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{{distinguish|O}}{{Restructure|date=January 2025}}{{inline}}
{{Infobox number
| number = 0
| cardinal = zero, {{nowrap|oh {{IPAc-en|ˈ|oʊ}}}}, nought, naught, none, nothing, null, nil, nada, zip, zilch, love 
| ordinal text = zeroth, noughth
| divisor = all numbers except itself
| roman unicode = Ↄ, ↄ
| lang1 = [[Arabic numerals|Arabic]] | lang1 symbol = {{resize|150%|٠, ۟, ۠}}
| lang2 = [[Bengali language|Bengali]]| lang2 symbol = {{resize|150%|০}}
| lang3 = [[Devanagari|Devanāgarī]]| lang3 symbol = {{resize|150%|०}}
| lang4 = [[Chinese numerals|Chinese]]| lang4 symbol =  零, 2000
| lang5 = [[Japanese numerals|Japanese]]| lang5 symbol =  零, 2000
| lang6 = [[Khmer numerals|Khmer]]| lang6 symbol = 2000
| lang7 = [[Thai numerals|Thai]]| lang7 symbol = 2000}}
'''Zero''' ('''0, ⁰''') is a [[wikt:unique|unique]] [[number]].<ref>{{cite book
|title=Principles of mathematics
|edition=2
|first1=Bertrand
|last1=Russell
|publisher=Forgotten Books
|year=1942
|isbn=1-4400-5416-9
|page=125
|url=https://books.google.com/books?id=63ooitcP2osC}}, [https://books.google.com/books?id=63ooitcP2osC&pg=PA125 Chapter 14, page 125]</ref> If there are zero things, then there is nothing at all. For example, if a person has zero [[Hat|hats]], that means they do not have any hats. The [[Roman numeral]] for zero is '''Ↄ''' or '''’'''.
==Symbol==
The symbol for the number zero is "0". It is the [[additive identity]] of common numbers.<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-09-22|website=Math Vault|language=en-US}}</ref> This means that if a number is added to 0, then that number would remain unchanged.<ref>{{Cite web|title=Zero|url=https://www.mathsisfun.com/numbers/zero.html|access-date=2020-09-22|website=www.mathsisfun.com}}</ref>{{better source|date=December 2024}}
==Mathematics==
* Adding zero to a number always gives that number. For example, adding zero to three gives three. In [[symbols]]:
 3 + 0 = 3
* Subtracting zero from a number always gives that number. For example, subtracting zero from three gives three. In symbols:
 3 − 0 = 3
* Subtracting a positive number from zero always makes that number negative (or, if a [[negative number]] is subtracted from zero, it makes the number positive). For example, subtracting three from zero gives negative three. In symbols:
 0 − 3 = −3
* Multiplying a number by zero always gives zero. For example, multiplying three by zero gives zero. In symbols:
 3 &times; 0 = 0
[[File:En-us-Zero.ogg|thumbnail|right|[[0 (number)]]]]
* Dividing zero by a different number always gives zero. For example, dividing zero by three gives zero. In symbols:
 0 ÷ 3 = 0
* [[division by zero|Dividing a number by zero]] has an undefined answer. For example, dividing three by zero has an undefined answer. In symbols:
 3 ÷ 0 has an undefined answer.
* Dividing zero by zero has an undefined answer. In symbols:
 0 ÷ 0 has an undefined answer.
The following table includes all of the above examples along with other operations in a condensed, generalized form (where ''x'' represents any number).
{| class="wikitable sortable"
!Operation
!Rule
!Example
|-
| Addition
|''x'' + 0 = ''x''
|3 + 0 = 3
|-
| rowspan="2" | Subtraction
|''x'' - 0 = ''x''
|3 - 0 = 3
|-
|0 - ''x'' = -''x''
|0 - 3 = -3
|-
| Multiplication
|''x'' × 0 = 0
|3 × 0 = 0
|-
| rowspan="2" | Division
|0 ÷ ''x'' = 0, when ''x'' ≠ 0
|0 ÷ 3 = 0
|-
|''x'' ÷ 0  is undefined
|3 ÷ 0 is undefined
|-
| rowspan="2" | Exponentiation
|0 ''<sup>x</sup>'' = 0, when x ≠ 0
|0<sup>3</sup> = 0
|-
|''x'' <sup>0</sup> = 1, when x ≠ 0
|3<sup>0</sup> = 1
|-
| Root
|√0 = 0
| 
|-
| Logarithm
|log<sub>''b''</sub>(0) is undefined
| 
|-
| Factorial
|0! = 1
| 
|-
| Sine
|sin 0º = 0
| 
|-
| Cosine
|cos 0º = 1
| 
|-
| Tangent
|tan 0º = 0
| 
|-
| Derivative
|0' = 0
| 
|-
| Integral
|∫ 0 d''x'' = 0 + ''C''
| 
|}

==History==
The [[Bakhshali manuscript]], discovered in 1881 near [[Peshawar]] has been identified as the earliest known text featuring the [[zero]] symbol (0), dating back to the 3rd or 4th century CE.<ref>{{Cite web|last=Agencies|date=2017-09-16|title=1,800-year-old dot is first ‘zero’, say researchers|url=https://www.dawn.com/news/1358036|access-date=2024-07-22|website=DAWN.COM|language=en}}</ref>{{better source|date=December 2024}} The [[Ancient Greece|ancient Greeks]] did not use zero as a [[number]], because they thought numbers represented [[Shape|shapes]]: <blockquote>To the [ancient] Greeks, who looked upon [[mathematics]] from a [[Geometry|geometric]] perspective, zero seemed absurd or unnecessary. When numbers or unknowns represented [[Length|lengths]], and squares represented [[Area|areas]], zero had no place. Why solve a problem that did not exist? If a length is zero, there is no [[line]]; if an area is zero, there is no object.<ref>{{Cite book|title=Mathematics: From Algebra to Algorithms, Adventures in Numbers|last=Willers|first=Michael|publisher=New Burlington Books|year=2021|isbn=978-1-80242-020-3|location=London, UK|pages=104}}</ref></blockquote>The idea of zero was first thought about in [[Babylon]], [[Indian subcontinent]] and in [[Central America]] at different times. Some places and countries did not know about zero, which may have made it harder for those people to do [[mathematics]]. For example, the year after [[1 BC]] is [[AD 1]] (there is no year zero). In India, zero was theorized in the seventh century by the [[Mathematician]] [[Aryabhata]].{{cn|date=December 2024}}

Over hundreds of years, the idea of zero was passed from country to country [[Greece]], [[Persia]] and the [[Arab world]]. The [[Europe]]ans learned about zero from the [[Arab]]s, and stopped using [[Roman numeral|Roman]] math. This is why numbers are called [[Hindu–Arabic numeral system|"Arabic numerals]]".{{cn|date=December 2024}}

==Computer science==
Zero is almost never used as a place number ([[Ordinal number]]). This means that it is not used like 1, 2, or 3 to indicate the order, or place, of something, like 1st, 2nd, or 3rd. An exception to this is seen in many programming languages.{{cn|date=December 2024}} Some other things about zero:<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Zero|url=https://mathworld.wolfram.com/Zero.html#:~:text=Zero%20is%20the%20integer%20denoted,as%20%22a%20zero%20of%20.%22|access-date=2020-09-22|website=mathworld.wolfram.com|language=en}}</ref>
* The number zero is a [[whole number]] (counting number).
* The number zero is not a [[positive number]].
* The number zero is not a [[negative number]], either.

Any number [[Division (mathematics)|divided]] by itself equals one, except if that number is zero. In symbols:

0 ÷ 0 = [[Division by zero|undefined]]
===Other applications===
In [[time]], zero means "[[now]]". For example, when a person is [[counting]] down the time to the start of something, such as a foot race or when a [[rocket]] takes off, the count is: "three, two, one, zero (or ''go'')". Zero is the exact time of the start of the race or when the rocket takes off into the sky.{{cn|date=December 2024}}

==  0 as a number ==

===Definition===
0 is the [[integer]] that precedes the positive [[1 (number)|1]], and follows [[-1 (number)|−1]]. In most numerical systems, 0 was identified before the idea of "negative integers" was accepted. It means "courageous one" in [[Hieroglyph|hieroglyphics]].{{cn|date=January 2025}} 

Zero is a number which means an amount of null size; that is, if the number of brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the numbers of pieces in two piles is zero, it means the two piles have an equal number of pieces.{{cn|date=January 2025}} 

Before counting starts, the result can be assumed to be zero; that is the number of items counted before one counts the first item, and counting the first item brings the result to one. And if there are no items to be counted, zero remains the final result.{{cn|date=January 2025}}
===Debates===
====Is zero a number ?====
While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing that one cannot have zero of something. Others say that if one has a bank balance of zero, one has a specific quantity of money in that account, namely none. It is that latter view which is accepted by mathematicians and most others.
====Is zero a natural number?====
A debate asking if '''0 is or not a natural number''' has been discussed over time.<ref>{{Cite journal|last=Harremoës|first=Peter|date=2011-02-02|title=Is Zero a Natural Number?|arxiv=1102.0418 }}</ref><ref>{{Cite journal|last=Fitch|first=Frederic B.|date=1949|title=On Natural Numbers, Integers, and Rationals|url=https://www.jstor.org/stable/2266507|journal=The Journal of Symbolic Logic|volume=14|issue=2|pages=81–84|doi=10.2307/2266507|jstor=2266507 |s2cid=27232089 |issn=0022-4812}}</ref>

The [[Encyclopædia Britannica]] defines 0 as a [[natural number]].<ref>{{Cite web|title=natural number {{!}} mathematics {{!}} Britannica|url=https://www.britannica.com/science/natural-number|access-date=2022-10-14|website=www.britannica.com|language=en}}</ref> [[Wolfram Research|Wolfram MathWorld´]]<nowiki/>says it isn't.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Natural Number|url=https://mathworld.wolfram.com/|access-date=2022-10-14|website=mathworld.wolfram.com|language=en}}</ref> The ''On-Line Encyclopedia of Integer Sequences'' objects.<ref>{{Cite web|title=A000027 - OEIS|url=https://oeis.org/A000027|access-date=2022-10-14|website=oeis.org}}</ref>{{dubious|date=January 2025}} ''The Princeton Companion to Mathematics'' defines 0 as a natural number.<ref>{{Cite book|url=https://books.google.com/books?id=ZOfUsvemJDMC&dq=is+0+a+natural+number+princeton&pg=PA17|title=The Princeton Companion to Mathematics|last1=Gowers|first1=Timothy|last2=Barrow-Green|first2=June|last3=Leader|first3=Imre|date=2010-07-18|publisher=Princeton University Press|isbn=978-1-4008-3039-8|language=en}}</ref>

Although the [[International Baccalaureate]] (IB) mathematics curriculum sees 0 as a natural number,<ref>{{Cite web|title=IB Maths Studies/Number and Algebra - Wikibooks, open books for an open world|url=https://en.wikibooks.org/wiki/IB_Maths_Studies/Number_and_Algebra|access-date=2022-10-14|website=en.wikibooks.org|language=en}}</ref> the ''[[Oxford University Press|Oxford Dictionary]]'' states,<ref>{{Cite web|title=Natural NUmber|url=https://www.oxfordlearnersdictionaries.com/definition/english/natural-number|website=Oxford Learner's Dictionary}}</ref>
{{Blockquote|text=a positive whole number such as 1, 2, or 3, and sometimes also zero.}}
===As year===
There was no [[year zero]] between 1 [[Anno Domini|BC]] and 1 [[Anno Domini|AD]]. More specifically, almost all [[Historian|historians]] leave out the ''year zero'' from the proleptic [[Gregorian calendar|proleptic Gregorian]] and [[Julian calendar]], but [[Astronomer|astronomers]] include it in these same calendars. However, the phrase ''Year Zero'' may be used to describe any event considered so important, that someone might want to start counting years all over again from zero.{{cn|date=January 2025}}

== 0 as a numeral ==
[[File:Text figures 036.svg|71px|right]]
The modern numeral 0 is normally written as a circle or (rounded) rectangle. In old-style fonts with text figures, 0 is usually the same height as a lowercase x.
{| style="clear: right" align="right"
| [[File:7-segment cdeg.svg|right|80px|Unusual appearance of the digit zero on seven-segment displays]] 
| [[File:7-segment abcdef.svg|right|80px|Usual appearance of the digit zero on seven-segment displays]]
|}
On the seven-segment displays of [[Calculator|calculators]], [[Watch|watches]] etc., 0 is usually written with six-line segments, though on some historical calculator models, it was written with four line segments. The four-segment 0 is not common. The ''number'' zero, as in the "zero brothers" example above, is not the same as the ''numeral'' or ''digit'' zero, used in [[Numeral system|numeral systems]] using [[positional notation]]. Successive positions of [[Digit|digits]] have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. A zero digit is not always necessary in a different positional number system. Something called bijective numeration{{clarification|date=January 2025}} is a possible example of a system without zeroes.{{cn|date=January 2025}}

==As numerical digit==
0 is also used as a [[numerical digit]] used to represent that number in [[numeral]]s. It is used to hold the place of that digit, because correct placing of digits affects a numeral's value. Examples:
* In the numeral 10, which stands for one times ten and zero units (or ones).
* In the numeral 100, which stands for one times a hundred plus zero tens plus zero units.

==Telling zero and the letter O apart==
[[File:Zero o comparison.svg|right|A comparison of the letter O and the number 0]]The number 0 and the letter O are both round, though of different widths. The difference is important on a computer. For one thing, a computer will not do arithmetic with the letter O, because it does not know that it should have been a zero.{{cn|date=December 2024}}

The oval-shaped zero and circular letter [[O]] came into use together on modern character displays. The zero with a dot in the centre seems to have begun as a choice on IBM 3270 controllers (this has the problem that it looks like the [[Greek alphabet|Greek letter]] [[Theta (letter)|theta]]). 

The slashed zero, looking like the letter O with a diagonal line drawn inside it, is used in old-style ASCII graphic sets that came from the default typewheel on the well-known ASR-33 Teletype. This format causes problems because it looks like the symbol <math>\emptyset</math>, representing the [[empty set]],<ref>{{Cite web|date=2020-04-11|title=Comprehensive List of Set Theory Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/set-theory-symbols/|access-date=2020-09-22|website=Math Vault|language=en-US}}</ref> as well as for certain Scandinavian languages which use Ø as a letter.
 
The rule which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more of a problem for [[Scandinavia|Scandinavians]], because it looks like two of their letters at the same time. Some Burroughs/Unisys computers display a zero with a backwards slash.

Yet, another convention common on early line [[computer printer|printers]] left zero without any extra dots or slashes but added a tail or hook to the letter O so that it resembled an inverted [[Q]] or cursive capital letter O.{{cn|date=December 2024}}[[File:Deutsches Kfz-Kennzeichen für Behördenfahrzeuge (Nummernbereich 3).jpg|thumbnail|right|A German licence plate showing zeroes]]The letters used on some European number plates for [[Automobile|cars]] make the two symbols look different. This is done by making the zero rather egg-shaped and the O more circular, but most of all by cutting open the zero on the upper right side, so the circle is not closed any more (as in [[:Image:Deutsches Kfz-Kennzeichen für Behördenfahrzeuge (Nummernbereich 3).jpg|German plates]]). The style of letters chosen is called ''[[:Image:FE-Schrift.svg|fälschungserschwerende Schrift]]'' (abbr.: ''FE Schrift''), meaning "script which is harder to falsify". 

However, those used in the [[United Kingdom]] do not make the letter o and the number 0 look different from each other, because there can never be any mistake if the letters are correctly spaced. In paper writing, one does not have to make the 0 (zero) and O (letter O) look different at all. Or you may add a slash across the zero in order to show the difference.

==Zeroes of a function==
{{Further|Function (mathematics)}}
If the function ''f''(''x'') = 0, then ''x'' is called a zero (or root) of the function ''f''.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Root|url=https://mathworld.wolfram.com/Root.html|access-date=2020-09-22|website=mathworld.wolfram.com|language=en}}</ref> For example, if the function ''f''(''x'') is ''x''<sup>2</sup> − 1, then the zeroes of the function are +1 and −1, because ''f''(+1) = (+1)<sup>2</sup> − 1 = 0, and ''f''(−1) = (−1)<sup>2</sup> − 1 = 0.

Zeroes of a function are used because they are another way to talk about solving an [[equation]], which is a main goal in [[algebra]]. If we want to solve an equation like ''x''<sup>2</sup> = 1, then we can subtract the right-hand side of the equation from both sides, in this case 1. Whatever we get on the left-hand side, in this case ''x''<sup>2</sup> − 1, can be called a function  ''f''(x). The right-hand side has to be zero, because we subtracted it from itself. So ''f''(x) = 0. Finding the zeroes of this function is the same as solving this equation. In the paragraph before, the zeroes of this function are +1 and −1, so they are the solutions of this equation. We got this equation by subtracting the same thing from both sides, so we also have solutions to the equation we started with, in this case ''x''<sup>2</sup> = 1. More generally, if we could find zeroes of functions, we could solve any equation.

==Related pages==
* [[Limit (mathematics)|Limit]]
* [[Infinity]]
* [[Set theory]]
* [[Real analysis]]
* [[Number theory]]
* [[Null vector]]
* [[Division by zero]]
* [[Bakhshali manuscript]]
* [[Mathematical analysis]]

== References ==

=== Citations ===
{{reflist}}

=== Sources ===
* Barrow, John D. (2001) ''The Book of Nothing'', Vintage. {{ISBN|0-09-928845-1}}.
* Diehl, Richard A. (2004) ''The Olmecs: America's First Civilization'', Thames & Hudson, London.
* Ifrah, Georges (2000) ''The Universal History of Numbers: From Prehistory to the Invention of the Computer'', Wiley. {{ISBN|0-471-39340-1}}.
* Kaplan, Robert (2000) ''The Nothing That Is: A Natural History of Zero'', Oxford: Oxford University Press.
* Seife, Charles (2000) ''Zero: The Biography of a Dangerous Idea'', Penguin USA (Paper). {{ISBN|0-14-029647-6}}.
* Tapan Kumar Das Gupta: "Der Ursprung des neuzeitlichen Zahlensystems - Entstehung und Verbreitung." Norderstedt 2013. {{ISBN|978-3-7322-4809-4}}.

== Other websites ==
* [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html A History of Zero] {{Webarchive|url=https://web.archive.org/web/20081204042054/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html |date=2008-12-04 }}
* [http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM Zero Saga]
* [http://www.neo-tech.com/zero/part6.html The Discovery of the Zero] {{Webarchive|url=https://web.archive.org/web/20090526045556/http://www.neo-tech.com/zero/part6.html |date=2009-05-26 }}
* [http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra] {{Webarchive|url=https://web.archive.org/web/20141009100628/http://www.ucs.louisiana.edu/~sxw8045/history.htm |date=2014-10-09 }}
* [http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF Why numbering should start at zero] by Edsger Dijkstra
* [http://www.purl.org/stefan_ram/pub/zero Numbering Starts with Zero]{{Dead link|date=May 2021 |bot=InternetArchiveBot |fix-attempted=yes }} same topic as Dijkstra's article, including some more aspects

[[Category:Integers]]