Editing Decimal (section)

== History ==
[[File:Qinghuajian, Suan Biao.jpg|thumb|upright|The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during the [[Warring States]] period in China.]]
Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.<ref>{{citation|first=Tobias|last=Dantzig|title=Number / The Language of Science |edition=4th |year=1954|publisher=The Free Press (Macmillan Publishing Co.) |isbn=0-02-906990-4|page=12}}</ref> Standardized weights used in the [[Indus Valley Civilisation]] ({{circa|3300–1300 BCE}}) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the ''Mohenjo-daro ruler'' – was divided into ten equal parts.<ref>Sergent, Bernard (1997), ''Genèse de l'Inde'' (in French), Paris: Payot, p. 113, {{ISBN|2-228-89116-9}}</ref><ref>{{cite journal | last1 = Coppa | first1 = A. | display-authors = etal | year = 2006 | title = Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population | bibcode = 2006Natur.440..755C | journal = Nature | volume = 440 | issue = 7085| pages = 755–56 | doi = 10.1038/440755a | pmid = 16598247 | s2cid = 6787162 }}</ref><ref>Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan ''Civilisation: A Contemporary Perspective'', New Delhi: Oxford and IBH Publishing Co., pp. 113–24</ref> [[Egyptian hieroglyphs]], in evidence since around 3000 BCE, used a purely decimal system,<ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp.&nbsp;200–13 (Egyptian Numerals)</ref> as did the [[Linear A]] script ({{circa|1800–1450 BCE}}) of the [[Minoan civilization|Minoans]]<ref>Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, {{isbn|978-0-486-42165-0}}, p.&nbsp;50</ref><ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 213–18 (Cretan numerals)</ref> and the [[Linear B]] script (c. 1400–1200 BCE) of the [[Mycenaean Greece|Mycenaeans]]. The [[Únětice culture]] in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.<ref>{{cite book |last1=Krause |first1=Harald |url=https://www.academia.edu/34550316 |title=Spangenbarrenhort Oberding |last2=Kutscher |first2=Sabrina |date=2017 |publisher=Museum Erding |isbn=978-3-9817606-5-1 |pages=238–243 |chapter=Spangenbarrenhort Oberding: Zusammenfassung und Ausblick}}</ref> The number system of [[classical Greece]] also used powers of ten, including an intermediate base of 5, as did [[Roman numerals]].<ref name="Greek numerals">{{Cite web |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |title=Greek numbers |access-date=2019-07-21 |archive-date=2019-07-21 |archive-url=https://web.archive.org/web/20190721085640/http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |url-status=live }}</ref> Notably, the polymath [[Archimedes]] (c. 287–212 BCE) invented a decimal positional system in his [[The Sand Reckoner|Sand Reckoner]] which was based on 10<sup>8</sup>.<ref name="Greek numerals"/><ref>[[Karl Menninger (mathematics)|Menninger, Karl]]: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp.&nbsp;150–53</ref> [[Hittites|Hittite]] hieroglyphs (since 15th century BCE) were also strictly decimal.<ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 218f. (The Hittite hieroglyphic system)</ref>

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1,000, 2,000, 3,000, 4,000, to 10,000.<ref>[[Lam Lay Yong]] et al. The Fleeting Footsteps pp. 137–39</ref>
The world's earliest positional decimal system was the Chinese [[rod calculus]].<ref name=Lam/>
[[File:Chounumerals.svg|thumb|right|280px|The world's earliest positional decimal system<br /> Upper row vertical form<br /> Lower row horizontal form]]

=== History of decimal fractions ===
[[File:Rod fraction.jpg|thumb|right|150px|counting rod decimal fraction 1/7]]
Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.<ref name=jnfractn1>{{Cite book | author=Joseph Needham | author-link=Joseph Needham | chapter = 19.2 Decimals, Metrology, and the Handling of Large Numbers |pages=82–90 | title = Science and Civilisation in China |volume=III, "Mathematics and the Sciences of the Heavens and the Earth" | title-link=Science and Civilisation in China | year = 1959 | publisher = Cambridge University Press}}</ref> Calculations with decimal fractions of lengths were [[Rod calculus#Decimal fraction|performed using positional counting rods]], as described in the 3rd–5th century CE ''[[Sunzi Suanjing]]''. The 5th century CE mathematician [[Zu Chongzhi]] calculated a 7-digit [[approximations of π|approximation of {{mvar|π}}]]. [[Qin Jiushao]]'s book ''[[Mathematical Treatise in Nine Sections]]'' (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.<ref>Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997  {{isbn|3-540-33782-2}}</ref> The number 0.96644 is denoted

:{{lang|zh|寸}}
:[[File:Counting rod 0.png|frameless|18px]] [[File:Counting rod h9 num.png|frameless|18px]] [[File:Counting rod v6.png|frameless|18px]] [[File:Counting rod h6.png|frameless|18px]] [[File:Counting rod v4.png|frameless|18px]] [[File:Counting rod h4.png|frameless|18px]].

Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.<ref name=Lam>[[Lam Lay Yong]], "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p. 38, Kurt Vogel notation</ref>

[[Al-Khwarizmi]] introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.<ref name=Lam/><ref>{{cite journal | last1 = Lay Yong | first1 = Lam | author-link = Lam Lay Yong | title = A Chinese Genesis, Rewriting the history of our numeral system | journal = Archive for History of Exact Sciences | volume = 38 | pages = 101–08 }}</ref>

Positional decimal fractions appear for the first time in a book by the Arab mathematician [[Abu'l-Hasan al-Uqlidisi]] written in the 10th century.<ref name=Berggren>{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam |editor-first=Victor J.|editor-last=Katz|publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=530 }}</ref> The Jewish mathematician [[Immanuel Bonfils]] used decimal fractions around 1350 but did not develop any notation to represent them.<ref>[[Solomon Gandz|Gandz, S.]]: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.</ref> The Persian mathematician [[Jamshid al-Kashi]] used, and claimed to have discovered, decimal fractions in the 15th century.<ref name=Berggren />

<div style="float: right;">[[File:Stevin-decimal notation.svg]]</div>
A forerunner of modern European decimal notation was introduced by [[Simon Stevin]] in the 16th century. Stevin's influential booklet ''[[De Thiende]]'' ("the art of tenths") was first published in Dutch in 1585 and translated into French as ''La Disme''.<ref name=van>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | publisher = Springer-Verlag | place = Berlin}}</ref>

[[John Napier]] introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.<ref name=constructionIA>{{cite book|title=[[Commons:File:The Construction of the Wonderful Canon of Logarithms.djvu|The Construction of the Wonderful Canon of Logarithms]]|first=John|last=Napier|translator-last1=Macdonald|translator-first1= William Rae|date=1889|orig-date=1620|publisher=Blackwood & Sons|publication-place=Edinburgh|via=Internet Archive|quote=In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.}}</ref>{{rp|p. 8, archive p. 32)}}

=== Natural languages ===
A method of expressing every possible [[natural number]] using a set of ten symbols emerged in India.<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/ |title=Indian numerals|work=Ancient Indian mathematics }}</ref> Several Indian languages show a straightforward decimal system. [[Dravidian languages]] have numbers between 10 and 20 expressed in a regular pattern of addition to 10.<ref>{{Citation |title=Appendix:Cognate sets for Dravidian languages |date=2024-09-25 |work=Wiktionary, the free dictionary |url=https://en.wiktionary.org/wiki/Appendix:Cognate_sets_for_Dravidian_languages |access-date=2024-11-09 |language=en}}</ref>

The [[Hungarian language]] also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").

A straightforward decimal rank system with a word for each order (10 {{lang|zh|十}}, 100 {{lang|zh|百}}, 1000 {{lang|zh|千}}, 10,000 {{lang|zh|万}}), and in which 11 is expressed as ''ten-one'' and 23 as ''two-ten-three'', and 89,345 is expressed as 8 (ten thousands) {{lang|zh|万}} 9 (thousand) {{lang|zh|千}} 3 (hundred) {{lang|zh|百}} 4 (tens) {{lang|zh|十}} 5 is found in [[Chinese language|Chinese]], and in [[Vietnamese language|Vietnamese]] with a few irregularities.  [[Japanese language|Japanese]], [[Korean language|Korean]], and [[Thai language|Thai]] have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is  "eleven" not "ten-one" or "one-teen".

Incan languages such as [[Quechuan languages|Quechua]] and [[Aymara language|Aymara]] have an almost straightforward decimal system, in which 11 is expressed as ''ten with one'' and 23 as ''two-ten with three''.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.<ref>{{Cite journal| last=Azar| first=Beth| year=1999| title=English words may hinder math skills development| url=http://www.apa.org/monitor/apr99/english.html  |journal=APA Monitor| volume=30| issue=4 |archive-url = https://web.archive.org/web/20071021015527/http://www.apa.org/monitor/apr99/english.html |archive-date = 2007-10-21}}</ref>

=== 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.