Editing 0 (section)

==Mathematics==
{{see also|Null (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]] 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|upright=1.4|A [[number line]] from −3 to 3, with 0 in the middle]]
The number 0 is the smallest [[nonnegative integer]], and the largest nonpositive integer. The [[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]], and hence a [[rational number]] and a [[real number]].{{sfn|Cheng|2017|pp=41, 48–53}} All rational numbers are [[algebraic number]]s, including 0. When the real numbers are extended to form the [[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 |author-link=André Weil |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]]. 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]] nor a [[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.)
[[File:AdditionZero.svg|alt=A collection of five dots and one of zero dots merge into one of five dots.|thumb|193x193px|5+0=5 illustrated with collections of dots.]]
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]] (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 [[Defined and undefined|undefined]], because 0 has no [[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]]". 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]].<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=etal |author-link2=Gilbert Strang |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]]'') is 0, and the product of 0 numbers (the ''[[empty product]]'') is 1. The [[factorial]] 0! evaluates to 1, as a special case of the empty product.<ref name=gkp>{{cite book|first1=Ronald L.|last1=Graham|author1-link=Ronald Graham |first2=Donald E.|last2=Knuth|author2-link=Donald Knuth|first3=Oren|last3=Patashnik|author3-link=Oren 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|upright=0.4|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]], 0 is the [[cardinality]] of the [[empty set]] (notated as "{ }", "<math display="inline">\emptyset</math>", or "∅"): 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.{{sfn|Cheng|2017|p=60}} 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 lowest [[ordinal number]], corresponding to the empty set viewed as a [[well-order|well-ordered set]]. In [[order theory]] (and especially its subfield [[lattice theory]]), 0 may denote the [[least element]] of a [[Lattice (order)|lattice]] or other [[partially ordered set]].

The role of 0 as additive identity generalizes beyond elementary algebra. In [[abstract algebra]], 0 is commonly used to denote a [[zero element]], which is the [[identity element]] for addition (if defined on the structure under consideration) and an [[absorbing element]] for multiplication (if defined). (Such elements may also be called [[zero element]]s.) Examples include identity elements of [[additive group]]s and [[vector space]]s. Another example is the '''zero function''' (or '''zero map''') on a domain {{mvar|D}}. This is the [[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 function|even]] and [[Odd function|odd]].

The number 0 is also used in several other ways within various branches of mathematics:
* A ''[[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 [[truth value]] false.
* In [[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]]s, which generalize the zero function.<ref>{{cite book|last=Riehl |first=Emily |title=Category Theory in Context |author-link=Emily Riehl |page=103 |url=https://math.jhu.edu/~eriehl/context/ |publisher=Dover |year=2016 |isbn=978-0-486-80903-8}}</ref>