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{{Short description|Number}}
{{Hatnote|This article is about the number.}} 
{{pp-semi-indef}}
{{Infobox number
|number=0
|cardinal=0, zero, {{nowrap|"oh" ({{IPAc-en|oʊ}})}}, nought, naught, nil
|ordinal=Zeroth, noughth, 0th
|latin prefix=nulli-
|lang1=[[Eastern Arabic numerals|Arabic]], Kurdish, [[Social:Persian language|Persian]], [[Social:Sindhi language|Sindhi]], Urdu
|lang1 symbol={{resize|150%|٠}}
|lang2=[[Indian numerals|Hindu numerals]]
|lang2 symbol={{resize|150%|०}}
|lang3=[[Chinese numerals|Chinese]]
|lang3 symbol=零, 〇
|lang4=[[Burmese numerals|Burmese]]
|lang4 symbol=၀
|lang5=[[Khmer numerals|Khmer]]
|lang5 symbol=០
|lang6=[[Thai numerals|Thai]]
|lang6 symbol=๐|lang7=Assamese, Bengali|lang7 symbol=০}}
'''0''' ('''zero''') is a [[Number|number]] representing an empty [[Quantity|quantity]]. Adding 0 to any number leaves that number unchanged. In mathematical terminology, 0 is the [[Additive identity|additive identity]] of the [[Integer|integer]]s, rational numbers, [[Real number|real number]]s, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 has the result 0, and consequently, [[Division by zero|division by zero]] has [[Undefined (mathematics)|no meaning in arithmetic]].

As a [[Numerical digit|numerical digit]], 0 plays a crucial role in [[Decimal|decimal]] notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "[[205 (number)|205]]" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as [[Binary number|binary]] and [[Hexadecimal|hexadecimal]]. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by [[Biography:Fibonacci|Fibonacci]]. It was independently used by the Maya.

Common [[Names for the number 0 in English|names for the number 0 in English]] include ''zero'', ''nought'', ''naught'' ({{IPAc-en|n|ɔː|t}}), and ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' ({{IPAc-en|oʊ}}). Informal or [[Social:Slang|slang]] terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' ({{IPAc-en|ɔː|t}}), and ''cipher'' have also been used.

==Etymology==
{{Main|Names for the number 0|Names for the number 0 in English}}

The word ''zero'' came into the English language via French {{lang|fr|zéro}} from the [[Social:Italian language|Italian]] {{lang|it|zero}}, a contraction of the Venetian {{lang|vec|zevero}} form of Italian {{lang|it|zefiro}} via ''ṣafira'' or ''ṣifr''.<ref>{{multiref2
| {{cite dictionary| first= Douglas| last= Harper |date=2011| entry-url=https://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none | entry= Zero |archive-url=https://web.archive.org/web/20170703014638/http://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none |archive-date=3 July 2017 | title= Etymonline Etymology Dictionary| quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", {{circa}} 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught"}}
|{{Cite book |last=Menninger |first=Karl |url=https://books.google.com/books?id=BFJHzSIj2u0C |title=Number Words and Number Symbols: A cultural history of numbers |publisher=Courier Dover Publications |year=1992 |isbn=978-0-486-27096-8 |pages=399–404 |access-date=5 January 2016 }}
| {{Cite web |date=December 2011 |title=zero, n. |url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false |url-status=live |archive-url=https://www.webcitation.org/65yd7ur9u?url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false |archive-date=7 March 2012 |access-date=4 March 2012 |website=OED Online |publisher=[[Organization:Oxford University Press|Oxford University Press]] |quote="French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr" }} }}</ref> In pre-Islamic time the word {{transliteration|ar|ṣifr}} (Arabic {{lang|ar|صفر}}) had the meaning "empty".<ref name=smithsonian/> {{transliteration|ar|Sifr}} evolved to mean zero when it was used to translate {{transliteration|sa|śūnya}} ({{lang-sa|शून्य}}) from India.<ref name="smithsonian">{{multiref2 
| Smithsonian Institution. {{Google books|0_UyAQAAMAAJ|Oriental Elements of Culture in the Occident|page=518}}. Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives. "Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya."
| {{cite book | first=Jan |last=Gullberg |date=1997|title=Mathematics: From the Birth of Numbers|publisher= W.W. Norton & Co.|isbn= 978-0-393-04002-9 | quote-page= 26|quote = ''Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.''}}
|{{ cite book | first=Robert|last= Logan |date=2010|title=The Poetry of Physics and the Physics of Poetry|publisher=World Scientific | isbn =978-981-4295-92-5|quote-page= 38|quote = The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.}} }}</ref> The first known English use of ''zero'' was in 1598.<ref>{{ cite dictionary| entry-url=http://www.merriam-webster.com/dictionary/zero|entry= Zero |archive-url=https://web.archive.org/web/20171206230446/https://www.merriam-webster.com/dictionary/zero |archive-date=6 December 2017 |title= Merriam Webster online Dictionary}} }}</ref>

The Italian mathematician [[Biography:Fibonacci|Fibonacci]] ({{Circa|1170|1250}}), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term ''zephyrum''. This became {{lang|it|zefiro}} in Italian, and was then contracted to {{lang|vec|zero}} in Venetian. The Italian word {{lang|it|zefiro}} was already in existence (meaning "west wind" from Latin and Greek {{lang|la|[[Religion:Zephyrus|Zephyrus]]}}) and may have influenced the spelling when transcribing Arabic {{transliteration|ar|ṣifr}}.<ref name="ifrah">{{harvnb|Ifrah|2000|p=589}}</ref>

===Modern usage===
Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" and "none" are often used. The British English words [[Names for the number 0 in English#%22Nought%22 and %22naught%22 versus %22ought%22 and %22aught%22|"nought" or "naught"]], and "nil" are also synonymous.<ref>{{Cite web |title=Collins - Free online dictionary |url=https://www.collinsdictionary.com/dictionary/english/nought}}</ref><ref>{{Cite web |title=Collins - Free online dictionary, thesaurus and reference materials - nill |url=https://www.collinsdictionary.com/dictionary/english/nil}}</ref>

It is often called "oh" in the context of reading out a string of digits, such as [[Telephone number|telephone number]]s, street addresses, credit card numbers, military time, or years. For example, the area code 201 may be pronounced "two oh one", and the year 1907 is often pronounced "nineteen oh seven". The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (such as Canadian postal codes) may exclude the use of the letter O.{{citation needed|date=December 2023}}

Slang words for zero include "zip", "zilch", "nada", and "scratch".<ref name="thesaurus">{{ cite web | url = http://thesaurus.com/browse/aught#visualthesaurus | title= 'Aught' synonyms | archive-url=https://web.archive.org/web/20140823071642/http://thesaurus.com/browse/aught#visualthesaurus |archive-date=23 August 2014 | work= Thesaurus.com | access-date=23 April 2013}}</ref>  In the context of sports, "nil" is sometimes used, especially in British English. Several sports have specific words for a score of zero, such as "love" in tennis – from French {{lang|fr|l'œuf}}, "the egg" – and "duck" in cricket, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero.<ref name=thesaurus />

==History==<!--Linked from History of zero (R to section)-->
===Ancient Near East===
{| style="float:right; clear:right; text-align:center; border: 1px solid" align=right cellspacing=0 cellpadding=8
|-
!nfr<br />&nbsp;
|heart with [[Biology:Trachea|trachea]]<br />beautiful, pleasant, good
|<hiero>F35</hiero>
|}
Ancient [[Egyptian numerals]] were of [[Decimal|base 10]].<ref>{{Cite web |title=Egyptian numerals |url=http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html |access-date=21 December 2019 |website=mathshistory.st-andrews.ac.uk |archive-date=15 November 2019 |archive-url=https://web.archive.org/web/20191115221313/http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html |url-status=live | author1=J J O'Connor |author2= E F Robertson|date= 2000|publisher= University of St Andrews}}</ref> They used hieroglyphs for the digits and were not [[Positional notation|positional]]. In one papyrus written around {{nowrap|1770 BC}}, a scribe recorded daily incomes and expenditures for the [[Social:Pharaoh|pharaoh]]'s court, using the ''[[Biology: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 [[History: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 |title=Zero |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/ |access-date=2021-09-07 |website=Maths History |language=en |archive-date=21 September 2021 |archive-url=https://web.archive.org/web/20210921191118/https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/| author1=J J O'Connor |author2= E F Robertson|date= 2000|publisher= University of St Andrews |url-status=live }}</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 [[History:Mesoamerican Long Count calendar|Mesoamerican Long Count calendar]] developed in south-central Mexico and Central America required the use of zero as a placeholder within its [[Vigesimal|vigesimal]] (base-20) positional numeral system. Many different glyphs, including the partial [[Quatrefoil|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 Olmecs.<ref>{{Cite news |last=Mortaigne |first=Véronique |date=28 November 2014 |title=The golden age of Mayan civilisation – exhibition review |work={{wipe|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}}, several centuries before the earliest known Long Count dates.{{citation needed|date=December 2023}}

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

===Classical antiquity===
The {{wipe|Ancient Greece}} 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 [[Biography:Charles Seife|Charles Seife]], the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in 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 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 [[Physics:Vacuum|vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Biography:Zeno of Elea|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|right|300px|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, [[Biography:Ptolemy|Ptolemy]], influenced by [[Biography:Hipparchus|Hipparchus]] and the [[History:Babylonia|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=[[Social:Dover Publications|Dover Publications]] |year=1969 |isbn=978-0-486-22332-2 |edition=2  |pages=[https://archive.org/details/exactsciencesant00neug/page/n30 13]–14, plate 2  |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 }}{{sps|date=November 2023}}</ref> in his work on mathematical astronomy called the ''Syntaxis Mathematica'', also known as the ''[[Astronomy:Almagest|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 |title=A history of Zero |url=http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html |publisher=MacTutor History of Mathematics |last2=Robertson |first2=E F |access-date=28 March 2020 |archive-date=7 April 2020 |archive-url=https://web.archive.org/web/20200407074239/http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html |url-status=live }}</ref> Ptolemy used it many times in his ''Almagest'' (VI.8) for the magnitude of [[Earth:Solar eclipse|solar]] and [[Astronomy:Lunar eclipse|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 [[Astronomy:Angular diameter|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|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 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.

The earliest use of zero in the calculation of the [[Religion:Computus|Julian Easter]] occurred before AD{{nbs}}311, at the first entry in a table of [[Physics:Epact|epact]]s as preserved in an Ethiopic document for the years 311 to 369, using a [[Social:Ge'ez|Ge'ez]] word for "none" (English translation is "0" elsewhere) alongside Ge'ez 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 [[Biography:Dionysius Exiguus|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 [[Religion:Computus|calculators of Easter]]. The initial "N" was used as a zero symbol in a table of Roman numerals by [[Biography:Bede|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.

===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|counting rods]], based on the example provided by ''A History of Mathematics''. An empty space is used to represent zero.<ref name="Hodgkin" />]]
The ''Sūnzĭ Suànjīng'', of unknown date but estimated to be dated from the 1st to {{nowrap|5th centuries AD}}, and Japanese records dated from the 18th century, describe how the {{nowrap| 4th century BC}} Chinese [[Counting rods|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> According to ''A History of Mathematics'', 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> The counting rod system is considered a [[Positional notation|positional notation]] system.<ref>{{harvnb|Shen|Crossley|Lun|1999|p= 12}}: "the ancient Chinese system is a place notation system"</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> [[Biography: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 borrowed from Indian sources or 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 |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 |author-link=Kim Plofker}}</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 numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to [[Engineering:Morse code|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 [[Social:Sanskrit|Sanskrit]] word ''[[Philosophy:Śūnyatā|śūnya]]'' explicitly to refer to zero.<ref name=plofker/>

The concept of zero as a written digit in the decimal place value notation was developed in [[Earth: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, three samples from the manuscript were shown by [[Physics:Radiocarbon dating|radiocarbon dating]] to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the [[Biology:Birch|birch]] bark fragments from different centuries forming the manuscript came to be packaged together.<ref name="Devlin 2017">{{Cite news |last=Devlin |first=Hannah |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 |last=Revell |first=Timothy |date=14 September 2017 |title=History of zero pushed back 500 years by ancient Indian text |work=New Scientist |url=https://www.newscientist.com/article/2147450-history-of-zero-pushed-back-500-years-by-ancient-indian-text |access-date=25 October 2017 |archive-date=25 October 2017 |archive-url=https://web.archive.org/web/20171025185119/https://www.newscientist.com/article/2147450-history-of-zero-pushed-back-500-years-by-ancient-indian-text |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>

The ''[[Religion:Lokavibhaga|Lokavibhāga]]'', a Jain text on [[Astronomy:Cosmology|cosmology]] surviving in a medieval Sanskrit translation of the [[Social:Prakrit|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, ''[[Philosophy:Śūnyatā|śūnya]]'' ("void, empty") is also used to refer to zero.{{sfnp|Ifrah|2000| p= 416}}

The ''[[Unsolved:Aryabhatiya|Aryabhatiya]]'' ({{circa}} 500), 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=J J O'Connor |author2=E F Robertson |title=Aryabhata the Elder |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |access-date=26 May 2013 |website=School of Mathematics and Statistics University of St Andrews, Scotland |archive-date=11 July 2015 |archive-url=https://web.archive.org/web/20150711055702/http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |url-status=live | date=2000 }}</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=William L. Hosch |pages=97–98 |access-date=26 September 2016  }}</ref>

Rules governing the use of zero appeared in Brahmagupta's ''[[Religion:Brāhmasphuṭasiddhānta|Brahmasputha Siddhanta]]'' (7th century), which states the sum of zero with itself as zero, and incorrectly describes [[Division by zero|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 | translator=  Henry Thomas Colebrooke |date=1817|place= London| publisher= John Murray |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====
[[File:Khmer Numerals - 605 from the Sambor inscriptions.jpg|thumb|right|alt=script from left to right with a one and a half rotation swirl, a large dot, and a stretched-bent swirl|The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.]]
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 [[Earth:Mekong|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| 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= The Mon-Khmer Studies Journal|volume= 25|date=1996|pages= 125–132 | url= http://sealang.net/sala/archives/pdf8/diller1996new.pdf }} }}</ref>

The first known use of special [[Glyph|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 876.<ref>{{Cite web |last=Casselman |first=Bill |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====
The [[Social:Arabic|Arabic]]-language inheritance of science was largely Greek,<ref>{{Cite book |last=Pannekoek |first=Anton |title=A History of Astronomy |publisher=George Allen & Unwin |year=1961 |page=165 |isbn = 9780045200023 | oclc=840043 | author-link=Anton Pannekoek}}</ref> followed by Hindu influences.<ref name="Durant">{{cite book | first= Will | last= Durant |date=1950|title=The Story of Civilization|volume= 4, The Age of Faith: Constantine to Dante – A.D. 325–1300|publisher= Simon & Schuster|isbn=978-0-9650007-5-8|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 }}</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 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 [[Social:Latin|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 [[Religion:Muslim|Muslim]]s, the Moors, together with knowledge of classical [[Astronomy:Astronomy|astronomy]] and instruments like the [[Astronomy:Astrolabe|astrolabe]]. [[Biography: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 [[Biography:Fibonacci|Fibonacci]] or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

<blockquote>After my father's appointment by his homeland as state official in the customs house of 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 [[Religion:Hinduism|Hindus]] [''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 [[Biography:Euclid|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 <!-- for the eventuality: |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 = 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|algorismus}} after the Persian mathematician al-Khwārizmī. One popular manual was written by [[Biography:Johannes de Sacrobosco|Johannes de Sacrobosco]] in the early 1200s and was one of the earliest scientific books to be 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/>

==Symbols and representations==
{{Main|Symbols for zero}}
[[File:Text figures 036.svg|71px|alt=horizontal guidelines with a zero touching top and bottom, a three dipping below, and a six cresting above the guidelines, from left to right|left]]
[[File:Oslo airport train station, Platform 0.jpg|thumb|Oslo airport train station, Platform 0]]
Today, the numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print [[Typeface|typeface]]s made the capital letter O more rounded than the narrower, elliptical digit 0.<ref name="bemer" /> Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.<ref name="bemer">{{Cite journal |last=Bemer |first=R. W. |year=1967 |title=Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh |journal=Communications of the ACM |volume=10 |issue=8 |pages=513–518 |doi=10.1145/363534.363563 |s2cid=294510}}</ref>

A [[Slashed zero|slashed zero]] (<math>0\!\!\!{/}</math>) is often used to distinguish the number from the letter (mostly in computing, navigation and in the military, for example). The digit 0 with a dot in the center seems to have originated as an option on [[Engineering:IBM 3270|IBM 3270]] displays and has continued with some modern computer typefaces such as [[Andalé Mono]], and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion.

==Mathematics==

The concept of zero plays multiple roles in mathematics: as a digit, it is an important part of positional notation for representing numbers, while it also plays an important role as a number in its own right in many algebraic settings.

=== As a digit ===
{{main|Positional notation}}

In positional number systems (such as the usual decimal notation for representing numbers), the digit 0 plays the role of a placeholder, indicating that certain powers of the base do not contribute.  For example, the decimal number 205 is the sum of two hundreds and five ones, with the 0 digit indicating that no tens are added.  The digit plays the same role in decimal fractions and in the [[Decimal representation|decimal representation]] of other real numbers (indicating whether any tenths, hundredths, thousandths, etc., are present) and in bases other than 10 (for example, in binary, where it indicates which powers of 2 are omitted).{{sfn|Reimer|2014|pp=156,199–204}}

===Elementary algebra===
[[File:Number line with numbers -3 to 3.svg|thumb|300x300px|A [[Number line|number line]] from −3 to 3, with 0 in the middle]]
The number 0 is the smallest nonnegative integer. The [[Natural number|natural number]] following 0 is 1 and no natural number precedes 0. The number 0 [[Natural number|may or may not be considered a natural number]],<ref>{{Cite book |last1=Bunt |first1=Lucas Nicolaas Hendrik |url=https://books.google.com/books?id=7xArILpcndYC |title=The historical roots of elementary mathematics |last2=Jones |first2=Phillip S. |last3=Bedient |first3=Jack D. |publisher=Courier Dover Publications |year=1976 |isbn=978-0-486-13968-5 |pages=254–255 |access-date=5 January 2016 |archive-date=23 June 2016 |archive-url=https://web.archive.org/web/20160623174716/https://books.google.com/books?id=7xArILpcndYC |url-status=live }}, [https://books.google.com/books?id=7xArILpcndYC&pg=PA255 Extract of pp. 254–255] {{Webarchive|url=https://web.archive.org/web/20160510195505/https://books.google.com/books?id=7xArILpcndYC&pg=PA255 |date=10 May 2016 }}</ref>{{sfn|Cheng|2017|p=32}} but it is an [[Integer|integer]], and hence a [[Rational number|rational number]] and a [[Real number|real number]].{{sfn|Cheng|2017|pp=41, 48–53}} All rational numbers are [[Algebraic number|algebraic number]]s, including 0. When the real numbers are extended to form the [[Complex number|complex number]]s, 0 becomes the [[Origin (mathematics)|origin]] of the complex plane.

The number 0 can be regarded as neither positive nor negative<ref>{{Cite web |author=Weisstein, Eric W. |title=Zero |url=http://mathworld.wolfram.com/Zero.html |access-date=4 April 2018 |website=Wolfram |language=en |archive-date=1 June 2013 |archive-url=https://web.archive.org/web/20130601190920/http://mathworld.wolfram.com/Zero.html |url-status=live }}</ref> or, alternatively, both positive and negative<ref>{{Cite book |last=Weil |first=André |url=https://books.google.com/books?id=NEHaBwAAQBAJ&pg=PA3 |title=Number Theory for Beginners |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4612-9957-8 |language=en |access-date=6 April 2021 |archive-date=14 June 2021 |archive-url=https://web.archive.org/web/20210614182810/https://books.google.com/books?id=NEHaBwAAQBAJ&pg=PA3 |url-status=live }}</ref> and is usually displayed as the central number in a [[Number line|number line]]. Zero is [[Parity (mathematics)|even]]<ref>[[Lemma (mathematics)|Lemma]] B.2.2, ''The integer 0 is even and is not odd'', in {{Cite book |last=Penner |first=Robert C. |url=https://archive.org/details/discretemathemat0000penn |title=Discrete Mathematics: Proof Techniques and Mathematical Structures |publisher=World Scientific |year=1999 |isbn=978-981-02-4088-2 |page=[https://archive.org/details/discretemathemat0000penn/page/34 34]}}</ref> (that is, a multiple of 2), and is also an integer multiple of any other integer, rational, or real number. It is neither a [[Prime number|prime number]] nor a [[Composite number|composite number]]: it is not prime because prime numbers are greater than 1 by definition, and it is not composite because it cannot be expressed as the product of two smaller natural numbers.<ref>{{Cite book |last=Reid |first=Constance |title-link=From Zero to Infinity |title=From zero to infinity: what makes numbers interesting |publisher=Mathematical Association of America |year=1992 |isbn=978-0-88385-505-8 |edition=4th |page= 23 |quote=zero neither prime nor composite}}</ref> (However, the singleton set {0} is a prime ideal in the [[Ring (mathematics)|ring]] of the integers.)

The following are some basic rules for dealing with the number 0. These rules apply for any real or complex number ''x'', unless otherwise stated.
* [[Addition]]: ''x'' + 0 = 0 + ''x'' = ''x''. That is, 0 is an [[Identity element|identity element]] (or neutral element) with respect to addition.
* [[Subtraction]]: ''x'' − 0 = ''x'' and 0 − ''x'' = −''x''.
* [[Multiplication]]: ''x'' · 0 = 0 · ''x'' = 0.
* [[Division (mathematics)|Division]]: {{sfrac|0|''x''}} = 0, for nonzero ''x''. But [[Division by zero|{{sfrac|''x''|0}}]] is undefined, because 0 has no [[Multiplicative inverse|multiplicative inverse]] (no real number multiplied by 0 produces 1), a consequence of the previous rule.{{sfn|Cheng|2017|p=47}}
* [[Exponentiation]]: ''x''<sup>0</sup> = {{sfrac|''x''|''x''}} = 1, except that [[Zero to the power of zero|the case ''x'' = 0]] is considered undefined in some contexts. For all positive real ''x'', {{nowrap|0<sup>''x''</sup> {{=}} 0}}.

The expression {{sfrac|0|0}}, which may be obtained in an attempt to determine the limit of an expression of the form {{sfrac|''f''(''x'')|''g''(''x'')}} as a result of applying the [[Limit of a function|lim]] operator independently to both operands of the fraction, is a so-called "[[Indeterminate form|indeterminate form]]". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of {{sfrac|''f''(''x'')|''g''(''x'')}}, if it exists, must be found by another method, such as [[L'Hôpital's rule|l'Hôpital's rule]].<ref>{{Cite book |last1=Herman |first1=Edwin |url=https://openstax.org/details/books/calculus-volume-1 |title=Calculus |volume=1 |last2=Strang |first2=Gilbert |date=2017 |publisher=OpenStax |isbn=978-1-938168-02-4 |location=Houston, Texas |oclc=1022848630 |display-authors=0 |access-date=26 July 2022 |archive-date=23 September 2022 |archive-url=https://web.archive.org/web/20220923230919/https://openstax.org/details/books/calculus-volume-1 |url-status=live  |pages=454–459}}</ref>

The sum of 0 numbers (the ''[[Empty sum|empty sum]]'') is 0, and the product of 0 numbers (the ''[[Empty product|empty product]]'') is 1. The [[Factorial|factorial]] 0! evaluates to 1, as a special case of the empty product.<ref name=gkp>{{cite book|first1=Ronald L.|last1=Graham|first2=Donald E.|last2=Knuth|first3=Oren|last3=Patashnik|date=1988|title=Concrete Mathematics|publisher=Addison-Wesley|location=Reading, MA|isbn=0-201-14236-8|title-link=Concrete Mathematics|page=111}}</ref>

===Other uses in mathematics===
[[File:Nullset.svg|thumb|100px|The empty set has zero elements]]

The role of 0 as the smallest counting number can be generalized or extended in various ways.  In [[Set theory|set theory]], 0 is the [[Cardinality|cardinality]] of the [[Empty set|empty set]]: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is ''[[Philosophy:Definition|defined]]'' to be the empty set.{{sfn|Cheng|2017|p=60}} When this is done, the empty set is the [[Von Neumann cardinal assignment|von Neumann cardinal assignment]] for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.

Also in set theory, 0 is the lowest [[Ordinal number|ordinal number]], corresponding to the empty set viewed as a [[Well-order|well-ordered set]].  In [[Order theory|order theory]] (and especially its subfield lattice theory), 0 may denote the least element of a [[Lattice (order)|lattice]] or other [[Partially ordered set|partially ordered set]].

The role of 0 as additive identity generalizes beyond elementary algebra.  In [[Abstract algebra|abstract algebra]], 0 is commonly used to denote a [[Zero element|zero element]], which is the [[Identity element|identity element]] for addition (if defined on the structure under consideration) and an [[Absorbing element|absorbing element]] for multiplication (if defined).  (Such elements may also be called [[Zero element|zero element]]s.)  Examples include identity elements of [[Additive group|additive group]]s and [[Vector space|vector space]]s.  Another example is the '''zero function''' (or '''zero map''') on a domain {{mvar|D}}.  This is the [[Constant function|constant function]] with 0 as its only possible output value, that is, it is the function {{mvar|f}} defined by {{math|''f''(''x'') {{=}} 0}} for all {{mvar|x}} in {{mvar|D}}.  As a function from the real numbers to the real numbers, the zero function is the only function that is both even and odd.

The number 0 is also used in several other ways within various branches of mathematics:
* A ''[[Zero of a function|zero of a function]]'' ''f'' is a point ''x'' in the domain of the function such that {{math|''f''(''x'') {{=}} 0}}.
* In propositional logic, 0 may be used to denote the [[Philosophy:Truth value|truth value]] false.
* In [[Probability theory|probability theory]], 0 is the smallest allowed value for the probability of any event.{{sfn|Kardar|2007|p=35}}
* [[Category theory]] introduces the idea of a zero object, often denoted 0, and the related concept of [[Zero morphism|zero morphism]]s, which generalize the zero function.<ref>{{cite book|last=Riehl |first=Emily |title=Category Theory in Context |page=103 |url=https://math.jhu.edu/~eriehl/context/ |publisher=Dover |year=2016 |isbn=978-0-486-80903-8}}</ref>

==Physics==
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an [[Physics:Thermodynamic temperature|absolute temperature]] (typically measured in [[Kelvin|kelvin]]s), [[Physics:Absolute zero|zero]] is the lowest possible value. ([[Physics:Negative temperature|Negative temperature]]s can be defined for some physical systems, but negative-temperature systems are not actually colder.) This is in contrast to temperatures on the Celsius scale, for example, where zero is arbitrarily defined to be at the [[Physics:Melting point|freezing point]] of water.<ref>{{cite book|first1=Andrew |last1=Rex |first2=C. B. P. |last2=Finn |title=Finn's Thermal Physics |edition=3rd |publisher=CRC Press |year=2017 |isbn=978-1-4987-1887-5 |pages=8–16}}</ref>{{sfn|Kardar|2007|pp=4–5,103–104}} Measuring sound intensity in [[Physics:Decibel|decibel]]s or [[Physics:Phon|phon]]s, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In [[Physics:Physics|physics]], the [[Physics:Zero-point energy|zero-point energy]] is the lowest possible energy that a [[Physics:Quantum mechanics|quantum mechanical]] [[Physics:Physical system|physical system]] may possess and is the energy of the [[Physics:Stationary state|ground state]] of the system.

==Computer science==
Modern computers store information in [[Binary code|binary]], that is, using an "alphabet" that contains only two symbols, usually chosen to be "0" and "1". Binary coding is convenient for [[Engineering:Digital electronics|digital electronics]], where "0" and "1" can stand for the absence or presence of electrical current in a wire.{{sfn|Woodford|2006|p=9}} [[Computer programming|Computer programmers]] typically use [[High-level programming language|high-level programming language]]s that are more easily intelligible to humans than the [[Machine code|binary instructions]] that are directly executed by the [[Central processing unit|central processing unit]]. 0 plays various important roles in high-level languages. For example, a [[Boolean data type|Boolean variable]] stores a value that is either ''true'' or ''false,'' and 0 is often the numerical representation of ''false.''{{sfn|Hill|2020|p=20}}

0 also plays a role in [[Array (data type)|array]] indexing. The most common practice throughout human history has been to start counting at one, and this is the practice in early classic programming languages such as [[Fortran]] and [[COBOL]]. However, in the late 1950s LISP introduced [[Zero-based numbering|zero-based numbering]] for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array of ''n'' items the sequence of array indices runs from 0 to {{nowrap|''n''−1}}.{{citation needed|date=December 2023}}

There can be confusion between 0- and 1-based indexing; for example, Java's JDBC indexes parameters from 1 although [[Java (programming language)|Java]] itself uses 0-based indexing.<ref>{{Cite web |title=ResultSet (Java Platform SE 8 ) |url=https://docs.oracle.com/javase/8/docs/api/java/sql/ResultSet.html |access-date=2022-05-09 |website=docs.oracle.com |archive-date=9 May 2022 |archive-url=https://web.archive.org/web/20220509185749/https://docs.oracle.com/javase/8/docs/api/java/sql/ResultSet.html |url-status=live }}</ref>

In C, a [[Byte|byte]] containing the value 0 serves to indicate where a [[String (computer science)|string]] of characters ends. Also, 0 is a standard way to refer to a [[Null pointer|null pointer]] in code.<ref>{{cite book|last=Reese |first=Richard M. |title=Understanding and Using C Pointers: Core Techniques for Memory Management |year=2013 |isbn=978-1-449-34455-9 |publisher=O'Reilly Media |url=https://books.google.com/books?id=-U155tRMLJgC&dq=C%20%22null%20pointer%22%200&pg=PT26}}</ref>

In databases, it is possible for a field not to have a value. It is then said to have a null value.<ref>{{Cite book |last1=Wu |first1=X. |url=https://books.google.com/books?id=SdLsCgAAQBAJ&q=%C2%A0In+databases%2C+it+is+possible+for+a+field+not+to+have+a+value+%28null%29&pg=PT197 |title=Knowledge-Base Assisted Database Retrieval Systems |last2=Ichikawa |first2=T. |last3=Cercone |first3=N. |date=25 October 1996 |publisher=World Scientific |isbn=978-981-4501-75-0 |language=en |access-date=7 November 2020 |archive-date=31 March 2022 |archive-url=https://web.archive.org/web/20220331032618/https://books.google.com/books?id=SdLsCgAAQBAJ&q=%C2%A0In+databases%2C+it+is+possible+for+a+field+not+to+have+a+value+%28null%29&pg=PT197 |url-status=live }}</ref> For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either ''true'' or ''false'', but it can be ''undetermined''. Any computation including a null value delivers a null result.<ref>{{cite web |url= https://www.ibm.com/docs/en/rbd/9.5.1?topic=parts-null-values-nullable-type |title= Null values and the nullable type |author= <!--Not stated--> |date= 12 December 2018 |website= IBM |access-date= 23 November 2021 |quote= In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service function. |archive-date= 23 November 2021 |archive-url= https://web.archive.org/web/20211123185142/https://www.ibm.com/docs/en/rbd/9.5.1?topic=parts-null-values-nullable-type |url-status= live }}</ref>

In mathematics, there is no "positive zero" or "negative zero" distinct from zero; both −0 and +0 represent exactly the same number. However, in some computer hardware [[Signed number representations|signed number representations]], zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives. This kind of dual representation is known as [[Signed zero|signed zero]], with the latter form sometimes called negative zero. These representations include the signed magnitude and [[Ones' complement|ones' complement]] binary integer representations (but not the [[Two's complement|two's complement]] binary form used in most modern computers), and most [[Floating-point arithmetic|floating-point]] number representations (such as IEEE 754 and [[IBM hexadecimal floating-point|IBM S/390]] floating-point formats).{{citation needed|date=December 2023}}

An epoch, in computing terminology, is the date and time associated with a zero timestamp. The Unix epoch begins the midnight before the first of January 1970.<ref>Paul DuBois.
[https://books.google.com/books?id=lFsaBAAAQBAJ "MySQL Cookbook: Solutions for Database Developers and Administrators"] {{Webarchive|url=https://web.archive.org/web/20170224134429/https://books.google.com/books?id=lFsaBAAAQBAJ |date=24 February 2017 }}
2014.
p. 204.</ref><ref>Arnold Robbins; Nelson Beebe.
[https://books.google.com/books?id=J9WbAgAAQBAJ "Classic Shell Scripting"] {{Webarchive|url=https://web.archive.org/web/20170224134147/https://books.google.com/books?id=J9WbAgAAQBAJ |date=24 February 2017 }}.
2005.
p. 274</ref><ref>Iztok Fajfar.
[https://books.google.com/books?id=eHq9CgAAQBAJ "Start Programming Using HTML, CSS, and JavaScript"] {{Webarchive|url=https://web.archive.org/web/20170224134155/https://books.google.com/books?id=eHq9CgAAQBAJ |date=24 February 2017 }}.
2015.
p. 160.</ref> The [[Software:Classic Mac OS|Classic Mac OS]] epoch and [[Software:Palm OS|Palm OS]] epoch begin the midnight before the first of January 1904.<ref>Darren R. Hayes.
[https://books.google.com/books?id=0qPfBQAAQBAJ "A Practical Guide to Computer Forensics Investigations"] {{Webarchive|url=https://web.archive.org/web/20170224134341/https://books.google.com/books?id=0qPfBQAAQBAJ |date=24 February 2017 }}.
2014.
p. 399</ref>

Many [[Application programming interface|APIs]] and [[Operating system|operating system]]s that require applications to return an integer value as an [[Exit status|exit status]] typically use zero to indicate success and non-zero values to indicate specific [[Software:Error code|error]] or warning conditions.{{citation needed|date=December 2023}}

Programmers often use a [[Slashed zero#Usage|slashed zero]] to avoid confusion with the letter "O".<ref>{{Cite web |date=18 August 2010 |title=Font Survey: 42 of the Best Monospaced Programming Fonts |url=https://www.codeproject.com/Articles/30040/Font-Survey-42-of-the-Best-Monospaced-Programming |url-status=live |access-date=22 July 2021 |website=codeproject.com|archive-url=https://web.archive.org/web/20120124051919/http://www.codeproject.com:80/Articles/30040/Font-Survey-42-of-the-Best-Monospaced-Programming |archive-date=24 January 2012 }}</ref>

== Other fields ==
===Biology===
In [[Biology:Zoology|comparative zoology]] and [[Cognitive science|cognitive science]], recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the [[Biology:Evolution|evolution]] of species.<ref>{{ cite periodical| last=Cepelewicz|first= Jordana   |url=https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/  | title=Animals Count and Use Zero. How Far Does Their Number Sense Go? |archive-url=https://web.archive.org/web/20210818042926/https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/ |archive-date=18 August 2021 | magazine= Quanta Magazine|date= August 9, 2021}}</ref>

===Dating systems===
{{Main|Year zero}}
In the [[History:Before Christ|BC]] [[History:Calendar era|calendar era]], the year 1{{nbsp}}BC is the first year before AD{{nbsp}}1; there is not a [[Year zero|year zero]]. By contrast, in [[Astronomy:Astronomical year numbering|astronomical year numbering]], the year 1{{nbsp}}BC is numbered 0, the year 2{{nbsp}}BC is numbered −1, and so forth.<ref>{{Cite book |last=Steel |first=Duncan |url=https://archive.org/details/markingtimeepicq00stee_0/page/113 |title=Marking Time: The epic quest to invent the perfect calendar |publisher=John Wiley & Sons |year=2000 |isbn=978-0-471-29827-4 |page=[https://archive.org/details/markingtimeepicq00stee_0/page/113 113] |quote=In the B.C./A.D. scheme there is no year zero. After 31 December 1{{nbsp}}BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers. | url-access=registration|oclc =1135427740 }}</ref>

==See also==
* Grammatical number
* [[Mathematical constant]]
* [[Number theory]]
* [[Peano axioms]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

==Bibliography==
{{Refbegin}}
* {{Cite book |last=Aczel |first=Amir D. |title=Finding Zero |publisher=Palgrave Macmillan |year=2015 |isbn=978-1-137-27984-2 |location=New York |url=https://archive.org/details/findingzeromathe0000acze}}
* {{Cite book |last=Asimov |first=Isaac |title=Asimov on Numbers |publisher=Pocket Books |year=1978 |isbn=978-0-671-82134-0 |location=New York |chapter=Nothing Counts |oclc=1105483009 |url=https://archive.org/details/asimovonnumbers00isaa}}
* {{Cite book |last=Barrow |first=John D. |title=The Book of Nothing |publisher=Vintage |year=2001 |isbn=0-09-928845-1 |url=https://archive.org/details/bookofnothing0000barr}}
* {{Cite book |first=Eugenia |last=Cheng |title=Beyond Infinity: An Expedition to the Outer Limits of Mathematics |publisher=Basic Books |year=2017 |isbn=978-1-5416-4413-7}}
* {{cite book|first=Mehran |last=Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-87342-0}}
* {{cite book|first=David |last=Reimer |title=Count Like an Egyptian |year=2014 |publisher=Princeton University Press |isbn=978-0-691-16012-2 }}
* {{Cite book |last=Woodford |first=Chris|url=https://books.google.com/books?id=My7Zr0aP2L8C&pg=PA9 |title=Digital Technology |date=2006 |publisher=Evans Brothers |isbn=978-0-237-52725-9 |access-date=24 March 2016 |archive-date=17 August 2019 |archive-url=https://web.archive.org/web/20190817150242/https://books.google.com/books?id=My7Zr0aP2L8C&pg=PA9 |url-status=live }}
* {{Cite book |last=Hill |first=Christian |title=Learning Scientific Programming with Python |publisher=Cambridge University Press |year=2020 |isbn= 978-1-10707541-2 |edition=2nd}}
{{Refend}}

=== Historical studies ===
{{Refbegin}}
* {{Cite book |last=Bourbaki |first=Nicolas |title=Elements of the History of Mathematics |publisher=Springer-Verlag |year=1998 |isbn=3-540-64767-8 |location=Berlin, Heidelberg, and New York |}}
* {{Cite book |last=Diehl |first=Richard A. |title=The Olmecs: America's First Civilization |publisher=Thames & Hudson |year=2004 |location=London |isbn=978-0-500-28503-9}}
* {{Cite book |last=Ifrah |first=Georges |title=The Universal History of Numbers: From Prehistory to the Invention of the Computer |publisher=Wiley |year=2000 |isbn=0-471-39340-1}}
* {{Cite book |last=Kaplan |first=Robert |title=The Nothing That Is: A Natural History of Zero |publisher=Oxford University Press |year=2000 |isbn=978-0-198-02945-8}}
* {{Cite book |last=Seife |first=Charles |title=Zero: The Biography of a Dangerous Idea |publisher=Penguin USA |year=2000 |isbn=0-14-029647-6 |}}
{{Refend}}

==External links==
* [http://www.huffingtonpost.com/amir-aczel/worlds-first-zero_b_3276709.html Searching for the World's First Zero]
* [https://web.archive.org/web/20081204042054/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html A History of Zero]
* [http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM Zero Saga]
* [https://web.archive.org/web/20141009100628/http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra]
* [[Biography:Edsger W. Dijkstra|Edsger W. Dijkstra]]: [http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF Why numbering should start at zero], EWD831 (PDF of a handwritten manuscript)
* {{MathWorld | urlname=0 | title=0}}
* {{Wikisource-inline|list=
** {{Cite Americana|wstitle=Zero |short=x |noicon=x}}
}}

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