Decillion || Thousand quintillion || Quintilliard 
|-
| 10<sup>36</sup>|| Undecillion || colspan="2" align="center" | Sextillion 
|-
| 10<sup>39</sup>|| Duodecillion || Thousand sextillion || Sextilliard 
|-
| 10<sup>42</sup>|| Tredecillion || colspan="2" align="center" | Septillion 
|-
| 10<sup>45</sup>|| Quattuordecillion || Thousand septillion || Septilliard 
|-
| 10<sup>48</sup>|| Quindecillion || colspan="2" align="center" | Octillion 
|-
| 10<sup>51</sup>|| Sexdecillion || Thousand octillion || Octilliard 
|-
| 10<sup>54</sup>|| Septendecillion || colspan="2" align="center" | Nonillion 
|-
| 10<sup>57</sup>|| Octodecillion || Thousand nonillion || Nonilliard 
|-
| 10<sup>60</sup>|| Novemdecillion || colspan="2" align="center" | Decillion 
|-
| 10<sup>63</sup>|| Vigintillion || Thousand decillion || Decilliard 
|-
| 10<sup>66</sup>|| Unvigintillion || colspan="2" align="center" | Undecillion 
|-
| 10<sup>69</sup>|| Duovigintillion || Thousand undecillion || Undecilliard 
|-
| 10<sup>72</sup>|| Trevigintillion || colspan="2" align="center" | Duodecillion 
|-
| 10<sup>75</sup>|| Quattuorvigintillion || Thousand duodecillion || Duodecilliard 
|-
| 10<sup>78</sup>|| Quinvigintillion || colspan="2" align="center" | Tredecillion 
|-
| 10<sup>81</sup>|| Sexvigintillion || Thousand tredecillion || Tredecilliard 
|-
| 10<sup>84</sup>|| Septenvigintillion || colspan="2" align="center" | Quattuordecillion 
|-
| 10<sup>87</sup>|| Octovigintillion || Thousand quattuordecillion || Quattuordecilliard 
|-
| 10<sup>90</sup>|| Novemvigintillion || colspan="2" align="center" | Quindecillion 
|-
| 10<sup>93</sup>|| Trigintillion || Thousand quindecillion || Quindecilliard 
|-
| 10<sup>96</sup>|| Untrigintillion || colspan="2" align="center" | Sexdecillion 
|-
| 10<sup>99</sup>|| Duotrigintillion || Thousand sexdecillion || Sexdecilliard 
|-
| 10<sup>120</sup>|| Novemtrigintillion || colspan="2" align="center" | Vigintillion
|-
| 10<sup>123</sup>|| Quadragintillion || Thousand vigintillion || Vigintilliard
|-
| 10<sup>153</sup>|| Quinquagintillion || Thousand quinvigintillion || Quinvigintilliard
|-
| 10<sup>180</sup>|| Novemquinquagintillion || colspan="2" align="center" | Trigintillion
|-
| 10<sup>183</sup>|| Sexagintillion || Thousand trigintillion || Trigintilliard
|-
| 10<sup>213</sup>|| Septuagintillion || Thousand quintrigintillion || Quintrigintilliard
|-
| 10<sup>240</sup>|| Novemseptuagintillion || colspan="2" align="center" | Quadragintillion
|-
| 10<sup>243</sup>|| Octogintillion || Thousand quadragintillion || Quadragintilliard
|-
| 10<sup>273</sup>|| Nonagintillion || Thousand quinquadragintillion || Quinquadragintilliard
|-
| 10<sup>300</sup>|| Novemnonagintillion || colspan="2" align="center" | Quinquagintillion
|-
| 10<sup>303</sup>||[[Centillion]]|| Thousand quinquagintillion || Quinquagintilliard
|-
| 10<sup>360</sup>|| Cennovemdecillion || colspan="2" align="center" | Sexagintillion
|-
| 10<sup>420</sup>|| Cennovemtrigintillion || colspan="2" align="center" | Septuagintillion
|-
| 10<sup>480</sup>|| Cennovemquinquagintillion || colspan="2" align="center" | Octogintillion
|-
| 10<sup>540</sup>|| Cennovemseptuagintillion || colspan="2" align="center" | Nonagintillion
|-
| 10<sup>600</sup>|| Cennovemnonagintillion || colspan="2" align="center" |[[Centillion]]
|-
| 10<sup>603</sup>|| Ducentillion || Thousand centillion ||[[Centilliard]]
|}
There is no consistent and widely accepted way to extend cardinals beyond [[centillion]] ([[centilliard]]).

=== {{anchor|Myriad system}} 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 [[Donald E. Knuth|Knuth]]-proposed system notation of numbers, named the -yllion system.<ref>{{Cite web|title=Large Numbers (page 2) at MROB|url=http://mrob.com/pub/math/largenum-2.html#yllion|access-date=2020-12-23|website=mrob.com|archive-date=2012-02-13|archive-url=https://web.archive.org/web/20120213070143/http://mrob.com/pub/math/largenum-2.html#yllion|url-status=live}}</ref> In this system, a new word is invented for every ''2<sup>n</sup>''-th power of ten. <!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not modify it, even if you modify the section title. -->

{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Value !! Myriad System Name !! Octad System Name !! Ancient Greek Myriad Scale !! [[Chinese numerals#Large numbers|Chinese Myriad Scale]]!! Chinese Long Scale !!Knuth-[[-yllion|proposed]]<br>System Name
|-
| 10<sup>0</sup>
| align="center" | One
| align="center" | One
| align="center" | εἷς (heîs)
|<span class="nounderlines">[[wikt:一|一]]</span>
|<span class="nounderlines">[[wikt:一|一]]</span>
| align="center" | One
|-
| 10<sup>1</sup>
| align="center" | Ten
| align="center" | Ten
| align="center" | δέκα (déka)
|<span class="nounderlines">[[wikt:十|十]]</span>
|<span class="nounderlines">[[wikt:十|十]]</span>
| align="center" | Ten
|-
| 10<sup>2</sup>
| align="center" | Hundred
| align="center" | Hundred
| align="center" | ἑκατόν (hekatón)
|<span class="nounderlines">[[wikt:百|百]]</span>
|<span class="nounderlines">[[wikt:百|百]]</span>
| align="center" | Hundred
|-
| 10<sup>3</sup>
| align="center" | Thousand
| align="center" | Thousand
| align="center" | χίλιοι (khī́lioi)
|<span class="nounderlines">[[wikt:千|千]]</span>
|<span class="nounderlines">[[wikt:千|千]]</span>
| align="center" | Ten hundred
|-
| 10<sup>4</sup>
| align="center" | Myriad
| align="center" | Myriad
| align="center" | μύριοι (mýrioi)
|<span class="nounderlines">[[wikt:萬|萬]]</span> (<span class="nounderlines">[[wikt:万|万]]</span>)
|<span class="nounderlines">[[wikt:萬|萬]]</span> (<span class="nounderlines">[[wikt:万|万]]</span>)
| align="center" | Myriad
|-
| 10<sup>5</sup>
| align="center" | Ten myriad
| align="center" | Ten myriad
| align="center" | δεκάκις μύριοι (dekákis mýrioi)
| 十萬 (十万)
| 十萬 (十万)
| align="center" | Ten myriad
|-
| 10<sup>6</sup>
| align="center" | Hundred myriad
| align="center" | Hundred myriad
| align="center" | ἑκατοντάκις μύριοι (hekatontákis mýrioi)
| 百萬 (百万)
| 百萬 (百万)
| align="center" | Hundred myriad
|-
| 10<sup>7</sup>
| align="center" | Thousand myriad
| align="center" | Thousand myriad
| align="center" | χιλιάκις μύριοι (khiliákis mýrioi)
| 千萬 (千万)
| 千萬 (千万)
| align="center" | Ten hundred myriad
|-
| 10<sup>8</sup>
| align="center" | Second myriad
| align="center" | Octad
| align="center" | μυριάκις μύριοι (muriákis mýrioi)
|<span class="nounderlines">[[wikt:億|億]]</span> (<span class="nounderlines">[[wikt:亿|亿]]</span>)
|<span class="nounderlines">[[wikt:億|億]]</span> (<span class="nounderlines">[[wikt:亿|亿]]</span>)
| align="center" | Myllion
|-
| 10<sup>9</sup>
| align="center" | Ten second myriad
| align="center" | Ten octad
| align="center" | δεκάκις μυριάκις μύριοι (dekákis muriákis múrioi)
|十億 (十亿)
|十億 (十亿)
| align="center" | Ten myllion
|-
| 10<sup>10</sup>
| align="center" | Hundred second myriad
| align="center" | Hundred octad
| align="center" | ἑκατοντάκις μυριάκις μύριοι (hekatontákis muriákis múrioi)
|百億 (百亿)
|百億 (百亿)
| align="center" | Hundred myllion
|-
| 10<sup>11</sup>
| align="center" | Thousand second myriad
| align="center" | Thousand octad
| align="center" | χῑλῐάκῐς μυριάκις μύριοι (khīliákis muriákis múrioi)
|千億 (千亿)
|千億 (千亿)
| align="center" | Ten hundred myllion
|-
| 10<sup>12</sup>
| align="center" | Third myriad
| align="center" | Myriad octad
| align="center" | μυριάκις μυριάκις μύριοι (muriákis muriákis mýrioi)
|<span class="nounderlines">[[wikt:兆|兆]]</span>
| 萬億 (万亿)
| align="center" | Myriad myllion
|-
| 10<sup>13</sup>
| align="center" | Ten third myriad
| align="center" | Ten myriad octad
| align="center" | δεκάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis mýrioi)
| 十兆
| 十萬億 (十万亿)
| align="center" | Ten myriad myllion
|-
| 10<sup>14</sup>
| align="center" | Hundred third myriad
| align="center" | Hundred myriad octad
| align="center" | ἑκατοντάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis mýrioi)
| 百兆
| 百萬億 (百万亿)
| align="center" | Hundred myriad myllion
|-
| 10<sup>15</sup>
| align="center" | Thousand third myriad
| align="center" | Thousand myriad octad
| align="center" | χιλιάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis mýrioi)
| 千兆
| 千萬億 (千万亿)
| align="center" | Ten hundred myriad myllion
|-
| 10<sup>16</sup>
| align="center" | Fourth myriad
| align="center" | Second octad
| align="center" | μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis mýrioi)
|<span class="nounderlines">[[wikt:京|京]]</span>
|<span class="nounderlines">[[wikt:兆|兆]]</span>
| align="center" | Byllion
|-
| 10<sup>17</sup>
| align="center" | Ten fourth myriad
| align="center" | Ten second octad
| align="center" | δεκάκις μυριάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis muriákis mýrioi)
| 十京
| 十兆
| align="center" | Ten byllion
|-
| 10<sup>18</sup>
| align="center" | Hundred fourth myriad
| align="center" | Hundred second octad
| align="center" | ἑκατοντάκις μυριάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis muriákis mýrioi)
| 百京
| 百兆
| align="center" | Hundred byllion
|-
| 10<sup>19</sup>
| align="center" | Thousand fourth myriad
| align="center" | Thousand second octad
| align="center" | χιλιάκις μυριάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis muriákis mýrioi)
| 千京
| 千兆
| align="center" | Ten hundred byllion
|-
| 10<sup>20</sup>
| align="center" | Fifth myriad
| align="center" | Myriad second octad
| align="center" | μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis mýrioi)
|<span class="nounderlines">[[wikt:垓|垓]]</span>
| 萬兆
| align="center" | Myriad byllion
|-
| 10<sup>21</sup>
| align="center" | Ten fifth myriad
| align="center" | Ten myriad second octad
| align="center" | δεκάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis muriákis muriákis mýrioi)
| 十垓
| 十萬兆
| align="center" | Ten myriad byllion
|-
| 10<sup>22</sup>
| align="center" | Hundred fifth myriad
| align="center" | Hundred myriad second octad
| align="center" | ἑκατοντάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis muriákis muriákis mýrioi)
| 百垓
| 百萬兆
| align="center" | Hundred myriad byllion
|-
| 10<sup>23</sup>
| align="center" | Thousand fifth myriad
| align="center" | Thousand myriad second octad
| align="center" | χιλιάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis muriákis muriákis mýrioi)
| 千垓
| 千萬兆
| align="center" | Ten hundred myriad byllion
|-
| 10<sup>24</sup>
| align="center" | Sixth myriad
| align="center" | Third octad
| align="center" | μυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis muriákis mýrioi)
|<span class="nounderlines">[[wikt:秭|秭]]</span> (in China); <span class="nounderlines">[[wikt:𥝱|𥝱]]</span> (in Japan)
| 億兆
| align="center" | Myllion byllion
|-
| 10<sup>28</sup>
| align="center" | Seventh myriad
| align="center" | Myriad third octad
|
|<span class="nounderlines">[[wikt:穰|穰]]</span>
| 萬億兆
| align="center" | Myriad myllion byllion
|-
| 10<sup>32</sup>
| align="center" | Eighth myriad
| align="center" | Fourth octad
|
|<span class="nounderlines">[[wikt:溝|溝]]</span> (<span class="nounderlines">[[wikt:沟|沟]]</span>)
|<span class="nounderlines">[[wikt:京|京]]</span>
| align="center" | Tryllion
|-
| 10<sup>36</sup>
| align="center" | Ninth myriad
| align="center" | Myriad fourth octad
|
|<span class="nounderlines">[[wikt:澗|澗]]</span> (<span class="nounderlines">[[wikt:涧|涧]]</span>)
| 萬京
| align="center" | Myriad tryllion
|-
| 10<sup>40</sup>
| align="center" | Tenth myriad
| align="center" | Fifth octad
|
|<span class="nounderlines">[[wikt:正|正]]</span>
| 億京
| align="center" | Myllion tryllion
|-
| 10<sup>44</sup>
| align="center" | Eleventh myriad
| align="center" | Myriad fifth octad
|
|<span class="nounderlines">[[wikt:載|載]]</span> (<span class="nounderlines">[[wikt:载|载]]</span>)
| 萬億京
| align="center" | Myriad myllion tryllion
|-
| 10<sup>48</sup>
| align="center" | Twelfth myriad
| align="center" | Sixth octad
|
|<span class="nounderlines">[[wikt:極|極]]</span> (<span class="nounderlines">[[wikt:极|极]]</span>) (in China and in Japan)
| 兆京
| align="center" | Byllion tryllion
|-
| 10<sup>52</sup>
| align="center" | Thirteenth myriad
| align="center" | Myriad sixth octad
|
|<span class="nounderlines">[[wikt:恆河沙|恆河沙]]</span> (<span class="nounderlines">[[wikt:恒河沙|恒河沙]]</span>) (in China)
| 萬兆京
| align="center" | Myriad byllion tryllion
|-
| 10<sup>56</sup>
| align="center" | Fourteenth myriad
| align="center" | Seventh octad
|
|<span class="nounderlines">[[wikt:阿僧祇|阿僧祇]]</span> (in China); <span class="nounderlines">[[wikt:恒河沙|恒河沙]]</span> (in Japan)
| 億兆京
| align="center" | Myllion byllion tryllion
|-
| 10<sup>60</sup>
| align="center" | Fifteenth myriad
| align="center" | Myriad seventh octad
|
|<span class="nounderlines">[[wikt:那由他|那由他]]</span>, <span class="nounderlines">[[wikt:那由多|那由多]]</span> (in China)
| 萬億兆京
| align="center" | Myriad myllion byllion tryllion
|-
| 10<sup>64</sup>
| align="center" | Sixteenth myriad
| align="center" | Eighth octad
|
|<span class="nounderlines">[[wikt:不可思議|不可思議]]</span> (<span class="nounderlines">[[wikt:不可思议|不可思议]]</span>) (in China), <span class="nounderlines">[[wikt:阿僧祇|阿僧祇]]</span> (in Japan)
|<span class="nounderlines">[[wikt:垓|垓]]</span>
| align="center" | Quadyllion
|-
| 10<sup>68</sup>
| align="center" | Seventeenth myriad
| align="center" | Myriad eighth octad
|
|<span class="nounderlines">[[wikt:無量大數|無量大數]]</span> (<span class="nounderlines">[[wikt:无量大数|无量大数]]</span>) (in China)
| 萬垓
| align="center" | Myriad quadyllion
|-
| 10<sup>72</sup>
| align="center" | Eighteenth myriad
| align="center" | Ninth octad
|
|<span class="nounderlines">[[wikt:那由他|那由他]]</span>, <span class="nounderlines">[[wikt:那由多|那由多]]</span> (in Japan)
| 億垓
| align="center" | Myllion quadyllion
|-
| 10<sup>80</sup>
| align="center" | Twentieth myriad
| align="center" | Tenth octad
|
|<span class="nounderlines">[[wikt:不可思議|不可思議]]</span> (in Japan)
| 兆垓
| align="center" | Byllion quadyllion
|-
| 10<sup>88</sup>
| align="center" | Twenty-second myriad
| align="center" | Eleventh octad
|
|<span class="nounderlines">[[wikt:無量大数|無量大数]]</span> (in Japan)
| 億兆垓
| align="center" | Myllion byllion quadyllion
|-
| 10<sup>128</sup>
| align="center" | Thirty-second myriad
| align="center" | Sixteenth octad
|
|
|<span class="nounderlines">[[wikt:秭|秭]]</span>
| align="center" | Quinyllion
|-
| 10<sup>256</sup>
| align="center" | Sixty-fourth myriad
| align="center" | Thirty-second octad
|
|
|<span class="nounderlines">[[wikt:穰|穰]]</span>
| align="center" | Sexyllion
|-
| 10<sup>512</sup>
| align="center" | 128th myriad
| align="center" | Sixty-fourth octad
|
|
|<span class="nounderlines">[[wikt:溝|溝]]</span> (<span class="nounderlines">[[wikt:沟|沟]]</span>)
| align="center" | Septyllion
|-
| 10<sup>1,024</sup>
| align="center" | 256th myriad
| align="center" | 128th octad
|
|
|<span class="nounderlines">[[wikt:澗|澗]]</span> (<span class="nounderlines">[[wikt:涧|涧]]</span>)
| align="center" | Octyllion
|-
| 10<sup>2,048</sup>
| align="center" | 512th myriad
| align="center" | 256th octad
|
|
|<span class="nounderlines">[[wikt:正|正]]</span>
| align="center" | Nonyllion
|-
| 10<sup>4,096</sup>
| align="center" | 1024th myriad
| align="center" | 512th octad
|
|
|<span class="nounderlines">[[wikt:載|載]]</span> (<span class="nounderlines">[[wikt:载|载]]</span>)
| align="center" | Decyllion
|-
| 10<sup>8,192</sup>
| align="center" | 2048th myriad
| align="center" | 1024th octad
|
|
|<span class="nounderlines">[[wikt:極|極]]</span> (<span class="nounderlines">[[wikt:极|极]]</span>)
| align="center" | Undecyllion
|-
| 10<sup>16,384</sup>
| align="center" | 4096th myriad
| align="center" | 2048th octad
|
|
|<span class="nounderlines">[[wikt:恆河沙|恆河沙]]</span> (<span class="nounderlines">[[wikt:恒河沙|恒河沙]]</span>)
| align="center" | Duodecyllion
|-
| 10<sup>32,768</sup>
| align="center" | 8192nd myriad
| align="center" | 4096th octad
|
|
|<span class="nounderlines">[[wikt:阿僧祇|阿僧祇]]</span>
| align="center" | Tredecyllion
|-
| 10<sup>65,536</sup>
| align="center" | 16384th myriad
| align="center" | 8192nd octad
|
|
|<span class="nounderlines">[[wikt:那由他|那由他]]</span>, <span class="nounderlines">[[wikt:那由多|那由多]]</span>
| align="center" | Quattuordecyllion
|-
| 10<sup>131,072</sup>
| align="center" | 32768th myriad
| align="center" | 16384th octad
|
|
|<span class="nounderlines">[[wikt:不可思議|不可思議]]</span> (<span class="nounderlines">[[wikt:不可思议|不可思议]]</span>)
| align="center" | Quindecyllion
|-
| 10<sup>262,144</sup>
| align="center" | 65536th myriad
| align="center" | 32768th octad
|
|
|<span class="nounderlines">[[wikt:無量大數|無量大數]]</span> (<span class="nounderlines">[[wikt:无量大数|无量大数]]</span>)
| align="center" | Sexdecyllion
|-
| 10<sup>524,288</sup>
| align="center" | 131072nd myriad
| align="center" | 65536th octad
|
|
|
| align="center" | Septendecyllion
|-
| 10<sup>1,048,576</sup>
| align="center" | 262144th myriad
| align="center" | 131072nd octad
|
|
|
| align="center" | Octodecyllion
|-
| 10<sup>2,097,152</sup>
| align="center" | 524288th myriad
| align="center" | 262144th octad
|
|
|
| align="center" | Novemdecyllion
|-
| 10<sup>4,194,304</sup>
| align="center" | 1048576th myriad
| align="center" | 524288th octad
|
|
|
| align="center" | Vigintyllion
|-
| 10<sup>2<sup>32</sup></sup>
| align="center" | 1073741824th myriad
| align="center" | 536870912nd octad
|
|
|
| align="center" | Trigintyllion
|-
| 10<sup>2<sup>42</sup></sup>
| align="center" | 1099511627776th myriad
| align="center" | 549755813888th octad
|
|
|
| align="center" | Quadragintyllion
|-
| 10<sup>2<sup>52</sup></sup>
|
|
|
|
|
| align="center" | Quinquagintyllion
|-
| 10<sup>2<sup>62</sup></sup>
|
|
|
|
|
| align="center" | Sexagintyllion
|-
| 10<sup>2<sup>72</sup></sup>
|
|
|
|
|
| align="center" | Septuagintyllion
|-
| 10<sup>2<sup>82</sup></sup>
|
|
|
|
|
| align="center" | Octogintyllion
|-
| 10<sup>2<sup>92</sup></sup>
|
|
|
|
|
| align="center" | Nonagintyllion
|-
| 10<sup>2<sup>102</sup></sup>
|
|
|
|
|
| align="center" | Centyllion
|-
| 10<sup>2<sup>1,002</sup></sup>
|
|
|
|
|
| align="center" | Millyllion
|-
| 10<sup>2<sup>10,002</sup></sup>
|
|
|
|
|
| align="center" | Myryllion
|-
|}

=={{anchor|Fractional numbers}} Fractional numerals==
<!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not modify it, even if you modify the section title. -->
{{See also|Fraction (mathematics)#Vocabulary|English numerals#Fractions and decimals}}
This is a table of English names for non-negative [[rational number]]s 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 [[Infinity|infinitely]] many ways, e.g. ''zero-point-one-two'' (0.12), ''twelve [[percent]]'' (12%), ''three twenty-fifths'' ({{sfrac|3|25}}), ''nine seventy-fifths'' ({{sfrac|9|75}}), ''six fiftieths'' ({{sfrac|6|50}}), ''twelve hundredths'' ({{sfrac|12|100}}), ''twenty-four two-hundredths'' ({{sfrac|24|200}}), etc.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Value !! Fraction !! Common names 
|-
| 1
| align="center" | {{sfrac|1|1}}
| One, Unity, Whole
|-
| 0.9
| align="center" | {{sfrac|9|10}}
| Nine tenths, [zero] point nine
|-
| {{gaps|0.833|333...}}
| align="center" | {{sfrac|5|6}}
| Five sixths
|-
| 0.8
| align="center" | {{sfrac|4|5}}
| Four fifths, eight tenths, [zero] point eight
|-
| 0.75
| align="center" | {{sfrac|3|4}}
| three quarters, three fourths, seventy-five hundredths, [zero] point seven five
|-
| 0.7
| align="center" | {{sfrac|7|10}}
| Seven tenths, [zero] point seven
|-
| {{gaps|0.666|666...}}
| align="center" | {{sfrac|2|3}}
| Two thirds
|-
| 0.6
| align="center" | {{sfrac|3|5}}
| Three fifths, six tenths, [zero] point six
|-
| 0.5
| align="center" | {{sfrac|1|2}}
| [[One half]], five tenths, [zero] point five
|-
| 0.4
| align="center" | {{sfrac|2|5}}
| Two fifths, four tenths, [zero] point four
|-
| {{gaps|0.333|333...}}
| align="center" | {{sfrac|1|3}}
| One third
|-
| 0.3
| align="center" | {{sfrac|3|10}}
| Three tenths, [zero] point three
|-
| 0.25
| align="center" | {{sfrac|1|4}}
| One quarter, one fourth, twenty-five hundredths, [zero] point two five
|-
| 0.2
| align="center" | {{sfrac|1|5}}
| One fifth, two tenths, [zero] point two
|-
| {{gaps|0.166|666...}}
| align="center" | {{sfrac|1|6}}
| One sixth
|-
| {{gaps|0.142|857|142|857...}}
| align="center" | {{sfrac|1|7}}
| One seventh
|-
| 0.125
| align="center" | {{sfrac|1|8}}
| One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
|-
| {{gaps|0.111|111...}}
| align="center" | {{sfrac|1|9}}
| One ninth
|-
| 0.1
| align="center" | {{sfrac|1|10}}
| One tenth, [zero] point one, One perdecime, one perdime
|-
| {{gaps|0.090|909...}}
| align="center" | {{sfrac|1|11}}
| One eleventh
|-
| 0.09
| align="center" | {{sfrac|9|100}}
| Nine hundredths, [zero] point zero nine
|-
| {{gaps|0.083|333...}}
| align="center" | {{sfrac|1|12}}
| One twelfth
|-
| 0.08
| align="center" | {{sfrac|2|25}}
| Two twenty-fifths, eight hundredths, [zero] point zero eight
|-
| {{gaps|0.076|923|076|923...}}
| align="center" | {{sfrac|1|13}}
| One thirteenth
|-
| {{gaps|0.071|428|571|428...}}
| align="center" | {{sfrac|1|14}}
| One fourteenth
|-
| {{gaps|0.066|666...}}
| align="center" | {{sfrac|1|15}}
| One fifteenth
|-
| 0.0625
| align="center" | {{sfrac|1|16}}
| One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
|-
| {{gaps|0.055|555...}}
| align="center" | {{sfrac|1|18}}
| One eighteenth
|-
| 0.05
| align="center" | {{sfrac|1|20}}
| One twentieth, five hundredths, [zero] point zero five
|-
| {{gaps|0.047|619|047|619...}}
| align="center" | {{sfrac|1|21}}
| One twenty-first
|-
| {{gaps|0.045|454|545...}}
| align="center" | {{sfrac|1|22}}
| One twenty-second
|-
| {{gaps|0.043|478|260|869|565|217|391|304|347...}}
| align="center" | {{sfrac|1|23}}
| One twenty-third
|-
| {{gaps|0.041|666...}}
| align="center" | {{sfrac|1|24}}
| One twenty-fourth
|-
| 0.04
| align="center" | {{sfrac|1|25}}
| One twenty-fifth, four hundredths, [zero] point zero four
|-
| {{gaps|0.033|333...}}
| align="center" | {{sfrac|1|30}}
| One thirtieth
|-
| 0.03125
| align="center" | {{sfrac|1|32}}
| One thirty-second, thirty one-hundred [and] twenty five hundred-thousandths, [zero] point zero three one two five
|-
| 0.03
| align="center" | {{sfrac|3|100}}
| Three hundredths, [zero] point zero three
|-
| 0.025
| align="center" | {{sfrac|1|40}}
| One fortieth, twenty-five thousandths, [zero] point zero two five
|-
| 0.02
| align="center" | {{sfrac|1|50}}
| One fiftieth, two hundredths, [zero] point zero two
|-
| {{gaps|0.016|666...}}
| align="center" | {{sfrac|1|60}}
| One sixtieth
|-
| 0.015625
| align="center" | {{sfrac|1|64}}
| One sixty-fourth, ten thousand fifty six-hundred [and] twenty-five millionths, [zero] point zero one five six two five 
|-
| {{gaps|0.012|345|679|012|345|679...}}
| align="center" | {{sfrac|1|81}}
| One eighty-first
|-
| {{gaps|0.010|101...}}
| align="center" | {{sfrac|1|99}}
| One ninety-ninth
|-
| 0.01
| align="center" | {{sfrac|1|100}}
| One hundredth, [zero] point zero one, One [[percent]]
|-
| {{gaps|0.009|900|990|099...}}
| align="center" | {{sfrac|1|101}}
| One hundred-first
|-
| {{gaps|0.008|264|462|809|917|355|371|900...}}
| align="center" | {{sfrac|1|121}}
| One over one hundred twenty-one
|-
| 0.001
| align="center" | {{sfrac|1|1000}}
| One thousandth, [zero] point zero zero one, One [[permille]]
|-
| {{gaps|0.000|277|777...}}
| align="center" | {{sfrac|1|3600}}
| One thirty-six hundredth
|-
| 0.0001
| align="center" | {{sfrac|1|{{gaps|10|000}}}}
| One ten-thousandth, [zero] point zero zero zero one, One myriadth, one permyria, one permyriad, one [[basis point]]
|-
| {{gaps|0.000|01}}
| align="center" | {{sfrac|1|{{gaps|100|000}}}}
| One hundred-thousandth, [zero] point zero zero zero zero one, One lakhth, one perlakh
|-
| {{gaps|0.000|001}}
| align="center" | {{sfrac|1|{{gaps|1|000|000}}}}
| One millionth, [zero] point zero zero zero zero zero one, One [[Parts per million|ppm]]
|-
| {{gaps|0.000|000|1}}
| align="center" | {{sfrac|1|{{gaps|10|000|000}}}}
| One ten-millionth, One crorth, one percrore
|-
| {{gaps|0.000|000|01}}
| align="center" | {{sfrac|1|{{gaps|100|000|000}}}}
| One hundred-millionth
|-
| {{gaps|0.000|000|001}}
| align="center" | {{sfrac|1|{{gaps|1|000|000|000}}}}
| One billionth (in some dialects), One [[Parts per billion|ppb]]
|-
| {{gaps|0.000|000|000|001}}
| align="center" | {{sfrac|1|{{gaps|1|000|000|000|000}}}}
| One trillionth, One ppt
|-
| 0
| align="center" | {{sfrac|0|1}}
| [[Names for the number 0 in English|Zero]], Nil
|-
|}

== Other specific quantity terms ==
{{See also|Non-numerical words for quantities}}
Various terms have arisen to describe commonly used measured quantities.
* [[1 (number)|Unit]]: 1 (based on a single entity of counting or measurement of an object or item)
* [[2 (number)|Pair]]: 2 (the base of the [[binary numeral system]])
* [[3 (number)|Leash]]: 3 (the base of the [[trinary numeral system]])
* [[Dozen]]: 12 (the base of the [[duodecimal]] numeral system)
* [[Baker's dozen]]: 13 (based on a group of thirteen objects or items)
* [[20 (number)|Score]]: 20 (the base of the [[vigesimal]] numeral system)
* [[60 (number)|Shock]]: 60 (the base of the [[sexagesimal]] numeral system)<ref>{{cite book |last1=Cardarelli |first1=François |title=Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins |date=2012 |publisher=Springer |isbn=978-1447100034 |page=585 |edition=Second}}</ref>
* [[Gross (unit)|Gross]]: (based on a group of 144 objects or items)
* [[Great gross]]: (based on a group of 1,728 objects or items)

==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 [[Nadahup languages|Nadëb]], pre-contact [[Mocoví language|Mocoví]] and [[Pilagá language|Pilagá]], [[Culina-Madijá language|Culina]] and pre-contact [[Jarawara language|Jarawara]], [[Jabutí language|Jabutí]], [[Canela language|Canela-Krahô]], [[Botocudo|Botocudo (Krenák)]], [[Chiquitano language|Chiquitano]], the [[Campa languages]], [[Arabela language|Arabela]], and [[Jivaroan languages|Achuar]].<ref>{{Cite web |url=http://www2.gslt.hum.gu.se/dissertations/hammarstrom.pdf |title=Hammarström (2009, page 197) "Rarities in numeral systems" |access-date=2010-06-16 |archive-url=https://web.archive.org/web/20120308070048/http://www2.gslt.hum.gu.se/dissertations/hammarstrom.pdf |archive-date=2012-03-08 |url-status=dead }}</ref> Some languages of Australia, such as [[Warlpiri language|Warlpiri]], do not have words for quantities above two,<ref>UCL Media Relations, [http://www.ucl.ac.uk/media/library/aboriginal "Aboriginal kids can count without numbers"] {{Webarchive|url=https://web.archive.org/web/20180620234700/http://www.ucl.ac.uk/media/library/aboriginal |date=2018-06-20 }}</ref><ref>{{cite journal |last1=Butterworth |first1=Brian |last2=Reeve |first2=Robert |last3=Reynolds |first3=Fiona |last4=Lloyd |first4=Delyth |title=Numerical thought with and without words: Evidence from indigenous Australian children |journal=PNAS |date=2 September 2008 |volume=105 |issue=35 |pages=13179–13184 |doi=10.1073/pnas.0806045105 |pmid=18757729 |pmc=2527348 |bibcode=2008PNAS..10513179B |quote=[Warlpiri] has three generic types of number words: singular, dual plural, and greater than dual plural. |doi-access=free }}</ref><ref>The Science Show, [http://www.abc.net.au/rn/scienceshow/stories/2008/2375526.htm Genetic anomaly could explain severe difficulty with arithmetic] {{Webarchive|url=https://web.archive.org/web/20100301113203/http://www.abc.net.au/rn/scienceshow/stories/2008/2375526.htm |date=2010-03-01 }}, Australian Broadcasting Corporation</ref> 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).<ref>Bernard Comrie, "[http://ling.cass.cn/pdf/TypNum_China_10ho.pdf The Typology of Numeral Systems] {{Webarchive|url=https://web.archive.org/web/20110514035109/http://ling.cass.cn//pdf/TypNum_China_10ho.pdf |date=2011-05-14 }}", p.&nbsp;3</ref>

===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 Strait Island languages|Torres Islands]]) to 23 ([[Eleman language|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.{{cn|date=August 2024}}

===2: binary===
{{main article|Binary numeral system}}
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===
{{main|Ternary numeral system#Practical usage}}
Ternary systems are based on the number 3, having practical usage in some analog logic, in baseball scoring and in [[Self-similarity|self–similar]] mathematical structures.

===4: quaternary===
{{Main|Quaternary numeral system}}
Quaternary systems are based on the number 4. Some [[Austronesian peoples|Austronesian]], [[Melanesians|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.<ref name="Ryan, Peter p&nbsp;219">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&nbsp;219.</ref> 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.<ref name="Ryan, Peter p&nbsp;219"/><ref>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}}.</ref>

===5: quinary===
{{Main|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).<ref name="Heath, Thomas 2003">Heath, Thomas, ''A Manual of Greek Mathematics'', Courier Dover: 2003. {{isbn|978-0-486-43231-1}} page, p:11</ref> 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. [[Nahua language|Aztec]] was a vigesimal (base-20) system with sub-base 5.

===6: senary===
{{Main|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 6<sup>6</sup>. Examples are [[Kanum language|Kanum]] and [[Kómnzo language|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.<ref name="Parkvall, M. 2008. p.291">Parkvall, M. ''Limits of Language'', 1st edn. 2008. p.291. {{ISBN|978-1-59028-210-6}}</ref>

===8: octal===
{{Main|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 tribe|Yuki]] and [[Pame people|Pame]] keep count by using the four spaces between their fingers rather than the fingers themselves.<ref>{{citation
| title=Ethnomathematics: A Multicultural View of Mathematical Ideas
| first=Marcia
| last=Ascher|author-link= Marcia Ascher
| year=1994
| publisher=Chapman & Hall
| isbn=0-412-98941-7
}}</ref>

===9: nonary===
Nonary systems are based on the number 9. It has been suggested that [[Nenets languages|Nenets]] has a base-nine system.<ref name="Parkvall, M. 2008. p.291"/>

===10: decimal===
{{Main|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 [[digit (anatomy)|digits]] per hand, ten in total.<ref name="Heath, Thomas 2003"/><ref>''Scientific American'' Munn& Co: 1968, vol 219: 219</ref> There are many regional variations including:

* Western system: based on [[one thousand|thousand]]s, with variants (see [[English numerals]])
* Indian system: [[crore]], [[lakh]] (see [[Indian numbering system]]. [[Indian numerals]])
* East Asian system: based on [[10000 (number)|ten-thousands]] (see below)

===12: duodecimal===
{{Main|Duodecimal}}
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 language|Janji]], [[Kahugu language|Kahugu]] and the Nimbia dialect of [[Gwandara language|Gwandara]].
* [[Melanesia]]{{Citation needed|date=June 2011}}
* reconstructed proto-[[Benue–Congo]]

Duodecimal numeric systems have some practical advantages over decimal. It is much easier to divide the base digit [[12 (number)|twelve]] (which is a [[highly composite number]]) by many important [[divisors]] in [[Market (economics)|market]] and trade settings, such as the numbers [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]] and [[6 (number)|6]].

Because of several measurements based on twelve,<ref>such as twelve months in a year, the twelve-hour clock, twelve inches to the foot, twelve [[Penny|pence]] to the [[shilling]]</ref> many Western languages have words for base-twelve units such as ''[[dozen]]'', ''[[Gross (unit)|gross]]'' and ''[[great gross]]'', which allow for rudimentary duodecimal [[nomenclature]], such as "two gross six dozen" for 360. [[Ancient Rome|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#Fractions|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===
{{Main|hexadecimal}}
Hexadecimal systems are based on the number 16.

The traditional [[Chinese units of measurement]] were base-16. For example, one jīn&nbsp;(斤) in the old system equals sixteen [[tael]]s. The [[suanpan]] (Chinese [[abacus]]) can be used to perform hexadecimal calculations such as additions and subtractions.<ref>{{Cite web|url=http://totton.idirect.com/soroban/Hex_as/|title=算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus|website=totton.idirect.com|access-date=2019-06-26|archive-date=2019-07-06|archive-url=https://web.archive.org/web/20190706221609/http://totton.idirect.com/soroban/Hex_as/|url-status=live}}</ref>

South Asian monetary systems were base-16. One rupee in Pakistan and India was divided into 16 annay. A single [[Indian anna|anna]] was subdivided into four [[paisa]] or twelve [[Pie (Indian coin)|pies]] (thus there were 64 paise or 192 pies in a rupee). The anna was [[Legal tender#Demonetization|demonetised]] as a currency unit when India [[Decimalisation|decimalised]] its currency in 1957, followed by Pakistan in 1961.

===20: vigesimal===
{{Main|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.<ref name="Heath, Thomas 2003"/><ref name="ReferenceA">Georges Ifrah, ''The Universal History of Numbers: The Modern Number System'', Random House, 2000: {{isbn|1-86046-791-1}}. 1262 pages</ref>
The system is in widespread use across the world. Some include the classical [[Mesoamerica]]n 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 language|Dzongkha]] in Bhutan.

Partial vigesimal systems are found in some European languages: [[Basque language|Basque]], [[Celtic languages]], [[French language|French]] (from Celtic), [[Danish language|Danish]], and [[Georgian language|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 ''[[20 (number)|score]]'' originates from [[tally stick]]s, 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 [[20 (number)|score]] of [[shilling|bob]]", referring to the 20 [[British shilling coin|shilling]]s in a [[pound sterling#Pre-decimal|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===
{{Main|Duotrigesimal}}
Duotrigesimal systems are based on the number 32. The [[Ngiti language|Ngiti]] ethnolinguistic group uses a base 32 numeral system.

===60: sexagesimal===
{{Main|Sexagesimal}}
Sexagesimal systems are based on the number 60. [[Ekari language|Ekari]] has a base-60 system. [[Sumer]]ia 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 [[degree (angle)|degrees, minutes, and seconds]].

===80: octogesimal===
Octogesimal systems are based on the number 80. [[Supyire language|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).

{{interlinear|indent=2
|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 [https://mpi-lingweb.shh.mpg.de/numeral Numeral Systems of the World's Languages] {{Webarchive|url=https://web.archive.org/web/20161221000001/https://mpi-lingweb.shh.mpg.de/numeral/ |date=2016-12-21 }} 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]]

===Related topics===
* [[Long and short scales]]
* [[Names of large numbers]]
* [[Numeral system]]
* [[Numeral prefix]]
* [[Names of small numbers]]

==Notes==
{{Reflist}}

==Further reading==
* Crespo Cantalapiedra, I. (2023). ''[https://zenodo.org/records/10225997 La diversidad en las lenguas: los numerales] {{Webarchive|url=https://web.archive.org/web/20240224123657/https://zenodo.org/records/10225997 |date=2024-02-24 }}''. Online book (in Spanish).
* {{cite book|author=James R. Hurford|title=The Linguistic Theory of Numerals|year=2010|orig-year=1975|publisher=Cambridge University Press|isbn=978-0-521-13368-5|author-link=James R. Hurford}}

{{Authority control}}

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[[Category:Numerals|*]]
[[Category:Names]]