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class="firstHeading mw-first-heading">Integer</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Integer" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Integer" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Language </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Integer" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-viewsource" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Integer&action=edit" data-event-name="menu.viewsource" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-viewsource"> View source </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer (computer science)</a>.</div> An integer is the <a href="/wiki/Number" title="Number">number</a> zero (<a href="/wiki/0" title="0">0</a>), a positive <a href="/wiki/Natural_number" title="Natural number">natural number</a> (1, 2, 3, . . .), or the negation of a positive natural number (<a href="/wiki/%E2%88%921" title="−1">−1</a>, −2, −3, . . .).<a href="#cite_note-1">[1]</a> The negations or <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverses</a> of the positive natural numbers are referred to as negative integers.<a href="#cite_note-2">[2]</a> The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all integers is often denoted by the <a href="/wiki/Boldface" class="mw-redirect" title="Boldface">boldface</a> Z or <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">.<a href="#cite_note-earliest-3">[3]</a><a href="#cite_note-Cameron1998-4">[4]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NumberLineIntegers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/NumberLineIntegers.svg/280px-NumberLineIntegers.svg.png" decoding="async" width="280" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/NumberLineIntegers.svg/420px-NumberLineIntegers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/NumberLineIntegers.svg/560px-NumberLineIntegers.svg.png 2x" data-file-width="400" data-file-height="50"></a><figcaption>The integers arranged on a <a href="/wiki/Number_line" title="Number line">number line</a></figcaption></figure> The set of natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> is a <a href="/wiki/Subset" title="Subset">subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">, which in turn is a subset of the set of all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">, itself a subset of the <a href="/wiki/Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">.<a href="#cite_note-8">[a]</a> Like the set of natural numbers, the set of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> is <a href="/wiki/Countable_set" title="Countable set">countably infinite</a>. An integer may be regarded as a real number that can be written without a <a href="/wiki/Fraction" title="Fraction">fractional component</a>. For example, 21, 4, 0, and −2048 are integers, while 9.75, <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠5+1/2⁠, 5/4, and √<a href="/wiki/Square_root_of_2" title="Square root of 2">2</a> are not.<a href="#cite_note-9">[8]</a> The integers form the smallest <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and the smallest <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> containing the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. In <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, the integers are sometimes qualified as rational integers to distinguish them from the more general <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. In fact, (rational) integers are algebraic integers that are also <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#History">1 History</a></li> <li class="toclevel-1 tocsection-2"><a href="#Algebraic_properties">2 Algebraic properties</a></li> <li class="toclevel-1 tocsection-3"><a href="#Order-theoretic_properties">3 Order-theoretic properties</a></li> <li class="toclevel-1 tocsection-4"><a href="#Construction">4 Construction</a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Traditional_development">4.1 Traditional development</a></li> <li class="toclevel-2 tocsection-6"><a href="#Equivalence_classes_of_ordered_pairs">4.2 Equivalence classes of ordered pairs</a></li> <li class="toclevel-2 tocsection-7"><a href="#Other_approaches">4.3 Other approaches</a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Computer_science">5 Computer science</a></li> <li class="toclevel-1 tocsection-9"><a href="#Cardinality">6 Cardinality</a></li> <li class="toclevel-1 tocsection-10"><a href="#See_also">7 See also</a></li> <li class="toclevel-1 tocsection-11"><a href="#Footnotes">8 Footnotes</a></li> <li class="toclevel-1 tocsection-12"><a href="#References">9 References</a></li> <li class="toclevel-1 tocsection-13"><a href="#Sources">10 Sources</a></li> <li class="toclevel-1 tocsection-14"><a href="#External_links">11 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="History">History</h2></div><section class="mf-section-1 collapsible-block" id="mf-section-1"> The word integer comes from the <a href="/wiki/Latin" title="Latin">Latin</a> <a href="https://en.wiktionary.org/wiki/integer#Latin" class="extiw" title="wikt:integer">integer</a> meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "<a href="https://en.wiktionary.org/wiki/entire" class="extiw" title="wikt:entire">Entire</a>" derives from the same origin via the <a href="/wiki/French_language" title="French language">French</a> word <a href="https://en.wiktionary.org/wiki/entier" class="extiw" title="wikt:entier">entier</a>, which means both entire and integer.<a href="#cite_note-10">[9]</a> Historically the term was used for a <a href="/wiki/Number" title="Number">number</a> that was a multiple of 1,<a href="#cite_note-11">[10]</a><a href="#cite_note-12">[11]</a> or to the whole part of a <a href="/wiki/Mixed_number" class="mw-redirect" title="Mixed number">mixed number</a>.<a href="#cite_note-13">[12]</a><a href="#cite_note-14">[13]</a> Only positive integers were considered, making the term synonymous with the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. The definition of integer expanded over time to include <a href="/wiki/Negative_number" title="Negative number">negative numbers</a> as their usefulness was recognized.<a href="#cite_note-negmath-15">[14]</a> For example <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in his 1765 <a href="/wiki/Elements_of_Algebra" title="Elements of Algebra">Elements of Algebra</a> defined integers to include both positive and negative numbers.<a href="#cite_note-16">[15]</a> The phrase the set of the integers was not used before the end of the 19th century, when <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> introduced the concept of <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a> and <a href="/wiki/Set_theory" title="Set theory">set theory</a>. The use of the letter Z to denote the set of integers comes from the <a href="/wiki/German_language" title="German language">German</a> word <a href="https://en.wiktionary.org/wiki/Zahlen" class="extiw" title="wikt:Zahlen">Zahlen</a> ("numbers")<a href="#cite_note-earliest-3">[3]</a><a href="#cite_note-Cameron1998-4">[4]</a> and has been attributed to <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>.<a href="#cite_note-17">[16]</a> The earliest known use of the notation in a textbook occurs in <a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Algèbre</a> written by the collective <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>, dating to 1947.<a href="#cite_note-earliest-3">[3]</a><a href="#cite_note-18">[17]</a> The notation was not adopted immediately, for example another textbook used the letter J<a href="#cite_note-19">[18]</a> and a 1960 paper used Z to denote the non-negative integers.<a href="#cite_note-20">[19]</a> But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.<a href="#cite_note-21">[20]</a> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is often annotated to denote various sets, with varying usage amongst different authors: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{+}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628778fcf14bd3629e9b9ebacffa172b0ad6ce41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{+}}"></noscript><span class="lazy-image-placeholder" style="width: 3.061ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628778fcf14bd3629e9b9ebacffa172b0ad6ce41" data-alt="{\displaystyle \mathbb {Z} ^{+}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{+}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cea652d9aba69b6accb9af09548981c2b4e24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{+}}"></noscript><span class="lazy-image-placeholder" style="width: 3.061ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cea652d9aba69b6accb9af09548981c2b4e24f" data-alt="{\displaystyle \mathbb {Z} _{+}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{>}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{>}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b674b889017afcd24e7b7d296f6d1746cefd2d94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{>}}"></noscript><span class="lazy-image-placeholder" style="width: 3.061ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b674b889017afcd24e7b7d296f6d1746cefd2d94" data-alt="{\displaystyle \mathbb {Z} ^{>}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for the positive integers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{0+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{0+}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a3b6ad192693a6aeb684d1876f82d852fcc228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.883ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{0+}}"></noscript><span class="lazy-image-placeholder" style="width: 3.883ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a3b6ad192693a6aeb684d1876f82d852fcc228" data-alt="{\displaystyle \mathbb {Z} ^{0+}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{\geq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{\geq }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1683538868e87548cd2a6dd28c28dfab75ff44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{\geq }}"></noscript><span class="lazy-image-placeholder" style="width: 3.061ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1683538868e87548cd2a6dd28c28dfab75ff44" data-alt="{\displaystyle \mathbb {Z} ^{\geq }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for non-negative integers, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{\neq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≠</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{\neq }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0413be677c88a720f7f2d2593da069808bbc252a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{\neq }}"></noscript><span class="lazy-image-placeholder" style="width: 3.061ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0413be677c88a720f7f2d2593da069808bbc252a" data-alt="{\displaystyle \mathbb {Z} ^{\neq }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for non-zero integers. Some authors use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{*}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45aa6a166e406bf9e192b7a32d6b140f27cd8538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{*}}"></noscript><span class="lazy-image-placeholder" style="width: 2.605ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45aa6a166e406bf9e192b7a32d6b140f27cd8538" data-alt="{\displaystyle \mathbb {Z} ^{*}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the <a href="/wiki/Group_of_units" class="mw-redirect" title="Group of units">group of units</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ). Additionally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></noscript><span class="lazy-image-placeholder" style="width: 2.609ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" data-alt="{\displaystyle \mathbb {Z} _{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is used to denote either the set of <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">integers modulo p</a> (i.e., the set of <a href="/wiki/Congruence_relation" title="Congruence relation">congruence classes</a> of integers), or the set of <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a>.<a href="#cite_note-edexcelc1-22">[21]</a><a href="#cite_note-23">[22]</a> The whole numbers were synonymous with the integers up until the early 1950s.<a href="#cite_note-24">[23]</a><a href="#cite_note-25">[24]</a><a href="#cite_note-26">[25]</a> In the late 1950s, as part of the <a href="/wiki/New_Math" title="New Math">New Math</a> movement,<a href="#cite_note-27">[26]</a> American elementary school teachers began teaching that whole numbers referred to the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, excluding negative numbers, while integer included the negative numbers.<a href="#cite_note-28">[27]</a><a href="#cite_note-29">[28]</a> The whole numbers remain ambiguous to the present day.<a href="#cite_note-30">[29]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Algebraic_properties">Algebraic properties</h2></div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Number-line.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/300px-Number-line.svg.png" decoding="async" width="300" height="20" class="mw-file-element" data-file-width="750" data-file-height="50"></noscript><span class="lazy-image-placeholder" style="width: 300px;height: 20px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/300px-Number-line.svg.png" data-width="300" data-height="20" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/450px-Number-line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/600px-Number-line.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Integers can be thought of as discrete, equally spaced points on an infinitely long <a href="/wiki/Number_line" title="Number line">number line</a>. In the above, non-<a href="/wiki/Sign_(mathematics)#Terminology_for_signs" title="Sign (mathematics)">negative</a> integers are shown in blue and negative integers in red.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"> Like the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under the <a href="/wiki/Binary_operation" title="Binary operation">operations</a> of addition and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , unlike the natural numbers, is also closed under <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>.<a href="#cite_note-31">[30]</a> The integers form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> which is the most basic one, in the following sense: for any ring, there is a unique <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from the integers into this ring. This <a href="/wiki/Universal_property" title="Universal property">universal property</a>, namely to be an <a href="/wiki/Initial_object" class="mw-redirect" title="Initial object">initial object</a> in the <a href="/wiki/Category_of_rings" title="Category of rings">category of rings</a>, characterizes the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not closed under <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, 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 a, b, and c: <table class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"> <caption>Properties of addition and multiplication on integers </caption> <tbody><tr> <th> </th> <th scope="col">Addition </th> <th scope="col">Multiplication </th></tr> <tr> <th scope="row"><a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">Closure</a>: </th> <td>a + b is an integer </td> <td>a × b is an integer </td></tr> <tr> <th scope="row"><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a>: </th> <td>a + (b + c) = (a + b) + c </td> <td>a × (b × c) = (a × b) × c </td></tr> <tr> <th scope="row"><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a>: </th> <td>a + b = b + a </td> <td>a × b = b × a </td></tr> <tr> <th scope="row">Existence of an <a href="/wiki/Identity_element" title="Identity element">identity element</a>: </th> <td>a + 0 = a </td> <td>a × 1 = a </td></tr> <tr> <th scope="row">Existence of <a href="/wiki/Inverse_element" title="Inverse element">inverse elements</a>: </th> <td>a + (−a) = 0 </td> <td>The only invertible integers (called <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a>) are –1 and 1. </td></tr> <tr> <th scope="row"><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a>: </th> <td colspan="2" align="center">a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c) </td></tr> <tr> <th scope="row">No <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a>: </th> <td></td> <td>If a × b = 0, then a = 0 or b = 0 (or both) </td></tr></tbody></table> The first five properties listed above for addition say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , under addition, is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>. It is also a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a>, since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The first four properties listed above for multiplication say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  under multiplication is a <a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoid</a>. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  together with addition and multiplication is a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> with <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">unity</a>. It is the prototype of all objects of such <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a>. Only those <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equalities</a> of <a href="/wiki/Algebraic_expression" title="Algebraic expression">expressions</a> are true in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <a href="/wiki/For_all" class="mw-redirect" title="For all">for all</a> values of variables, which are true in any unital commutative ring. Certain non-zero integers map to <a href="/wiki/Additive_identity" title="Additive identity">zero</a> in certain rings. The lack of <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a> in the integers (last property in the table) means that the commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. The lack of multiplicative inverses, which is equivalent to the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not closed under division, means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. The smallest field containing the integers as a <a href="/wiki/Subring" title="Subring">subring</a> is the field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. The process of constructing the rationals from the integers can be mimicked to form the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of any integral domain. And back, starting from an <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number field</a> (an extension of rational numbers), its <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> can be extracted, which includes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as its <a href="/wiki/Subring" title="Subring">subring</a>. Although ordinary division is not defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the division "with remainder" is defined on them. It is called <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of b. The integer q is called the quotient and r is called the <a href="/wiki/Remainder" title="Remainder">remainder</a> of the division of a by b. The <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> for computing <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisors</a> works by a sequence of Euclidean divisions. The above says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a>. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, and any positive integer can be written as the products of <a href="/wiki/Prime_number" title="Prime number">primes</a> in an <a href="/wiki/Essentially_unique" title="Essentially unique">essentially unique</a> way.<a href="#cite_note-32">[31]</a> This is the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Order-theoretic_properties">Order-theoretic properties</h2></div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="/wiki/Total_order" title="Total order">totally ordered set</a> without <a href="/wiki/Upper_and_lower_bounds" title="Upper and lower bounds">upper or lower bound</a>. The ordering of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < .... An integer is positive if it is greater than <a href="/wiki/0" title="0">zero</a>, 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: <ol><li>If a < b and c < d, then a + c < b + d</li> <li>If a < b and 0 < c, then ac < bc</li></ol> Thus it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  together with the above ordering is an <a href="/wiki/Ordered_ring" title="Ordered ring">ordered ring</a>. The integers are the only nontrivial <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> whose positive elements are <a href="/wiki/Well-ordered" class="mw-redirect" title="Well-ordered">well-ordered</a>.<a href="#cite_note-33">[32]</a> This is equivalent to the statement that any <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="/wiki/Valuation_ring" title="Valuation ring">valuation ring</a> is either a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>—or a <a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Construction">Construction</h2></div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"><h3 id="Traditional_development">Traditional development</h3></div> In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>, and the negations of the natural numbers. This can be formalized as follows.<a href="#cite_note-34">[33]</a> First construct the set of natural numbers according to the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>, call this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Then construct a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle P^{-}}"></noscript><span class="lazy-image-placeholder" style="width: 3.332ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" data-alt="{\displaystyle P^{-}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  which is <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and in one-to-one correspondence with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  via a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For example, take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle P^{-}}"></noscript><span class="lazy-image-placeholder" style="width: 3.332ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" data-alt="{\displaystyle P^{-}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  to be the <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,n)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e432fa1ab0d210a280dc37e0b7118565ed55946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (1,n)}"></noscript><span class="lazy-image-placeholder" style="width: 5.4ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e432fa1ab0d210a280dc37e0b7118565ed55946" data-alt="{\displaystyle (1,n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =n\mapsto (1,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =n\mapsto (1,n)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d453d0cfbd77129b34032455b63590a96e28714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.021ex; height:2.843ex;" alt="{\displaystyle \psi =n\mapsto (1,n)}"></noscript><span class="lazy-image-placeholder" style="width: 15.021ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d453d0cfbd77129b34032455b63590a96e28714" data-alt="{\displaystyle \psi =n\mapsto (1,n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Finally let 0 be some object not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle P^{-}}"></noscript><span class="lazy-image-placeholder" style="width: 3.332ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e7f2104dcc4156c6db49c5fe233e2f9e3fd91" data-alt="{\displaystyle P^{-}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , for example the ordered pair (0,0). Then the integers are defined to be the union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\cup P^{-}\cup \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∪</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\cup P^{-}\cup \{0\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373648483c45baf3647c3bfb81f55a8fd22e11d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.73ex; height:3.009ex;" alt="{\displaystyle P\cup P^{-}\cup \{0\}}"></noscript><span class="lazy-image-placeholder" style="width: 13.73ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373648483c45baf3647c3bfb81f55a8fd22e11d7" data-alt="{\displaystyle P\cup P^{-}\cup \{0\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The traditional arithmetic operations can then be defined on the integers in a <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> fashion, for each of positive numbers, negative numbers, and zero. For example <a href="/wiki/Negation" title="Negation">negation</a> is defined as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mi>P</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -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}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419206f23502e746a700939a1a4e57017144a0c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:28.45ex; height:8.843ex;" alt="{\displaystyle -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}}}"></noscript><span class="lazy-image-placeholder" style="width: 28.45ex;height: 8.843ex;vertical-align: -3.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419206f23502e746a700939a1a4e57017144a0c6" data-alt="{\displaystyle -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}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  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.<a href="#cite_note-35">[34]</a> <div class="mw-heading mw-heading3"><h3 id="Equivalence_classes_of_ordered_pairs">Equivalence classes of ordered pairs</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relative_numbers_representation.svg" class="mw-file-description"><noscript><img alt="Representation of equivalence classes for the numbers −5 to 5" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Relative_numbers_representation.svg/330px-Relative_numbers_representation.svg.png" decoding="async" width="330" height="318" class="mw-file-element" data-file-width="512" data-file-height="494"></noscript><span class="lazy-image-placeholder" style="width: 330px;height: 318px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Relative_numbers_representation.svg/330px-Relative_numbers_representation.svg.png" data-alt="Representation of equivalence classes for the numbers −5 to 5" data-width="330" data-height="318" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Relative_numbers_representation.svg/495px-Relative_numbers_representation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Relative_numbers_representation.svg/660px-Relative_numbers_representation.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Red points represent ordered pairs of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. Linked red points are equivalence classes representing the blue integers at the end of the line.</figcaption></figure> In modern set-theoretic mathematics, a more abstract construction<a href="#cite_note-36">[35]</a><a href="#cite_note-37">[36]</a> allowing one to define arithmetical operations without any case distinction is often used instead.<a href="#cite_note-38">[37]</a> The integers can thus be formally constructed as the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> (a,b).<a href="#cite_note-Campbell-1970-p83-39">[38]</a> The intuition is that (a,b) stands for the result of subtracting b from a.<a href="#cite_note-Campbell-1970-p83-39">[38]</a> To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> ~ on these pairs with the following rule: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\sim (c,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\sim (c,d)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9eed2e08692e7ce42d13803af10157e550d6328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.235ex; height:2.843ex;" alt="{\displaystyle (a,b)\sim (c,d)}"></noscript><span class="lazy-image-placeholder" style="width: 13.235ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9eed2e08692e7ce42d13803af10157e550d6328" data-alt="{\displaystyle (a,b)\sim (c,d)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> precisely when <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d=b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+d=b+c}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452fe2e66cb2e61da56ebdb3f8b8521557709cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+d=b+c}"></noscript><span class="lazy-image-placeholder" style="width: 13.229ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452fe2e66cb2e61da56ebdb3f8b8521557709cf6" data-alt="{\displaystyle a+d=b+c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;<a href="#cite_note-Campbell-1970-p83-39">[38]</a> by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>:=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cad410af4dd3435f2f9d35c9775cf0bf2f12b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.577ex; height:2.843ex;" alt="{\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]}"></noscript><span class="lazy-image-placeholder" style="width: 33.577ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cad410af4dd3435f2f9d35c9775cf0bf2f12b4" data-alt="{\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>:=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo>,</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f43120f0df09f5fcbc7dea5f5a9eac4acfabe37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.866ex; height:2.843ex;" alt="{\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]}"></noscript><span class="lazy-image-placeholder" style="width: 36.866ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f43120f0df09f5fcbc7dea5f5a9eac4acfabe37" data-alt="{\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> The negation (or additive inverse) of an integer is obtained by reversing the order of the pair: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -[(a,b)]:=[(b,a)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>:=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -[(a,b)]:=[(b,a)]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679f1ca47c5c2989051e3707e676604071f65efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.282ex; height:2.843ex;" alt="{\displaystyle -[(a,b)]:=[(b,a)]}"></noscript><span class="lazy-image-placeholder" style="width: 18.282ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679f1ca47c5c2989051e3707e676604071f65efc" data-alt="{\displaystyle -[(a,b)]:=[(b,a)]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> Hence subtraction can be defined as the addition of the additive inverse: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>:=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9a769afd448161d58b61f84bc7eb1b601ea448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.577ex; height:2.843ex;" alt="{\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]}"></noscript><span class="lazy-image-placeholder" style="width: 33.577ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9a769afd448161d58b61f84bc7eb1b601ea448" data-alt="{\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> The standard ordering on the integers is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(a,b)]<[(c,d)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo><</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(a,b)]<[(c,d)]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f5d9d3f2ef39e8cf90692589f0b97945454536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.822ex; height:2.843ex;" alt="{\displaystyle [(a,b)]<[(c,d)]}"></noscript><span class="lazy-image-placeholder" style="width: 15.822ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f5d9d3f2ef39e8cf90692589f0b97945454536" data-alt="{\displaystyle [(a,b)]<[(c,d)]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d<b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo><</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+d<b+c}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b0cf531600818efb3a8967fde95d2ed774e867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+d<b+c}"></noscript><span class="lazy-image-placeholder" style="width: 13.229ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b0cf531600818efb3a8967fde95d2ed774e867" data-alt="{\displaystyle a+d<b+c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> 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 (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (i.e., the natural numbers are <a href="/wiki/Embedding" title="Embedding">embedded</a> into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since –0 = 0. Thus, [(a,b)] is denoted by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>a</mi> <mo>≥</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>a</mi> <mo><</mo> <mi>b</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b\end{cases}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a3ad7ae93b9136af5dd636c43f7c5678257ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.414ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b\end{cases}}}"></noscript><span class="lazy-image-placeholder" style="width: 21.414ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a3ad7ae93b9136af5dd636c43f7c5678257ada" data-alt="{\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b\end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> 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 <a href="/wiki/Group_representation" title="Group representation">representation</a> of the integers as {..., −2, −1, 0, 1, 2, ...} . Some examples are: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59af091a8f28053ca4a7a7508509918bd31c9d8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:50.97ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 50.97ex;height: 15.509ex;vertical-align: -7.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59af091a8f28053ca4a7a7508509918bd31c9d8f" data-alt="{\displaystyle {\begin{aligned}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{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> <div class="mw-heading mw-heading3"><h3 id="Other_approaches">Other approaches</h3></div> In theoretical computer science, other approaches for the construction of integers are used by <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem provers</a> and <a href="/wiki/Rewriting" title="Rewriting">term rewrite engines</a>. Integers are represented as <a href="/wiki/Term_algebra" title="Term algebra">algebraic terms</a> built using a few basic operations (e.g., zero, succ, pred) and using <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, which are assumed to be already constructed (using the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano approach</a>). There exist at least ten such constructions of signed integers.<a href="#cite_note-40">[39]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></noscript><span class="lazy-image-placeholder" style="width: 5.328ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" data-alt="{\displaystyle (x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  that takes as arguments two natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and returns an integer (equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}"></noscript><span class="lazy-image-placeholder" style="width: 5.326ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265" data-alt="{\displaystyle x-y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ). 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 <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistant</a> <a href="/wiki/Isabelle_(proof_assistant)" title="Isabelle (proof assistant)">Isabelle</a>; 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. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Computer_science">Computer science</h2></div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer (computer science)</a></div> An integer is often a primitive <a href="/wiki/Data_type" title="Data type">data type</a> in <a href="/wiki/Computer_language" title="Computer language">computer languages</a>. However, integer data types can only represent a <a href="/wiki/Subset" title="Subset">subset</a> of all integers, since practical computers are of finite capacity. Also, in the common <a href="/wiki/Two%27s_complement" title="Two's complement">two's complement</a> representation, the inherent definition of <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> distinguishes between "negative" and "non-negative" rather than "negative, positive, and 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 <a href="/wiki/Algol68" class="mw-redirect" title="Algol68">Algol68</a>, <a href="/wiki/C_(computer_language)" class="mw-redirect" title="C (computer language)">C</a>, <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="/wiki/Object_Pascal" title="Object Pascal">Delphi</a>, etc.). Variable-length representations of integers, such as <a href="/wiki/Bignum" class="mw-redirect" title="Bignum">bignums</a>, 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 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10). </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Cardinality">Cardinality</h2></div><section class="mf-section-6 collapsible-block" id="mf-section-6"> The set of integers is <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a>, meaning it is possible to pair each integer with a unique natural number. An example of such a pairing is <dl><dd>(0, 1), (1, 2), (−1, 3), (2, 4), (−2, 5), (3, 6), . . . ,(1 − k, 2k − 1), (k, 2k ), . . .</dd></dl> More technically, the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is said to equal ℵ0 (<a href="/wiki/Aleph_number" title="Aleph number">aleph-null</a>). The pairing between elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is called a <a href="/wiki/Bijection" title="Bijection">bijection</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="See_also">See also</h2></div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><noscript><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" data-file-width="128" data-file-height="128"></noscript><span class="lazy-image-placeholder" style="width: 28px;height: 28px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" data-alt="icon" data-width="28" data-height="28" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-class="mw-file-element"> </a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <ul><li><a href="/wiki/Canonical_representation_of_a_positive_integer" class="mw-redirect" title="Canonical representation of a positive integer">Canonical factorization of a positive integer</a></li> <li><a href="/wiki/Complex_integer" class="mw-redirect" title="Complex integer">Complex integer</a></li> <li><a href="/wiki/Hyperinteger" title="Hyperinteger">Hyperinteger</a></li> <li><a href="/wiki/Integer_complexity" title="Integer complexity">Integer complexity</a></li> <li><a href="/wiki/Integer_lattice" title="Integer lattice">Integer lattice</a></li> <li><a href="/wiki/Integer_part" class="mw-redirect" title="Integer part">Integer part</a></li> <li><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequence</a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function">Integer-valued function</a></li> <li><a href="/wiki/Mathematical_symbols" class="mw-redirect" title="Mathematical symbols">Mathematical symbols</a></li> <li><a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">Parity (mathematics)</a></li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integer</a></li></ul> <table style="margin:2em; border:2px solid silver; font-size:95%; border-collapse:collapse"> <tbody><tr> <td> <table style="margin:4px; border:2px solid silver"> <tbody><tr> <td> <table style="margin:1em"> <caption><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number systems</a> </caption> <tbody><tr> <td><a href="/wiki/Complex_number" title="Complex number">Complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {C} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" data-alt="{\displaystyle :\;\mathbb {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Real_number" title="Real number">Real</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" data-alt="{\displaystyle :\;\mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Rational_number" title="Rational number">Rational</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Q} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }"></noscript><span class="lazy-image-placeholder" style="width: 3.745ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" data-alt="{\displaystyle :\;\mathbb {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a class="mw-selflink selflink">Integer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 3.487ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" data-alt="{\displaystyle :\;\mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Natural_number" title="Natural number">Natural</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" data-alt="{\displaystyle :\;\mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>: 0 </td></tr> <tr> <td><a href="/wiki/One" class="mw-redirect" title="One">One</a>: 1 </td></tr> <tr> <td><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> </td></tr> <tr> <td><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Fraction" title="Fraction">Fraction</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a> </td></tr> <tr> <td><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a> </td></tr> <tr> <td><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> </td> <td> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Irrational_number" title="Irrational number">Irrational</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic irrational</a> </td></tr> <tr> <td><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a> </td></tr> <tr> <td><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="Footnotes">Footnotes</h2></div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-lower-alpha"> <ol class="references"> <li id="cite_note-8"><a href="#cite_ref-8">^</a> More precisely, each system is <a href="/wiki/Embedding" title="Embedding">embedded</a> in the next, isomorphically mapped to a subset.<a href="#cite_note-5">[5]</a> 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.<a href="#cite_note-6">[6]</a> Such a convention is "a matter of choice", yet not.<a href="#cite_note-7">[7]</a> </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="References">References</h2></div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=PZIdcYCCf2kC&dq=integer&pg=PA280">Science and Technology Encyclopedia</a>. University of Chicago Press. September 2000. p. 280. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-74267-0" title="Special:BookSources/978-0-226-74267-0"><bdi>978-0-226-74267-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Technology+Encyclopedia&rft.pages=280&rft.pub=University+of+Chicago+Press&rft.date=2000-09&rft.isbn=978-0-226-74267-0&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPZIdcYCCf2kC%26dq%3Dinteger%26pg%3DPA280&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHillmanAlexanderson1963" class="citation book cs1">Hillman, Abraham P.; Alexanderson, Gerald L. (1963). <a rel="nofollow" class="external text" href="https://archive.org/details/algebratrigonome0000hill/page/42/mode/2up">Algebra and trigonometry;</a>. Boston: Allyn and Bacon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+and+trigonometry%3B&rft.place=Boston&rft.pub=Allyn+and+Bacon&rft.date=1963&rft.aulast=Hillman&rft.aufirst=Abraham+P.&rft.au=Alexanderson%2C+Gerald+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebratrigonome0000hill%2Fpage%2F42%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-earliest-3">^ <a href="#cite_ref-earliest_3-0">a</a> <a href="#cite_ref-earliest_3-1">b</a> <a href="#cite_ref-earliest_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller2010" class="citation web cs1">Miller, Jeff (29 August 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100131022510/http://jeff560.tripod.com/nth.html">"Earliest Uses of Symbols of Number Theory"</a>. Archived from <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/nth.html">the original</a> on 31 January 2010. Retrieved 20 September 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earliest+Uses+of+Symbols+of+Number+Theory&rft.date=2010-08-29&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fnth.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-Cameron1998-4">^ <a href="#cite_ref-Cameron1998_4-0">a</a> <a href="#cite_ref-Cameron1998_4-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Jephson_Cameron1998" class="citation book cs1">Peter Jephson Cameron (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4">Introduction to Algebra</a>. Oxford University Press. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850195-4" title="Special:BookSources/978-0-19-850195-4"><bdi>978-0-19-850195-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161208142220/https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4">Archived</a> from the original on 8 December 2016. Retrieved 15 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algebra&rft.pages=4&rft.pub=Oxford+University+Press&rft.date=1998&rft.isbn=978-0-19-850195-4&rft.au=Peter+Jephson+Cameron&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsyYYl-NVM5IC%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParteeMeulenWall1990" class="citation book cs1">Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qV7TUuaYcUIC&pg=PA80">Mathematical Methods in Linguistics</a>. Springer Science & Business Media. pp. 78–82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-277-2245-4" title="Special:BookSources/978-90-277-2245-4"><bdi>978-90-277-2245-4</bdi></a>. <q>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.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+in+Linguistics&rft.pages=78-82&rft.pub=Springer+Science+%26+Business+Media&rft.date=1990-04-30&rft.isbn=978-90-277-2245-4&rft.aulast=Partee&rft.aufirst=Barbara+H.&rft.au=Meulen%2C+Alice+ter&rft.au=Wall%2C+Robert+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqV7TUuaYcUIC%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWohlgemuth2014" class="citation book cs1">Wohlgemuth, Andrew (10 June 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PEP_AwAAQBAJ&pg=PA237">Introduction to Proof in Abstract Mathematics</a>. Courier Corporation. p. 237. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-14168-8" title="Special:BookSources/978-0-486-14168-8"><bdi>978-0-486-14168-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Proof+in+Abstract+Mathematics&rft.pages=237&rft.pub=Courier+Corporation&rft.date=2014-06-10&rft.isbn=978-0-486-14168-8&rft.aulast=Wohlgemuth&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPEP_AwAAQBAJ%26pg%3DPA237&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolkinghorne2011" class="citation book cs1">Polkinghorne, John (19 May 2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DCqQDwAAQBAJ&pg=PA68">Meaning in Mathematics</a>. OUP Oxford. p. 68. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-162189-5" title="Special:BookSources/978-0-19-162189-5"><bdi>978-0-19-162189-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Meaning+in+Mathematics&rft.pages=68&rft.pub=OUP+Oxford&rft.date=2011-05-19&rft.isbn=978-0-19-162189-5&rft.aulast=Polkinghorne&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDCqQDwAAQBAJ%26pg%3DPA68&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrep2019" class="citation book cs1">Prep, Kaplan Test (4 June 2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6l_sDwAAQBAJ&pg=PA708">GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT</a>. Simon and Schuster. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5062-4844-8" title="Special:BookSources/978-1-5062-4844-8"><bdi>978-1-5062-4844-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=GMAT+Complete+2020%3A+The+Ultimate+in+Comprehensive+Self-Study+for+GMAT&rft.pub=Simon+and+Schuster&rft.date=2019-06-04&rft.isbn=978-1-5062-4844-8&rft.aulast=Prep&rft.aufirst=Kaplan+Test&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6l_sDwAAQBAJ%26pg%3DPA708&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1995" class="citation book cs1">Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NlQL97qBSZkC">Quantification in Natural Languages</a>. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-3352-4" title="Special:BookSources/978-0-7923-3352-4"><bdi>978-0-7923-3352-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A-Quantifiers+and+Scope&rft.btitle=Quantification+in+Natural+Languages&rft.place=Dordrecht%2C+The+Netherlands%3B+Boston%2C+MA&rft.pages=262&rft.pub=Kluwer+Academic+Publishers&rft.date=1995&rft.isbn=978-0-7923-3352-4&rft.aulast=Evans&rft.aufirst=Nick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNlQL97qBSZkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmedleyRoseRose1845" class="citation book cs1">Smedley, Edward; Rose, Hugh James; Rose, Henry John (1845). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZVI_AQAAMAAJ&pg=PA537">Encyclopædia Metropolitana</a>. B. Fellowes. p. 537. <q>An integer is a multiple of unity</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclop%C3%A6dia+Metropolitana&rft.pages=537&rft.pub=B.+Fellowes&rft.date=1845&rft.aulast=Smedley&rft.aufirst=Edward&rft.au=Rose%2C+Hugh+James&rft.au=Rose%2C+Henry+John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZVI_AQAAMAAJ%26pg%3DPA537&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFEncyclopaedia_Britannica1771">Encyclopaedia Britannica 1771</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA367">367</a> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPisanoBoncompagni1202" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Fibonacci" title="Fibonacci">Pisano, Leonardo</a>; Boncompagni, Baldassarre (transliteration) (1202). <a rel="nofollow" class="external text" href="https://bibdig.museogalileo.it/tecanew/opera?bid=1072400&seq=30">Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij</a> [The Book of Calculation] (Manuscript) (in Latin). Translated by Sigler, Laurence E. Museo Galileo. p. 30. <q>Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant.</q> [And the fractions are always put after the whole, thus first the integer is written, and then the fraction]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Incipit+liber+Abbaci+compositus+to+Lionardo+filio+Bonaccii+Pisano+in+year+Mccij&rft.pages=30&rft.pub=Museo+Galileo&rft.date=1202&rft.aulast=Pisano&rft.aufirst=Leonardo&rft.au=Boncompagni%2C+Baldassarre+%28transliteration%29&rft_id=https%3A%2F%2Fbibdig.museogalileo.it%2Ftecanew%2Fopera%3Fbid%3D1072400%26seq%3D30&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFEncyclopaedia_Britannica1771">Encyclopaedia Britannica 1771</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA83">83</a> </li> <li id="cite_note-negmath-15"><a href="#cite_ref-negmath_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartinez2014" class="citation book cs1">Martinez, Alberto (2014). Negative Math. Princeton University Press. pp. 80–109.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Negative+Math&rft.pages=80-109&rft.pub=Princeton+University+Press&rft.date=2014&rft.aulast=Martinez&rft.aufirst=Alberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1771" class="citation book cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1771). <a rel="nofollow" class="external text" href="https://archive.org/details/1770LEULERVollstandigeAnleitungZurAlgebraVol1/page/n31/mode/2up">Vollstandige Anleitung Zur Algebra</a> [Complete Introduction to Algebra] (in German). Vol. 1. p. 10. <q>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.</q> [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.]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vollstandige+Anleitung+Zur+Algebra&rft.pages=10&rft.date=1771&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2F1770LEULERVollstandigeAnleitungZurAlgebraVol1%2Fpage%2Fn31%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1 cs1-prop-long-vol"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z-7kAAAAMAAJ">The University of Leeds Review</a>. Vol. 31–32. University of Leeds. 1989. p. 46. <q>Incidentally, Z comes from "Zahl": the notation was created by Hilbert.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+University+of+Leeds+Review&rft.pages=46&rft.pub=University+of+Leeds.&rft.date=1989&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZ-7kAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1951" class="citation book cs1 cs1-prop-foreign-lang-source">Bourbaki, Nicolas (1951). <a rel="nofollow" class="external text" href="https://archive.org/details/algebrebour00bour/page/26/mode/2up">Algèbre, Chapter 1</a> (in French) (2nd ed.). Paris: Hermann. p. 27. <q>Le symétrisé de N se note Z; ses éléments sont appelés entiers rationnels.</q> [The group of differences of N is denoted by Z; its elements are called the rational integers.]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre%2C+Chapter+1&rft.place=Paris&rft.pages=27&rft.edition=2nd&rft.pub=Hermann&rft.date=1951&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebrebour00bour%2Fpage%2F26%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirkhoff1948" class="citation book cs1">Birkhoff, Garrett (1948). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.166886/page/n63/mode/2up">Lattice Theory</a> (Revised ed.). American Mathematical Society. p. 63. <q>the set J of all integers</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lattice+Theory&rft.pages=63&rft.edition=Revised&rft.pub=American+Mathematical+Society&rft.date=1948&rft.aulast=Birkhoff&rft.aufirst=Garrett&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.166886%2Fpage%2Fn63%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSociety1960" class="citation book cs1">Society, Canadian Mathematical (1960). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uMAXOmLTCGsC&dq=integer%20set%20Z&pg=PA374">Canadian Journal of Mathematics</a>. Canadian Mathematical Society. p. 374. <q>Consider the set Z of non-negative integers</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Canadian+Journal+of+Mathematics&rft.pages=374&rft.pub=Canadian+Mathematical+Society&rft.date=1960&rft.aulast=Society&rft.aufirst=Canadian+Mathematical&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuMAXOmLTCGsC%26dq%3Dinteger%2520set%2520Z%26pg%3DPA374&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBezuszka1961" class="citation book cs1">Bezuszka, Stanley (1961). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dhJPAQAAMAAJ&q=integer+set+Z">Contemporary Progress in Mathematics: Teacher Supplement [to] Part 1 and Part 2</a>. Boston College. p. 69. <q>Modern Algebra texts generally designate the set of integers by the capital letter Z.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Progress+in+Mathematics%3A+Teacher+Supplement+%5Bto%5D+Part+1+and+Part+2&rft.pages=69&rft.pub=Boston+College&rft.date=1961&rft.aulast=Bezuszka&rft.aufirst=Stanley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdhJPAQAAMAAJ%26q%3Dinteger%2Bset%2BZ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-edexcelc1-22"><a href="#cite_ref-edexcelc1_22-0">^</a> Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMathews1892" class="citation book cs1">Mathews, George Ballard (1892). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iQ_vAAAAMAAJ&pg=PA2">Theory of Numbers</a>. Deighton, Bell and Company. p. 2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Numbers&rft.pages=2&rft.pub=Deighton%2C+Bell+and+Company&rft.date=1892&rft.aulast=Mathews&rft.aufirst=George+Ballard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiQ_vAAAAMAAJ%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBetz1934" class="citation book cs1">Betz, William (1934). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RzNCAAAAIAAJ">Junior Mathematics for Today</a>. Ginn. <q>The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Junior+Mathematics+for+Today&rft.pub=Ginn&rft.date=1934&rft.aulast=Betz&rft.aufirst=William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRzNCAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeck1950" class="citation book cs1">Peck, Lyman C. (1950). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tclXAAAAYAAJ&q=integers+whole+numbers">Elements of Algebra</a>. McGraw-Hill. p. 3. <q>The numbers which so arise are called positive whole numbers, or positive integers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Algebra&rft.pages=3&rft.pub=McGraw-Hill&rft.date=1950&rft.aulast=Peck&rft.aufirst=Lyman+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtclXAAAAYAAJ%26q%3Dintegers%2Bwhole%2Bnumbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayden1981" class="citation thesis cs1">Hayden, Robert (1981). <a rel="nofollow" class="external text" href="https://dr.lib.iastate.edu/handle/20.500.12876/80303">A history of the "new math" movement in the United States</a> (PhD). Iowa State University. p. 145. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.31274%2Frtd-180813-5631">10.31274/rtd-180813-5631</a>. <q>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).</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=A+history+of+the+%22new+math%22+movement+in+the+United+States&rft.inst=Iowa+State+University&rft.date=1981&rft_id=info%3Adoi%2F10.31274%2Frtd-180813-5631&rft.aulast=Hayden&rft.aufirst=Robert&rft_id=https%3A%2F%2Fdr.lib.iastate.edu%2Fhandle%2F20.500.12876%2F80303&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=OO9RAQAAIAAJ&pg=PA14">The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook</a>. National Council of Teachers of Mathematics. 1959. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780608166186" title="Special:BookSources/9780608166186"><bdi>9780608166186</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Growth+of+Mathematical+Ideas%2C+Grades+K-12%3A+24th+Yearbook&rft.pages=14&rft.pub=National+Council+of+Teachers+of+Mathematics&rft.date=1959&rft.isbn=9780608166186&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOO9RAQAAIAAJ%26pg%3DPA14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeans1963" class="citation book cs1">Deans, Edwina (1963). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bAUJAQAAMAAJ&pg=PA42">Elementary School Mathematics: New Directions</a>. U.S. Department of Health, Education, and Welfare, Office of Education. p. 42.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+School+Mathematics%3A+New+Directions&rft.pages=42&rft.pub=U.S.+Department+of+Health%2C+Education%2C+and+Welfare%2C+Office+of+Education&rft.date=1963&rft.aulast=Deans&rft.aufirst=Edwina&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbAUJAQAAMAAJ%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.ahdictionary.com/word/search.html?q=whole+number">"entry: whole number"</a>. The American Heritage Dictionary. HarperCollins.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+American+Heritage+Dictionary&rft.atitle=entry%3A+whole+number&rft_id=https%3A%2F%2Fwww.ahdictionary.com%2Fword%2Fsearch.html%3Fq%3Dwhole%2Bnumber&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/integer">"Integer | mathematics"</a>. Encyclopedia Britannica. Retrieved 11 August 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=Integer+%7C+mathematics&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Finteger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1993" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1993). Algebra (3rd ed.). Addison-Wesley. pp. 86–87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-55540-0" title="Special:BookSources/978-0-201-55540-0"><bdi>978-0-201-55540-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=86-87&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1993&rft.isbn=978-0-201-55540-0&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner2012" class="citation book cs1">Warner, Seth (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185">Modern Algebra</a>. Dover Books on Mathematics. Courier Corporation. Theorem 20.14, p. 185. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13709-4" title="Special:BookSources/978-0-486-13709-4"><bdi>978-0-486-13709-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150906083836/https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185">Archived</a> from the original on 6 September 2015. Retrieved 29 April 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Algebra&rft.series=Dover+Books+on+Mathematics&rft.pages=Theorem+20.14%2C+p.-185&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-13709-4&rft.aulast=Warner&rft.aufirst=Seth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTqHDAgAAQBAJ%26pg%3DPA185&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988">. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson1985" class="citation book cs1">Mendelson, Elliott (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/numbersystemsfou0000mend/page/152/mode/2up">Number systems and the foundations of analysis</a>. Malabar, Fla. : R.E. Krieger Pub. Co. p. 153. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89874-818-5" title="Special:BookSources/978-0-89874-818-5"><bdi>978-0-89874-818-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+systems+and+the+foundations+of+analysis&rft.pages=153&rft.pub=Malabar%2C+Fla.+%3A+R.E.+Krieger+Pub.+Co.&rft.date=1985&rft.isbn=978-0-89874-818-5&rft.aulast=Mendelson&rft.aufirst=Elliott&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumbersystemsfou0000mend%2Fpage%2F152%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson2008" class="citation book cs1">Mendelson, Elliott (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3domViIV7HMC&pg=PA86">Number Systems and the Foundations of Analysis</a>. Dover Books on Mathematics. Courier Dover Publications. p. 86. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45792-5" title="Special:BookSources/978-0-486-45792-5"><bdi>978-0-486-45792-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161208233040/https://books.google.com/books?id=3domViIV7HMC&pg=PA86">Archived</a> from the original on 8 December 2016. Retrieved 15 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Systems+and+the+Foundations+of+Analysis&rft.series=Dover+Books+on+Mathematics&rft.pages=86&rft.pub=Courier+Dover+Publications&rft.date=2008&rft.isbn=978-0-486-45792-5&rft.aulast=Mendelson&rft.aufirst=Elliott&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3domViIV7HMC%26pg%3DPA86&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988">. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> Ivorra Castillo: Álgebra </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKramervon_Pippich2017" class="citation book cs1">Kramer, Jürg; von Pippich, Anna-Maria (2017). From Natural Numbers to Quaternions (1st ed.). Switzerland: Springer Cham. pp. 78–81. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-69429-0">10.1007/978-3-319-69429-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-69427-6" title="Special:BookSources/978-3-319-69427-6"><bdi>978-3-319-69427-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Natural+Numbers+to+Quaternions&rft.place=Switzerland&rft.pages=78-81&rft.edition=1st&rft.pub=Springer+Cham&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-319-69429-0&rft.isbn=978-3-319-69427-6&rft.aulast=Kramer&rft.aufirst=J%C3%BCrg&rft.au=von+Pippich%2C+Anna-Maria&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrobisher1999" class="citation book cs1">Frobisher, Len (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126">Learning to Teach Number: A Handbook for Students and Teachers in the Primary School</a>. The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. p. 126. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7487-3515-0" title="Special:BookSources/978-0-7487-3515-0"><bdi>978-0-7487-3515-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161208121843/https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126">Archived</a> from the original on 8 December 2016. Retrieved 15 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learning+to+Teach+Number%3A+A+Handbook+for+Students+and+Teachers+in+the+Primary+School&rft.series=The+Stanley+Thornes+Teaching+Primary+Maths+Series&rft.pages=126&rft.pub=Nelson+Thornes&rft.date=1999&rft.isbn=978-0-7487-3515-0&rft.aulast=Frobisher&rft.aufirst=Len&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKwJQIt4jQHUC%26pg%3DPA126&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988">. </li> <li id="cite_note-Campbell-1970-p83-39">^ <a href="#cite_ref-Campbell-1970-p83_39-0">a</a> <a href="#cite_ref-Campbell-1970-p83_39-1">b</a> <a href="#cite_ref-Campbell-1970-p83_39-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCampbell,_Howard_E.1970" class="citation book cs1">Campbell, Howard E. (1970). <a rel="nofollow" class="external text" href="https://archive.org/details/structureofarith00camp/page/83">The structure of arithmetic</a>. Appleton-Century-Crofts. p. <a rel="nofollow" class="external text" href="https://archive.org/details/structureofarith00camp/page/83">83</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-390-16895-5" title="Special:BookSources/978-0-390-16895-5"><bdi>978-0-390-16895-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+structure+of+arithmetic&rft.pages=83&rft.pub=Appleton-Century-Crofts&rft.date=1970&rft.isbn=978-0-390-16895-5&rft.au=Campbell%2C+Howard+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstructureofarith00camp%2Fpage%2F83&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaravel2017" class="citation conference cs1">Garavel, Hubert (2017). <a rel="nofollow" class="external text" href="https://hal.inria.fr/hal-01667321">On the Most Suitable Axiomatization of Signed Integers</a>. Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016). Lecture Notes in Computer Science. Vol. 10644. Springer. pp. 120–134. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-72044-9_9">10.1007/978-3-319-72044-9_9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-72043-2" title="Special:BookSources/978-3-319-72043-2"><bdi>978-3-319-72043-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180126125528/https://hal.inria.fr/hal-01667321">Archived</a> from the original on 26 January 2018. Retrieved 25 January 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=On+the+Most+Suitable+Axiomatization+of+Signed+Integers&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=120-134&rft.pub=Springer&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-319-72044-9_9&rft.isbn=978-3-319-72043-2&rft.aulast=Garavel&rft.aufirst=Hubert&rft_id=https%3A%2F%2Fhal.inria.fr%2Fhal-01667321&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="Sources">Sources</h2></div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1986" class="citation book cs1"><a href="/wiki/Eric_Temple_Bell" title="Eric Temple Bell">Bell, E.T.</a> (1986). <a href="/wiki/Men_of_Mathematics" title="Men of Mathematics">Men of Mathematics</a>. New York: Simon & Schuster. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-671-46400-0" title="Special:BookSources/0-671-46400-0"><bdi>0-671-46400-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Men+of+Mathematics&rft.place=New+York&rft.pub=Simon+%26+Schuster&rft.date=1986&rft.isbn=0-671-46400-0&rft.aulast=Bell&rft.aufirst=E.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988">)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerstein1975" class="citation book cs1">Herstein, I.N. (1975). Topics in Algebra (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-01090-1" title="Special:BookSources/0-471-01090-1"><bdi>0-471-01090-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Algebra&rft.edition=2nd&rft.pub=Wiley&rft.date=1975&rft.isbn=0-471-01090-1&rft.aulast=Herstein&rft.aufirst=I.N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneBirkhoff1999" class="citation book cs1"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1999). Algebra (3rd ed.). American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-1646-2" title="Special:BookSources/0-8218-1646-2"><bdi>0-8218-1646-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=3rd&rft.pub=American+Mathematical+Society&rft.date=1999&rft.isbn=0-8218-1646-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEncyclopaedia_Britannica1771" class="citation book cs1">A Society of Gentlemen in Scotland (1771). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d50qAQAAMAAJ">Encyclopaedia Britannica</a>. Edinburgh.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+Britannica&rft.place=Edinburgh&rft.date=1771&rft.au=A+Society+of+Gentlemen+in+Scotland&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dd50qAQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="External_links">External links</h2></div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="512" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 40px;height: 40px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" data-alt="" data-width="40" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-class="mw-file-element"> </div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/Special:Search/integer" class="extiw" title="wiktionary:Special:Search/integer">integer</a> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Integer">"Integer"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Integer&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DInteger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.positiveintegers.org">The Positive Integers – divisor tables and numeral representation tools</a></li> <li><a rel="nofollow" class="external text" href="http://oeis.org/">On-Line Encyclopedia of Integer Sequences</a> cf <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Integer.html">"Integer"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Integer&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FInteger.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger" class="Z3988"></li></ul> This article incorporates material from Integer on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>. <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output 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absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Integer&oldid=1257841897">https://en.wikipedia.org/w/index.php?title=Integer&oldid=1257841897</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Integer&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 16 November 2024, at 22:21 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Heelgetal" title="Heelgetal – Afrikaans" lang="af" hreflang="af" data-title="Heelgetal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ganze_Zahl" title="Ganze Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Ganze Zahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="पूर्ण संख्या – Angika" lang="anp" hreflang="anp" data-title="पूर्ण संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%D9%8A%D8%AD" title="عدد صحيح – Arabic" lang="ar" hreflang="ar" data-title="عدد صحيح" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_entero" title="Numero entero – Aragonese" lang="an" hreflang="an" data-title="Numero entero" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B1%D5%B4%D5%A2%D5%B8%D5%B2%D5%BB_%D5%A9%D5%AB%D6%82" title="Ամբողջ թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Ամբողջ թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%85%E0%A6%96%E0%A6%A3%E0%A7%8D%E0%A6%A1_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="অখণ্ড সংখ্যা – Assamese" lang="as" hreflang="as" data-title="অখণ্ড সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_enteru" title="Númberu enteru – Asturian" lang="ast" hreflang="ast" data-title="Númberu enteru" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Tam_%C9%99d%C9%99dl%C9%99r" title="Tam ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Tam ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AA%D8%A7%D9%85_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="تام ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="تام ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%82%E0%A6%B0%E0%A7%8D%E0%A6%A3_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="পূর্ণ সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="পূর্ণ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%A9ng-s%C3%B2%CD%98" title="Chéng-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Chéng-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%91%D3%A9%D1%82%D3%A9%D0%BD_%D2%BB%D0%B0%D0%BD" title="Бөтөн һан – Bashkir" lang="ba" hreflang="ba" data-title="Бөтөн һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8D%D0%BB%D1%8B_%D0%BB%D1%96%D0%BA" title="Цэлы лік – Belarusian" lang="be" hreflang="be" data-title="Цэлы лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A6%D1%8D%D0%BB%D1%8B_%D0%BB%D1%96%D0%BA" title="Цэлы лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Цэлы лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Integer" title="Integer – Central Bikol" lang="bcl" hreflang="bcl" data-title="Integer" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A6%D1%8F%D0%BB%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Цяло число – Bulgarian" lang="bg" hreflang="bg" data-title="Цяло число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Bosnian" lang="bs" hreflang="bs" data-title="Cijeli broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kevan_daveel" title="Kevan daveel – Breton" lang="br" hreflang="br" data-title="Kevan daveel" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_enter" title="Nombre enter – Catalan" lang="ca" hreflang="ca" data-title="Nombre enter" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%83%D0%BB%D0%BB%D0%B8_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Тулли хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Тулли хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cel%C3%A9_%C4%8D%C3%ADslo" title="Celé číslo – Czech" lang="cs" hreflang="cs" data-title="Celé číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhamba_mhumburu" title="Nhamba mhumburu – Shona" lang="sn" hreflang="sn" data-title="Nhamba mhumburu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfanrif" title="Cyfanrif – Welsh" lang="cy" hreflang="cy" data-title="Cyfanrif" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Heltal" title="Heltal – Danish" lang="da" hreflang="da" data-title="Heltal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ganze_Zahl" title="Ganze Zahl – German" lang="de" hreflang="de" data-title="Ganze Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%A4isarv" title="Täisarv – Estonian" lang="et" hreflang="et" data-title="Täisarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BA%CE%AD%CF%81%CE%B1%CE%B9%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Ακέραιος αριθμός – Greek" lang="el" hreflang="el" data-title="Ακέραιος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_entero" title="Número entero – Spanish" lang="es" hreflang="es" data-title="Número entero" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Entjero" title="Entjero – Esperanto" lang="eo" hreflang="eo" data-title="Entjero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_oso" title="Zenbaki oso – Basque" lang="eu" hreflang="eu" data-title="Zenbaki oso" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%DB%8C%D8%AD" title="عدد صحیح – Persian" lang="fa" hreflang="fa" data-title="عدد صحیح" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Heiltal" title="Heiltal – Faroese" lang="fo" hreflang="fo" data-title="Heiltal" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Entier_relatif" title="Entier relatif – French" lang="fr" hreflang="fr" data-title="Entier relatif" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sl%C3%A1nuimhir" title="Slánuimhir – Irish" lang="ga" hreflang="ga" data-title="Slánuimhir" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_enteiro" title="Número enteiro – Galician" lang="gl" hreflang="gl" data-title="Número enteiro" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%91%D3%80%D0%B0%D1%80%D1%87%D1%87%D0%B0_%D1%82%D0%B0%D1%8C%D1%80%D0%B0%D1%85%D1%8C" title="БӀарчча таьрахь – Ingush" lang="inh" hreflang="inh" data-title="БӀарчча таьрахь" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B4%E6%95%B8" title="整數 – Gan" lang="gan" hreflang="gan" data-title="整數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AB%82%E0%AA%B0%E0%AB%8D%E0%AA%A3%E0%AA%BE%E0%AA%82%E0%AA%95_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE%E0%AA%93" title="પૂર્ણાંક સંખ્યાઓ – Gujarati" lang="gu" hreflang="gu" data-title="પૂર્ણાંક સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D2%AF%D0%BA%D0%BB_%D1%82%D0%BE%D0%B9%D0%B3" title="Бүкл тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Бүкл тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%88%98" title="정수 – Korean" lang="ko" hreflang="ko" data-title="정수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Helu_piha" title="Helu piha – Hawaiian" lang="haw" hreflang="haw" data-title="Helu piha" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B4%D5%A2%D5%B8%D5%B2%D5%BB_%D5%A9%D5%AB%D5%BE" title="Ամբողջ թիվ – Armenian" lang="hy" hreflang="hy" data-title="Ամբողջ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%BE%E0%A4%82%E0%A4%95" title="पूर्णांक – Hindi" lang="hi" hreflang="hi" data-title="पूर्णांक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Cy%C5%82a_li%C4%8Dba" title="Cyła ličba – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Cyła ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Croatian" lang="hr" hreflang="hr" data-title="Cijeli broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Integro" title="Integro – Ido" lang="io" hreflang="io" data-title="Integro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_bulat" title="Bilangan bulat – Indonesian" lang="id" hreflang="id" data-title="Bilangan bulat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_integre" title="Numero integre – Interlingua" lang="ia" hreflang="ia" data-title="Numero integre" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-integer" title="I-integer – Xhosa" lang="xh" hreflang="xh" data-title="I-integer" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Heilt%C3%B6lur" title="Heiltölur – Icelandic" lang="is" hreflang="is" data-title="Heiltölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_intero" title="Numero intero – Italian" lang="it" hreflang="it" data-title="Numero intero" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A9%D7%9C%D7%9D" title="מספר שלם – Hebrew" lang="he" hreflang="he" data-title="מספר שלם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Wilangan_bulat" title="Wilangan bulat – Javanese" lang="jv" hreflang="jv" data-title="Wilangan bulat" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%97%E1%83%94%E1%83%9A%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="მთელი რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="მთელი რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D2%AF%D1%82%D1%96%D0%BD_%D1%81%D0%B0%D0%BD" title="Бүтін сан – Kazakh" lang="kk" hreflang="kk" data-title="Бүтін сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Nambakamili" title="Nambakamili – Swahili" lang="sw" hreflang="sw" data-title="Nambakamili" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Anty%C3%A9_r%C3%A9latif" title="Antyé rélatif – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Antyé rélatif" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Tamhejmar" title="Tamhejmar – Kurdish" lang="ku" hreflang="ku" data-title="Tamhejmar" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D2%AF%D1%82%D2%AF%D0%BD_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80" title="Бүтүн сандар – Kyrgyz" lang="ky" hreflang="ky" data-title="Бүтүн сандар" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%96%E0%BB%89%E0%BA%A7%E0%BA%99" title="ຈຳນວນຖ້ວນ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນຖ້ວນ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_integer" title="Numerus integer – Latin" lang="la" hreflang="la" data-title="Numerus integer" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vesels_skaitlis" title="Vesels skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Vesels skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ganz_Zuel" title="Ganz Zuel – Luxembourgish" lang="lb" hreflang="lb" data-title="Ganz Zuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sveikasis_skai%C4%8Dius" title="Sveikasis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Sveikasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Gans_getal" title="Gans getal – Limburgish" lang="li" hreflang="li" data-title="Gans getal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_intera" title="Numero intera – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero intera" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/mulna%27u" title="mulna'u – Lojban" lang="jbo" hreflang="jbo" data-title="mulna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_intreg" title="Numer intreg – Lombard" lang="lmo" hreflang="lmo" data-title="Numer intreg" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Eg%C3%A9sz_sz%C3%A1mok" title="Egész számok – Hungarian" lang="hu" hreflang="hu" data-title="Egész számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BB_%D0%B1%D1%80%D0%BE%D1%98" title="Цел број – Macedonian" lang="mk" hreflang="mk" data-title="Цел број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsimivaky" title="Isa tsimivaky – Malagasy" lang="mg" hreflang="mg" data-title="Isa tsimivaky" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%82%E0%B5%BC%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="പൂർണ്ണസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="പൂർണ്ണസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Integer" title="Integer – Maltese" lang="mt" hreflang="mt" data-title="Integer" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="पूर्ण संख्या – Marathi" lang="mr" hreflang="mr" data-title="पूर्ण संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Integer" title="Integer – Malay" lang="ms" hreflang="ms" data-title="Integer" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/N%C3%BAmaro_anteiro" title="Númaro anteiro – Mirandese" lang="mwl" hreflang="mwl" data-title="Númaro anteiro" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D2%AF%D1%85%D1%8D%D0%BB_%D1%82%D0%BE%D0%BE" title="Бүхэл тоо – Mongolian" lang="mn" hreflang="mn" data-title="Бүхэл тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Integer" title="Integer – Fijian" lang="fj" hreflang="fj" data-title="Integer" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geheel_getal" title="Geheel getal – Dutch" lang="nl" hreflang="nl" data-title="Geheel getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B4%E6%95%B0" title="整数 – Japanese" lang="ja" hreflang="ja" data-title="整数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Hial_taal" title="Hial taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Hial taal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Heltall" title="Heltall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Heltall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Heiltal" title="Heiltal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Heiltal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_enti%C3%A8r" title="Nombre entièr – Occitan" lang="oc" hreflang="oc" data-title="Nombre entièr" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Intiijarii" title="Lakkoofsa Intiijarii – Oromo" lang="om" hreflang="om" data-title="Lakkoofsa Intiijarii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Butun_sonlar" title="Butun sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Butun sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%82%E0%A8%B0%E0%A8%A8_%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86" title="ਪੂਰਨ ਸੰਖਿਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੂਰਨ ਸੰਖਿਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%86%D9%B9%DB%8C%D8%AC%D8%B1" title="انٹیجر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="انٹیجر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Intiga" title="Intiga – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Intiga" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_antregh" title="Nùmer antregh – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer antregh" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Hele_Tall" title="Hele Tall – Low German" lang="nds" hreflang="nds" data-title="Hele Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_ca%C5%82kowite" title="Liczby całkowite – Polish" lang="pl" hreflang="pl" data-title="Liczby całkowite" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_inteiro" title="Número inteiro – Portuguese" lang="pt" hreflang="pt" data-title="Número inteiro" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_%C3%AEntreg" title="Număr întreg – Romanian" lang="ro" hreflang="ro" data-title="Număr întreg" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BB%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Целое число – Russian" lang="ru" hreflang="ru" data-title="Целое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Integer" title="Integer – Northern Sotho" lang="nso" hreflang="nso" data-title="Integer" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Northern Sotho" class="interlanguage-link-target">Sesotho sa Leboa</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_plot%C3%AB" title="Numrat e plotë – Albanian" lang="sq" hreflang="sq" data-title="Numrat e plotë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_rilativu" title="Nùmmuru rilativu – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru rilativu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%92%E0%B6%9B%E0%B7%92%E0%B6%BD" title="නිඛිල – Sinhala" lang="si" hreflang="si" data-title="නිඛිල" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Integer" title="Integer – Simple English" lang="en-simple" hreflang="en-simple" data-title="Integer" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cel%C3%A9_%C4%8D%C3%ADslo" title="Celé číslo – Slovak" lang="sk" hreflang="sk" data-title="Celé číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Celo_%C5%A1tevilo" title="Celo število – Slovenian" lang="sl" hreflang="sl" data-title="Celo število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Co%C5%82kowito_n%C5%AFmera" title="Cołkowito nůmera – Silesian" lang="szl" hreflang="szl" data-title="Cołkowito nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Abyoone" title="Abyoone – Somali" lang="so" hreflang="so" data-title="Abyoone" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%AA%DB%95%D9%88%D8%A7%D9%88" title="ژمارەی تەواو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی تەواو" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BE_%D0%B1%D1%80%D0%BE%D1%98" title="Цео број – Serbian" lang="sr" hreflang="sr" data-title="Цео број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Cijeli broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kokonaisluku" title="Kokonaisluku – Finnish" lang="fi" hreflang="fi" data-title="Kokonaisluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Heltal" title="Heltal – Swedish" lang="sv" hreflang="sv" data-title="Heltal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Buumbilang" title="Buumbilang – Tagalog" lang="tl" hreflang="tl" data-title="Buumbilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%B4%E0%AF%81_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="முழு எண் – Tamil" lang="ta" hreflang="ta" data-title="முழு எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B1%82%E0%B0%B0%E0%B1%8D%E0%B0%A3_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="పూర్ణ సంఖ్య – Telugu" lang="te" hreflang="te" data-title="పూర్ణ సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="จำนวนเต็ม – Thai" lang="th" hreflang="th" data-title="จำนวนเต็ม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D2%B3%D0%BE%D0%B8_%D0%B1%D1%83%D1%82%D1%83%D0%BD" title="Ададҳои бутун – Tajik" lang="tg" hreflang="tg" data-title="Ададҳои бутун" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tam_say%C4%B1" title="Tam sayı – Turkish" lang="tr" hreflang="tr" data-title="Tam sayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Bitin_sanlar" title="Bitin sanlar – Turkmen" lang="tk" hreflang="tk" data-title="Bitin sanlar" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D1%96%D0%BB%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Ціле число – Ukrainian" lang="uk" hreflang="uk" data-title="Ціле число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B5%D8%AD%DB%8C%D8%AD_%D8%B9%D8%AF%D8%AF" title="صحیح عدد – Urdu" lang="ur" hreflang="ur" data-title="صحیح عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_nguy%C3%AAn" title="Số nguyên – Vietnamese" lang="vi" hreflang="vi" data-title="Số nguyên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Terveharv" title="Terveharv – Võro" lang="vro" hreflang="vro" data-title="Terveharv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B4%E6%95%B8" title="整數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="整數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Geh%C3%AAel_getal" title="Gehêel getal – West Flemish" lang="vls" hreflang="vls" data-title="Gehêel getal" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Buok" title="Buok – Waray" lang="war" hreflang="war" data-title="Buok" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B4%E6%95%B0" title="整数 – Wu" lang="wuu" hreflang="wuu" data-title="整数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%90%D7%A0%D7%A6%D7%A2_%D7%A6%D7%90%D7%9C" title="גאנצע צאל – Yiddish" lang="yi" hreflang="yi" data-title="גאנצע צאל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_odidi" title="Nọ́mbà odidi – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà odidi" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B4%E6%95%B8" title="整數 – Cantonese" lang="yue" hreflang="yue" data-title="整數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Sv%C4%93kas%C4%97s_skaitlios" title="Svēkasės skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Svēkasės skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B4%E6%95%B0" title="整数 – Chinese" lang="zh" hreflang="zh" data-title="整数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Integer" title="Integer – Iban" lang="iba" hreflang="iba" data-title="Integer" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target">Jaku Iban</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 16 November 2024, at 22:21 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a 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data-event-name="switch_to_desktop">Desktop</a></li> </ul> </div> </footer> </div> </div> <div class="mw-notification-area" data-mw="interface"></div>  <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-rpq2w","wgBackendResponseTime":242,"wgPageParseReport":{"limitreport":{"cputime":"1.253","walltime":"1.623","ppvisitednodes":{"value":7632,"limit":1000000},"postexpandincludesize":{"value":349236,"limit":2097152},"templateargumentsize":{"value":7488,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":222754,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1049.745 1 -total"," 32.59% 342.094 2 Template:Reflist"," 24.42% 256.371 33 Template:Cite_book"," 13.92% 146.079 14 Template:Navbox"," 11.46% 120.351 2 Template:Sidebar_with_collapsible_lists"," 10.74% 112.730 1 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