Editing Numeral (linguistics) (section)

==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)}]’