Editing 0 (section)

===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>