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{{Dablink|"Zero" redirects here. For other uses, see [[Zero (disambiguation)]]. "Nought" redirects here. For the Rogue Trooper characters, see [[Norts]].}}

{| class="infobox nowraplinks" style="width: 20em;"
|-
! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 0
|-
| colspan="2" | {{numbers (digits)}}
|-
| [[Cardinal number|Cardinal]]
| 0, zero, "oh" ({{IPA-en|ˈoʊ|pron}}), nought, naught, nil, null
|-
| [[Ordinal number (linguistics)|Ordinal]] || 0th, [[zeroth]], noughth
|-
| [[Factorization]] || <math> 0 </math>
|-
| [[Divisor]]s || all numbers
|-
| [[Roman numerals|Roman numeral]] || N/A
|-
| [[Arabic language|Arabic]] || style="font-size:150%" | ٠
|-
| [[Bengali language|Bengali]] || style="font-size:150%" | ০
|-
| [[Devanagari|Devanāgarī]] || style="font-size:150%" | ०
|-
| [[Chinese language|Chinese]] || 〇，零
|-
| [[Japanese numerals|Japanese numeral]] || 〇，零
|-
| [[Khmer numerals|Khmer]] || ០
|-
| [[Thai numerals|Thai]] || ๐
|-
| [[Binary numeral system|Binary]] || 0
|-
| [[Octal]] || 0
|-
| [[Duodecimal]] || 0
|-
| [[Hexadecimal]] || 0
|}

'''0''' ('''zero''') is both a [[number]] and the [[numerical digit]] used to represent that number in [[numeral system|numerals]]. It plays a central role in [[mathematics]] as the [[additive identity]] of the [[integer]]s, [[real number]]s, and many other [[algebra]]ic structures. As a digit, zero is used as a placeholder in [[positional notation|place value systems]]. In the [[English language]], zero may also be called '''oh''', '''null''', '''nil''', or '''nought'''.<ref>{{cite book|editor=Catherine Soanes| others=Maurice Waite, Sara Hawker|title=The Oxford Dictionary, Thesaurus and Wordpower Guide| format=Hardback|accessdate=2007-12-21|edition=2nd|year=2001|publisher=[[Oxford University Press]]|location=[[New York|New York, United States]]|isbn=978-0-19-860393-3}}</ref>

==0 as a number==
'''0''' is the [[integer]] preceding [[1 (number)|1]]. In most systems, 0 was identified before the idea of negative things that go lower than zero was accepted. [[Evenness of zero|Zero is an even number]].<ref>[[Lemma (mathematics)|Lemma]] B.2.2, ''The integer 0 is even and is not odd'', in {{cite book|first=Robert C.|last=Penner|year=1999|title=Discrete Mathematics: Proof Techniques and Mathematical Structures| publisher=World Scientific|isbn=9810240880|pages=34}}</ref> 0 is neither positive nor negative. By some definitions 0 is also a [[natural number]], and then the only natural number not to be positive.

Zero is a number which quantifies a count or an amount of [[Empty set|null]] size. Almost all [[historian]]s omit the [[year zero]] from the [[proleptic Gregorian calendar|proleptic Gregorian]] and [[proleptic Julian calendar|Julian calendar]]s, but [[astronomer]]s include it in these same calendars. However, the phrase [[Year Zero (political notion)|Year Zero]] may be used to describe any event considered so significant that it serves as a new base point in time.

==0 as a digit==
[[Image:Text figures 036.svg|71px|left]]
The modern numerical digit 0 is usually written as a circle, an ellipse, or a rounded rectangle. In most modern [[typeface]]s, the height of the 0 character is the same as the other digits. However, in typefaces with [[text figures]], the character is often shorter ([[x-height]]).

{| align="right"
| [[Image:7-segment cdeg.svg|80px|Unusual smaller appearance of zero on seven-segment displays]]
| [[Image:7-segment abcdef.svg|80px|Usual appearance of zero on seven-segment displays]]
|}

On the [[seven-segment display]]s of calculators, watches, and household appliances, 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments.

The value, or ''number'', zero is not the same as the ''digit'' zero, used in [[numeral system]]s using [[positional notation]]. Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system, for example, in the number 02.

In rare instances, a leading 0 may distinguish a number. This appears in [[roulette]] in the United States, where '00' is distinct from '0' (a wager on '0' will not win if the ball lands in '00', and vice versa). Sports where competitors are numbered follow this as well; a [[stock car]] numbered '07' would be considered distinct from one numbered '7'. This is most common with single-digit numbers.

===Distinguishing the digit 0 from the letter O===
[[Image:Zero o comparison.svg|A comparison of the letter O and the number 0.]]

Traditionally, many print typefaces made the capital letter [[O]] more rounded than narrower, elliptical digit 0.<ref name="bemer" /> [[Typewriter]]s 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 [[Visual display unit|displays]].<ref name="bemer">R. W. Bemer. "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". ''Communications of the ACM'', Volume 10, Issue 8 (August 1967), pp.&nbsp;513–518.</ref>

The digit 0 with a dot in the centre seems to have originated as an option on [[IBM 3270]] displays. Its appearance has continued with the [[Microsoft Windows]] typeface [[Andalé Mono]]. One variation used a short vertical bar instead of the dot. This could be confused with the [[Greek alphabet|Greek letter]] [[Theta]] on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character and very little used anyway.

An alternative, the [[slashed zero]] (looking similar to the letter O except for the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style [[ASCII]] graphic sets descended from the default typewheel on the [[ASR-33 Teletype]]. This form is similar to the symbol <math>\emptyset</math>, or "∅" ([[Unicode]] character U+2205), representing the [[empty set]], as well as to the letter [[Ø]] used in several [[North Germanic languages|Scandinavian languages]]. Some [[Burroughs Corporation|Burroughs]]/[[Unisys]] equipment displays a digit 0 with a ''reversed'' slash.
The opposing convention that has the letter O ''with'' a slash and the digit 0 ''without'' was advocated by SHARE, a prominent [[IBM]] user group,<ref name="bemer"/> and recommended by IBM for writing [[Fortran|FORTRAN]] programs,<ref name="einarsson">Bo Einarsson and Yurij Shokin. ''Fortran 90 for the Fortran 77 Programmer''. [http://www.nsc.liu.se/~boein/f77to90/a7.html Appendix 7: "The historical development of Fortran"]</ref> and by a few other early mainframe makers; this is even more problematic for [[Scandinavia]]ns because it means two of their letters collide. Others advocated the opposite convention,<ref name="bemer" /> including IBM for writing [[Algol]] programs.<ref name="einarsson" /> Another convention used on some early [[line printer]]s left digit 0 unornamented but added a tail or hook to the capital O so that it resembled an inverted [[Q]] or cursive capital letter-O (<math>\mathcal O</math>).<ref name="bemer" />

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). The [[Texas Instruments TI-99/4A]] computer featured a more angular capital O and a more rounded digit 0, whereas others made the choice the other way around.
[[Image:Deutsches Kfz-Kennzeichen für Behördenfahrzeuge (Nummernbereich 3).jpg|thumb|German license plate with slit zeros]]

The typeface used on most [[Europe]]an [[vehicle registration plate]]s distinguishes the two symbols partially in this manner (having a more rectangular or wider shape for the capital O than the digit 0), but in several countries a further distinction is made by slitting open the digit 0 on the upper right side (as in [[Vehicle registration plates of Germany|German plates]] using the ''[[FE-Schrift|fälschungserschwerende Schrift]]'', "harder-to-falsify script").

Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. For example, [http://southwest.com/content/travel_center/retrieveCheckinDoc.html confirmation numbers] used by [[Southwest Airlines]] use only the capital letters O and I instead of the digits 0 and 1, while [[Canadian postal code]]s use only the digits 1 and 0 and never the capital letters O and I, although letters and numbers always alternate.

==Names==
{{main|Names for the number 0}}
The word "'''zero'''" came via [[French language|French]] ''zéro'' from [[Venetian language|Venetian]] ''zero'', which (together with ''[[wikt:cipher|cipher]]'') came via [[Italian language|Italian]] ''zefiro'' from Arabic صفر, ''ṣafira'' = "it was empty", ''ṣifr'' = "zero", "[[nothing]]", which was used to translate [[Sanskrit]] ''{{IAST|[[śūnyatā|śūnya]]}}'' ( शून्य ), meaning ''void'' or ''empty''.

Italian ''zefiro'' already meant "west wind" from Latin and Greek ''[[Anemoi|zephyrus]]''; this may have influenced the spelling when transcribing Arabic ''şifr''.<ref name="ifrah">Georges Ifrah. ''The Universal History of Numbers: From Prehistory to the Invention of the Computer''. Wiley (2000). ISBN 0-471-39340-1.</ref> The Italian mathematician [[Fibonacci]] (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Hindu decimal system to Europe, used the term ''zephyrum''. This became ''zefiro'' in Italian, which was contracted to ''zero'' in Venetian.

As the [[Hinduism|Hindu]] decimal zero and its new mathematics spread from the Arab world to Europe in the [[Middle Ages]], words derived from ''ṣifr'' and ''zephyrus'' came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a "... cifre en algorisme", i.e., an "arithmetical nothing"."<ref name="ifrah"/> From ''ṣifr'' also came French ''chiffre'' = "digit", "figure", "number", ''chiffrer'' = "to calculate or compute", ''chiffré'' = "encrypted". Today, the word in Arabic is still ''ṣifr'', and cognates of ''ṣifr'' are common in the languages of Europe and southwest Asia.

==History==
===Early history===
By the middle of the [[2nd millennium BCE]], the [[Babylonian mathematics]] had a sophisticated [[sexagesimal]] positional numeral system. The lack of a positional value (or zero) was indicated by a ''space'' between sexagesimal numerals. By [[300 BCE]], a punctuation symbol (two slanted wedges) was co-opted as a [[Free variables and bound variables|placeholder]] in the same [[Babylonian numerals|Babylonian system]]. In a tablet unearthed at [[Kish (Sumer)|Kish]] (dating from about 700 BCE), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.<ref name="multiref1">Kaplan, Robert. (2000). ''The Nothing That Is: A Natural History of Zero''. Oxford: Oxford University Press.</ref>

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

Records show that the [[Ancient Greece|ancient Greeks]] seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to [[philosophy|philosophical]] and, by the Medieval period, religious arguments about the nature and existence of zero and the [[vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in large part on the uncertain interpretation of zero.

The concept of zero as a number and not merely a symbol for separation is attributed to [[India]]
where by the [[9th century]] CE practical calculations were carried out using zero, which was treated like any other number, even in case of division.<ref name="bourbaki46">Bourbaki, Nicolas (1998). ''Elements of the History of Mathematics''. Berlin, Heidelberg, and New York: Springer-Verlag. 46. ISBN 3540647678.</ref><ref name="ebcal">''Britannica Concise Encyclopedia'' (2007), entry ''algebra'' </ref> The Indian scholar [[Pingala]] (circa [[5th century BCE|5th]]-[[2nd century BCE]]) used [[binary numeral system|binary numbers]] in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to [[Morse code]].<ref> [http://home.ica.net/~roymanju/Binary.htm Binary Numbers in Ancient India]</ref><ref> [http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers] (pdf, 145KB)</ref> He and his contemporary Indian scholars used the Sanskrit word ''[[Śūnyatā|śūnya]]'' to refer to zero or ''void''.

=== History of zero ===<!-- This section is linked from [[Olmec]] -->
[[Image:Estela C de Tres Zapotes.jpg|thumb|The back of Olmec Stela C from [[Tres Zapotes]], the second oldest Long Count date yet discovered. The numerals 7.16.6.16.18 translate to September, 32 BCE (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of [[Isthmian script|Epi-Olmec script]].]]

The [[Mesoamerican Long Count calendar]] developed in south-central [[Mexico]] and [[Central America]] required the use of zero as a place-holder within its [[vigesimal]] (base-20) positional numeral system. Many different glyphs, including this partial quatrefoil—[[Image:MAYA-g-num-0-inc-v1.svg]]—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 BCE.<ref>No long count date actually using the number 0 has been found before the 3rd century CE, 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> Since the eight earliest Long Count dates appear outside the Maya homeland,<ref>Diehl, p. 186</ref> it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the [[Olmec]]s. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BCE, several centuries before the earliest known Long Count dates.

Although zero became an integral part of [[Maya numerals]], it did not influence [[Old World]] numeral systems.

The use of a blank on a counting board to represent 0 dated back in India to 4th century BCE.<ref> Robert Temple, ''The Genius of China, A place for zero''; ISBN 1-85375-292-4</ref>

In [[China]], [[counting rods]] were used for calculation since the [[4th century BC|4th century BCE]]. Chinese mathematicians understood negative numbers and zero, though they had no symbol for the latter,<ref>{{Citation|title=Sangi o koeta otoko (The man who exceeded counting rods)| last=Wáng|first=Qīngxiáng|isbn=4-88595-226-3|publisher=Tōyō Shoten| place=Tokyo|year=1999}}</ref> until the work of [[Song Dynasty]] mathematician [[Qin Jiushao]] in 1247 established a symbol for zero in China.<ref>Needham, Joseph (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd. Page 43.</ref> ''[[The Nine Chapters on the Mathematical Art]]'', which was mainly composed in the [[1st century|1st century CE]], stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."<ref>The statement in Chinese, found in Chapter 8 of ''The Nine Chapters on the Mathematical Art'' is 正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。The word 無入 used here, for which ''zero'' is the standard translation by mathematical historians, literally means: ''no entry''. The full Chinese text can be found at [[:wikisource:zh:九章算術]].</ref>

By [[130]], [[Ptolemy]], influenced by [[Hipparchus]] and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic [[Greek numerals]]. Because it was used alone, not just as a placeholder, this [[Greek numerals#Hellenistic zero|Hellenistic zero]] was perhaps the first documented use of a ''number'' zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)&mdash;they were not used for the integral part of a number. In later [[Byzantine Empire|Byzantine]] manuscripts of Ptolemy's ''Syntaxis Mathematica'' (also known as the ''Almagest''), the Hellenistic zero had morphed into the Greek letter [[omicron]] (otherwise meaning 70).

Another zero was used in tables alongside [[Roman numerals#zero|Roman numerals]] by [[525]] (first known use by [[Dionysius Exiguus]]), but as a word, ''nulla'' meaning "nothing," not as a symbol. When division produced zero as a remainder, ''nihil'', also meaning "nothing," was used. These medieval zeros were used by all future medieval [[computus|computists]] (calculators of [[Easter]]). An isolated use of the initial, N, was used in a table of Roman numerals by [[Bede]] or a colleague about [[725]], a zero symbol.

In 498 CE, Indian mathematician and astronomer [[Aryabhata]] stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal-based place value notation.<ref>''Aryabhatiya of Aryabhata'', translated by Walter Eugene Clark.</ref>

The oldest known text to use a decimal [[positional notation|place-value system]], including a zero, is the Jain text from India entitled the '''Lokavibhâga''', dated 458 CE. This text uses Sanskrit numeral words for the digits, with words such as the Sanskrit word for ''void'' for zero.<ref>Ifrah, Georges (2000), p.&nbsp;416.</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 [[Chaturbhuja Temple]] at [[Gwalior]] in India, dated 876 CE.<ref> [http://www.ams.org/featurecolumn/archive/india-zero.html Feature Column from the AMS<!-- Bot generated title -->]</ref><ref>Ifrah, Georges (2000), p.&nbsp;400.</ref> There are many documents on copper plates, with the same small ''o'' in them, dated back as far as the sixth century CE, but their authenticity may be doubted.<ref name="multiref1"/>

The Arabic numerals and the positional number system were introduced to the [[Islamic Golden Age|Islamic civilization]] by [[Al-Khwarizmi]].{{Fact|date=March 2009}} Al-Khwarizmi's book on arithmetic synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero.

It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the [[Western world]] through [[Latin]] translations of his ''Arithmetic''.

=== Rules of Brahmagupta ===
The rules governing the use of zero appeared for the first time in [[Brahmagupta]]'s book ''[[Brahmasphutasiddhanta|Brahmasputha Siddhanta]] (The Opening of the Universe)'',<ref name="brahmagupta"> [http://books.google.com/books?id=A3cAAAAAMAAJ&printsec=frontcover&dq=brahmagupta ''Algebra with Arithmetic of Brahmagupta and Bhaskara''], translated to English by Henry Thomas Colebrooke, London1817</ref> written in [[628]]. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahmagupta:<ref name="brahmagupta" />

* The sum of zero and a negative number is negative.
* The sum of zero and a positive number is positive.
* The sum of zero and zero is zero.
* The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
* A positive or negative number [[Division by zero|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.

In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign [[NaN]], which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is done.

=== Zero as a decimal digit ===
:''See also: [[History of the Hindu-Arabic numeral system]].''

Positional notation without the use of zero (using an empty space in tabular arrangements, or the word ''kha'' "emptiness") is known to have been in use in India from the [[6th century]]. The earliest certain use of zero as a ''decimal'' positional digit dates to the [[5th century]] mention in the text [[Lokavibhaga]]. The glyph for the zero digit was written in the shape of a dot, and consequently called ''[[bindu]]'' ("dot"). The dot had been used in Greece during earlier ciphered numeral periods.

The [[Hindu-Arabic numeral system]] (base 10) reached Europe in the 11th century, via the [[Iberian Peninsula]] through [[Spain|Spanish]] [[Muslim]]s, the [[Moors]], together with knowledge of [[astronomy]] and instruments like the [[astrolabe]], first imported by [[Pope Sylvester II|Gerbert of Aurillac]]. For this reason, the numerals came to be known in Europe as "[[Arabic numerals]]". The Italian mathematician [[Fibonacci]] or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

<blockquote>
After my father's appointment by 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 [[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 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 ... any number may be written.<ref>Sigler, L., ''Fibonacci’s Liber Abaci''. English translation, Springer, 2003.</ref><ref>Grimm, R.E., "The Autobiography of Leonardo Pisano", ''[[Fibonacci Quarterly]]'' '''11'''/1 (February 1973), pp. 99-104.</ref>
</blockquote>

Here Leonardo of Pisa uses the phrase "sign 0," indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called ''[[Algorism|algorimus]]'' after the Persian mathematician al-Khwarizmi. The most popular was written by [[Johannes de Sacrobosco]], about 1235 and was one of the earliest scientific books to be ''printed'' in [[1488]]. Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the [[Roman numerals]]. In the [[16th century]], they became commonly used in Europe.

== In mathematics ==
===Elementary algebra===
The number 0 is the least [[Negative and non-negative numbers|non-negative]] integer. The [[natural number]] following 0 is 1 and no natural number precedes 0. The number 0 [[Natural number#History of natural numbers and the status of zero|may or may not be considered a natural number]], but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number).

The number 0 is neither positive nor negative, neither a [[prime number]] nor a [[composite number]], nor is it a [[unit (ring theory)|unit]]. It is, however, [[Parity (mathematics)|even]] (see [[evenness of zero]]).

The following are some basic (elementary) 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]] (or neutral element) with respect to [[addition]].
* Subtraction: ''x'' − 0 = ''x'' and 0 − ''x'' = −''x''.
* Multiplication: ''x'' · 0 = 0 · ''x'' = 0.
* Division: {{frac|0|''x''}} = 0, for nonzero ''x''. But {{frac|''x''|0}} is [[Defined and undefined|undefined]], because 0 has no multiplicative inverse, a consequence of the previous rule; see [[division by zero]]. In the real numbers, for positive ''x'', as ''y'' in {{frac|''x''|''y''}} approaches 0 from the positive side, the quotient increases indefinitely toward positive [[infinity]], but as ''y'' approaches 0 from the negative side, the quotient tends toward negative infinity. In other words,

<math>x > 0 \Rightarrow \lim_{y \to 0^+} {x \over y} = +\infty</math>

and

<math>x > 0 \Rightarrow \lim_{y \to 0^-} {x \over y} = -\infty</math>.

* Exponentiation: ''x''<sup>0</sup> = <sup>''x''</sup>/<sub>''x''</sub> = 1, except that the case ''x'' = 0 may be left undefined in some contexts; see [[Exponentiation#Zero to the zero power|Zero to the zero power]]. For all positive real ''x'', 0<sup>''x''</sup> = 0.

The expression {{frac|0|0}}, which may be obtained in an attempt to determine the limit of an expression of the form {{frac|''f''(''x'')|''g''(''x'')}} as a result of applying the [[limit (mathematics)|lim]] operator independently to both operands of the fraction, is a so-called "[[indeterminate form]]". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of {{frac|''f''(''x'')|''g''(''x'')}}, if it exists, must be found by another method, such as [[l'Hôpital's rule]].

[[empty sum|The sum of 0 numbers]] is 0, and [[empty product|the product of 0 numbers]] is 1. The [[factorial]] 0! evaluates to 1.

===Other branches of mathematics===
*In [[set theory]], 0 is the [[cardinality]] of the 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 ''[[definition|defined]]'' to be the empty set. When this is done, the empty set is the [[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 least [[ordinal number]], corresponding to the empty set viewed as a [[well-order|well-ordered set]].
*In [[propositional calculus|propositional logic]], 0 may be used to denote the [[truth value]] false.
*In [[abstract algebra]], 0 is commonly used to denote a [[Algebraic structure|zero element]], which is a [[Identity element|neutral element]] for addition (if defined on the structure under consideration) and an [[absorbing element]] for multiplication (if defined).
*In [[lattice (order)|lattice theory]], 0 may denote the [[Greatest element|bottom element]] of a [[Lattice (order)|bounded lattice]].
*In [[category theory]], 0 is sometimes used to denote an [[initial and terminal objects|initial object]] of a [[category (mathematics)|category]].

=== Other uses of ''zero'' in mathematics ===
* A [[Root (mathematics)|zero of a function]] ''f'' is a point ''x'' in the domain of the function such that <span style="white-space:nowrap;">''f''(''x'') = 0</span>. When there are finitely many zeros these are called the [[root (mathematics)|roots]] of the function. See also [[zero (complex analysis)]] for zeros of a [[holomorphic function]].
* The zero function (or zero map) on a domain ''D'' is the [[constant function]] with 0 as its only possible output value, i.e., the function ''f'' defined by <span style="white-space:nowrap;">''f''(''x'') = 0</span> for all ''x'' in ''D''. A particular zero function is a [[zero morphism]] in category theory; e.g., a zero map is the identity in the additive group of functions. The [[determinant]] on non-invertible [[Matrix (mathematics)|square matrices]] is a zero map.

== In science ==
=== 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, on the [[Kelvin]] temperature scale, zero is the coldest possible temperature ([[negative temperature]]s exist but are not actually colder), whereas on the [[Celsius]] scale, zero is arbitrarily defined to be at the [[Melting point|freezing point]] of water. Measuring sound intensity in [[decibel]]s or [[phon]]s, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In [[physics]], the [[zero-point energy]] is the lowest possible [[energy]] that a [[quantum mechanics|quantum mechanical]] [[physical system]] may possess and is the energy of the [[Stationary state|ground state]] of the system.

=== Chemistry ===
Zero has been proposed as the [[atomic number]] of the theoretical element [[tetraneutron]]. It has been shown that a cluster of four [[neutron]]s may be stable enough to be considered an [[atom]] in its own right. This would create an [[chemical element|element]] with no [[proton]]s and no charge on its [[atomic nucleus|nucleus]].

As early as 1926, Professor Andreas von Antropoff coined the term [[neutronium]] for a conjectured form of [[matter]] made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the [[periodic table]]. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements. It is at the centre of the [[Chemical Galaxy]] (2005).

=== Medical science ===
* [[Patient zero]] is the initial [[patient]] in the [[sample (statistics)|population sample]] of an [[epidemiology|epidemiological]] investigation.

== In computer science ==
=== Numbering from 1 or 0? ===
The most common practice throughout human history has been to start counting at one. Nevertheless, in [[computer science]] zero is often used as the starting point. For example, in almost all old [[programming language]]s, an [[Array data type|array]] starts from 1 by [[default (computer science)|default]]. As programming languages have developed, it has become more common that an array starts from zero by default, the "first" index in the array being 0. In particular, the popularity of the [[C (programming language)|C]] programming language in the 1980s has made this approach common.

One advantage of this convention is in the use of [[modular arithmetic]]. Every integer is [[Congruence relation|congruent]] modulo ''N'' to one of the numbers 0, 1, 2, ..., {{nowrap|''N'' − 1}}, where {{nowrap|''N'' ≥ 1}}. Because of this, many arithmetic concepts (such as hash tables) are more elegantly expressed in code when the array starts at zero.

A second advantage of zero-based array indexes is that this can improve efficiency under certain circumstances. To illustrate, suppose ''a'' is the [[memory address]] of the first element of an array, and ''i'' is the index of the desired element. In this fairly typical scenario, it is quite common to want the address of the desired element. If the index numbers count from 1, the desired address is computed by this expression:

:<math>a + s \times (i-1) \,\!</math>

where ''s'' is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this:

:<math>a + s \times i \,\!</math>

This simpler expression can be more efficient to compute in certain situations.

Note, however, that a language wishing to index arrays from 1 could simply adopt the convention that every "array address" is represented by ''a′'' = ''a'' – ''s''; that is, rather than using the address of the first array element, such a language would use the address of an imaginary element located immediately before the first actual element. The indexing expression for a 1-based index would be the following:

:<math>a' + s \times i \,\!</math>

Hence, the efficiency benefit of zero-based indexing is not inherent, but is an artifact of the decision to represent an array by the address of its first element.

A third advantage is that ranges are more elegantly expressed as the half-open [[Interval (mathematics)|interval]], [0,''n''), as opposed to the closed interval, [1,''n''], because empty ranges often occur as input to algorithms (which would be tricky to express with the closed interval without resorting to obtuse conventions like [1,0]). On the other hand, closed intervals occur in mathematics because it is often necessary to calculate the terminating condition (which would be impossible in some cases because the half-open interval isn't always a [[closed set]]) which would have a subtraction by 1 everywhere.

This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of ''n'' objects will be {{nowrap|''n'' – 1}} and referred to the ''n''th element. For this reason, the first element is often referred to as the ''[[zeroth]]'' element to eliminate any possible doubt (though, strictly speaking, this is unnecessary and arguably incorrect,{{Clarify me|date=March 2009}} since the meanings of the ordinal numbers are not ambiguous).

=== Null value ===
In databases a field can have a [[null (computer programming)|null value]]. This is equivalent to the field not having a value. 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 [[Ternary logic|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. Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded.

=== Null pointer ===
A ''[[Pointer (computing)|null pointer]]'' is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at [[compile time]] when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).

(Note that on most common architectures, the null pointer is represented internally the same way an integer of the same byte width having a value of zero is represented, so C compilers on such systems perform no actual conversion.)

=== Negative zero ===
{{main|Signed zero}}
In mathematics <math>-0 = 0 = +0</math>, both [[−0 (number)|−0]] and +0 represent the exact same number, i.e., there is no “negative zero” distinct from zero. In some [[signed number representations]] (but not the [[two's complement]] representation used to represent integers in most computers today) and most [[floating point]] number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as [[−0 (number)|negative zero]].

== In other fields ==
[[Image:ICS Zero.svg|thumb|[[International maritime signal flags|International maritime signal flag]] for 0]]

*In some countries and some company phone networks, dialing 0 on a telephone places a call for [[operator assistance]].
*In [[Braille]], the numeral 0 has the same dot configuration as the letter [[J]].
*[[DVD]]s that can be played in any region are sometimes referred to as being "[[region 0]]"
*In classical music, 0 is very rarely used as a number for a composition: [[Anton Bruckner]] wrote a [[Symphony No. 0 (Bruckner)|Symphony No. 0 in D minor]] and a [[Study Symphony|Symphony No. 00]]; [[Alfred Schnittke]] also wrote a Symphony No. 0.
*[[Roulette]] wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
*A chronological prequel of a series may be numbered as 0.
*In [[Formula One]], if the reigning [[List of Formula One World Drivers' Champions|World Champion]] no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in [[1993]] and [[1994]], with [[Damon Hill]] driving car 0, due to the reigning World Champion ([[Nigel Mansell]] and [[Alain Prost]] respectively) not competing in the championship.
*In the educational series [[Schoolhouse Rock!]], the song ''My Hero, Zero'' is about the use of zero as a placeholder. The song explains that by appending zeros to a number, it is multiplied by 10 for each one added. This enables mathematicians to create numbers as large as needed.

== Quotations ==
<blockquote>
The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the [[Nirvana]] into [[dynamo]]s. No single mathematical creation has been more potent for the general on-go of intelligence and power. &mdash; [[G. B. Halsted]]
</blockquote>

<blockquote>
Dividing by zero...allows you to prove, mathematically, anything in the universe. You can prove that 1+1=42, and from there you can prove that J. Edgar Hoover is a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.) &mdash; [[Charles Seife]], ''Zero: The Biography of a Dangerous Idea''
</blockquote>

<blockquote>
...a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions. &mdash; [[Pierre-Simon Laplace]]
</blockquote>

<blockquote>
The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. &mdash; [[Alfred North Whitehead]]
</blockquote>

<blockquote>
...a fine and wonderful refuge of the divine spirit – almost an amphibian between being and non-being. &mdash; [[Gottfried Leibniz]]
</blockquote>

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

== Notes ==
{{reflist|2}}

== References ==
{{FOLDOC}}
{{refbegin}}
* [[John D. Barrow|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.
* [[Charles Seife|Seife, Charles]] (2000) ''Zero: The Biography of a Dangerous Idea'', Penguin USA (Paper). ISBN 0-14-029647-6.
* [[Nicolas Bourbaki|Bourbaki, Nicolas]] (1998). ''Elements of the History of Mathematics''. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3540647678.
*Isaac Asimov article "nothing counts" in "Asimov on Numbers" Pocket Books, 1978
{{refend}}

==External links==
{{Wikisource1911Enc|Zero}}
* [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]
* [http://www.neo-tech.com/zero/part6.html The Discovery of the Zero]
* [http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra]
* [[Edsger W. Dijkstra]]: [http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF Why numbering should start at zero], 192 ([[Portable Document Format|PDF]] of a handwritten manuscript)
* [http://www.schoolhouserock.tv/My.html "My Hero Zero"] Educational children's song in [[Schoolhouse Rock!]]

[[Category:Zero| ]]
[[Category:Elementary arithmetic]]
[[Category:Integers|00]]
[[Category:Nothing]]
[[Category:Indian inventions]]

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Revision as of 02:12, 9 September 2009

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