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Dit is echter niet verplicht." class=""><span>Account aanmaken</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speciaal:Aanmelden&returnto=Geheel+getal" title="U wordt van harte uitgenodigd om aan te melden, maar dit is niet verplicht [o]" accesskey="o" class=""><span>Aanmelden</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Meer opties" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Persoonlijke hulpmiddelen" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Persoonlijke hulpmiddelen</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Gebruikersmenu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_nl.wikipedia.org&uselang=nl"><span>Doneren</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciaal:GebruikerAanmaken&returnto=Geheel+getal" title="Registreer u vooral en meld u aan. Dit is echter niet verplicht."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Account aanmaken</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciaal:Aanmelden&returnto=Geheel+getal" title="U wordt van harte uitgenodigd om aan te melden, maar dit is niet verplicht [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Aanmelden</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pagina's voor uitgelogde redacteuren <a href="/wiki/Help:Inleiding" aria-label="Meer leren over bewerken"><span>meer lezen</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speciaal:MijnBijdragen" title="Bijdragen IP-adres [y]" accesskey="y"><span>Bijdragen</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speciaal:MijnOverleg" title="Overlegpagina van de anonieme gebruiker van dit IP-adres [n]" accesskey="n"><span>Overleg</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Inhoud" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Inhoud</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">verbergen</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Top</div> </a> </li> <li id="toc-Definitie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definitie</span> </div> </a> <ul id="toc-Definitie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenschappen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eigenschappen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Eigenschappen</span> </div> </a> <ul id="toc-Eigenschappen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructie_vanuit_de_natuurlijke_getallen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructie_vanuit_de_natuurlijke_getallen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Constructie vanuit de natuurlijke getallen</span> </div> </a> <ul id="toc-Constructie_vanuit_de_natuurlijke_getallen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kardinaliteit" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kardinaliteit"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kardinaliteit</span> </div> </a> <ul id="toc-Kardinaliteit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Meer_gehele_getallen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Meer_gehele_getallen"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Meer gehele getallen</span> </div> </a> <ul id="toc-Meer_gehele_getallen-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Inhoud" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Inhoudsopgave omschakelen" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Inhoudsopgave omschakelen</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Geheel getal</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ga naar een artikel in een andere taal. Beschikbaar in 131 talen" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-131" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">131 talen</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ganze_Zahl" title="Ganze Zahl – Zwitserduits" lang="gsw" hreflang="gsw" data-title="Ganze Zahl" data-language-autonym="Alemannisch" data-language-local-name="Zwitserduits" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_entero" title="Numero entero – Aragonees" lang="an" hreflang="an" data-title="Numero entero" data-language-autonym="Aragonés" data-language-local-name="Aragonees" class="interlanguage-link-target"><span>Aragonés</span></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"><span>अंगिका</span></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 – Arabisch" lang="ar" hreflang="ar" data-title="عدد صحيح" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></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 – Assamees" lang="as" hreflang="as" data-title="অখণ্ড সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamees" class="interlanguage-link-target"><span>অসমীয়া</span></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 – Asturisch" lang="ast" hreflang="ast" data-title="Númberu enteru" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></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 – Azerbeidzjaans" lang="az" hreflang="az" data-title="Tam ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" class="interlanguage-link-target"><span>Azərbaycanca</span></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"><span>تۆرکجه</span></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 – Basjkiers" lang="ba" hreflang="ba" data-title="Бөтөн һан" data-language-autonym="Башҡортса" data-language-local-name="Basjkiers" class="interlanguage-link-target"><span>Башҡортса</span></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 – Samogitisch" lang="sgs" hreflang="sgs" data-title="Svēkasės skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitisch" class="interlanguage-link-target"><span>Žemaitėška</span></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"><span>Bikol Central</span></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 – Belarussisch" lang="be" hreflang="be" data-title="Цэлы лік" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></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"><span>Беларуская (тарашкевіца)</span></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 – Bulgaars" lang="bg" hreflang="bg" data-title="Цяло число" data-language-autonym="Български" data-language-local-name="Bulgaars" class="interlanguage-link-target"><span>Български</span></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 – Bengaals" lang="bn" hreflang="bn" data-title="পূর্ণ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bengaals" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kevan_daveel" title="Kevan daveel – Bretons" lang="br" hreflang="br" data-title="Kevan daveel" data-language-autonym="Brezhoneg" data-language-local-name="Bretons" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Bosnisch" lang="bs" hreflang="bs" data-title="Cijeli broj" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_enter" title="Nombre enter – Catalaans" lang="ca" hreflang="ca" data-title="Nombre enter" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></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 – Soranî" lang="ckb" hreflang="ckb" data-title="ژمارەی تەواو" data-language-autonym="کوردی" data-language-local-name="Soranî" class="interlanguage-link-target"><span>کوردی</span></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 – Tsjechisch" lang="cs" hreflang="cs" data-title="Celé číslo" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></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 – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Тулли хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" class="interlanguage-link-target"><span>Чӑвашла</span></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"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Heltal" title="Heltal – Deens" lang="da" hreflang="da" data-title="Heltal" data-language-autonym="Dansk" data-language-local-name="Deens" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ganze_Zahl" title="Ganze Zahl – Duits" lang="de" hreflang="de" data-title="Ganze Zahl" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></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 – Grieks" lang="el" hreflang="el" data-title="Ακέραιος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integer" title="Integer – Engels" lang="en" hreflang="en" data-title="Integer" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></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"><span>Esperanto</span></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 – Spaans" lang="es" hreflang="es" data-title="Número entero" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%A4isarv" title="Täisarv – Estisch" lang="et" hreflang="et" data-title="Täisarv" data-language-autonym="Eesti" data-language-local-name="Estisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_oso" title="Zenbaki oso – Baskisch" lang="eu" hreflang="eu" data-title="Zenbaki oso" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></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 – Perzisch" lang="fa" hreflang="fa" data-title="عدد صحیح" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kokonaisluku" title="Kokonaisluku – Fins" lang="fi" hreflang="fi" data-title="Kokonaisluku" data-language-autonym="Suomi" data-language-local-name="Fins" class="interlanguage-link-target"><span>Suomi</span></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"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Integer" title="Integer – Fijisch" lang="fj" hreflang="fj" data-title="Integer" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijisch" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Heiltal" title="Heiltal – Faeröers" lang="fo" hreflang="fo" data-title="Heiltal" data-language-autonym="Føroyskt" data-language-local-name="Faeröers" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Entier_relatif" title="Entier relatif – Frans" lang="fr" hreflang="fr" data-title="Entier relatif" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Hial_taal" title="Hial taal – Noord-Fries" lang="frr" hreflang="frr" data-title="Hial taal" data-language-autonym="Nordfriisk" data-language-local-name="Noord-Fries" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sl%C3%A1nuimhir" title="Slánuimhir – Iers" lang="ga" hreflang="ga" data-title="Slánuimhir" data-language-autonym="Gaeilge" data-language-local-name="Iers" class="interlanguage-link-target"><span>Gaeilge</span></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 – Ganyu" lang="gan" hreflang="gan" data-title="整數" data-language-autonym="贛語" data-language-local-name="Ganyu" class="interlanguage-link-target"><span>贛語</span></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"><span>Kriyòl gwiyannen</span></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 – Galicisch" lang="gl" hreflang="gl" data-title="Número enteiro" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></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"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Helu_piha" title="Helu piha – Hawaïaans" lang="haw" hreflang="haw" data-title="Helu piha" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaïaans" class="interlanguage-link-target"><span>Hawaiʻi</span></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 – Hebreeuws" lang="he" hreflang="he" data-title="מספר שלם" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></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"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Kroatisch" lang="hr" hreflang="hr" data-title="Cijeli broj" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></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 – Oppersorbisch" lang="hsb" hreflang="hsb" data-title="Cyła ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="Oppersorbisch" class="interlanguage-link-target"><span>Hornjoserbsce</span></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 – Hongaars" lang="hu" hreflang="hu" data-title="Egész számok" data-language-autonym="Magyar" data-language-local-name="Hongaars" class="interlanguage-link-target"><span>Magyar</span></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 – Armeens" lang="hy" hreflang="hy" data-title="Ամբողջ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></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"><span>Արեւմտահայերէն</span></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"><span>Interlingua</span></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"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_bulat" title="Bilangan bulat – Indonesisch" lang="id" hreflang="id" data-title="Bilangan bulat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></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 – Ingoesjetisch" lang="inh" hreflang="inh" data-title="БӀарчча таьрахь" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingoesjetisch" class="interlanguage-link-target"><span>ГӀалгӀай</span></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"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Heilt%C3%B6lur" title="Heiltölur – IJslands" lang="is" hreflang="is" data-title="Heiltölur" data-language-autonym="Íslenska" data-language-local-name="IJslands" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_intero" title="Numero intero – Italiaans" lang="it" hreflang="it" data-title="Numero intero" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></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 – Japans" lang="ja" hreflang="ja" data-title="整数" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Intiga" title="Intiga – Jamaicaans Creools" lang="jam" hreflang="jam" data-title="Intiga" data-language-autonym="Patois" data-language-local-name="Jamaicaans Creools" class="interlanguage-link-target"><span>Patois</span></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"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Wilangan_bulat" title="Wilangan bulat – Javaans" lang="jv" hreflang="jv" data-title="Wilangan bulat" data-language-autonym="Jawa" data-language-local-name="Javaans" class="interlanguage-link-target"><span>Jawa</span></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 – Georgisch" lang="ka" hreflang="ka" data-title="მთელი რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></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 – Kazachs" lang="kk" hreflang="kk" data-title="Бүтін сан" data-language-autonym="Қазақша" data-language-local-name="Kazachs" class="interlanguage-link-target"><span>Қазақша</span></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 – Koreaans" lang="ko" hreflang="ko" data-title="정수" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Tamhejmar" title="Tamhejmar – Koerdisch" lang="ku" hreflang="ku" data-title="Tamhejmar" data-language-autonym="Kurdî" data-language-local-name="Koerdisch" class="interlanguage-link-target"><span>Kurdî</span></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 – Kirgizisch" lang="ky" hreflang="ky" data-title="Бүтүн сандар" data-language-autonym="Кыргызча" data-language-local-name="Kirgizisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_integer" title="Numerus integer – Latijn" lang="la" hreflang="la" data-title="Numerus integer" data-language-autonym="Latina" data-language-local-name="Latijn" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ganz_Zuel" title="Ganz Zuel – Luxemburgs" lang="lb" hreflang="lb" data-title="Ganz Zuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgs" class="interlanguage-link-target"><span>Lëtzebuergesch</span></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"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Gans_getal" title="Gans getal – Limburgs" lang="li" hreflang="li" data-title="Gans getal" data-language-autonym="Limburgs" data-language-local-name="Limburgs" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_intreg" title="Numer intreg – Lombardisch" lang="lmo" hreflang="lmo" data-title="Numer intreg" data-language-autonym="Lombard" data-language-local-name="Lombardisch" class="interlanguage-link-target"><span>Lombard</span></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 – Laotiaans" lang="lo" hreflang="lo" data-title="ຈຳນວນຖ້ວນ" data-language-autonym="ລາວ" data-language-local-name="Laotiaans" class="interlanguage-link-target"><span>ລາວ</span></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 – Litouws" lang="lt" hreflang="lt" data-title="Sveikasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Litouws" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vesels_skaitlis" title="Vesels skaitlis – Lets" lang="lv" hreflang="lv" data-title="Vesels skaitlis" data-language-autonym="Latviešu" data-language-local-name="Lets" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsimivaky" title="Isa tsimivaky – Malagassisch" lang="mg" hreflang="mg" data-title="Isa tsimivaky" data-language-autonym="Malagasy" data-language-local-name="Malagassisch" class="interlanguage-link-target"><span>Malagasy</span></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 – Macedonisch" lang="mk" hreflang="mk" data-title="Цел број" data-language-autonym="Македонски" data-language-local-name="Macedonisch" class="interlanguage-link-target"><span>Македонски</span></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"><span>മലയാളം</span></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 – Mongools" lang="mn" hreflang="mn" data-title="Бүхэл тоо" data-language-autonym="Монгол" data-language-local-name="Mongools" class="interlanguage-link-target"><span>Монгол</span></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"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Integer" title="Integer – Maleis" lang="ms" hreflang="ms" data-title="Integer" data-language-autonym="Bahasa Melayu" data-language-local-name="Maleis" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Integer" title="Integer – Maltees" lang="mt" hreflang="mt" data-title="Integer" data-language-autonym="Malti" data-language-local-name="Maltees" class="interlanguage-link-target"><span>Malti</span></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 – Mirandees" lang="mwl" hreflang="mwl" data-title="Númaro anteiro" data-language-autonym="Mirandés" data-language-local-name="Mirandees" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Hele_Tall" title="Hele Tall – Nedersaksisch" lang="nds" hreflang="nds" data-title="Hele Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Nedersaksisch" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Heiltal" title="Heiltal – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Heiltal" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Heltall" title="Heltall – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Heltall" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Integer" title="Integer – Noord-Sotho" lang="nso" hreflang="nso" data-title="Integer" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Noord-Sotho" class="interlanguage-link-target"><span>Sesotho sa Leboa</span></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 – Occitaans" lang="oc" hreflang="oc" data-title="Nombre entièr" data-language-autonym="Occitan" data-language-local-name="Occitaans" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Intiijarii" title="Lakkoofsa Intiijarii – Afaan Oromo" lang="om" hreflang="om" data-title="Lakkoofsa Intiijarii" data-language-autonym="Oromoo" data-language-local-name="Afaan Oromo" class="interlanguage-link-target"><span>Oromoo</span></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"><span>ਪੰਜਾਬੀ</span></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 – Pools" lang="pl" hreflang="pl" data-title="Liczby całkowite" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></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 – Piëmontees" lang="pms" hreflang="pms" data-title="Nùmer antregh" data-language-autonym="Piemontèis" data-language-local-name="Piëmontees" class="interlanguage-link-target"><span>Piemontèis</span></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"><span>پنجابی</span></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 – Portugees" lang="pt" hreflang="pt" data-title="Número inteiro" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></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 – Roemeens" lang="ro" hreflang="ro" data-title="Număr întreg" data-language-autonym="Română" data-language-local-name="Roemeens" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="goed artikel"><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 – Russisch" lang="ru" hreflang="ru" data-title="Целое число" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></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 – Siciliaans" lang="scn" hreflang="scn" data-title="Nùmmuru rilativu" data-language-autonym="Sicilianu" data-language-local-name="Siciliaans" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Cijeli_broj" title="Cijeli broj – Servo-Kroatisch" lang="sh" hreflang="sh" data-title="Cijeli broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Servo-Kroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></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 – Singalees" lang="si" hreflang="si" data-title="නිඛිල" data-language-autonym="සිංහල" data-language-local-name="Singalees" class="interlanguage-link-target"><span>සිංහල</span></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"><span>Simple English</span></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 – Slowaaks" lang="sk" hreflang="sk" data-title="Celé číslo" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Celo_%C5%A1tevilo" title="Celo število – Sloveens" lang="sl" hreflang="sl" data-title="Celo število" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" class="interlanguage-link-target"><span>Slovenščina</span></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"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Abyoone" title="Abyoone – Somalisch" lang="so" hreflang="so" data-title="Abyoone" data-language-autonym="Soomaaliga" data-language-local-name="Somalisch" class="interlanguage-link-target"><span>Soomaaliga</span></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 – Albanees" lang="sq" hreflang="sq" data-title="Numrat e plotë" data-language-autonym="Shqip" data-language-local-name="Albanees" class="interlanguage-link-target"><span>Shqip</span></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 – Servisch" lang="sr" hreflang="sr" data-title="Цео број" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Heltal" title="Heltal – Zweeds" lang="sv" hreflang="sv" data-title="Heltal" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></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"><span>Kiswahili</span></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 – Silezisch" lang="szl" hreflang="szl" data-title="Cołkowito nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silezisch" class="interlanguage-link-target"><span>Ślůnski</span></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"><span>தமிழ்</span></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"><span>తెలుగు</span></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 – Tadzjieks" lang="tg" hreflang="tg" data-title="Ададҳои бутун" data-language-autonym="Тоҷикӣ" data-language-local-name="Tadzjieks" class="interlanguage-link-target"><span>Тоҷикӣ</span></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"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Bitin_sanlar" title="Bitin sanlar – Turkmeens" lang="tk" hreflang="tk" data-title="Bitin sanlar" data-language-autonym="Türkmençe" data-language-local-name="Turkmeens" class="interlanguage-link-target"><span>Türkmençe</span></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"><span>Tagalog</span></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 – Turks" lang="tr" hreflang="tr" data-title="Tam sayı" data-language-autonym="Türkçe" data-language-local-name="Turks" class="interlanguage-link-target"><span>Türkçe</span></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 – Oekraïens" lang="uk" hreflang="uk" data-title="Ціле число" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</span></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"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Butun_sonlar" title="Butun sonlar – Oezbeeks" lang="uz" hreflang="uz" data-title="Butun sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Oezbeeks" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></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 – Vietnamees" lang="vi" hreflang="vi" data-title="Số nguyên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></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-Vlaams" lang="vls" hreflang="vls" data-title="Gehêel getal" data-language-autonym="West-Vlams" data-language-local-name="West-Vlaams" class="interlanguage-link-target"><span>West-Vlams</span></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"><span>Winaray</span></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 – Wuyu" lang="wuu" hreflang="wuu" data-title="整数" data-language-autonym="吴语" data-language-local-name="Wuyu" class="interlanguage-link-target"><span>吴语</span></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 – Kalmuks" lang="xal" hreflang="xal" data-title="Бүкл тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmuks" class="interlanguage-link-target"><span>Хальмг</span></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"><span>IsiXhosa</span></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 – Jiddisch" lang="yi" hreflang="yi" data-title="גאנצע צאל" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></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"><span>Yorùbá</span></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 – Chinees" lang="zh" hreflang="zh" data-title="整数" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</span></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 – Klassiek Chinees" lang="lzh" hreflang="lzh" data-title="整數" data-language-autonym="文言" data-language-local-name="Klassiek Chinees" class="interlanguage-link-target"><span>文言</span></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 – Minnanyu" lang="nan" hreflang="nan" data-title="Chéng-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnanyu" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></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 – Kantonees" lang="yue" hreflang="yue" data-title="整數" data-language-autonym="粵語" data-language-local-name="Kantonees" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12503#sitelinks-wikipedia" title="Taalkoppelingen bewerken" class="wbc-editpage">Koppelingen bewerken</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Naamruimten"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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href="/wiki/Rationaal_getal" title="Rationaal getal">Rationale getallen</a></li> <li><a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">Reële getallen</a></li> <li><a href="/wiki/Complex_getal" title="Complex getal">Complexe getallen</a></li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternionen</a></li> <li><a href="/wiki/P-adisch_getal" title="P-adisch getal"><i>p</i>-adische getallen</a></li> <li><a href="/w/index.php?title=Hyperre%C3%ABel_getal&action=edit&redlink=1" class="new" title="Hyperreëel getal (de pagina bestaat niet)">Hyperreële getallen</a></li> <li><a href="/wiki/Surre%C3%ABel_getal" title="Surreëel getal">Surreële getallen</a></li> <li><a href="/wiki/Transfiniet_getal" title="Transfiniet getal">Transfiniete getallen</a></li></ul> </td></tr> <tr> <th class="infobox-kop notheme" colspan="3"> </th></tr> <tr> <td colspan="3"> <ul><li><a href="/wiki/Irrationaal_getal" title="Irrationaal getal">Irrationale getallen</a></li> <li><a href="/wiki/Algebra%C3%AFsch_getal" title="Algebraïsch getal">Algebraïsche getallen</a></li> <li><a href="/wiki/Transcendent_getal" title="Transcendent getal">Transcendente getallen</a></li> <li><a href="/wiki/Imaginair_getal" title="Imaginair getal">Imaginaire getallen</a></li></ul> </td></tr> </tbody></table> <p>De <b>gehele</b> of (op de <a href="/wiki/Basisschool" class="mw-redirect" title="Basisschool">basisschool</a> in Nederland) <b>hele getallen</b> zijn alle <a href="/wiki/Getal_(wiskunde)" title="Getal (wiskunde)">getallen</a> in de <a href="/wiki/Rij_(wiskunde)" title="Rij (wiskunde)">rij</a> </p> <dl><dd>…, −3, −2, −1, 0, 1, 2, 3, …</dd></dl> <p>die voortgezet wordt door er steeds 1 bij te tellen of er 1 af te trekken. De gehele getallen omvatten <a href="/wiki/0_(cijfer)" title="0 (cijfer)">0</a>, de <a href="/wiki/Natuurlijk_getal" title="Natuurlijk getal">natuurlijke getallen</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> dus de getallen waarmee wordt geteld, en de tegengestelden daarvan, de negatieve gehele getallen. </p><p>Een geheel getal heet 'geheel' omdat het niet <a href="/wiki/Breuk_(wiskunde)" title="Breuk (wiskunde)">gebroken</a> is en zonder <a href="/wiki/Positiestelsel" title="Positiestelsel">cijfers achter de komma</a> kan worden geschreven. De getallen 21, 4 en −121 zijn bijvoorbeeld gehele getallen, terwijl 9,75, 5½ en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c878efdf227cf70df28aa7d43cea0069e6f515e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {12}}}"></span> geen gehele getallen zijn. De <a href="/wiki/Verzameling_(wiskunde)" title="Verzameling (wiskunde)">verzameling</a> gehele getallen is een <a href="/wiki/Deelverzameling" title="Deelverzameling">deelverzameling</a> van de <a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">reële getallen</a>, en wordt meestal voorgesteld door een vet gedrukte <b>Z</b> of het <a href="/wiki/Symbool" title="Symbool">symbool</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> (<a href="/wiki/Unicode" title="Unicode">Unicode</a> U+2124 ℤ), wat voor <i>Zahlen</i>, het <a href="/wiki/Duits" title="Duits">Duits</a> voor getallen, staat.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>De <a href="/wiki/Wiskunde" title="Wiskunde">wiskundetak</a> die zich met de studie bezighoudt naar de eigenschappen van de gehele getallen, noemt men de <a href="/wiki/Getaltheorie" title="Getaltheorie">getaltheorie</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitie">Definitie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geheel_getal&veaction=edit&section=1" title="Bewerk dit kopje: Definitie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Geheel_getal&action=edit&section=1" title="De broncode bewerken van de sectie: Definitie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De gehele getallen kunnen worden gedefinieerd als de elementen van de kleinste verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> met de eigenschappen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110c69811fdcceac47cad4190e2e4b3d71f214ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.553ex; height:2.176ex;" alt="{\displaystyle 0\in \mathbb {Z} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {Z} \implies z+1\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {Z} \implies z+1\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9625131c4fb94bd2da8fc1016fc6e16d59f670c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.347ex; height:2.343ex;" alt="{\displaystyle z\in \mathbb {Z} \implies z+1\in \mathbb {Z} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {Z} \implies z-1\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {Z} \implies z-1\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03a517ed27a21c05009b3fa727d0aef0f61f742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.347ex; height:2.343ex;" alt="{\displaystyle z\in \mathbb {Z} \implies z-1\in \mathbb {Z} }"></span></dd></dl> <p>Voor de representatie van gehele getallen in de <a href="/wiki/Computer" title="Computer">computer</a> maakt men gebruik van het datatype <a href="/wiki/Integer_(informatica)" title="Integer (informatica)">integer</a>. Het is echter belangrijk daarbij op te merken dat deze twee niet hetzelfde zijn. Het datatype integer is, aangezien een integer een beperkte hoeveelheid geheugen inneemt, een <a href="/wiki/Eindige_verzameling" title="Eindige verzameling">eindige verzameling</a>, terwijl de gehele getallen een <a href="/wiki/Oneindige_verzameling" title="Oneindige verzameling">oneindige verzameling</a> vormen. </p> <div class="mw-heading mw-heading2"><h2 id="Eigenschappen">Eigenschappen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geheel_getal&veaction=edit&section=2" title="Bewerk dit kopje: Eigenschappen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Geheel_getal&action=edit&section=2" title="De broncode bewerken van de sectie: Eigenschappen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>De verzameling gehele getallen is <a href="/wiki/Gesloten_verzameling" title="Gesloten verzameling">gesloten</a> onder <a href="/wiki/Optellen" title="Optellen">optellen</a>, <a href="/wiki/Aftrekken_(wiskunde)" title="Aftrekken (wiskunde)">aftrekken</a> en <a href="/wiki/Vermenigvuldigen" title="Vermenigvuldigen">vermenigvuldigen</a>: elke optelling, aftrekking of vermenigvuldiging van twee gehele getallen levert opnieuw een geheel getal. De verzameling is niet gesloten onder de bewerking <a href="/wiki/Delen" title="Delen">delen</a>: niet elke deling van twee gehele getallen levert opnieuw een geheel getal op, bijvoorbeeld 1/2 is een <a href="/wiki/Rationaal_getal" title="Rationaal getal">rationaal getal</a>. De gehele getallen vormen een <a href="/wiki/Ring_(wiskunde)" title="Ring (wiskunde)">ring</a>.</li> <li>De elementen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> hebben een bepaalde volgorde. Strikter geformuleerd: de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> wordt <a href="/wiki/Totale_orde" title="Totale orde">totaal geordend</a> door de <a href="/wiki/Relatie_(wiskunde)" title="Relatie (wiskunde)">relatie</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> (kleiner dan) en bevat in die ordening zowel oneindig stijgende als oneindig dalende ketens.</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots <-2<-1<0<1<2<\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo><</mo> <mo>−</mo> <mn>2</mn> <mo><</mo> <mo>−</mo> <mn>1</mn> <mo><</mo> <mn>0</mn> <mo><</mo> <mn>1</mn> <mo><</mo> <mn>2</mn> <mo><</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots <-2<-1<0<1<2<\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8482a11eaa12b3cc3d4aa07890acb595563dd8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:33.466ex; height:2.343ex;" alt="{\displaystyle \ldots <-2<-1<0<1<2<\ldots }"></span></dd></dl></dd> <dd>Deze orde heeft de eigenschappen: <ul><li>als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c<d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo><</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c<d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826f129a304df73d8c5c71cc0ea8787fad9d40ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle c<d}"></span>, dan is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+c<b+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo><</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c<b+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffff35f8aa2508180de411cb546968bc63bd6c27" 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+c<b+d}"></span></li> <li>als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e428b0dcf13d97aa75f3e6167290764263bd07e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle 0<c}"></span>, dan is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac<bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>c</mi> <mo><</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ac<bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0265433404d1a45f8a1cdda0343945d9dbed5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle ac<bc}"></span></li></ul></dd></dl> <ul><li>Bij iedere twee gehele getallen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, waarvan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad073253b4c817f2ec7e3dd7517b7f89a8e581dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 0}"></span> is, zijn altijd twee unieke gehele getallen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> te vinden, met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r<|b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r<|b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fc64c6bf589852efd9096b83210cc89be7bdb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.699ex; height:2.843ex;" alt="{\displaystyle 0\leq r<|b|}"></span>, zodat:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=bq+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=bq+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff37cbe024f8cc6ff961323bde02fb9c5d32066e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.284ex; height:2.509ex;" alt="{\displaystyle a=bq+r}"></span></dd></dl></dd> <dd>In bovenstaande stelling heet het getal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> het <a href="/wiki/Quoti%C3%ABnt" title="Quotiënt">quotiënt</a> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> de <a href="/wiki/Rest" title="Rest">rest</a> van de deling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Deze vorm van delen heet <a href="/wiki/Geheeltallige_deling" title="Geheeltallige deling">geheeltallige deling</a>.</dd> <dd>Als in bovenstaande stelling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span>, is de <a href="/wiki/Breuk_(wiskunde)" title="Breuk (wiskunde)">breuk</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/b=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/b=q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d1ab4394909747c49b7a55defe03a8522e2f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.558ex; height:2.843ex;" alt="{\displaystyle a/b=q}"></span>, dus geheel. Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034cc599221cc81da7ebd4c9090e1a988809b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.31ex; height:2.676ex;" alt="{\displaystyle r\neq 0}"></span>, is de breuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/b=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/b=q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d1ab4394909747c49b7a55defe03a8522e2f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.558ex; height:2.843ex;" alt="{\displaystyle a/b=q}"></span> geen geheel, maar een <a href="/wiki/Rationaal_getal" title="Rationaal getal">rationaal getal</a>, met een geheel deel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> en een gebroken of fractioneel deel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r/b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cbb898a3dd99fcf2a8ac27874e03f386f336ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.209ex; height:2.843ex;" alt="{\displaystyle r/b}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Constructie_vanuit_de_natuurlijke_getallen">Constructie vanuit de natuurlijke getallen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geheel_getal&veaction=edit&section=3" title="Bewerk dit kopje: Constructie vanuit de natuurlijke getallen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Geheel_getal&action=edit&section=3" title="De broncode bewerken van de sectie: Constructie vanuit de natuurlijke getallen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De gehele getallen kunnen ook geconstrueerd worden met behulp van de natuurlijke getallen. Zij vormen daarvan de <a href="/wiki/Grothendieck-groep" title="Grothendieck-groep">grothendieck-groep</a>. </p><p>Op het <a href="/wiki/Cartesisch_product" title="Cartesisch product">cartesisch product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578dc62c2bb6d5e2b2624c6b58b02787df469372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{2}}"></span> wordt een <a href="/wiki/Equivalentierelatie" title="Equivalentierelatie">equivalentierelatie</a> gedefinieerd door: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span></dd></dl> <p>als </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d=c+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+d=c+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d19425ec3e783fe87252dd283121987a5f014c" 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=c+b}"></span></dd></dl> <p>met de implicatie dat het paar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> staat voor het gehele getal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span>. </p><p>De gehele getallen bestaan uit de equivalentieklassen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{[(a,b)]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</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 fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{[(a,b)]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef760e1883bf4b27e390448da1ceca605c18224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.689ex; height:2.843ex;" alt="{\displaystyle \{[(a,b)]\}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} =\mathbb {N} ^{2}/\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} =\mathbb {N} ^{2}/\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5c3f41186fe0de60448c2f0b9369fc832232c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.997ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} =\mathbb {N} ^{2}/\sim }"></span>,</dd></dl> <p>met als optelling: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3e7ed2867994c88cab704ab991184c0407be68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.931ex; height:2.843ex;" alt="{\displaystyle [(a,b)]+[(c,d)]=[(a+c,b+d)]}"></span>,</dd></dl> <p>en als vermenigvuldiging: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <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> <mo>=</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\cdot (c,d)=(ac+bd,ad+bc)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7ba58df2f7028e020bf6e9443e64ff35436430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.338ex; height:2.843ex;" alt="{\displaystyle (a,b)\cdot (c,d)=(ac+bd,ad+bc)}"></span></dd></dl> <p>De gehele getallen zijn geordend door: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <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)<(c,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fba65cff00870cc2e2df5154ca26e9b70d7991" 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)<(c,d)}"></span> als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d<c+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo><</mo> <mi>c</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+d<c+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a759f1ba16c3db0acac5bd8e99ab9335ff8f05" 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<c+b}"></span>.</dd></dl> <p>Iedere equivalentieklasse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> heeft een eenduidige vertegenwoordiger met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> van de vorm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd8a7c3a302914ba5ae7cac4d8df11b59943934" 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 (n,0)}"></span> als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5d3957d5f94566507526017e4ebb67c02efe81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\geq b}"></span>, of van de vorm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9516a58dc3bd99e060e9ec8565620a57a3a9e" 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 (0,n)}"></span> als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>. De equivalentieklasse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(n,0)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(n,0)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47dba03ba7f4b82eccf1668b6fca4ac85f6d44f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.694ex; height:2.843ex;" alt="{\displaystyle [(n,0)]}"></span> wordt met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> geïdentificeerd en voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920e98ff3dd1cb41e01f76243300450c958d5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 0}"></span> als <i>positief</i> geheel getal aangeduid, en de equivalentieklasse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(0,n)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(0,n)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d56db0bdc77dfe66d209a8072b429e4eeb18f7c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.694ex; height:2.843ex;" alt="{\displaystyle [(0,n)]}"></span> wordt met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00139753ecf4fe00a10a17bd5620b70a61b29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}"></span> aangegeven en <i>negatief</i> geheel getal genoemd. </p> <div class="mw-heading mw-heading2"><h2 id="Kardinaliteit">Kardinaliteit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geheel_getal&veaction=edit&section=4" title="Bewerk dit kopje: Kardinaliteit" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Geheel_getal&action=edit&section=4" title="De broncode bewerken van de sectie: Kardinaliteit"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De gehele getallen kunnen afgeteld worden, anders gezegd: de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> is <a href="/wiki/Gelijkmachtigheid" title="Gelijkmachtigheid">gelijkmachtig</a> aan de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> van natuurlijke getallen, dus <a href="/wiki/Oneindigheid#Aftelbaar_oneindig" title="Oneindigheid">aftelbaar oneindig</a>. Beide verzamelingen bevatten als het ware "evenveel" <a href="/wiki/Element_(wiskunde)" title="Element (wiskunde)">elementen</a>, hoewel de natuurlijke getallen toch maar een deel van de gehele getallen vormen. De <a href="/wiki/Kardinaliteit" title="Kardinaliteit">kardinaliteit</a> van de gehele getallen wordt aangegeven met het symbool <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span> (<a href="/wiki/Alef-getal" title="Alef-getal">aleph-null</a>). Dat de gehele getallen kunnen worden afgeteld, kan als volgt worden aangetoond: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}0&1&-1&2&-2&3&-3&4&-4&\ldots \\1&2&3&4&5&6&7&8&9&\ldots \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}0&1&-1&2&-2&3&-3&4&-4&\ldots \\1&2&3&4&5&6&7&8&9&\ldots \end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30332acc16f7d6774b96713e10b5191c95b559a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.073ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}0&1&-1&2&-2&3&-3&4&-4&\ldots \\1&2&3&4&5&6&7&8&9&\ldots \end{matrix}}}"></span></dd></dl> <p>Op deze manier worden de gehele getallen door de <a href="/wiki/Bijectie" title="Bijectie">bijectie</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {Z} \to \mathbb {N} \setminus {\{0\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {Z} \to \mathbb {N} \setminus {\{0\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb69bc591b2c3be36d4029e831bc77ee635c398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.74ex; height:2.843ex;" alt="{\displaystyle f:\mathbb {Z} \to \mathbb {N} \setminus {\{0\}}}"></span> een-op-een op de natuurlijke getallen, zonder 0, afgebeeld met </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}2|x|+1,&{\mbox{als }}x\leq 0\\2x,&{\mbox{als }}x>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>≤</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </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 f(x)={\begin{cases}2|x|+1,&{\mbox{als }}x\leq 0\\2x,&{\mbox{als }}x>0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11eca98ff08821d2c47016da747f711453019147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.666ex; height:6.176ex;" alt="{\displaystyle f(x)={\begin{cases}2|x|+1,&{\mbox{als }}x\leq 0\\2x,&{\mbox{als }}x>0\end{cases}}}"></span></dd></dl> <p>De bijectie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {Z} \to \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {Z} \to \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0503c9f84baf243b4ca6bb8b7279f13ce18aa57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.896ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {Z} \to \mathbb {N} }"></span> met </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\begin{cases}2|x|,&{\mbox{als }}x<0\\0,&{\mbox{als }}x=0\\2x-1,&{\mbox{als }}x>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </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 g(x)={\begin{cases}2|x|,&{\mbox{als }}x<0\\0,&{\mbox{als }}x=0\\2x-1,&{\mbox{als }}x>0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/778ab751759c288b7b48a8e87adbd1206b947ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:28.532ex; height:8.509ex;" alt="{\displaystyle g(x)={\begin{cases}2|x|,&{\mbox{als }}x<0\\0,&{\mbox{als }}x=0\\2x-1,&{\mbox{als }}x>0\end{cases}}}"></span></dd></dl> <p>beeldt de gehele getallen op alle natuurlijke getallen af, met 0. </p><p>Door de definitie van kardinale gelijkheid hebben de twee verzamelingen dezelfde kardinaliteit. </p> <div class="mw-heading mw-heading2"><h2 id="Meer_gehele_getallen">Meer gehele getallen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geheel_getal&veaction=edit&section=5" title="Bewerk dit kopje: Meer gehele getallen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Geheel_getal&action=edit&section=5" title="De broncode bewerken van de sectie: Meer gehele getallen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De <a href="/wiki/Geheel_getal_van_Gauss" title="Geheel getal van Gauss">Gauss-gehele getallen</a> en de <a href="/wiki/Geheel_getal_van_Eisenstein" title="Geheel getal van Eisenstein">Eisenstein-gehele getallen</a> zijn twee verschillende uitbreidingen van de gehele getallen naar de <a href="/wiki/Complex_getal" title="Complex getal">complexe getallen</a>. </p> <div class="toccolours appendix" role="presentation" style="font-size:90%; margin:1em 0 -0.5em; clear:both;"> <div><span style="font-weight:bold">Voetnoten</span></div> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Dit is ervan afhankelijk dat 0 wel of niet bij de natuurlijke getallen wordt gerekend.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r67679320">.mw-parser-output .taalaanduiding{font-family:sans-serif;font-size:85%;cursor:help;color:var(--color-subtle,#555)}.mw-parser-output .taalaanduiding span{border-bottom:1px dotted var(--color-subtle,#555)}</style><span class="taalaanduiding" title="Taal: Engels">(<span>en</span>) </span> <span style="font-variant:small-caps;">Jeff Miller</span>, <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>.</span> </li> </ol></div></div> </div> <style data-mw-deduplicate="TemplateStyles:r67837862">.mw-parser-output .navigatie{position:relative;clear:both;overflow:auto;margin:1em auto -0.5em;padding:2px;background-color:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);text-align:center;font-size:87%}.mw-parser-output .navigatie-bewerken{margin-left:0.5em}.mw-parser-output .navigatie-bewerken .mw-ui-icon::before{background-size:0.9em}.mw-parser-output .navigatie-afb-links,.mw-parser-output .navigatie-afb-rechts{position:absolute}.mw-parser-output .navigatie-afb-rechts{right:2px}.mw-parser-output .navigatie-afb-groot{float:right;padding-left:0.5em}.mw-parser-output .navigatie-titel{background-color:#ddeef8;padding:2px 5.5em;font-weight:bold}.mw-parser-output .navigatie-inhoud{padding:0.5em}.mw-parser-output .navigatie-inhoud p:first-child{margin:0}.mw-parser-output .navigatie div[style*="background-color"],.mw-parser-output .navigatie div[style*="background"]{color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navigatie-titel{background-color:var(--background-color-interactive,#eaecf0)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navigatie-titel{background-color:var(--background-color-interactive,#eaecf0)!important}}</style> <div class="navigatie" role="navigation" aria-labelledby="Bijzondere_getallen"> <div class="navigatie-afb-links noviewer"><div class="navigatie-bewerken show-autoconfirmed nomobile"><span class="nowrap 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/OOjs_UI_icon_speechBubbles-ltr.svg/24px-OOjs_UI_icon_speechBubbles-ltr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/OOjs_UI_icon_speechBubbles-ltr.svg/32px-OOjs_UI_icon_speechBubbles-ltr.svg.png 2x" data-file-width="20" data-file-height="20" /></span></span> · <span typeof="mw:File"><a href="//nl.wikipedia.org/w/index.php?title=Sjabloon:Navigatie_bijzondere_getallen&action=edit" title="Sjabloon bewerken"><img alt="Sjabloon bewerken" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr.svg/16px-OOjs_UI_icon_edit-ltr.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr.svg/24px-OOjs_UI_icon_edit-ltr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr.svg/32px-OOjs_UI_icon_edit-ltr.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></span></div></div> <div id="Bijzondere_getallen" class="navigatie-titel">Bijzondere getallen</div> <div class="navigatie-inhoud"> <style data-mw-deduplicate="TemplateStyles:r67785531">.mw-parser-output .navigatie-tabel{margin:0 auto 0 auto;text-align:left}.mw-parser-output .navigatie-tabel .links{text-align:right}.mw-parser-output .navigatie-tabel .rechts{padding-left:0.5em}@media screen and (max-width:640px){.mw-parser-output .navigatie-tabel tr{display:grid;grid-template-columns:1fr;width:100%;margin-bottom:0.5em}.mw-parser-output .navigatie-tabel tr:last-of-type{margin-bottom:0}.mw-parser-output .navigatie-tabel .links{text-align:unset}.mw-parser-output .navigatie-tabel .rechts{padding:unset}}</style><table class="navigatie-tabel vatop" cellpadding="0" cellspacing="0" style=""><tbody><tr><td class="links" style=""><span class="nowrap"><b><a href="/wiki/Wiskundige_constante" title="Wiskundige constante">Wiskundige constanten</a>:</b></span></td><td class="rechts"><a href="/wiki/E_(wiskunde)" title="E (wiskunde)">e</a> · <a href="/wiki/Constante_van_Euler-Mascheroni" title="Constante van Euler-Mascheroni">constante van Euler-Mascheroni</a> · <a href="/wiki/Constante_van_Gelfond" title="Constante van Gelfond">constante van Gelfond</a> · <a href="/wiki/Gulden_snede" title="Gulden snede">gulden getal</a> · <a href="/wiki/6174_(getal)" title="6174 (getal)">constante van Kaprekar</a> · <a href="/wiki/Getal_van_Graham" title="Getal van Graham">getal van Graham</a> · <a href="/wiki/Getal_van_Skewes" title="Getal van Skewes">getal van Skewes</a> · <a href="/wiki/Pi_(wiskunde)" title="Pi (wiskunde)">pi</a></td></tr><tr><td class="links"><b><a href="/wiki/Verzameling_(wiskunde)" title="Verzameling (wiskunde)">Verzamelingen</a>:</b></td><td class="rechts"><a href="/wiki/Algebra%C3%AFsch_getal" title="Algebraïsch getal">algebraïsch getal</a> · <a href="/wiki/Bevriende_getallen" title="Bevriende getallen">bevriende getallen</a> · <a href="/wiki/Bijna_perfect_getal" title="Bijna perfect getal">bijna perfect getal</a> · <a href="/wiki/Complex_getal" title="Complex getal">complex getal</a> · <a href="/wiki/Evenwichtig_priemgetal" title="Evenwichtig priemgetal">evenwichtig priemgetal</a> · <a href="/wiki/Fermatgetal" title="Fermatgetal">fermatgetal</a> · <a href="/wiki/Gebrekkig_getal" title="Gebrekkig getal">gebrekkig getal</a> · <a class="mw-selflink selflink">geheel getal</a> · <a href="/wiki/Kaprekargetal" title="Kaprekargetal">kaprekargetal</a> · <a href="/wiki/Mersennepriemgetal" title="Mersennepriemgetal">mersennepriemgetal</a> · <a href="/wiki/Natuurlijk_getal" title="Natuurlijk getal">natuurlijk getal</a> · <a href="/wiki/Overvloedig_getal" title="Overvloedig getal">overvloedig getal</a> · <a href="/wiki/Palindroomgetal" title="Palindroomgetal">palindroomgetal</a> · <a href="/wiki/Palindroompriemgetal" title="Palindroompriemgetal">palindroompriemgetal</a> · <a href="/wiki/Perfect_getal" title="Perfect getal">perfect getal</a> · <a href="/wiki/Plastisch_getal" title="Plastisch getal">plastisch getal</a> · <a href="/wiki/Praktisch_getal" title="Praktisch getal">praktisch getal</a> · <a href="/wiki/Priemgetal" title="Priemgetal">priemgetal</a> · <a href="/wiki/Priemtweeling" title="Priemtweeling">priemtweeling</a> · <a href="/wiki/Rationaal_getal" title="Rationaal getal">rationaal getal</a> · <a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">reëel getal</a> · <a href="/wiki/Rekenkundig_getal" title="Rekenkundig getal">rekenkundig getal</a> · <a href="/wiki/Samengesteld_getal" title="Samengesteld getal">samengesteld getal</a> · <a href="/wiki/Semiperfect_getal" title="Semiperfect getal">semiperfect getal</a> · <a href="/wiki/Sphenisch_getal" title="Sphenisch getal">sphenisch getal</a> · <a href="/wiki/Vreemd_getal" title="Vreemd getal">vreemd getal</a></td></tr></tbody></table> </div></div> <div class="interProject commons mw-list-item" style="display:none;"><a 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