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{{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.

== History ==

The word integer comes from the [[Latin]] [[wikt:integer#Latin|''integer'']] meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). "[[wikt:entire|Entire]]" derives from the same origin via the [[French language|French]] word ''[[wikt:entier|entier]]'', which means both ''entire'' and ''integer''.<ref>{{cite book |first=Nick |last=Evans |contribution=A-Quantifiers and Scope |editor-first=Emmon W. |editor-last=Bach |title=Quantification in Natural Languages |isbn=978-0-7923-3352-4 |year=1995 |pages=262 |url=https://books.google.com/books?id=NlQL97qBSZkC |location=Dordrecht, The Netherlands; Boston, MA |publisher=Kluwer Academic Publishers}}</ref> Historically the term was used for a [[number]] that was a multiple of 1,<ref>{{cite book |last1=Smedley |first1=Edward |last2=Rose |first2=Hugh James |last3=Rose |first3=Henry John |title=Encyclopædia Metropolitana |date=1845 |publisher=B. Fellowes |page=537 |url=https://books.google.com/books?id=ZVI_AQAAMAAJ&pg=PA537 |language=en|quote=An integer is a multiple of unity}}</ref><ref>{{harvnb|Encyclopaedia Britannica|1771|p=[https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA367 367]}}</ref> or to the whole part of a [[mixed number]].<ref>{{cite book| title = Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij | type=Manuscript | trans-title=The Book of Calculation | last1 = Pisano | first1 = Leonardo  | author1-link=Fibonacci | publisher = Museo Galileo | date = 1202 | url = https://bibdig.museogalileo.it/tecanew/opera?bid=1072400&seq=30 |lang=la|last2=Boncompagni|first2=Baldassarre (transliteration)|translator-last = Sigler |translator-first = Laurence E.|page=30|quote=Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant.|trans-quote=And the fractions are always put after the whole, thus first the integer is written, and then the fraction}}</ref><ref>{{harvnb|Encyclopaedia Britannica|1771|p=[https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA83 83]}}</ref> Only positive integers were considered, making the term synonymous with the [[natural number]]s. The definition of integer expanded over time to include [[negative number]]s as their usefulness was recognized.<ref name="negmath">{{cite book|last=Martinez|first=Alberto|title=Negative Math|pages=80–109|date=2014|publisher=Princeton University Press}}</ref> For example [[Leonhard Euler]] in his 1765 ''[[Elements of Algebra]]'' defined integers to include both positive and negative numbers.<ref>{{cite book |last1=Euler |first1=Leonhard |title=Vollstandige Anleitung Zur Algebra|lang=de|trans-title=Complete Introduction to Algebra|volume=1|date=1771 |url=https://archive.org/details/1770LEULERVollstandigeAnleitungZurAlgebraVol1/page/n31/mode/2up|page=10|quote=Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen, um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden.|trans-quote=All these numbers, both positive and negative, are called whole numbers, which are either greater or lesser than nothing. We call them whole numbers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak.}}</ref>

The phrase ''the set of the integers'' was not used before the end of the 19th century, when [[Georg Cantor]] introduced the concept of [[infinite set]]s and [[set theory]]. The use of the letter Z to denote the set of integers comes from the [[German language|German]] word ''[[wikt:Zahlen|Zahlen]]'' ("numbers")<ref name="earliest">{{cite web |url=http://jeff560.tripod.com/nth.html |title=Earliest Uses of Symbols of Number Theory |access-date=2010-09-20 |date=2010-08-29 |first=Jeff |last=Miller |archive-url=https://web.archive.org/web/20100131022510/http://jeff560.tripod.com/nth.html |archive-date=2010-01-31 |url-status=dead }}</ref><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> and has been attributed to [[David Hilbert]].<ref>{{cite book |title=The University of Leeds Review |date=1989 |publisher=University of Leeds. |page=46 |url=https://books.google.com/books?id=Z-7kAAAAMAAJ|language=en|volume=31-32|quote=Incidentally, Z comes from "Zahl": the notation was created by Hilbert.}}</ref> The earliest known use of the notation in a textbook occurs in [[Éléments de mathématique|Algèbre]] written by the collective [[Nicolas Bourbaki]], dating to 1947.<ref name="earliest"/><ref>{{cite book |last1=Bourbaki |first1=Nicolas |title=Algèbre, Chapter 1 |date=1951|edition=2nd |publisher=Hermann|location=Paris |page=27|language=fr|url=https://archive.org/details/algebrebour00bour/page/26/mode/2up|quote=Le symétrisé de '''N''' se note '''Z'''; ses éléments sont appelés entiers rationnels.|trans-quote=The group of differences of '''N''' is denoted by '''Z'''; its elements are called the rational integers.}}</ref> The notation was not adopted immediately. For example, another textbook used the letter J,<ref>{{cite book |last1=Birkhoff |first1=Garrett |title=Lattice Theory |date=1948 |publisher=American Mathematical Society |page=63 |edition=Revised |url=https://archive.org/details/in.ernet.dli.2015.166886/page/n63/mode/2up|quote=the set ''J'' of all integers}}</ref> and a 1960 paper used Z to denote the non-negative integers.<ref>{{cite book |last1=Society |first1=Canadian Mathematical |title=Canadian Journal of Mathematics |date=1960 |publisher=Canadian Mathematical Society |page=374 |url=https://books.google.com/books?id=uMAXOmLTCGsC&dq=integer%20set%20Z&pg=PA374 |language=en|quote=Consider the set ''Z'' of non-negative integers}}</ref> But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.<ref>{{cite book |last1=Bezuszka |first1=Stanley |title=Contemporary Progress in Mathematics: Teacher Supplement [to] Part 1 and Part 2 |date=1961 |publisher=Boston College |page=69 |url=https://books.google.com/books?id=dhJPAQAAMAAJ&q=integer+set+Z |language=en|quote=Modern Algebra texts generally designate the set of integers by the capital letter Z.}}</ref>

The symbol <math>\mathbb{Z}</math> is often annotated to denote various sets, with varying usage amongst different authors: <math>\mathbb{Z}^+</math>, <math>\mathbb{Z}_+</math>, or <math>\mathbb{Z}^{>}</math> for the positive integers, <math>\mathbb{Z}^{0+}</math> or <math>\mathbb{Z}^{\geq}</math> for non-negative integers, and <math>\mathbb{Z}^{\neq}</math> for non-zero integers. Some authors use <math>\mathbb{Z}^{*}</math> for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the [[group of units]] of <math>\mathbb{Z}</math>). Additionally, <math>\mathbb{Z}_{p}</math> is used to denote either the set of [[integers modulo n|integers modulo {{math|''p''}}]] (i.e., the set of [[congruence relation|congruence classes]] of integers), or the set of [[p-adic integer|{{math|''p''}}-adic integers]].<ref name=edexcelc1>Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008</ref><ref>LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.</ref>

{{anchor|Whole numbers}}
The ''whole numbers'' were synonymous with the integers up until the early 1950s.<ref>{{cite book |last1=Mathews |first1=George Ballard |title=Theory of Numbers |date=1892 |publisher=Deighton, Bell and Company |page=2 |url=https://books.google.com/books?id=iQ_vAAAAMAAJ&pg=PA2 |language=en}}</ref><ref>{{cite book |last1=Betz |first1=William |title=Junior Mathematics for Today |date=1934 |publisher=Ginn |url=https://books.google.com/books?id=RzNCAAAAIAAJ |language=en |quote=The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.}}</ref><ref>{{cite book |last1=Peck |first1=Lyman C. |title=Elements of Algebra |date=1950 |publisher=McGraw-Hill |page=3 |url=https://books.google.com/books?id=tclXAAAAYAAJ&q=integers+whole+numbers |language=en |quote=The numbers which so arise are called positive whole numbers, or positive integers.}}</ref> In the late 1950s, as part of the [[New Math]] movement,<ref>{{cite thesis|type=PhD |url= https://dr.lib.iastate.edu/handle/20.500.12876/80303 |title=A history of the "new math" movement in the United States|date=1981|last=Hayden|first=Robert|publisher=Iowa State University |doi=10.31274/rtd-180813-5631|page=145|quote=A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).|doi-access=free}}</ref> American elementary school teachers began teaching that ''whole numbers'' referred to the [[natural number]]s, excluding negative numbers, while ''integer'' included the negative numbers.<ref>{{cite book |title=The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook |date=1959 |publisher=National Council of Teachers of Mathematics |page=14 |isbn=9780608166186 |url=https://books.google.com/books?id=OO9RAQAAIAAJ&pg=PA14 |language=en}}</ref><ref>{{cite book |last1=Deans |first1=Edwina |title=Elementary School Mathematics: New Directions |date=1963 |publisher=U.S. Department of Health, Education, and Welfare, Office of Education |page=42 |url=https://books.google.com/books?id=bAUJAQAAMAAJ&pg=PA42 |language=en}}</ref> The ''whole numbers'' remain ambiguous to the present day.<ref>{{cite web |title=entry: whole number |url=https://www.ahdictionary.com/word/search.html?q=whole+number |website=The American Heritage Dictionary |publisher=HarperCollins}}</ref>

== Algebraic properties ==
[[File:Number-line.svg|right|thumb|300px|Integers can be thought of as discrete, equally spaced points on an infinitely long [[number line]]. In the above, non-[[Sign (mathematics)#Terminology for signs|negative]] integers are shown in blue and negative integers in red.]]
{{Group theory sidebar |Discrete}}
{{Ring theory sidebar}}

Like the [[natural numbers]], <math>\mathbb{Z}</math> is [[closure (mathematics)|closed]] under the [[binary operation|operations]] of addition and [[multiplication]], that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, {{num|0}}), <math>\mathbb{Z}</math>, unlike the natural numbers, is also closed under [[subtraction]].<ref>{{Cite web|title=Integer {{!}} mathematics|url=https://www.britannica.com/science/integer|access-date=2020-08-11|website=Encyclopedia Britannica|language=en}}</ref>

The integers form a [[ring (mathematics)|ring]] which is the most basic one, in the following sense: for any ring, there is a unique [[ring homomorphism]] from the integers into this ring. This [[universal property]], namely to be an [[initial object]] in the [[category of rings]], characterizes the ring&nbsp;<math>\mathbb{Z}</math>.

<math>\mathbb{Z}</math> is not closed under [[division (mathematics)|division]], since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under [[exponentiation]], the integers are not (since the result can be a fraction when the exponent is negative).

The following table lists some of the basic properties of addition and multiplication for any integers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}:
{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+Properties of addition and multiplication on integers
!
!scope="col" |Addition
!scope="col" |Multiplication
|-
!scope="row" |[[Closure (mathematics)|Closure]]:
|{{math|''a'' + ''b''}} is an integer
|{{math|''a'' × ''b''}} is an integer
|-
!scope="row"|[[Associativity]]:
|{{math|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}}
|{{math|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}
|-
!scope="row" |[[Commutativity]]:
|{{math|''a'' + ''b'' {{=}} ''b'' + ''a''}}
|{{math|''a'' × ''b'' {{=}} ''b'' × ''a''}}
|-
!scope="row" |Existence of an [[identity element]]:
|{{math|''a'' + 0 {{=}} ''a''}}
|{{math|''a'' × 1 {{=}} ''a''}}
|-
!scope="row" |Existence of [[inverse element]]s:
|{{math|''a'' + (−''a'') {{=}} 0}}
|The only invertible integers (called [[Unit (ring theory)|units]]) are –1 and 1.
|-
!scope="row" |[[Distributivity]]:
|colspan=2 align=center |{{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}} and {{math|(''a'' + ''b'') × ''c'' {{=}} (''a'' × ''c'') + (''b'' × ''c'')}}
|-
!scope="row" |No [[zero divisor]]s:
| || | If {{math|''a'' × ''b'' {{=}} 0}}, then {{math|''a'' {{=}} 0}} or {{math|''b'' {{=}} 0}} (or both)
|}

The first five properties listed above for addition say that <math>\mathbb{Z}</math>, under addition, is an [[abelian group]]. It is also a [[cyclic group]], since every non-zero integer can be written as a finite sum {{nowrap|1 + 1 + ... + 1}} or {{nowrap|(−1) + (−1) + ... + (−1)}}. In fact, <math>\mathbb{Z}</math> under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is [[group isomorphism|isomorphic]] to <math>\mathbb{Z}</math>.

The first four properties listed above for multiplication say that <math>\mathbb{Z}</math> under multiplication is a [[commutative monoid]]. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that <math>\mathbb{Z}</math> under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that <math>\mathbb{Z}</math> together with addition and multiplication is a [[commutative ring]] with [[multiplicative identity|unity]]. It is the prototype of all objects of such [[algebraic structure]]. Only those [[equality (mathematics)|equalities]] of [[algebraic expression|expressions]] are true in&nbsp;<math>\mathbb{Z}</math> [[for all]] values of variables, which are true in any unital commutative ring. Certain non-zero integers map to [[additive identity|zero]] in certain rings.

The lack of [[zero divisor]]s in the integers (last property in the table) means that the commutative ring&nbsp;<math>\mathbb{Z}</math> is an [[integral domain]].

The lack of multiplicative inverses, which is equivalent to the fact that <math>\mathbb{Z}</math> is not closed under division, means that <math>\mathbb{Z}</math> is ''not'' a [[field (mathematics)|field]]. The smallest field containing the integers as a [[subring]] is the field of [[rational number]]s. The process of constructing the rationals from the integers can be mimicked to form the [[field of fractions]] of any integral domain. And back, starting from an [[algebraic number field]] (an extension of rational numbers), its [[ring of integers]] can be extracted, which includes <math>\mathbb{Z}</math> as its [[subring]].

Although ordinary division is not defined on <math>\mathbb{Z}</math>, the division "with remainder" is defined on them. It is called [[Euclidean division]], and possesses the following important property: given two integers {{math|''a''}} and {{math|''b''}} with {{math|''b'' ≠ 0}}, there exist unique integers {{math|''q''}} and {{math|''r''}} such that {{math|''a'' {{=}} ''q'' × ''b'' + ''r''}} and {{math|0 ≤ ''r'' <  {{!}}''b''{{!}}}}, where {{math|{{!}}''b''{{!}}}} denotes the [[absolute value]] of {{math|''b''}}. The integer {{math|''q''}} is called the ''quotient'' and {{math|''r''}} is called the ''[[remainder]]'' of the division of {{math|''a''}} by {{math|''b''}}. The [[Euclidean algorithm]] for computing [[greatest common divisor]]s works by a sequence of Euclidean divisions.

The above says that <math>\mathbb{Z}</math> is a [[Euclidean domain]]. This implies that <math>\mathbb{Z}</math> is a [[principal ideal domain]], and any positive integer can be written as the products of [[prime number|primes]] in an [[essentially unique]] way.<ref>{{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Algebra |edition=3rd |publisher=Addison-Wesley |year=1993 |isbn=978-0-201-55540-0 |pages=86–87}}</ref> This is the [[fundamental theorem of arithmetic]].

==Order-theoretic properties==
<math>\mathbb{Z}</math> is a [[total order|totally ordered set]] without [[upper and lower bounds|upper or lower bound]]. The ordering of <math>\mathbb{Z}</math> is given by:
{{math|:... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ...}}.
An integer is ''positive'' if it is greater than [[0|zero]], and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:
# If {{math|''a'' < ''b''}} and {{math|''c'' < ''d''}}, then {{math|''a'' + ''c'' < ''b'' + ''d''}}
# If {{math|''a'' < ''b''}} and {{math|0 < ''c''}}, then {{math|''ac'' < ''bc''}}

Thus it follows that <math>\mathbb{Z}</math> together with the above ordering is an [[ordered ring]].

The integers are the only nontrivial [[totally ordered]] [[abelian group]] whose positive elements are [[well-ordered]].<ref>{{cite book |title=Modern Algebra |series=Dover Books on Mathematics |first=Seth |last=Warner |publisher=Courier Corporation |year=2012 |isbn=978-0-486-13709-4 |at=Theorem 20.14, p.&nbsp;185 |url=https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185 |access-date=2015-04-29 |archive-url=https://web.archive.org/web/20150906083836/https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185|archive-date=2015-09-06 |url-status=live}}.</ref> This is equivalent to the statement that any [[Noetherian ring|Noetherian]] [[valuation ring]] is either a [[Field (mathematics)|field]]—or a [[discrete valuation ring]].

==Construction==
=== Traditional development ===
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, [[zero]], and the negations of the natural numbers. This can be formalized as follows.<ref>{{cite book |last1=Mendelson |first1=Elliott |title=Number systems and the foundations of analysis |date=1985 |publisher=Malabar, Fla. : R.E. Krieger Pub. Co. |isbn=978-0-89874-818-5 |page=153 |url=https://archive.org/details/numbersystemsfou0000mend/page/152/mode/2up}}</ref> First construct the set of natural numbers according to the [[Peano axioms]], call this <math>P</math>. Then construct a set <math>P^-</math> which is [[Disjoint sets|disjoint]] from <math>P</math> and in one-to-one correspondence with <math>P</math> via a function <math>\psi</math>. For example, take <math>P^-</math> to be the [[ordered pair]]s <math>(1,n)</math> with the mapping <math>\psi = n \mapsto (1,n)</math>. Finally let 0 be some object not in <math>P</math> or <math>P^-</math>, for example the ordered pair (0,0). Then the integers are defined to be the union <math>P \cup P^- \cup \{0\}</math>. 

The traditional arithmetic operations can then be defined on the integers in a [[piecewise]] fashion, for each of positive numbers, negative numbers, and zero. For example [[negation]] is defined as follows:

<math display="block">
-x = \begin{cases}
  \psi(x), & \text{if } x \in P \\
  \psi^{-1}(x), & \text{if } x \in P^- \\
  0, & \text{if } x = 0
\end{cases}
</math>

The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.<ref>{{cite book |title=Number Systems and the Foundations of Analysis |series=Dover Books on Mathematics |first=Elliott |last=Mendelson |publisher=Courier Dover Publications |year=2008 |isbn=978-0-486-45792-5 |page=86 |url=https://books.google.com/books?id=3domViIV7HMC&pg=PA86 |access-date=2016-02-15 |archive-url=https://web.archive.org/web/20161208233040/https://books.google.com/books?id=3domViIV7HMC&pg=PA86 |archive-date=2016-12-08|url-status=live}}.</ref> 

=== Equivalence classes of ordered pairs ===
[[File:Relative numbers representation.svg|thumb|alt=Representation of equivalence classes for the numbers −5 to 5
|Red points represent ordered pairs of [[natural number]]s. Linked red points are equivalence classes representing the blue integers at the end of the line.|upright=1.5]]

In modern set-theoretic mathematics, a more abstract construction<ref>Ivorra Castillo: ''Álgebra''</ref><ref>{{Cite book |last1=Kramer |first1=Jürg |title=From Natural Numbers to Quaternions |last2=von Pippich |first2=Anna-Maria |publisher=Springer Cham |year=2017 |isbn=978-3-319-69427-6 |edition=1st |location=Switzerland |pages=78–81 |language=en |doi=10.1007/978-3-319-69429-0}}</ref> allowing one to define arithmetical operations without any case distinction is often used instead.<ref>{{cite book |title=Learning to Teach Number: A Handbook for Students and Teachers in the Primary School |series=The Stanley Thornes Teaching Primary Maths Series |first=Len |last=Frobisher |publisher=Nelson Thornes |year=1999 |isbn=978-0-7487-3515-0 |page=126 |url=https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126 |access-date=2016-02-15 |archive-url=https://web.archive.org/web/20161208121843/https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126 |archive-date=2016-12-08 |url-status=live}}.</ref> The integers can thus be formally constructed as the [[equivalence class]]es of [[ordered pair]]s of [[natural number]]s {{math|(''a'',''b'')}}.<ref name="Campbell-1970-p83">{{cite book |author=Campbell, Howard E. |title=The structure of arithmetic |publisher=Appleton-Century-Crofts |year=1970 |isbn=978-0-390-16895-5 |page=[https://archive.org/details/structureofarith00camp/page/83 83] |url-access=registration |url=https://archive.org/details/structureofarith00camp/page/83 }}</ref>

The intuition is that {{math|(''a'',''b'')}} stands for the result of subtracting {{math|''b''}} from {{math|''a''}}.<ref name="Campbell-1970-p83"/> To confirm our expectation that {{nowrap|1 − 2}} and {{nowrap|4 − 5}} denote the same number, we define an [[equivalence relation]] {{math|~}} on these pairs with the following rule:
:<math>(a,b)\sim(c,d) </math>
precisely when
:<math>a+d=b+c </math>.

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;<ref name="Campbell-1970-p83"/> by using {{math|[(''a'',''b'')]}} to denote the equivalence class having {{math|(''a'',''b'')}} as a member, one has:
:<math>[(a,b)]+[(c,d)]:=[(a+c,b+d)]</math>.
:<math>[(a,b)]\cdot[(c,d)]:=[(ac+bd,ad+bc)]</math>.

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:<math>-[(a,b)]:=[(b,a)]</math>.

Hence subtraction can be defined as the addition of the additive inverse:
:<math>[(a,b)]-[(c,d)]:=[(a+d,b+c)]</math>.

The standard ordering on the integers is given by:
:<math>[(a,b)]<[(c,d)]</math> [[if and only if]] <math>a+d<b+c</math>.

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form {{math|(''n'',0)}} or {{math|(0,''n'')}} (or both at once). The natural number {{math|''n''}} is identified with the class {{math|[(''n'',0)]}} (i.e., the natural numbers are [[embedding|embedded]] into the integers by map sending {{math|''n''}} to {{math|[(''n'',0)]}}), and the class {{math|[(0,''n'')]}} is denoted {{math|−''n''}} (this covers all remaining classes, and gives the class {{math|[(0,0)]}} a second time since –0&nbsp;=&nbsp;0.

Thus, {{math|[(''a'',''b'')]}} is denoted by
:<math>\begin{cases}a-b,&\mbox{if }a\ge b\\-(b-a),&\mbox{if }a<b\end{cases}</math>

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.

This notation recovers the familiar [[group representation|representation]] of the integers as {{math|{..., −2, −1, 0, 1, 2, ...} }}.

Some examples are:
:<math>\begin{align}0&=[(0,0)]&=[(1,1)]&=\cdots& &=[(k,k)]\\1&=[(1,0)]&=[(2,1)]&=\cdots&&=[(k+1,k)]\\-1&=[(0,1)]&=[(1,2)]&=\cdots&&=[(k,k+1)]\\2&=[(2,0)]&=[(3,1)]&=\cdots&&=[(k+2,k)]\\-2&=[(0,2)]&= [(1,3)]&=\cdots&&=[(k,k+2)]\end{align}</math>

=== Other approaches ===

In theoretical computer science, other approaches for the construction of integers are used by [[Automated theorem proving|automated theorem provers]] and [[Rewriting|term rewrite engines]]. Integers are represented as [[Term algebra|algebraic terms]] built using a few basic operations (e.g.,  '''zero''', '''succ''', '''pred''') and using [[natural number]]s, which are assumed to be already constructed (using the [[Peano axioms|Peano approach]]).

There exist at least ten such constructions of signed integers.<ref>{{cite conference |last=Garavel |first=Hubert |title=On the Most Suitable Axiomatization of Signed Integers |conference=Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016) |year=2017 |publisher=Springer |series=Lecture Notes in Computer Science |volume=10644 |pages=120–134 |doi=10.1007/978-3-319-72044-9_9 |isbn=978-3-319-72043-2 |url=https://hal.inria.fr/hal-01667321 |access-date=2018-01-25 |archive-url=https://web.archive.org/web/20180126125528/https://hal.inria.fr/hal-01667321 |archive-date=2018-01-26 |url-status=live }}</ref> These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2), and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.

The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation '''pair'''<math>(x,y)</math> that takes as arguments two natural numbers <math>x</math> and <math>y</math>, and returns an integer (equal to <math>x-y</math>). This operation is not free since the integer 0 can be written '''pair'''(0,0), or '''pair'''(1,1), or '''pair'''(2,2), etc.. This technique of construction is used by the [[proof assistant]] [[Isabelle (proof assistant)|Isabelle]]; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.

==Computer science==
{{Main|Integer (computer science)}}
An integer is often a primitive [[data type]] in [[computer language]]s. However, integer data types can only represent a [[subset]] of all integers, since practical computers are of finite capacity. Also, in the common [[two's complement]] representation, the inherent definition of [[sign (mathematics)|sign]] distinguishes between "negative" and "non-negative" rather than "negative, positive, and&nbsp;0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted ''int'' or Integer in several programming languages (such as [[Algol68]], [[C (computer language)|C]], [[Java (programming language)|Java]], [[Object Pascal|Delphi]], etc.).

Variable-length representations of integers, such as [[bignum]]s, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of&nbsp;2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or&nbsp;10).

==Cardinality==
The set of integers is [[countably infinite]], meaning it is possible to pair each integer with a unique natural number. An example of such a pairing is

:{{math|(0,&thinsp;1), (1,&thinsp;2), (−1,&thinsp;3), (2,&thinsp;4), (−2,&thinsp;5), }} {{math|(3,&thinsp;6), .&hairsp;.&hairsp;.&hairsp;,(1&hairsp;−&thinsp;''k'',&thinsp;2''k''&thinsp;−&hairsp;1), (''k'',&thinsp;2''k''&hairsp;), .&hairsp;.&hairsp;.}}

More technically, the [[cardinality]] of <math>\mathbb{Z}</math> is said to equal {{math|ℵ{{sub|0}}}} ([[Aleph number|aleph-null]]). The pairing between elements of <math>\mathbb{Z}</math> and <math>\mathbb{N}</math> is called a [[bijection]].

== See also ==
{{Portal|Mathematics}}
* [[Canonical representation of a positive integer|Canonical factorization of a positive integer]]
* [[Complex integer]]
* [[Hyperinteger]]
* [[Integer complexity]]
* [[Integer lattice]]
* [[Integer part]]
* [[Integer sequence]]
* [[Integer-valued function]]
* [[Mathematical symbols]]
* [[Parity (mathematics)]]
* [[Profinite integer]]
{{Classification of numbers}}

== Footnotes ==
{{notelist|1}}
<!-- {{reflist|group=note}} -- Please use "ref group=lower-alpha" or template "{ { efn |" for footnotes. -->

== References ==
{{reflist}}

== Sources ==
{{refbegin}}
* {{cite book |author-link=Eric Temple Bell |last=Bell |first=E.T. |title=[[Men of Mathematics]] |location=New York |publisher=Simon & Schuster |date=1986 |isbn=0-671-46400-0}})
* {{cite book |last=Herstein |first=I.N. |title=Topics in Algebra |publisher=Wiley |edition=2nd |date=1975 |isbn=0-471-01090-1}}
* {{cite book |author-link1=Saunders Mac Lane |last1=Mac Lane |first1=Saunders |author-link2=Garrett Birkhoff |last2=Birkhoff |first2=Garrett |title=Algebra |publisher=American Mathematical Society |edition=3rd |date=1999 |isbn=0-8218-1646-2}}
* {{cite book |author1=A Society of Gentlemen in Scotland |title=Encyclopaedia Britannica |date=1771 |location=Edinburgh |url=https://books.google.com/books?id=d50qAQAAMAAJ |language=en|ref = {{harvid|Encyclopaedia Britannica|1771}}}}
{{refend}}

== External links ==
{{Wiktionary}}
* {{Springer|title=Integer|id=p/i051290}}
* [http://www.positiveintegers.org The Positive Integers – divisor tables and numeral representation tools]
* [http://oeis.org/ On-Line Encyclopedia of Integer Sequences] cf [[OEIS]]
* {{mathworld|Integer}}

{{PlanetMath attribution|id=403|title=Integer}}

{{Integers}}
{{Number systems}}
{{Rational numbers}}

{{Authority control}}

[[Category:Elementary mathematics]]
[[Category:Abelian group theory]]
[[Category:Ring theory]]
[[Category:Integers| ]]
[[Category:Elementary number theory]]
[[Category:Algebraic number theory]]
[[Category:Sets of real numbers]]