Editing Integer (section)

{{Short description|Number in {..., –2, –1, 0, 1, 2, ...} }}
{{about|integers in mathematics|integers as a data type|Integer (computer science)}}
{{Use dmy dates|date=July 2020}}
[[File:NumberLineIntegers.svg|thumb|upright=1.25|The integers arranged on a [[number line]]]]

An '''integer''' is the [[number]] zero ([[0]]), a positive [[natural number]] (1, 2, 3, ...), or the negation of a positive natural number ([[−1]], −2, −3, ...).<ref>{{cite book |title=Science and Technology Encyclopedia |date=September 2000 |publisher=University of Chicago Press |isbn=978-0-226-74267-0 |page=280 |url=https://books.google.com/books?id=PZIdcYCCf2kC&dq=integer&pg=PA280 |language=en}}</ref> The negations or [[additive inverse]]s of the positive natural numbers are referred to as '''negative integers'''.<ref>{{cite book |last1=Hillman |first1=Abraham P. |last2=Alexanderson |first2=Gerald L. |title=Algebra and trigonometry; |date=1963 |publisher=Allyn and Bacon |location=Boston |url=https://archive.org/details/algebratrigonome0000hill/page/42/mode/2up}}</ref> The [[set (mathematics)|set]] of all integers is often denoted by the [[boldface]] {{math|'''Z'''}} or [[blackboard bold]] {{nobr|<math>\mathbb{Z}</math>.<ref name="earliest"/><ref name="Cameron1998">{{cite book |author=Peter Jephson Cameron |title=Introduction to Algebra |url=https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 |year=1998 |publisher=Oxford University Press |isbn=978-0-19-850195-4 |page=4 |access-date=2016-02-15 |archive-url=https://web.archive.org/web/20161208142220/https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 |archive-date=2016-12-08 |url-status=live }}</ref>}}

The set of natural numbers <math>\mathbb{N}</math> is a [[subset]] of <math>\mathbb{Z}</math>, which in turn is a subset of the set of all [[rational number]]s <math>\mathbb{Q}</math>, itself a subset of the [[real number]]s <math>\mathbb{R}</math>.{{efn|More precisely, each system is [[Embedding|embedded]] in the next, isomorphically mapped to a subset.<ref>{{cite book |last1=Partee |first1=Barbara H. |last2=Meulen |first2=Alice ter |last3=Wall |first3=Robert E. |title=Mathematical Methods in Linguistics |date=30 April 1990 |publisher=Springer Science & Business Media |isbn=978-90-277-2245-4 |pages=78–82 |url=https://books.google.com/books?id=qV7TUuaYcUIC&pg=PA80 |language=en |quote=The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.}}</ref> The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.<ref>{{cite book |last1=Wohlgemuth |first1=Andrew |title=Introduction to Proof in Abstract Mathematics |date=10 June 2014 |publisher=Courier Corporation |isbn=978-0-486-14168-8 |page=237 |url=https://books.google.com/books?id=PEP_AwAAQBAJ&pg=PA237 |language=en}}</ref> Such a convention is "a matter of choice", yet not.<ref>{{cite book |last1=Polkinghorne |first1=John |title=Meaning in Mathematics |date=19 May 2011 |publisher=OUP Oxford |isbn=978-0-19-162189-5 |page=68 |url=https://books.google.com/books?id=DCqQDwAAQBAJ&pg=PA68 |language=en}}</ref>}} Like the set of natural numbers, the set of integers <math>\mathbb{Z}</math> is [[Countable set|countably infinite]]. An integer may be regarded as a real number that can be written without a [[fraction|fractional component]]. For example, 21, 4, 0, and −2048 are integers, while 9.75, {{sfrac|5|1|2}}, 5/4, and [[Square root of 2|{{sqrt|2}}]] are not.<ref>{{cite book |last1=Prep |first1=Kaplan Test |title=GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT |date=4 June 2019 |publisher=Simon and Schuster |isbn=978-1-5062-4844-8 |url=https://books.google.com/books?id=6l_sDwAAQBAJ&pg=PA708 |language=en}}</ref>

The integers form the smallest [[Group (mathematics)|group]] and the smallest [[ring (mathematics)|ring]] containing the [[natural number]]s. In [[algebraic number theory]], the integers are sometimes qualified as '''rational integers''' to distinguish them from the more general [[algebraic integer]]s. In fact, (rational) integers are algebraic integers that are also [[rational number]]s.