Editing 0 (section)

===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}}