Editing 0 (section)

==History==<!--Linked from [[History of zero]] (R to section)-->

===Ancient Near East===
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Ancient [[Egyptian numerals]] were of [[decimal|base 10]].<ref>{{Cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=2000 |title=Egyptian numerals |url=http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html |url-status=live |archive-url=https://web.archive.org/web/20191115221313/http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html |archive-date=15 November 2019 |access-date=21 December 2019 |website=[[mathshistory.st-andrews.ac.uk]] |publisher=University of St Andrews}}</ref> They used [[hieroglyphs]] for the digits and were not [[positional notation|positional]]. In [[Papyrus Boulaq 18|one papyrus]] written around {{nowrap|1770 BC}}, a scribe recorded daily incomes and expenditures for the [[pharaoh]]'s court, using the ''[[Nefer|nfr]]'' hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. Egyptologist [[Alan Gardiner]] suggested that the ''nfr'' hieroglyph was being used as a symbol for zero. The same symbol was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line.<ref>{{cite journal|first=Beatrice |last=Lumpkin |doi=10.1007/BF03024613 |title=Mathematics Used in Egyptian Construction and Bookkeeping |journal=The Mathematical Intelligencer |year=2002 |volume=24 |number=2 |pages=20–25|s2cid=120648746 }}</ref>

By the middle of the 2nd millennium BC, [[Babylonian mathematics]] had a sophisticated [[sexagesimal|base 60]] positional numeral system. The lack of a positional value (or zero) was indicated by a ''space'' between [[sexagesimal]] numerals. In a tablet unearthed at [[Kish (Sumer)|Kish]] (dating to as early as {{nowrap|700 BC}}), the scribe Bêl-bân-aplu used three hooks as a [[Free variables and bound variables|placeholder]] in the same [[Babylonian numerals|Babylonian system]].{{sfn|Kaplan|2000}} By {{nowrap|300 BC}}, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.<ref>{{Cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=2000 |title=Zero |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/ |url-status=live |archive-url=https://web.archive.org/web/20210921191118/https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/ |archive-date=21 September 2021 |access-date=2021-09-07 |website=Maths History |publisher=University of St Andrews |language=en}}</ref><ref>{{Cite web |title=Babylonian mathematics | date= 2016 |url=https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?printable=1&id=1976 |access-date=2021-09-07 |website= The Open University |archive-date=7 September 2021 |archive-url=https://web.archive.org/web/20210907135356/https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?printable=1&id=1976 |url-status=live }}</ref>

The Babylonian positional numeral system differed from the later [[Hindu–Arabic numeral system|Hindu–Arabic system]] in that it did not explicitly specify the magnitude of the leading sexagesimal digit, so that for example the lone digit 1 ([[File:Babylonian 1.svg|20px]]) might represent any of 1, 60, 3600 = 60<sup>2</sup>, etc., similar to the significand of a [[floating-point number]] but without an explicit exponent, and so only distinguished implicitly from context. The zero-like placeholder mark was only ever used in between digits, but never alone or at the end of a number.{{sfn|Reimer|2014|p=172}}

===Pre-Columbian Americas===
[[File:Cero maya.svg|thumb|Maya numeral zero]]
The [[Mesoamerican Long Count calendar]] developed in south-central Mexico and Central America required the use of zero as a placeholder within its [[vigesimal]] (base-20) positional numeral system. Many different glyphs, including the partial [[quatrefoil]] were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, [[Chiapas]]) has a date of 36&nbsp;BC.{{efn|No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.}}<ref>{{Cite web |title=Cyclical views of time |url=https://www.mexicolore.co.uk/aztecs/calendar/cyclical-views-of-time |access-date=2024-01-20 |website=www.mexicolore.co.uk}}</ref>

Since the eight earliest Long Count dates appear outside the Maya homeland,{{sfnp|Diehl|2004| p= 186}} it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the [[Olmec]]s.<ref>{{Cite news |last=Mortaigne |first=Véronique |date=28 November 2014 |title=The golden age of Mayan civilisation – exhibition review |work=[[The Guardian]] |url=https://www.theguardian.com/culture/2014/nov/28/mayan-civilisation-paris-exhibition |url-status=live |access-date=10 October 2015 |archive-url=https://web.archive.org/web/20141128222215/http://www.theguardian.com/culture/2014/nov/28/mayan-civilisation-paris-exhibition |archive-date=28 November 2014}}</ref> Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the {{nowrap|4th century BC}},<ref>{{Citation |last=Cyphers |first=Ann |title=The Olmec, 1800–400 BCE |date=2014 |work=The Cambridge World Prehistory |pages=1005–1025 |editor-last=Renfrew |editor-first=Colin |url=https://www.cambridge.org/core/books/cambridge-world-prehistory/olmec-1800400-bce/2C66AF7B3D041260EE2BFC94DF085029 |access-date=2024-08-13 |place=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-11993-1 |editor2-last=Bahn |editor2-first=Paul}}.</ref> several centuries before the earliest known Long Count dates.<ref>{{Cite magazine |title=Expedition Magazine {{!}} Time, Kingship, and the Maya Universe Maya Calendars |url=https://www.penn.museum/sites/expedition/time-kingship-and-the-maya-universe-maya-calendars/ |access-date=2024-08-13 |magazine=Expedition Magazine |language=en}}</ref>

Although zero became an integral part of [[Maya numerals]], with a different, empty [[tortoise]]-like "[[Plastron|shell shape]]" used for many depictions of the "zero" numeral, it is assumed not to have influenced [[Old World]] numeral systems.{{citation needed|date=December 2023}}

[[Quipu]], a knotted cord device, used in the [[Inca Empire]] and its predecessor societies in the [[Andes|Andean]] region to record accounting and other digital data, is encoded in a [[decimal|base ten]] positional system. Zero is represented by the absence of a knot in the appropriate position.<ref>{{Cite web |last=Leon |first=Manuel de |date=2022-12-20 |title=Knots representing numbers: The mathematics of the Incas |url=https://english.elpais.com/science-tech/2022-12-20/knots-representing-numbers-the-mathematics-of-the-incas.html |access-date=2024-06-05 |website=EL PAÍS English |language=en-us}}</ref>

===Classical antiquity===
The [[Ancient Greece|ancient Greeks]] had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it.<ref>{{Cite web |last=Wallin |first=Nils-Bertil |date=19 November 2002 |title=The History of Zero |url=http://yaleglobal.yale.edu/about/zero.jsp |archive-url=https://web.archive.org/web/20160825124525/http://yaleglobal.yale.edu/about/zero.jsp |archive-date=25 August 2016 |access-date=1 September 2016 |website=YaleGlobal online |publisher=The Whitney and Betty Macmillan Center for International and Area Studies at Yale.}}</ref> According to mathematician [[Charles Seife]], the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in [[Ancient Greek astronomy|astronomy]] after 500 BC, representing it with the lowercase Greek letter ''ό'' (''όμικρον'': [[omicron]]). However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into [[Greek numerals]]. Greeks seemed to have a philosophical opposition to using zero as a number.<ref name="Seife2000">{{cite book | first = Charles | last = Seife | date = 1 September 2000 | title = Zero: The Biography of a Dangerous Idea | publisher = Penguin | page = 39 | isbn = 978-0-14-029647-1 | oclc = 1005913932 | url = https://books.google.com/books?id=obJ70nxVYFUC | access-date = 30 April 2022 | author-link= Charles Seife }}</ref> Other scholars give the Greek partial adoption of the Babylonian zero a later date, with neuroscientist Andreas Nieder giving a date of after 400 BC and mathematician Robert Kaplan dating it after the [[Wars of Alexander the Great|conquests of Alexander]].<ref name="Nieder2019">{{cite book | first = Andreas | last = Nieder | date = 19 November 2019 | title = A Brain for Numbers: The Biology of the Number Instinct | publisher = MIT Press | page = 286 | isbn = 978-0-262-35432-5 | url = https://books.google.com/books?id=x4y5DwAAQBAJ&pg=PA286 | access-date = 30 April 2022   }}</ref>{{sfn|Kaplan|2000|p=17}}

Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the [[medieval]] period, religious arguments about the nature and existence of zero and the [[vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in large part on the uncertain interpretation of zero.<ref>{{cite encyclopedia |last=Huggett |first=Nick |title=Zeno's Paradoxes |date=2019 |url=https://plato.stanford.edu/archives/win2019/entries/paradox-zeno/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Winter 2019 |publisher=Metaphysics Research Lab, Stanford University |access-date=2020-08-09 |archive-date=10 January 2021 |archive-url=https://web.archive.org/web/20210110135804/https://plato.stanford.edu/archives/win2019/entries/paradox-zeno/ |url-status=live }}</ref>

[[File:P. Lund, Inv. 35a.jpg|thumb|upright=1.4|alt=Fragment of papyrus with clear Greek script, lower-right corner suggests a tiny zero with a double-headed arrow shape above it|Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus]]
By AD{{nbsp}}150, [[Ptolemy]], influenced by [[Hipparchus]] and the [[Babylonia]]ns, was using a symbol for zero ({{overset|—|°}})<ref>{{Cite book |last=Neugebauer |first=Otto |url=https://archive.org/details/exactsciencesant00neug |title=The Exact Sciences in Antiquity |publisher=[[Dover Publications]] |year=1969 |isbn=978-0-486-22332-2 |edition=2  |pages=[https://archive.org/details/exactsciencesant00neug/page/n30 13]–14, plate 2  |author-link=Otto E. Neugebauer |orig-date=1957 |url-access=registration}}</ref><ref name="Mercier">{{cite web |last=Mercier |first=Raymond |title=Consideration of the Greek symbol 'zero' |url=http://www.raymondm.co.uk/prog/GreekZeroSign.pdf |work=Home of Kairos |access-date=28 March 2020 |archive-date=5 November 2020 |archive-url=https://web.archive.org/web/20201105113109/http://www.raymondm.co.uk/prog/GreekZeroSign.pdf |url-status=live }}{{self-published inline|date=November 2023}}</ref> in his work on [[mathematical astronomy]] called the ''Syntaxis Mathematica'', also known as the ''[[Almagest]]''.<ref name="Ptolemy">{{cite book |last=Ptolemy |title=Ptolemy's Almagest |pages=306–307 |year=1998 |orig-date=1984, {{circa}}150 |publisher=[[Princeton University Press]] |isbn=0-691-00260-6 |author-link=Ptolemy |translator-last=Toomer | title-link= Almagest |translator-first=G. J. |translator-link=Gerald J. Toomer}}</ref> This [[Greek numerals#Hellenistic zero|Hellenistic zero]] was perhaps the earliest documented use of a numeral representing zero in the Old World.<ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |title=A history of Zero |url=http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html |url-status=live |archive-url=https://web.archive.org/web/20200407074239/http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html |archive-date=7 April 2020 |access-date=28 March 2020 |publisher=MacTutor History of Mathematics}}</ref> Ptolemy used it many times in his ''Almagest'' (VI.8) for the magnitude of [[solar eclipse|solar]] and [[lunar eclipse]]s. It represented the value of both [[digit (unit)|digit]]s and [[Minute and second of arc|minutes]] of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the [[angular diameter]] of the Sun. Minutes of immersion was tabulated from 0{{prime}}0{{pprime}} to 31{{prime}}20{{pprime}} to 0{{prime}}0{{pprime}}, where 0{{prime}}0{{pprime}} used the symbol as a placeholder in two positions of his [[sexagesimal]] positional numeral system,{{efn|Each place in Ptolemy's sexagesimal system was written in [[Greek numerals]] from {{nowrap|0 to 59}}, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20.}} while the combination meant a zero angle. Minutes of immersion was also a continuous function {{nowrap|{{sfrac|1|12}} 31{{prime}}20{{pprime}} {{radic|d(24−d)}}}} (a triangular pulse with [[convex lens|convex]] sides), where d was the digit function and 31{{prime}}20{{pprime}} was the sum of the radii of the Sun's and Moon's discs.<ref name="Pedersen">{{cite book |last=Pedersen |first=Olaf |title=A Survey of the Almagest |  series= Sources and Studies in the History of Mathematics and Physical Sciences | doi= 10.1007/978-0-387-84826-6_7 |pages=232–235 |year=2010 |orig-date=1974 |publisher=Springer |isbn=978-0-387-84825-9 |author-link=Olaf Pedersen | editor= Alexander Jones}}</ref> Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. Over time, Ptolemy's zero tended to increase in size and lose the [[overline]], sometimes depicted as a large elongated 0-like omicron "Ο" or as omicron with overline "ō" instead of a dot with overline.<ref>{{cite web |title=Proposal to encode the Greek Zero in the UCS |url=https://www.unicode.org/L2/L2004/04054-greek-zero.pdf |date=2024-07-31 |archive-url=http://web.archive.org/web/20221007235444/https://unicode.org/L2/L2004/04054-greek-zero.pdf |archive-date=2022-10-07 |url-status=live}}</ref>

The earliest use of zero in the calculation of the [[Computus|Julian Easter]] occurred before AD{{spaces}}311, at the first entry in a table of [[epact]]s as preserved in an [[Ethiopia|Ethiopic]] document for the years 311 to 369, using a [[Geʽez]] word for "none" (English translation is "0" elsewhere) alongside Geʽez numerals (based on Greek numerals), which was translated from an equivalent table published by the [[Church of Alexandria]] in [[Medieval Greek]].<ref name="Neugebauer">{{cite book |last=Neugebauer |first=Otto |title=Ethiopic Astronomy and Computus |pages=25, 53, 93, 183, Plate I |year=2016 |publisher=Red Sea Press |orig-date=1979 |edition=Red Sea Press |isbn=978-1-56902-440-9 |author-link=Otto Neugebauer}}. The pages in this edition have numbers six less than the same pages in the original edition.</ref> This use was repeated in 525 in an equivalent table, that was translated via the Latin {{lang|la|nulla}} ("none") by [[Dionysius Exiguus]], alongside [[Roman numerals#Zero|Roman numerals]].<ref name="Dionysius">{{cite web |last=Deckers |first=Michael |title=Cyclus Decemnovennalis Dionysii |trans-title= Nineteen Year Cycle of Dionysius |url=http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |year=2003 |orig-date=525 |archive-url=https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |archive-date=15 January 2019}}</ref> When division produced zero as a remainder, ''nihil'', meaning "nothing", was used. These medieval zeros were used by all future medieval [[computus|calculators of Easter]]. The initial "N" was used as a zero symbol in a table of Roman numerals by [[Bede]]—or his colleagues—around AD{{nbsp}}725.<ref name="zero">C. W. Jones, ed., ''Opera Didascalica'', vol. 123C in ''Corpus Christianorum, Series Latina''.</ref>

In most [[History of mathematics|cultures]], 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.{{Citation needed |date=June 2024}}

===China===
[[File:Zero in Rod Calculus.png|thumb|right|alt=Five illustrated boxes from left to right contain a T-shape, an empty box, three vertical bars, three lower horizontal bars with an inverted wide T-shape above, and another empty box. Numerals underneath left to right are six, zero, three, nine, and zero|This is a depiction of zero expressed in Chinese [[counting rods]], based on the example provided by ''A History of Mathematics''. An empty space is used to represent zero.<ref name="Hodgkin" />]]
The ''[[Sunzi Suanjing|Sūnzĭ Suànjīng]]'', of unknown date but estimated to be dated from the 1st to {{nowrap|5th centuries AD}}, describe how the{{nowrap| 4th century BC}} Chinese [[counting rods]] system enabled one to perform decimal calculations. As noted in the ''[[Xiahou Yang Suanjing]]'' (425–468 AD), to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places.<ref>{{MacTutor|class=HistTopics|id=Chinese_numerals |title=Chinese numerals |date=January 2004}}</ref> The rods gave the decimal representation of a number, with an empty space denoting zero.<ref name="Hodgkin">{{Cite book |last=Hodgkin |first=Luke |url=https://archive.org/details/historyofmathema0000hodg |title=A History of Mathematics: From Mesopotamia to Modernity |date=2005 |publisher=Oxford University Press |isbn=978-0-19-152383-0 |page=[https://archive.org/details/historyofmathema0000hodg/page/85 85] |url-access=registration}}</ref><ref>{{Cite web |title=Chinese numerals |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Chinese_numerals/ |access-date=2024-04-28 |website=Maths History |language=en}}</ref> The counting rod system is a [[positional notation]] system.<ref>{{harvnb|Shen|Crossley|Lun|1999|p= 12}}: "the ancient Chinese system is a place notation system"</ref><ref>{{Citation |last=Eberhard-Bréard |first=Andrea |title=Mathematics in China |date=2008 |pages=1371–1378 |editor-last=Selin |editor-first=Helaine |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-1-4020-4425-0_9453 |isbn=978-1-4020-4425-0 |encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures}}.</ref>

Zero was not treated as a number at that time, but as a "vacant position".<ref name="Crossley">{{Cite book |last1=Shen | first1=Kangshen |url=https://books.google.com/books?id=eiTJHRGTG6YC |title=The Nine Chapters on the Mathematical Art: Companion and Commentary |first2=John N.|last2= Crossley |first3=Anthony W.-C. | last3=Lun | author1-mask= Shen Kanshen |publisher=Oxford University Press |year=1999 |isbn=978-0-19-853936-0 |page=35 |quote=zero was regarded as a number in India ... whereas the Chinese employed a vacant position}}</ref> [[Qin Jiushao|Qín Jiǔsháo]]'s 1247 ''[[Mathematical Treatise in Nine Sections]]'' is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero.<ref name="Qin">{{Cite web |title=Mathematics in the Near and Far East |url=http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |website=grmath4.phpnet.us |page=262 |access-date=7 June 2012 |archive-date=4 November 2013 |archive-url=https://web.archive.org/web/20131104120005/http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |url-status=live }}</ref> The origin of this symbol is unknown; it may have been produced by modifying a square symbol.<ref>{{cite book |first=Jean-Claude |last=Martzloff |translator-first1=Stephen S. |translator-last1=Wilson |title=A History of Chinese Mathematics |publisher=Springer |year=2007 |isbn=978-3-540-33783-6 |page=208}}</ref> Chinese authors had been familiar with the idea of negative numbers by the [[Han dynasty]] {{nowrap|(2nd century AD)}}, as seen in ''[[The Nine Chapters on the Mathematical Art]]''.<ref name="struik33">Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications. pp. 32–33. "''In these matrices we find negative numbers, which appear here for the first time in history.''"</ref>

===India===

[[Pingala]] ({{Circa|3rd}} or 2nd century BC),<ref name="plofker">{{Cite book |last=Plofker |first=Kim |author-link=Kim Plofker |title=Mathematics in India |title-link=Mathematics in India (book) |publisher=Princeton University Press |year=2009 |isbn=978-0-691-12067-6 |pages=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA54 54–56] |quote=In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)<sup>7</sup> = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.}}</ref> a [[Sanskrit prosody]] scholar,<ref>{{Cite book |author=Vaman Shivaram Apte |chapter-url=https://books.google.com/books?id=4ArxvCxV1l4C&pg=PA648 | title= The Student's Sanskrit-English Dictionary |chapter=Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India |publisher=Motilal Banarsidass |year=1970 |isbn=978-81-208-0045-8 |pages=648–649 |access-date=21 April 2017 }}</ref> used [[binary numeral system|binary sequences]], in the form of short and long syllables (the latter equal in length to two short syllables), to identify the possible valid Sanskrit [[Metre (poetry)#Disyllables|meter]]s, a notation similar to [[Morse code]].<ref>{{Cite web |title=Math for Poets and Drummers: The Mathematics of Rhythm | first=Rachel| last= Hall | publisher=Saint Joseph's University | date=February 15, 2005 | type=slideshow|url=http://people.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf |archive-url=https://web.archive.org/web/20190122014628/http://people.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf |archive-date=22 January 2019 |access-date=20 December 2015 |url-status= dead }}</ref> Pingala used the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' explicitly to refer to zero.<ref name=plofker/>

[[File:Bakhshali manuscript zero detail.jpg|thumb|Bakhshali manuscript, with the numeral "zero" represented by a black dot; its date is uncertain.<ref name="Devlin 2017"/>]]

The concept of zero as a written digit in the '''''decimal''''' place value notation was developed in [[Indian subcontinent|India]].<ref name="bourbaki46">{{harvnb|Bourbaki|1998|p=46}}.</ref> A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the [[Bakhshali manuscript]], a practical manual on arithmetic for merchants.<ref name="Weiss">{{Cite news |last=Weiss |first=Ittay |date=20 September 2017 |title=Nothing matters: How India's invention of zero helped create modern mathematics |work=The Conversation |url=https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232 |access-date=12 July 2018 |archive-date=12 July 2018 |archive-url=https://web.archive.org/web/20180712124031/https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232 |url-status=live }}</ref> In 2017, researchers at the [[Bodleian Library]] reported [[radiocarbon dating]] results for three samples from the manuscript, indicating that they came from three different centuries: from AD 224–383, AD 680–779, and AD 885–993. It is not known how the [[birch]] bark fragments from different centuries forming the manuscript came to be packaged together. If the writing on the oldest birch bark fragments is as old as those fragments, it represents South Asia's oldest recorded use of a zero symbol. However, it is possible that the writing dates instead to the time period of the youngest fragments, AD 885–993. The latter dating has been argued to be more consistent with the sophisticated use of zero within the document, as portions of it appear to show zero being employed as a number in its own right, rather than only as a positional placeholder.<ref name="Devlin 2017">{{Cite news |last=Devlin |first=Hannah |author-link=Hannah Devlin |date=13 September 2017 |title=Much ado about nothing: ancient Indian text contains earliest zero symbol |work=The Guardian |url=https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol |access-date=14 September 2017 |issn=0261-3077 |archive-date=20 November 2017 |archive-url=https://web.archive.org/web/20171120225416/https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol |url-status=live }}</ref><ref>{{Cite news |date=14 September 2017 |title=Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero' |work=Bodleian Library |url=http://www.bodleian.ox.ac.uk/bodley/news/2017/sep-14 |access-date=25 October 2017 |archive-date=14 September 2017 |archive-url=https://web.archive.org/web/20170914215604/http://www.bodleian.ox.ac.uk/bodley/news/2017/sep-14 |url-status=live }}</ref><ref>{{Cite journal |last1=Plofker |first1=Kim |author-link1=Kim Plofker |last2=Keller |first2=Agathe |last3=Hayashi |first3=Takao |author-link3=Takao Hayashi |last4=Montelle |first4=Clemency |author-link4=Clemency Montelle |last5=Wujastyk |first5=Dominik |date=2017-10-06 |title=The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating |url=https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22 |journal=History of Science in South Asia |language=en |volume=5 |issue=1 |pages=134–150 |doi=10.18732/H2XT07 |doi-access=free }}</ref>

The ''[[Lokavibhaga|Lokavibhāga]]'', a [[Jain]] text on [[cosmology]] surviving in a medieval Sanskrit translation of the [[Prakrit]] original, which is internally dated to AD 458 ([[Saka era]] 380), uses a decimal [[positional notation|place-value system]], including a zero. In this text, ''[[Śūnyatā|śūnya]]'' ("void, empty") is also used to refer to zero.{{sfnp|Ifrah|2000| p= 416}}

The ''[[Aryabhatiya]]'' ({{circa}} 499), states ''sthānāt sthānaṁ daśaguṇaṁ syāt'' "from place to place each is ten times the preceding".<ref name="Aryab">''Aryabhatiya of Aryabhata'', translated by [[Walter Eugene Clark]].</ref><ref>{{Cite web |author=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=2000 |title=Aryabhata the Elder |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |url-status=live |archive-url=https://web.archive.org/web/20150711055702/http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |archive-date=11 July 2015 |access-date=26 May 2013 |website=School of Mathematics and Statistics, University of St. Andrews |publication-place=Scotland}}</ref><ref>{{Cite book |url=https://books.google.com/books?id=cuN7rH6RzikC&pg=PA97 |title=The Britannica Guide to Numbers and Measurement (Math Explained) |date=15 August 2010 |publisher=The Rosen Publishing Group |isbn=978-1-61530-108-9 |editor=Hosch |editor-first=William L. |pages=97–98 |access-date=26 September 2016}}</ref>

Rules governing the use of zero appeared in [[Brahmagupta]]'s ''[[Brāhmasphuṭasiddhānta|Brahmasputha Siddhanta]]'' (7th century), which states the sum of zero with itself as zero, and incorrectly describes [[division by zero]] in the following way:<ref name="brahmagupta">{{cite book |url=https://archive.org/details/algebrawitharith00brahuoft |title=Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bháscara |date=1817 |publisher=John Murray |place=London, England |translator=Henry Thomas Colebrooke |oclc=1039515732}}</ref>{{sfn|Kaplan|2000|p=[https://archive.org/details/nothingthatisnat00kapl/page/68 68–75]}}

<blockquote>A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.</blockquote>

====Epigraphy====
{{multiple image
| perrow = 2
| total_width = 330
| align = right
| header = Sambor Inscription
| image1 = First zero 1.jpg
| caption1 = 
| image2 = Khmer Numerals - 605 from the Sambor inscriptions.jpg
| caption2 =  
| image3 = 03-National Museum of Cambodia-nX-1.jpg
| caption3 = 
| footer = The oldest, firmly dated use of zero as a decimal figure, found on the Sambor Inscription. The number "605" is written in [[Khmer numerals]] (top), referring to the year it was made: [[Shaka era|605 Saka era]] (683 CE). The fragment, inscribed in [[Old Khmer]], was once part of a temple doorway, and was found in [[Kratié province]], [[Cambodia]].
}}

A black dot is used as a decimal placeholder in the [[Bakhshali manuscript]], portions of which date from AD 224–993.<ref name = "Devlin 2017" />

There are numerous copper plate inscriptions, with the same small {{sc|o}} in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.{{sfn|Kaplan|2000}}

A stone tablet found in the ruins of a temple near Sambor on the [[Mekong]], [[Kratié Province]], [[Cambodia]], includes the inscription of "605" in [[Khmer numerals]] (a set of numeral glyphs for the [[Hindu–Arabic numeral system]]). The number is the year of the inscription in the [[Saka era]], corresponding to a date of AD 683.<ref name="Cambodia">{{multiref2|{{cite journal| author-link=George Cœdès|last=Cœdès |first=George |title=A propos de l'origine des chiffres arabes |journal= Bulletin of the School of Oriental Studies, University of London |volume= 6 | number= 2|date= 1931|pages= 323–328 |jstor=607661 | publisher= Cambridge University Press | language=fr | doi= 10.1017/S0041977X00092806|s2cid=130482979 }}|{{cite journal|last= Diller|first= Anthony | title=New Zeros and Old Khmer|journal=[[Mon-Khmer Studies]]|volume= 25|date=1996|pages= 125–132 | url= http://sealang.net/sala/archives/pdf8/diller1996new.pdf }} }}</ref>

The first known use of special [[glyph]]s for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the [[Chaturbhuj Temple, Gwalior]], in India, dated AD 876.<ref>{{Cite web |last=Casselman |first=Bill |author-link=Bill Casselman (mathematician) |title=All for Nought |url=http://www.ams.org/samplings/feature-column/fcarc-india-zero |website=ams.org |publisher=University of British Columbia), American Mathematical Society |access-date=20 December 2015 |archive-date=6 December 2015 |archive-url=https://web.archive.org/web/20151206184352/http://www.ams.org/samplings/feature-column/fcarc-india-zero |url-status=live }}</ref>{{sfnp|Ifrah|2000|p=400}}

===Middle Ages===
====Transmission to Islamic culture====
{{See also|History of the Hindu–Arabic numeral system}}
The [[Arabic]]-language inheritance of science was largely [[Greece|Greek]],<ref>{{Cite book |last=Pannekoek |first=Anton |title=A History of Astronomy |publisher=George Allen & Unwin |year=1961 |page=165 | oclc=840043 | author-link=Anton Pannekoek | url= https://archive.org/details/historyofastrono0000pann}}</ref> followed by Hindu influences.<ref name="Durant">{{cite book | first= Will | last= Durant |date=1950|title=The Story of Civilization, Volume IV, The Age of Faith: Constantine to Dante – A.D. 325–1300|publisher= Simon & Schuster |quote-page= 241 | quote=The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the ''Siddhantas'' – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables. | author-link=Will Durant |url=https://archive.org/details/ageoffaithahisto012288mbp}}</ref> In 773, at [[Al-Mansur]]'s behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.

In AD 813, astronomical tables were prepared by a [[Persian people|Persian]] mathematician, [[Muḥammad ibn Mūsā al-Khwārizmī]], using Hindu numerals;<ref name="Durant" /> and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.<ref>{{Cite book |last=Brezina |first=Corona |url=https://books.google.com/books?id=955jPgAACAAJ |title=Al-Khwarizmi: The Inventor of Algebra |publisher=The Rosen Publishing Group |year=2006 |isbn=978-1-4042-0513-0 |access-date=26 September 2016 }}</ref> This book was later translated into [[Latin]] in the 12th century under the title ''Algoritmi de numero Indorum''. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "[[Algorithm]]" or "[[Algorism]]" started to acquire a meaning of any arithmetic based on decimals.<ref name="Durant" />

[[Muhammad ibn Ahmad al-Khwarizmi]], in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ''ṣifr''.<ref>{{harvnb|Durant|1950|p=241}}: "In 976, Muhammad ibn Ahmad, in his ''Keys of the Sciences'', remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ''ṣifr'', "empty" whence our cipher."</ref>

====Transmission to Europe====
The [[Hindu–Arabic numeral system]] (base 10) reached Western Europe in the 11th century, via [[Al-Andalus]], through Spanish [[Muslim]]s, the [[Moors]], together with knowledge of [[classical astronomy]] and instruments like the [[astrolabe]]. [[Pope Sylvester II|Gerbert of Aurillac]] is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician [[Fibonacci]] or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

<blockquote>After my father's appointment by [[Republic of Pisa|his homeland]] as state official in the customs house of [[Béjaïa|Bugia]] for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the [[algorism]], as well as the art of [[Pythagoras]], I considered as almost a mistake in respect to the method of the [[Hinduism|Hindus]] [{{lang|la-x-medieval|Modus Indorum}}]. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of [[Euclid]]'s geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the [[Latin people]] might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0{{nbsp}}... any number may be written.<ref>{{multiref2|{{cite book | translator-last=Sigler|translator-first= Laurence E.| title= Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation |publisher= Springer|date= 2003| isbn =978-1-4613-0079-3 | doi=10.1007/978-1-4613-0079-3 |series= Sources and Studies in the History of Mathematics and Physical Sciences|last1= Sigler|first1= Laurence}}|{{cite periodical| last=Grimm | first= Richard E. | title=The Autobiography of Leonardo Pisano|  magazine =[[Fibonacci Quarterly]]| volume= 11 | number=1 |date=February 1973|pages= 99–104 |archive-url=https://web.archive.org/web/20231126180044/https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=318a17253f745e2af400eb2ebb4dc4e762560a5b | archive-date= 26 November 2023 |url-status=live | url = https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=318a17253f745e2af400eb2ebb4dc4e762560a5b}}|{{Cite book |last=Hansen |first=Alice |url=https://books.google.com/books?id=COJsbuUI1h8C&pg=PT31 |title=Primary Mathematics: Extending Knowledge in Practice |date=2008 |publisher=SAGE | doi = 10.4135/9781446276532|isbn=978-0-85725-233-3 |language=en |access-date=7 November 2020 |archive-date=7 March 2021 |archive-url=https://web.archive.org/web/20210307234959/https://books.google.com/books?id=COJsbuUI1h8C&q=%22Therefore%2C+embracing+more+stringently+that+method+of+the+Hindus%2C+and+taking+stricter+pains+in+its+study%2C+while+adding+certain+things+from+my+own+understanding+and+inserting+also+certain+things+from+the+niceties+of+Euclid%27s+geometric+art.%22&pg=PT31 |url-status=live }} }}</ref></blockquote>

From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called {{lang|la-x-medieval|algorismus}} after the Persian mathematician [[al-Khwārizmī]]. One popular manual was written by [[Johannes de Sacrobosco]] in the early 1200s and was one of the earliest scientific books to be [[History of printing|printed]], in 1488.<ref name=Karpinski1911>{{cite book |first1=D. E. |last1=Smith |first2=L. C. |last2=Karpinski |year=1911 |chapter=The spread of the <nowiki>[Hindu–Arabic]</nowiki> numerals in Europe |title=The Hindu–Arabic Numerals |pages=134–136 |publisher=Ginn and Company |via=Internet Archive |chapter-url=https://archive.org/stream/hinduarabicnume02karpgoog#page/n145/mode/1up
}}</ref><ref>{{cite journal |last1=Pedersen |first1=Olaf |date=1985 |title=In Quest of Sacrobosco |journal=Journal for the History of Astronomy |volume=16 |issue=3 |pages=175–221 |doi=10.1177/002182868501600302  |bibcode=1985JHA....16..175P |s2cid=118227787 }}</ref> The practice of calculating on paper using Hindu–Arabic numerals only gradually displaced calculation by abacus and recording with [[Roman numerals]].{{sfn|Ifrah|2000|pp=588–590}} In the 16th century, Hindu–Arabic numerals became the predominant numerals used in Europe.<ref name=Karpinski1911/>