Decimal: Difference between revisions

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|Sindhu Valley civilization]] ({{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 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 10⁸.<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>

Starting from the 2nd century BCBCE, some Chinese units for length were based on divisions into ten; by the 3rd century AD,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

[[Al-Khwarizmi|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|Jamshīd al-Kāshī]] used, and claimed to have discovered, decimal fractions in the 15th century.<ref name=Berggren />

Incan languages such as [[QuechuaQuechuan 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''.