Editing Decimal (section)

=== Other bases ===
{{main | Positional notation}}
{{Fundamental info units}}
Some cultures do, or did, use other bases of numbers.
* [[Pre-Columbian]] [[Mesoamerica]]n cultures such as the [[Maya numerals|Maya]] used a [[vigesimal|base-20]] system (perhaps based on using all twenty fingers and [[toe]]s).
* The [[Yuki tribe|Yuki]] language in [[California]] and the Pamean languages<ref>{{Cite journal
 | last=Avelino
 | first=Heriberto
 | title=The typology of Pame number systems and the limits of Mesoamerica as a linguistic area
 | journal=Linguistic Typology
 | year=2006
 | volume=10
 | issue=1
 | pages=41–60
 | url=http://linguistics.berkeley.edu/~avelino/Avelino_2006.pdf |archive-url=https://web.archive.org/web/20060712201924/http://www.linguistics.berkeley.edu/~avelino/Avelino_2006.pdf |archive-date=2006-07-12 |url-status=live
 | doi=10.1515/LINGTY.2006.002
 | s2cid=20412558
 }}</ref> in [[Mexico]] have [[octal]] ([[radix|base]]-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.<ref>{{cite news|jstor=2686959|title=Ethnomathematics: A Multicultural View of Mathematical Ideas|author=Marcia Ascher|author-link= Marcia Ascher |publisher=The College Mathematics Journal}}</ref>
* The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were.<ref>{{citation
 | last = McClean | first = R. J.
 | date = July 1958
 | doi = 10.1111/j.1468-0483.1958.tb00018.x
 | issue = 4
 | journal = German Life and Letters
 | quote = Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system
 | pages = 293–99
 | title = Observations on the Germanic numerals
 | volume = 11}}.</ref><ref>{{citation
 | last = Voyles | first = Joseph
 | date = October 1987
 | issue = 4
 | journal = The Journal of English and Germanic Philology
 | jstor = 27709904
 | pages = 487–95
 | title = The cardinal numerals in pre-and proto-Germanic
 | volume = 86}}.</ref>  Where this counting system is known, it is based on the "[[long hundred]]" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's ''Introduction to Old Norse''<ref>Gordon's [https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon Introduction to Old Norse] {{Webarchive|url=https://web.archive.org/web/20160415205641/https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon |date=2016-04-15 }} p.&nbsp;293</ref> gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. Goodare<ref>{{cite journal
 | last = Goodare | first = Julian
 | date = November 1994
 | doi = 10.9750/psas.123.395.418
 | journal = Proceedings of the Society of Antiquaries of Scotland
 | pages = 395–418
 | title = The long hundred in medieval and early modern Scotland
 | volume = 123}}</ref> details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.<ref>{{Cite journal|last=Stevenson|first=W.H.|date=1890|title=The Long Hundred and its uses in England|journal=Archaeological Review|volume=December 1889|pages=313–22}}</ref><ref>{{Cite book|last=Poole, Reginald Lane|title=The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911|date=2006|publisher=Lawbook Exchange|isbn=1-58477-658-7|location=Clark, NJ|oclc=76960942}}</ref>
* Many or all of the [[Chumashan languages]] originally used a [[quaternary numeral system|base-4]] counting system, in which the names for numbers were structured according to multiples of 4 and [[hexadecimal|16]].<ref>There is a surviving list of [[Ventureño language]] number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in ''Native American Mathematics'', edited by Michael P. Closs (1986), {{isbn|0-292-75531-7}}.</ref>
* Many languages<ref name="Hammarstrom 2010">{{Cite book
| contribution=Rarities in Numeral Systems
| first=Harald
| last=Hammarström
| editor1-first=Jan
| editor1-last=Wohlgemuth
| editor2-first=Michael
| editor2-last=Cysouw
| title=Rethinking Universals: How rarities affect linguistic theory
| date=17 May 2007
| location=Berlin
| publisher=Mouton de Gruyter
| series=Empirical Approaches to Language Typology
| volume=45
| publication-date=2010
| url=http://www.cs.chalmers.se/~harald2/rarapaper.pdf
| url-status=dead
| archive-url=https://web.archive.org/web/20070819214057/http://www.cs.chalmers.se/~harald2/rarapaper.pdf
| archive-date=19 August 2007
}}</ref> use [[quinary|quinary (base-5)]] number systems, including [[Gumatj language|Gumatj]], [[Nunggubuyu language|Nunggubuyu]],<ref>{{Cite journal
 |title        = Facts and fallacies of aboriginal number systems
 |last         = Harris
 |first        = John
 |editor-last  = Hargrave
 |editor-first = Susanne
 |pages        = 153–81
 |year         = 1982
 |journal      = Work Papers of SIL-AAB Series B
 |volume       = 8
 |url          = http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf
 |url-status      = dead
 |archive-url   = https://web.archive.org/web/20070831202737/http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf
 |archive-date  = 2007-08-31
}}</ref> [[Kuurn Kopan Noot language|Kuurn Kopan Noot]]<ref>Dawson, J. "[https://archive.org/details/australianabori00dawsgoog ''Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria''] (1881), p.&nbsp;xcviii.</ref> and [[Saraveca]]. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
* Some [[Nigeria]]ns use [[duodecimal]] systems.<ref>{{Cite conference
| title=Decimal vs. Duodecimal: An interaction between two systems of numeration
| last=Matsushita
| first=Shuji
| conference=2nd Meeting of the AFLANG, October 1998, Tokyo
| year=1998
| url=http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| archive-url=https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| archive-date=2008-10-05
| access-date=2011-05-29
}}</ref>  So did some small communities in India and Nepal, as indicated by their languages.<ref>{{Cite book
| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
| first=Martine
| last=Mazaudon
| title=La Pluralité
| editor-first=Jacques
| editor-last=François
| year=2002
| pages=91–119
| publisher=Peeters
| place=Leuven
| isbn=90-429-1295-2
| url=http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
| access-date=2014-09-12
| archive-date=2016-03-28
| archive-url=https://web.archive.org/web/20160328145817/http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
| url-status=dead
}}</ref>
* The [[Huli language]] of [[Papua New Guinea]] is reported to have [[pentadecimal|base-15]] numbers.<ref>{{Cite journal
 | last=Cheetham
 | first=Brian
 | title=Counting and Number in Huli
 | journal=Papua New Guinea Journal of Education
 | year=1978
 | volume=14
 | pages=16–35
 | url=http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm
 | archive-url=https://web.archive.org/web/20070928061238/http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm
 | archive-date=2007-09-28
}}</ref>  ''Ngui'' means 15, ''ngui ki'' means 15 × 2 = 30, and ''ngui ngui'' means 15 × 15 = 225.
* [[Umbu-Ungu language|Umbu-Ungu]], also known as Kakoli, is reported to have [[base 24|base-24]] numbers.<ref>{{Cite journal
 |last1        = Bowers
 |first1       = Nancy
 |last2       = Lepi
 |first2      = Pundia
 |title       = Kaugel Valley systems of reckoning
 |year        = 1975
 |journal     = Journal of the Polynesian Society
 |volume      = 84
 |issue       = 3
 |pages       = 309–24
 |url         = http://www.ethnomath.org/resources/bowers-lepi1975.pdf
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20110604091351/http://www.ethnomath.org/resources/bowers-lepi1975.pdf
 |archive-date = 2011-06-04
}}</ref> ''Tokapu'' means 24, ''tokapu talu'' means 24 × 2 = 48, and ''tokapu tokapu'' means 24 × 24 = 576.
* [[Ngiti language|Ngiti]] is reported to have a [[base 32|base-32]] number system with base-4 cycles.<ref name="Hammarstrom 2010"/>
* The [[Ndom language]] of [[Papua New Guinea]] is reported to have [[base-6]] numerals.<ref>{{ Citation | last=Owens | first=Kay | title=The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania | journal=Mathematics Education Research Journal | year=2001 | volume=13 | issue=1 | pages=47–71 | url=http://www.uog.ac.pg/glec/Key/Kay/owens131.htm | doi=10.1007/BF03217098 | bibcode=2001MEdRJ..13...47O | s2cid=161535519 | url-status=dead | archive-url=https://web.archive.org/web/20150926003303/http://www.uog.ac.pg/glec/Key/Kay/owens131.htm | archive-date=2015-09-26 }}</ref>  ''Mer'' means 6, ''mer an thef'' means 6 × 2 = 12, ''nif'' means 36, and ''nif thef'' means 36×2 = 72.