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کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں [o]" accesskey="o" class=""><span>داخل ہوں</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="مزید اختیارات" > <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="ذاتی آلات" > <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">ذاتی آلات</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <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_ur.wikipedia.org&uselang=ur"><span>عطیہ دیجیے</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%AA%D8%AE%D9%84%DB%8C%D9%82_%DA%A9%DA%BE%D8%A7%D8%AA%DB%81&returnto=%D8%B9%D8%AF%D8%AF" title="کھاتہ بنانے یا اس میں داخل ہونے کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>کھاتہ بنائیں</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%AF%D8%A7%D8%AE%D9%84_%DB%81%D9%88%DA%BA&returnto=%D8%B9%D8%AF%D8%AF" title="کھاتے میں داخل ہونے کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>داخل ہوں</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"> Pages for logged out editors <a href="/wiki/%D9%85%D8%B9%D8%A7%D9%88%D9%86%D8%AA:%D8%AA%D8%B9%D8%A7%D8%B1%D9%81" aria-label="ترمیم کاری کے متعلق مزید تفصیلات"><span>مزید تفصیلات</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/%D8%AE%D8%A7%D8%B5:%D9%85%DB%8C%D8%B1%DB%8C_%D8%B4%D8%B1%D8%A7%DA%A9%D8%AA%DB%8C%DA%BA" title="اس آئی پی پتے سے انجام دی جانے والی تمام ترامیم کی فہرست [y]" accesskey="y"><span>شراکتیں</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D9%85%DB%8C%D8%B1%D8%A7_%D8%AA%D8%A8%D8%A7%D8%AF%D9%84%DB%81_%D8%AE%DB%8C%D8%A7%D9%84" title="اس آئی پی پتے کی ترامیم سے متعلق گفتگو [n]" accesskey="n"><span>تبادلۂ خیال</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="ویب گاہ"> <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="مندرجات" 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">مندرجات</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">بغلی پٹی پر جائیں</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">چھپائیں</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">دیباچہ</div> </a> </li> <li id="toc-صحیح_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#صحیح_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>صحیح عدد</span> </div> </a> <ul id="toc-صحیح_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-مکمل_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#مکمل_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>مکمل عدد</span> </div> </a> <ul id="toc-مکمل_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-قدرتی_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#قدرتی_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>قدرتی عدد</span> </div> </a> <ul id="toc-قدرتی_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-تقسیم_الخوارزم" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#تقسیم_الخوارزم"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>تقسیم الخوارزم</span> </div> </a> <ul id="toc-تقسیم_الخوارزم-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-جفت_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#جفت_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>جفت عدد</span> </div> </a> <ul id="toc-جفت_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-طاق_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#طاق_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>طاق عدد</span> </div> </a> <ul id="toc-طاق_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-عادِ_اعظم" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#عادِ_اعظم"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>عادِ اعظم</span> </div> </a> <button aria-controls="toc-عادِ_اعظم-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>عادِ اعظم کے ذیلی قطعہ پر جائیں</span> </button> <ul id="toc-عادِ_اعظم-sublist" class="vector-toc-list"> <li id="toc-عاد_اعظم_الخوارزم" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#عاد_اعظم_الخوارزم"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>عاد اعظم الخوارزم</span> </div> </a> <ul id="toc-عاد_اعظم_الخوارزم-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-نظریہ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#نظریہ"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>نظریہ</span> </div> </a> <ul id="toc-نظریہ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-ذواضعاف_اقل" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ذواضعاف_اقل"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>ذواضعاف اقل</span> </div> </a> <button aria-controls="toc-ذواضعاف_اقل-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>ذواضعاف اقل کے ذیلی قطعہ پر جائیں</span> </button> <ul id="toc-ذواضعاف_اقل-sublist" class="vector-toc-list"> <li id="toc-نظریہ_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#نظریہ_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>نظریہ</span> </div> </a> <ul id="toc-نظریہ_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-مرکب_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#مرکب_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>مرکب عدد</span> </div> </a> <ul id="toc-مرکب_عدد-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-مفرد_عدد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#مفرد_عدد"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>مفرد عدد</span> </div> </a> <button aria-controls="toc-مفرد_عدد-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>مفرد عدد کے ذیلی قطعہ پر جائیں</span> </button> <ul id="toc-مفرد_عدد-sublist" class="vector-toc-list"> <li id="toc-حساب_کا_بنیادی_نظریہ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#حساب_کا_بنیادی_نظریہ"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>حساب کا بنیادی نظریہ</span> </div> </a> <ul id="toc-حساب_کا_بنیادی_نظریہ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-نظریہ_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#نظریہ_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>نظریہ</span> </div> </a> <ul id="toc-نظریہ_3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-نظریہ_4" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#نظریہ_4"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>نظریہ</span> </div> </a> <ul id="toc-نظریہ_4-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-مفرد_عدد_کی_چھاننی" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#مفرد_عدد_کی_چھاننی"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.4</span> <span>مفرد عدد کی چھاننی</span> </div> </a> <ul id="toc-مفرد_عدد_کی_چھاننی-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-مفرد_عدد_کی_پہچان" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#مفرد_عدد_کی_پہچان"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.5</span> <span>مفرد عدد کی پہچان</span> </div> </a> <ul id="toc-مفرد_عدد_کی_پہچان-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-توزیع_مفرد_اعداد" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#توزیع_مفرد_اعداد"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.6</span> <span>توزیع مفرد اعداد</span> </div> </a> <ul id="toc-توزیع_مفرد_اعداد-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-مزید_دیکھیے" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#مزید_دیکھیے"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>مزید دیکھیے</span> </div> </a> <ul id="toc-مزید_دیکھیے-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="مندرجات" 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="فہرست عناوین دیکھیں" > <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">فہرست عناوین دیکھیں</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">عدد</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="Go to an article in another language. Available in 189 languages" > <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-189" 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">189 زبانیں</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF" title="عدد – عربی" lang="ar" hreflang="ar" data-title="عدد" data-language-autonym="العربية" data-language-local-name="عربی" class="interlanguage-link-target"><span>العربية</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" title="عدد – فارسی" lang="fa" hreflang="fa" data-title="عدد" data-language-autonym="فارسی" data-language-local-name="فارسی" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Number" title="Number – انگریزی" lang="en" hreflang="en" data-title="Number" data-language-autonym="English" data-language-local-name="انگریزی" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Getal" title="Getal – افریقی" lang="af" hreflang="af" data-title="Getal" data-language-autonym="Afrikaans" data-language-local-name="افریقی" 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/Zahl" title="Zahl – سوئس جرمن" lang="gsw" hreflang="gsw" data-title="Zahl" data-language-autonym="Alemannisch" data-language-local-name="سوئس جرمن" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%81%E1%8C%A5%E1%88%AD" title="ቁጥር – امہاری" lang="am" hreflang="am" data-title="ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="امہاری" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – انگیکا" lang="anp" hreflang="anp" data-title="संख्या" data-language-autonym="अंगिका" data-language-local-name="انگیکا" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/R%C4%ABm" title="Rīm – Old English" lang="ang" hreflang="ang" data-title="Rīm" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero" title="Numero – اراگونیز" lang="an" hreflang="an" data-title="Numero" data-language-autonym="Aragonés" data-language-local-name="اراگونیز" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%A1%DC%A2%DC%9D%DC%A2%DC%90" title="ܡܢܝܢܐ – Aramaic" lang="arc" hreflang="arc" data-title="ܡܢܝܢܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B9%D5%AB%D6%82" title="Թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="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-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/Numiru" title="Numiru – ارومانی" lang="rup" hreflang="rup" data-title="Numiru" data-language-autonym="Armãneashti" data-language-local-name="ارومانی" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-frp mw-list-item"><a href="https://frp.wikipedia.org/wiki/Nombro" title="Nombro – Arpitan" lang="frp" hreflang="frp" data-title="Nombro" data-language-autonym="Arpetan" data-language-local-name="Arpitan" class="interlanguage-link-target"><span>Arpetan</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="সংখ্যা – آسامی" lang="as" hreflang="as" data-title="সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="آسامی" 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" title="Númberu – اسٹوریائی" lang="ast" hreflang="ast" data-title="Númberu" data-language-autonym="Asturianu" data-language-local-name="اسٹوریائی" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-atj mw-list-item"><a href="https://atj.wikipedia.org/wiki/Akitasowin" title="Akitasowin – اٹیکامیکو" lang="atj" hreflang="atj" data-title="Akitasowin" data-language-autonym="Atikamekw" data-language-local-name="اٹیکامیکو" class="interlanguage-link-target"><span>Atikamekw</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapy" title="Papapy – گُارانی" lang="gn" hreflang="gn" data-title="Papapy" data-language-autonym="Avañe'ẽ" data-language-local-name="گُارانی" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99d" title="Ədəd – آذربائیجانی" lang="az" hreflang="az" data-title="Ədəd" data-language-autonym="Azərbaycanca" data-language-local-name="آذربائیجانی" 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%B9%D8%AF%D8%AF" title="عدد – South Azerbaijani" lang="azb" hreflang="azb" data-title="عدد" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="সংখ্যা – بنگلہ" lang="bn" hreflang="bn" data-title="সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="بنگلہ" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Wilangan" title="Wilangan – Banjar" lang="bjn" hreflang="bjn" data-title="Wilangan" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-ba%CC%8Dk" title="Sò͘-ba̍k – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-ba̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D0%B0%D0%BD" title="Һан – باشکیر" lang="ba" hreflang="ba" data-title="Һан" data-language-autonym="Башҡортса" data-language-local-name="باشکیر" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA" title="Лік – بیلاروسی" lang="be" hreflang="be" data-title="Лік" data-language-autonym="Беларуская" data-language-local-name="بیلاروسی" 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%9B%D1%96%D0%BA" title="Лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="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-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A8%E0%A4%82%E0%A4%AC%E0%A4%B0" title="नंबर – Bhojpuri" lang="bh" hreflang="bh" data-title="नंबर" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Bilang" title="Bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Bilang" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – بلغاری" lang="bg" hreflang="bg" data-title="Число" data-language-autonym="Български" data-language-local-name="بلغاری" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Zoih" title="Zoih – Bavarian" lang="bar" hreflang="bar" data-title="Zoih" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%82%E0%BE%B2%E0%BD%84%E0%BD%A6%E0%BC%8B%E0%BD%80%E0%BC%8D" title="གྲངས་ཀ། – تبتی" lang="bo" hreflang="bo" data-title="གྲངས་ཀ།" data-language-autonym="བོད་ཡིག" data-language-local-name="تبتی" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Broj" title="Broj – بوسنیائی" lang="bs" hreflang="bs" data-title="Broj" data-language-autonym="Bosanski" data-language-local-name="بوسنیائی" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver" title="Niver – بریٹن" lang="br" hreflang="br" data-title="Niver" data-language-autonym="Brezhoneg" data-language-local-name="بریٹن" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE" title="Тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="بہترین مضمون"><a href="https://ca.wikipedia.org/wiki/Nombre" title="Nombre – کیٹالان" lang="ca" hreflang="ca" data-title="Nombre" data-language-autonym="Català" data-language-local-name="کیٹالان" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B8%D1%81%D0%B5%D0%BF" title="Хисеп – چوواش" lang="cv" hreflang="cv" data-title="Хисеп" data-language-autonym="Чӑвашла" data-language-local-name="چوواش" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8C%C3%ADslo" title="Číslo – چیک" lang="cs" hreflang="cs" data-title="Číslo" data-language-autonym="Čeština" data-language-local-name="چیک" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Numero" title="Numero – Chavacano" lang="cbk" hreflang="cbk" data-title="Numero" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhamba" title="Nhamba – شونا" lang="sn" hreflang="sn" data-title="Nhamba" data-language-autonym="ChiShona" data-language-local-name="شونا" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-tum mw-list-item"><a href="https://tum.wikipedia.org/wiki/Chipendero" title="Chipendero – ٹمبوکا" lang="tum" hreflang="tum" data-title="Chipendero" data-language-autonym="ChiTumbuka" data-language-local-name="ٹمبوکا" class="interlanguage-link-target"><span>ChiTumbuka</span></a></li><li class="interlanguage-link interwiki-cho mw-list-item"><a href="https://cho.wikipedia.org/wiki/Hohltina" title="Hohltina – چاکٹاؤ" lang="cho" hreflang="cho" data-title="Hohltina" data-language-autonym="Chahta anumpa" data-language-local-name="چاکٹاؤ" class="interlanguage-link-target"><span>Chahta anumpa</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Numeru" title="Numeru – کوراسیکن" lang="co" hreflang="co" data-title="Numeru" data-language-autonym="Corsu" data-language-local-name="کوراسیکن" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif" title="Rhif – ویلش" lang="cy" hreflang="cy" data-title="Rhif" data-language-autonym="Cymraeg" data-language-local-name="ویلش" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-dag mw-list-item"><a href="https://dag.wikipedia.org/wiki/Kalinli" title="Kalinli – Dagbani" lang="dag" hreflang="dag" data-title="Kalinli" data-language-autonym="Dagbanli" data-language-local-name="Dagbani" class="interlanguage-link-target"><span>Dagbanli</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tal" title="Tal – ڈینش" lang="da" hreflang="da" data-title="Tal" data-language-autonym="Dansk" data-language-local-name="ڈینش" 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/Zahl" title="Zahl – جرمن" lang="de" hreflang="de" data-title="Zahl" data-language-autonym="Deutsch" data-language-local-name="جرمن" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dty mw-list-item"><a href="https://dty.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%BE" title="अंका – Doteli" lang="dty" hreflang="dty" data-title="अंका" data-language-autonym="डोटेली" data-language-local-name="Doteli" class="interlanguage-link-target"><span>डोटेली</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arv" title="Arv – اسٹونین" lang="et" hreflang="et" data-title="Arv" data-language-autonym="Eesti" data-language-local-name="اسٹونین" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Αριθμός – یونانی" lang="el" hreflang="el" data-title="Αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="یونانی" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer" title="Nómmer – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero" title="Número – ہسپانوی" lang="es" hreflang="es" data-title="Número" data-language-autonym="Español" data-language-local-name="ہسپانوی" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nombro" title="Nombro – ایسپرانٹو" lang="eo" hreflang="eo" data-title="Nombro" data-language-autonym="Esperanto" data-language-local-name="ایسپرانٹو" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/N%C3%BAmiru" title="Númiru – Extremaduran" lang="ext" hreflang="ext" data-title="Númiru" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki" title="Zenbaki – باسکی" lang="eu" hreflang="eu" data-title="Zenbaki" data-language-autonym="Euskara" data-language-local-name="باسکی" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Ginti" title="Ginti – Fiji Hindi" lang="hif" hreflang="hif" data-title="Ginti" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Tal" title="Tal – فیروئیز" lang="fo" hreflang="fo" data-title="Tal" data-language-autonym="Føroyskt" data-language-local-name="فیروئیز" 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/Nombre" title="Nombre – فرانسیسی" lang="fr" hreflang="fr" data-title="Nombre" data-language-autonym="Français" data-language-local-name="فرانسیسی" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Getal" title="Getal – مغربی فریسیئن" lang="fy" hreflang="fy" data-title="Getal" data-language-autonym="Frysk" data-language-local-name="مغربی فریسیئن" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ff mw-list-item"><a href="https://ff.wikipedia.org/wiki/Limle" title="Limle – فولہ" lang="ff" hreflang="ff" data-title="Limle" data-language-autonym="Fulfulde" data-language-local-name="فولہ" class="interlanguage-link-target"><span>Fulfulde</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir" title="Uimhir – آئرش" lang="ga" hreflang="ga" data-title="Uimhir" data-language-autonym="Gaeilge" data-language-local-name="آئرش" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/%C3%80ireamh" title="Àireamh – سکاٹش گیلک" lang="gd" hreflang="gd" data-title="Àireamh" data-language-autonym="Gàidhlig" data-language-local-name="سکاٹش گیلک" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero" title="Número – گالیشیائی" lang="gl" hreflang="gl" data-title="Número" data-language-autonym="Galego" data-language-local-name="گالیشیائی" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8" title="數 – gan" lang="gan" hreflang="gan" data-title="數" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D0%B3" title="Тойг – کالمیک" lang="xal" hreflang="xal" data-title="Тойг" data-language-autonym="Хальмг" data-language-local-name="کالمیک" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98_(%EC%88%98%ED%95%99)" title="수 (수학) – کوریائی" lang="ko" hreflang="ko" data-title="수 (수학)" data-language-autonym="한국어" data-language-local-name="کوریائی" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Lamba" title="Lamba – ہؤسا" lang="ha" hreflang="ha" data-title="Lamba" data-language-autonym="Hausa" data-language-local-name="ہؤسا" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/N%C4%81_helu" title="Nā helu – ہوائی" lang="haw" hreflang="haw" data-title="Nā helu" data-language-autonym="Hawaiʻi" data-language-local-name="ہوائی" class="interlanguage-link-target"><span>Hawaiʻi</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%AB%D5%BE" title="Թիվ – آرمینیائی" lang="hy" hreflang="hy" data-title="Թիվ" data-language-autonym="Հայերեն" data-language-local-name="آرمینیائی" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – ہندی" lang="hi" hreflang="hi" data-title="संख्या" data-language-autonym="हिन्दी" data-language-local-name="ہندی" 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/Broj" title="Broj – کراتی" lang="hr" hreflang="hr" data-title="Broj" data-language-autonym="Hrvatski" data-language-local-name="کراتی" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Nombro" title="Nombro – ایڈو" lang="io" hreflang="io" data-title="Nombro" data-language-autonym="Ido" data-language-local-name="ایڈو" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Numero" title="Numero – ایلوکو" lang="ilo" hreflang="ilo" data-title="Numero" data-language-autonym="Ilokano" data-language-local-name="ایلوکو" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan" title="Bilangan – انڈونیثیائی" lang="id" hreflang="id" data-title="Bilangan" data-language-autonym="Bahasa Indonesia" data-language-local-name="انڈونیثیائی" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero" title="Numero – بین لسانیات" lang="ia" hreflang="ia" data-title="Numero" data-language-autonym="Interlingua" data-language-local-name="بین لسانیات" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9D%D1%8B%D0%BC%C3%A6%D1%86" title="Нымæц – اوسیٹک" lang="os" hreflang="os" data-title="Нымæц" data-language-autonym="Ирон" data-language-local-name="اوسیٹک" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/INANI" title="INANI – ژوسا" lang="xh" hreflang="xh" data-title="INANI" data-language-autonym="IsiXhosa" data-language-local-name="ژوسا" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Inombolo" title="Inombolo – زولو" lang="zu" hreflang="zu" data-title="Inombolo" data-language-autonym="IsiZulu" data-language-local-name="زولو" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tala_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Tala (stærðfræði) – آئس لینڈک" lang="is" hreflang="is" data-title="Tala (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="آئس لینڈک" 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" title="Numero – اطالوی" lang="it" hreflang="it" data-title="Numero" data-language-autonym="Italiano" data-language-local-name="اطالوی" class="interlanguage-link-target"><span>Italiano</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" title="מספר – عبرانی" lang="he" hreflang="he" data-title="מספר" data-language-autonym="עברית" data-language-local-name="عبرانی" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Wilangan_(mat%C3%A9matika)" title="Wilangan (matématika) – جاوی" lang="jv" hreflang="jv" data-title="Wilangan (matématika)" data-language-autonym="Jawa" data-language-local-name="جاوی" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/%C3%91%CA%8A%CA%8A_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Ñʊʊ (tʊʊzʊʊ) – Kabiye" lang="kbp" hreflang="kbp" data-title="Ñʊʊ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಸಂಖ್ಯೆ – کنّاڈا" lang="kn" hreflang="kn" data-title="ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="کنّاڈا" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="რიცხვი – جارجیائی" lang="ka" hreflang="ka" data-title="რიცხვი" data-language-autonym="ქართული" data-language-local-name="جارجیائی" 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%A1%D0%B0%D0%BD" title="Сан – قزاخ" lang="kk" hreflang="kk" data-title="Сан" data-language-autonym="Қазақша" data-language-local-name="قزاخ" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niver" title="Niver – کورنش" lang="kw" hreflang="kw" data-title="Niver" data-language-autonym="Kernowek" data-language-local-name="کورنش" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba" title="Namba – سواحلی" lang="sw" hreflang="sw" data-title="Namba" data-language-autonym="Kiswahili" data-language-local-name="سواحلی" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Nonm" title="Nonm – ہیتی" lang="ht" hreflang="ht" data-title="Nonm" data-language-autonym="Kreyòl ayisyen" data-language-local-name="ہیتی" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nomm" title="Nomm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nomm" 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-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar" title="Hejmar – کردش" lang="ku" hreflang="ku" data-title="Hejmar" data-language-autonym="Kurdî" data-language-local-name="کردش" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lbe mw-list-item"><a href="https://lbe.wikipedia.org/wiki/%D0%90%D1%8C%D0%B4%D0%B0%D0%B4" title="Аьдад – Lak" lang="lbe" hreflang="lbe" data-title="Аьдад" data-language-autonym="Лакку" data-language-local-name="Lak" class="interlanguage-link-target"><span>Лакку</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" title="ຈຳນວນ – لاؤ" lang="lo" hreflang="lo" data-title="ຈຳນວນ" data-language-autonym="ລາວ" data-language-local-name="لاؤ" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus" title="Numerus – لاطینی" lang="la" hreflang="la" data-title="Numerus" data-language-autonym="Latina" data-language-local-name="لاطینی" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skaitlis" title="Skaitlis – لیٹوین" lang="lv" hreflang="lv" data-title="Skaitlis" data-language-autonym="Latviešu" data-language-local-name="لیٹوین" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Zuel" title="Zuel – لکسمبرگیش" lang="lb" hreflang="lb" data-title="Zuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="لکسمبرگیش" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skai%C4%8Dius" title="Skaičius – لیتھوینین" lang="lt" hreflang="lt" data-title="Skaičius" data-language-autonym="Lietuvių" data-language-local-name="لیتھوینین" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Getal" title="Getal – لیمبرگش" lang="li" hreflang="li" data-title="Getal" data-language-autonym="Limburgs" data-language-local-name="لیمبرگش" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero" title="Numero – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero" 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-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/namcu" title="namcu – لوجبان" lang="jbo" hreflang="jbo" data-title="namcu" data-language-autonym="La .lojban." data-language-local-name="لوجبان" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Ennamba" title="Ennamba – گینڈا" lang="lg" hreflang="lg" data-title="Ennamba" data-language-autonym="Luganda" data-language-local-name="گینڈا" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer" title="Numer – لومبارڈ" lang="lmo" hreflang="lmo" data-title="Numer" data-language-autonym="Lombard" data-language-local-name="لومبارڈ" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%A1m" title="Szám – ہنگیرین" lang="hu" hreflang="hu" data-title="Szám" data-language-autonym="Magyar" data-language-local-name="ہنگیرین" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95" title="अंक – میتھیلی" lang="mai" hreflang="mai" data-title="अंक" data-language-autonym="मैथिली" data-language-local-name="میتھیلی" class="interlanguage-link-target"><span>मैथिली</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%98" title="Број – مقدونیائی" lang="mk" hreflang="mk" data-title="Број" data-language-autonym="Македонски" data-language-local-name="مقدونیائی" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa" title="Isa – ملاگاسی" lang="mg" hreflang="mg" data-title="Isa" data-language-autonym="Malagasy" data-language-local-name="ملاگاسی" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="സംഖ്യ – مالایالم" lang="ml" hreflang="ml" data-title="സംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="مالایالم" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – مراٹهی" lang="mr" hreflang="mr" data-title="संख्या" data-language-autonym="मराठी" data-language-local-name="مراٹهی" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%A3" title="რიცხუ – Mingrelian" lang="xmf" hreflang="xmf" data-title="რიცხუ" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%A7%D8%B4%D9%85%D8%A7%D8%B1%DA%A9" title="اشمارک – مزندرانی" lang="mzn" hreflang="mzn" data-title="اشمارک" data-language-autonym="مازِرونی" data-language-local-name="مزندرانی" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor" title="Nombor – مالے" lang="ms" hreflang="ms" data-title="Nombor" data-language-autonym="Bahasa Melayu" data-language-local-name="مالے" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/N%C3%BAmaro" title="Númaro – میرانڈیز" lang="mwl" hreflang="mwl" data-title="Númaro" data-language-autonym="Mirandés" data-language-local-name="میرانڈیز" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE" title="Тоо – منگولین" lang="mn" hreflang="mn" data-title="Тоо" data-language-autonym="Монгол" data-language-local-name="منگولین" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="ကိန်း – برمی" lang="my" hreflang="my" data-title="ကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="برمی" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nah mw-list-item"><a href="https://nah.wikipedia.org/wiki/Tlapohualli" title="Tlapohualli – Nahuatl" lang="nah" hreflang="nah" data-title="Tlapohualli" data-language-autonym="Nāhuatl" data-language-local-name="Nahuatl" class="interlanguage-link-target"><span>Nāhuatl</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Naba" title="Naba – فجی" lang="fj" hreflang="fj" data-title="Naba" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="فجی" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Getal_(wiskunde)" title="Getal (wiskunde) – ڈچ" lang="nl" hreflang="nl" data-title="Getal (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="ڈچ" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-cr mw-list-item"><a href="https://cr.wikipedia.org/wiki/%E1%90%8A%E1%91%AD%E1%90%A6%E1%91%96%E1%93%B1%E1%90%A3" title="ᐊᑭᐦᑖᓱᐣ – Cree" lang="cr" hreflang="cr" data-title="ᐊᑭᐦᑖᓱᐣ" data-language-autonym="Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ" data-language-local-name="Cree" class="interlanguage-link-target"><span>Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%99%E0%A5%8D%E0%A4%95" title="अङ्क – نیپالی" lang="ne" hreflang="ne" data-title="अङ्क" data-language-autonym="नेपाली" data-language-local-name="نیپالی" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83" title="ल्याः – نیواری" lang="new" hreflang="new" data-title="ल्याः" data-language-autonym="नेपाल भाषा" data-language-local-name="نیواری" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0" title="数 – جاپانی" lang="ja" hreflang="ja" data-title="数" data-language-autonym="日本語" data-language-local-name="جاپانی" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Taal" title="Taal – شمالی فریزئین" lang="frr" hreflang="frr" data-title="Taal" data-language-autonym="Nordfriisk" data-language-local-name="شمالی فریزئین" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tall" title="Tall – نارویجین بوکمل" lang="nb" hreflang="nb" data-title="Tall" data-language-autonym="Norsk bokmål" data-language-local-name="نارویجین بوکمل" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tal" title="Tal – نارویجین نینورسک" lang="nn" hreflang="nn" data-title="Tal" data-language-autonym="Norsk nynorsk" data-language-local-name="نارویجین نینورسک" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Neunm%C3%A9tho" title="Neunmétho – Norman" lang="nrf" hreflang="nrf" data-title="Neunmétho" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target"><span>Nouormand</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Nombre" title="Nombre – Novial" lang="nov" hreflang="nov" data-title="Nombre" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre" title="Nombre – آکسیٹان" lang="oc" hreflang="oc" data-title="Nombre" data-language-autonym="Occitan" data-language-local-name="آکسیٹان" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%A8%D0%BE%D1%82%D0%BF%D0%B0%D0%BB" title="Шотпал – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Шотпал" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Son" title="Son – ازبیک" lang="uz" hreflang="uz" data-title="Son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ازبیک" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਅੰਕ – پنجابی" lang="pa" hreflang="pa" data-title="ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="پنجابی" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%D9%85%D8%A8%D8%B1" title="نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF" title="عدد – پشتو" lang="ps" hreflang="ps" data-title="عدد" data-language-autonym="پښتو" data-language-local-name="پشتو" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Nomba" title="Nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Tahl" title="Tahl – ادنی جرمن" lang="nds" hreflang="nds" data-title="Tahl" data-language-autonym="Plattdüütsch" data-language-local-name="ادنی جرمن" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba" title="Liczba – پولش" lang="pl" hreflang="pl" data-title="Liczba" data-language-autonym="Polski" data-language-local-name="پولش" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero" title="Número – پُرتگالی" lang="pt" hreflang="pt" data-title="Número" data-language-autonym="Português" data-language-local-name="پُرتگالی" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/San" title="San – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="San" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r" title="Număr – رومینین" lang="ro" hreflang="ro" data-title="Număr" data-language-autonym="Română" data-language-local-name="رومینین" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupay" title="Yupay – کویچوآ" lang="qu" hreflang="qu" data-title="Yupay" data-language-autonym="Runa Simi" data-language-local-name="کویچوآ" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A7%D1%96%D1%81%D0%BB%D0%BE" title="Чісло – Rusyn" lang="rue" hreflang="rue" data-title="Чісло" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – روسی" lang="ru" hreflang="ru" data-title="Число" data-language-autonym="Русский" data-language-local-name="روسی" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%90%D1%85%D1%81%D0%B0%D0%B0%D0%BD" title="Ахсаан – ساکھا" lang="sah" hreflang="sah" data-title="Ахсаан" data-language-autonym="Саха тыла" data-language-local-name="ساکھا" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83" title="संख्याः – سنسکرت" lang="sa" hreflang="sa" data-title="संख्याः" data-language-autonym="संस्कृतम्" data-language-local-name="سنسکرت" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sg mw-list-item"><a href="https://sg.wikipedia.org/wiki/N%C3%B6m%C3%B6r%C3%B6" title="Nömörö – ساںغو" lang="sg" hreflang="sg" data-title="Nömörö" data-language-autonym="Sängö" data-language-local-name="ساںغو" class="interlanguage-link-target"><span>Sängö</span></a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Nomoro" title="Nomoro – شمالی سوتھو" lang="nso" hreflang="nso" data-title="Nomoro" data-language-autonym="Sesotho sa Leboa" data-language-local-name="شمالی سوتھو" class="interlanguage-link-target"><span>Sesotho sa Leboa</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri" title="Numri – البانی" lang="sq" hreflang="sq" data-title="Numri" data-language-autonym="Shqip" data-language-local-name="البانی" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru" title="Nùmmuru – سیسیلین" lang="scn" hreflang="scn" data-title="Nùmmuru" data-language-autonym="Sicilianu" data-language-local-name="سیسیلین" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Number" title="Number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Number" 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-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D9%86%DA%AF" title="انگ – سندھی" lang="sd" hreflang="sd" data-title="انگ" data-language-autonym="سنڌي" data-language-local-name="سندھی" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C4%8C%C3%ADslo_(matematika)" title="Číslo (matematika) – سلوواک" lang="sk" hreflang="sk" data-title="Číslo (matematika)" data-language-autonym="Slovenčina" data-language-local-name="سلوواک" 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/%C5%A0tevilo" title="Število – سلووینیائی" lang="sl" hreflang="sl" data-title="Število" data-language-autonym="Slovenščina" data-language-local-name="سلووینیائی" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BC%D1%A7" title="Чисмѧ – چرچ سلاوک" lang="cu" hreflang="cu" data-title="Чисмѧ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="چرچ سلاوک" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/N%C5%AFmera" title="Nůmera – Silesian" lang="szl" hreflang="szl" data-title="Nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_(cillanaad)" title="Thiin (cillanaad) – صومالی" lang="so" hreflang="so" data-title="Thiin (cillanaad)" data-language-autonym="Soomaaliga" data-language-local-name="صومالی" class="interlanguage-link-target"><span>Soomaaliga</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" title="ژمارە – سینٹرل کردش" lang="ckb" hreflang="ckb" data-title="ژمارە" data-language-autonym="کوردی" data-language-local-name="سینٹرل کردش" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%98" title="Број – سربین" lang="sr" hreflang="sr" data-title="Број" data-language-autonym="Српски / srpski" data-language-local-name="سربین" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Broj" title="Broj – سربو-کروئیشین" lang="sh" hreflang="sh" data-title="Broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="سربو-کروئیشین" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Wilangan" title="Wilangan – سنڈانیز" lang="su" hreflang="su" data-title="Wilangan" data-language-autonym="Sunda" data-language-local-name="سنڈانیز" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luku" title="Luku – فینیش" lang="fi" hreflang="fi" data-title="Luku" data-language-autonym="Suomi" data-language-local-name="فینیش" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tal" title="Tal – سویڈش" lang="sv" hreflang="sv" data-title="Tal" data-language-autonym="Svenska" data-language-local-name="سویڈش" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilang" title="Bilang – ٹیگا لوگ" lang="tl" hreflang="tl" data-title="Bilang" data-language-autonym="Tagalog" data-language-local-name="ٹیگا لوگ" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="எண் – تمل" lang="ta" hreflang="ta" data-title="எண்" data-language-autonym="தமிழ்" data-language-local-name="تمل" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Am%E1%B8%8Dan" title="Amḍan – تشلحيت" lang="shi" hreflang="shi" data-title="Amḍan" data-language-autonym="Taclḥit" data-language-local-name="تشلحيت" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Am%E1%B8%8Dan" title="Amḍan – قبائلی" lang="kab" hreflang="kab" data-title="Amḍan" data-language-autonym="Taqbaylit" data-language-local-name="قبائلی" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-roa-tara mw-list-item"><a href="https://roa-tara.wikipedia.org/wiki/Numere" title="Numere – Tarantino" lang="nap-x-tara" hreflang="nap-x-tara" data-title="Numere" data-language-autonym="Tarandíne" data-language-local-name="Tarantino" class="interlanguage-link-target"><span>Tarandíne</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BD" title="Сан – تاتار" lang="tt" hreflang="tt" data-title="Сан" data-language-autonym="Татарча / tatarça" data-language-local-name="تاتار" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="సంఖ్య – تیلگو" lang="te" hreflang="te" data-title="సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="تیلگو" class="interlanguage-link-target"><span>తెలుగు</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" title="จำนวน – تھائی" lang="th" hreflang="th" data-title="จำนวน" data-language-autonym="ไทย" data-language-local-name="تھائی" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-ti mw-list-item"><a href="https://ti.wikipedia.org/wiki/%E1%89%81%E1%8C%BD%E1%88%AA" title="ቁጽሪ – ٹگرینیا" lang="ti" hreflang="ti" data-title="ቁጽሪ" data-language-autonym="ትግርኛ" data-language-local-name="ٹگرینیا" class="interlanguage-link-target"><span>ትግርኛ</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" title="Адад – تاجک" lang="tg" hreflang="tg" data-title="Адад" data-language-autonym="Тоҷикӣ" data-language-local-name="تاجک" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tcy mw-list-item"><a href="https://tcy.wikipedia.org/wiki/%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಸಂಖ್ಯೆ – Tulu" lang="tcy" hreflang="tcy" data-title="ಸಂಖ್ಯೆ" data-language-autonym="ತುಳು" data-language-local-name="Tulu" class="interlanguage-link-target"><span>ತುಳು</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Say%C4%B1" title="Sayı – ترکی" lang="tr" hreflang="tr" data-title="Sayı" data-language-autonym="Türkçe" data-language-local-name="ترکی" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/San" title="San – ترکمان" lang="tk" hreflang="tk" data-title="San" data-language-autonym="Türkmençe" data-language-local-name="ترکمان" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%9B%D1%8B%D0%B4" title="Лыд – ادمورت" lang="udm" hreflang="udm" data-title="Лыд" data-language-autonym="Удмурт" data-language-local-name="ادمورت" class="interlanguage-link-target"><span>Удмурт</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – یوکرینیائی" lang="uk" hreflang="uk" data-title="Число" data-language-autonym="Українська" data-language-local-name="یوکرینیائی" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/N%C3%B9maro" title="Nùmaro – Venetian" lang="vec" hreflang="vec" data-title="Nùmaro" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Lugu" title="Lugu – Veps" lang="vep" hreflang="vep" data-title="Lugu" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91" title="Số – ویتنامی" lang="vi" hreflang="vi" data-title="Số" data-language-autonym="Tiếng Việt" data-language-local-name="ویتنامی" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Arv" title="Arv – Võro" lang="vro" hreflang="vro" data-title="Arv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-guc mw-list-item"><a href="https://guc.wikipedia.org/wiki/Nuumerokana" title="Nuumerokana – Wayuu" lang="guc" hreflang="guc" data-title="Nuumerokana" data-language-autonym="Wayuunaiki" data-language-local-name="Wayuu" class="interlanguage-link-target"><span>Wayuunaiki</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B8" title="數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ihap" title="Ihap – وارے" lang="war" hreflang="war" data-title="Ihap" data-language-autonym="Winaray" data-language-local-name="وارے" 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%B0" title="数 – وو چائینیز" lang="wuu" hreflang="wuu" data-title="数" data-language-autonym="吴语" data-language-local-name="وو چائینیز" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Nomboro" title="Nomboro – زونگا" lang="ts" hreflang="ts" data-title="Nomboro" data-language-autonym="Xitsonga" data-language-local-name="زونگا" class="interlanguage-link-target"><span>Xitsonga</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A6%D7%90%D7%9C" title="צאל – یدش" lang="yi" hreflang="yi" data-title="צאל" data-language-autonym="ייִדיש" data-language-local-name="یدش" 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" title="Nọ́mbà – یوروبا" lang="yo" hreflang="yo" data-title="Nọ́mbà" data-language-autonym="Yorùbá" data-language-local-name="یوروبا" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8" title="數 – کینٹونیز" lang="yue" hreflang="yue" data-title="數" data-language-autonym="粵語" data-language-local-name="کینٹونیز" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Skaitlios" title="Skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0" title="数 – چینی" lang="zh" hreflang="zh" data-title="数" data-language-autonym="中文" data-language-local-name="چینی" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur" title="Lumur – ایبان" lang="iba" hreflang="iba" data-title="Lumur" data-language-autonym="Jaku Iban" data-language-local-name="ایبان" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%8E%E2%B4%B9%E2%B4%B0%E2%B5%8F" title="ⴰⵎⴹⴰⵏ – اسٹینڈرڈ مراقشی تمازیقی" lang="zgh" hreflang="zgh" data-title="ⴰⵎⴹⴰⵏ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="اسٹینڈرڈ مراقشی تمازیقی" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</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/Q11563#sitelinks-wikipedia" title="ترمیم بین 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</div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> عمومی </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D9%85%D8%B1%D8%A8%D9%88%D8%B7_%D8%B5%D9%81%D8%AD%D8%A7%D8%AA/%D8%B9%D8%AF%D8%AF" title="اُن تمام ویکی صفحات کی فہرست جو اس صفحہ سے مربوط ہیں [j]" accesskey="j"><span>مربوط صفحات</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D9%85%D8%AA%D8%B9%D9%84%D9%82%DB%81_%D8%AD%D8%A7%D9%84%DB%8C%DB%81_%D8%AA%D8%A8%D8%AF%DB%8C%D9%84%DB%8C%D8%A7%DA%BA/%D8%B9%D8%AF%D8%AF" rel="nofollow" title="اِس صفحہ سے مربوط صفحات میں حالیہ تبدیلیاں [k]" accesskey="k"><span>متعلقہ تبدیلیاں</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D8%AE%D8%B5%D9%88%D8%B5%DB%8C_%D8%B5%D9%81%D8%AD%D8%A7%D8%AA" title="جملہ خصوصی صفحات کی فہرست [q]" 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title="متعلقہ ڈیٹا سے منسلک کریں [g]" accesskey="g"><span>ڈیٹا آئٹم</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button 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title="فائل:AADAAD-JAMAAT-2.PNG"><span class="mw-file-element mw-broken-media" data-width="245">فائل:AADAAD-JAMAAT-2.PNG</span></a><figcaption>اعداد کی شجری جماعت بندی کا ایک اظہار۔</figcaption></figure> <p><b>عدد</b> (جمع: اعداد ؛ انگریزی: number) اصل میں ایک <a href="/w/index.php?title=%D8%AC%D9%90%D8%B1%D9%85_%D9%85%D8%AC%D8%B1%D8%AF&action=edit&redlink=1" class="new" title="جِرم مجرد (صفحہ موجود نہیں)">جِرم مجرد (abstract object)</a> ہے جس کو <a href="/w/index.php?title=%D8%B4%D9%85%D8%A7%D8%B1&action=edit&redlink=1" class="new" title="شمار (صفحہ موجود نہیں)">شمار</a> و <a href="/wiki/%D9%BE%DB%8C%D9%85%D8%A7%D8%A6%D8%B4" title="پیمائش">پیمائش</a> کرنے کے لیے استعمال کیا جاتا ہے جبکہ وہ <a href="/wiki/%D8%B9%D9%84%D8%A7%D9%85%D8%A7%D8%AA_%D9%88_%D8%A7%D8%B4%D8%A7%D8%B1%D8%A7%D8%AA" class="mw-redirect" title="علامات و اشارات">علامت</a> جو کسی عدد کو ظاہر کرنے کے لیے اختیار کی جاتی ہے اسے <a href="/w/index.php?title=%D8%B9%D8%AF%DB%8C%D8%AF&action=edit&redlink=1" class="new" title="عدید (صفحہ موجود نہیں)">عدید (numeral)</a> کہا جاتا ہے، لیکن عام روزمرہ کے استعمال میں عدد سے مراد لکھی جانے والی علامت اور جرم مجرد (یعنی اس جرم کا شمار یا پیمائش) دونوں کی لی جاتی ہے۔ </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="صحیح_عدد"><span id=".D8.B5.D8.AD.DB.8C.D8.AD_.D8.B9.D8.AF.D8.AF"></span>صحیح عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=1" title="تدوینِ قطعہ: صحیح عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%B5%D8%AD%DB%8C%D8%AD_%D8%B9%D8%AF%D8%AF" title="صحیح عدد">صحیح عدد</a> </p><p><i>صحیح اعداد</i> گنتی کے تمام مثبت اور منفی اعداد، بشمول صفر،کے مجموعہ کو کہتے ہیں۔ انگریزی میں انھیں انٹیجر (integer) کہا جاتا ہے۔ اس مجموعہ کو ہم یوں لکھ سکتے ہیں: <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} =\{\cdots ,-5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5,\cdots \}}"> <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> <mo fence="false" stretchy="false">{</mo> <mo>⋯</mo> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mo>+</mo> <mn>4</mn> <mo>,</mo> <mo>+</mo> <mn>5</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} =\{\cdots ,-5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccba233c2cd79f389ef684a848d4ac4239aa0320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.083ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} =\{\cdots ,-5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5,\cdots \}}"></span> </p><p>محمود حسين علي </p> <div class="mw-heading mw-heading2"><h2 id="مکمل_عدد"><span id=".D9.85.DA.A9.D9.85.D9.84_.D8.B9.D8.AF.D8.AF"></span>مکمل عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=2" title="تدوینِ قطعہ: مکمل عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D9%85%DA%A9%D9%85%D9%84_%D8%B9%D8%AF%D8%AF" class="mw-redirect" title="مکمل عدد">مکمل عدد</a> </p><p>غیر منفی صحیح اعداد کو مکمل اعداد کہتے ہیں۔ انگریزی میں انھیں whole numbers کہا جاتا ہے۔ مکمل اعداد کے مجموعہ کو یوں لکھا جا سکتا ہے: <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 {W} =\{0,+1,+2,+3,+4,+5,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">W</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mo>+</mo> <mn>4</mn> <mo>,</mo> <mo>+</mo> <mn>5</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {W} =\{0,+1,+2,+3,+4,+5,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332695c9605f800b376d41075fd2dfd7cba7bdb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.689ex; height:2.843ex;" alt="{\displaystyle \mathbb {W} =\{0,+1,+2,+3,+4,+5,\cdots \}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="قدرتی_عدد"><span id=".D9.82.D8.AF.D8.B1.D8.AA.DB.8C_.D8.B9.D8.AF.D8.AF"></span>قدرتی عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=3" title="تدوینِ قطعہ: قدرتی عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/w/index.php?title=%D9%85%D8%AD%D9%85%D9%88%D8%AF_%D8%AD%D8%B3%D9%8A%D9%86_%D8%B9%D9%84%D9%89&action=edit&redlink=1" class="new" title="محمود حسين على (صفحہ موجود نہیں)">محمود حسين على</a> في تنفيذ الرسم الهندسة سائق معتمد مثبت صحیح اعداد کو قدرتی اعداد کہا جاتا ہے۔ انگریزی میں انھیں natural numbers کہا جاتا ہے۔ قدرتی اعداد کے مجموعہ یوں لکھا جا سکتا ہے: <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} =\{+1,+2,+3,+4,+5,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mo>+</mo> <mn>4</mn> <mo>,</mo> <mo>+</mo> <mn>5</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} =\{+1,+2,+3,+4,+5,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c638299190fcbfd3647f8cdff7318b7a0dfb06f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.847ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} =\{+1,+2,+3,+4,+5,\cdots \}}"></span> </p><p>کبھی صفر کو بھی قدرتی اعداد میں شامل سمجھا جاتا ہے۔ </p> <div class="mw-heading mw-heading2"><h2 id="تقسیم_الخوارزم"><span id=".D8.AA.D9.82.D8.B3.DB.8C.D9.85_.D8.A7.D9.84.D8.AE.D9.88.D8.A7.D8.B1.D8.B2.D9.85"></span>تقسیم الخوارزم</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=4" title="تدوینِ قطعہ: تقسیم الخوارزم"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>فرض کرو کہ صحیح اعداد <i>a</i> اور <i>b</i> ہیں، <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04895307c8d25d492e7b4c8116697dcbb0217fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.071ex; height:2.176ex;" alt="{\displaystyle \ a>0}"></span>۔ پھر ایسے منفرد صحیح اعداد <i>q</i> اور <i>r</i> موجود ہیں، <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<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56db7a45425976538639b15b21eeb618e441167c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.638ex; height:2.343ex;" alt="{\displaystyle 0\leq r<a}"></span> ،جبکہ <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=aq+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ b=aq+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87ac83169bdbce09dd075884a1b5e3ab973f183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.865ex; height:2.509ex;" alt="{\displaystyle \ b=aq+r}"></span> <br /> مثال: اگر <i>b=511</i>، <i>a=5</i>، پھر <i>q=101</i>، <i>r=6</i>، کیونکہ <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 511=5\times 101+6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>511</mn> <mo>=</mo> <mn>5</mn> <mo>×</mo> <mn>101</mn> <mo>+</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 511=5\times 101+6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3326e37301d07679cb3a73a86ee996d42122f221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.079ex; height:2.343ex;" alt="{\displaystyle 511=5\times 101+6}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="جفت_عدد"><span id=".D8.AC.D9.81.D8.AA_.D8.B9.D8.AF.D8.AF"></span>جفت عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=5" title="تدوینِ قطعہ: جفت عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%AC%D9%81%D8%AA_%D8%B9%D8%AF%D8%AF" title="جفت عدد">جفت عدد</a> </p><p>ایسا صحیح عدد جو 2 سے (پورا) تقسیم ہو <i>جفت</i> کہلاتا ہے۔ انگریزی میں even کہتے ہیں۔ جفت اعداد کا مجموعہ <br /> <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 {E} =\{\cdots ,-8,-6,-4,-2,0,2,4,6,8,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>⋯</mo> <mo>,</mo> <mo>−</mo> <mn>8</mn> <mo>,</mo> <mo>−</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} =\{\cdots ,-8,-6,-4,-2,0,2,4,6,8,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4444cadb4ebead1a6deb653d1f9e2b44b57f3685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.841ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} =\{\cdots ,-8,-6,-4,-2,0,2,4,6,8,\cdots \}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="طاق_عدد"><span id=".D8.B7.D8.A7.D9.82_.D8.B9.D8.AF.D8.AF"></span>طاق عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=6" title="تدوینِ قطعہ: طاق عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%B7%D8%A7%D9%82_%D8%B9%D8%AF%D8%AF" title="طاق عدد">طاق عدد</a> </p><p>ایسا صحیح عدد جو 2 سے (پورا) تقسیم نہ ہو، <i>طاق</i> کہلاتا ہے۔ انگریزی میں odd کہتے ہیں۔ طاق اعداد کا مجموعہ <br /> <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 {O} =\{\cdots ,-9,-7,-5,-3,-1,1,3,5,7,9,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>⋯</mo> <mo>,</mo> <mo>−</mo> <mn>9</mn> <mo>,</mo> <mo>−</mo> <mn>7</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} =\{\cdots ,-9,-7,-5,-3,-1,1,3,5,7,9,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb8b33d5680376a70e56a095e7f811796c53bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.104ex; height:2.843ex;" alt="{\displaystyle \mathbb {O} =\{\cdots ,-9,-7,-5,-3,-1,1,3,5,7,9,\cdots \}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="عادِ_اعظم"><span id=".D8.B9.D8.A7.D8.AF.D9.90_.D8.A7.D8.B9.D8.B8.D9.85"></span>عادِ اعظم</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=7" title="تدوینِ قطعہ: عادِ اعظم"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%B9%D8%A7%D8%AF_%D8%A7%D8%B9%D8%B8%D9%85" title="عاد اعظم">عاد اعظم</a> </p><p>سب سے بڑا صحیح عدد جو دو صحیح اعداد کو پورا تقسیم کرے، ان دو اعداد کا <i>عادِ اعظم</i> کہلاتا ہے۔ انگریزی میں عاداعظم کو greatest common divisor (gcd) کہتے ہیں۔ مثال کے طور پر 30 اور 42 کا عاد اعظم 6 ہے، کیونکہ <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}30=5\times 3\times 2\\42=7\times 3\times 2\\\gcd(30,42)=3\times 2=6\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>30</mn> <mo>=</mo> <mn>5</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>42</mn> <mo>=</mo> <mn>7</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>30</mn> <mo>,</mo> <mn>42</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}30=5\times 3\times 2\\42=7\times 3\times 2\\\gcd(30,42)=3\times 2=6\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d764bf5f597f260ad73f2bacea6594cda13bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.257ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}30=5\times 3\times 2\\42=7\times 3\times 2\\\gcd(30,42)=3\times 2=6\end{matrix}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="عاد_اعظم_الخوارزم"><span id=".D8.B9.D8.A7.D8.AF_.D8.A7.D8.B9.D8.B8.D9.85_.D8.A7.D9.84.D8.AE.D9.88.D8.A7.D8.B1.D8.B2.D9.85"></span>عاد اعظم الخوارزم</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=8" title="تدوینِ قطعہ: عاد اعظم الخوارزم"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>اگرچہ دو صحیح اعداد کا عاد اعظم ان اعداد کے ضربی جُز دیکھ کر معلوم کیا جا سکتا ہے، مگر جب اعداد بڑے ہوں تو ضربی جز نکالنا مشکل ہو جاتا ہے۔ عادِاعظم نکالنے کا ایک تیز طریقہ "تقسیم الخوارزم" کے ذریعہ ہے۔ اس الخوارزم کو عموماً یکلڈ کا الخوارزم کہا جاتا ہے۔ <br /> یکلڈ الخوارزم: فرض کرو کہ <i>a</i> اور <i>b</i> صحیح اعداد ہیں، <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04895307c8d25d492e7b4c8116697dcbb0217fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.071ex; height:2.176ex;" alt="{\displaystyle \ a>0}"></span>۔ تقسیم الخوازم یکے بعد دیگرے استعمال کرو: <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}b=aq_{1}+r_{1}\,,\,0\leq r_{1}<a,\\a=r_{1}q_{2}+r_{2}\,,\,0\leq r_{2}<r_{1},\\r_{1}=r_{2}q_{3}+r_{3}\,,\,0\leq r_{3}<r_{3},\\\vdots \\r_{n-2}=r_{n-1}q_{n}+r_{n}\,,\,0\leq r_{3}<r_{3},\\r_{n-1}=r_{n}q_{n+1}\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> <mi>b</mi> <mo>=</mo> <mi>a</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>a</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}b=aq_{1}+r_{1}\,,\,0\leq r_{1}<a,\\a=r_{1}q_{2}+r_{2}\,,\,0\leq r_{2}<r_{1},\\r_{1}=r_{2}q_{3}+r_{3}\,,\,0\leq r_{3}<r_{3},\\\vdots \\r_{n-2}=r_{n-1}q_{n}+r_{n}\,,\,0\leq r_{3}<r_{3},\\r_{n-1}=r_{n}q_{n+1}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ab0315bfc6992bbe832a5551fd7f903eb2955d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:33.968ex; height:20.176ex;" alt="{\displaystyle {\begin{matrix}b=aq_{1}+r_{1}\,,\,0\leq r_{1}<a,\\a=r_{1}q_{2}+r_{2}\,,\,0\leq r_{2}<r_{1},\\r_{1}=r_{2}q_{3}+r_{3}\,,\,0\leq r_{3}<r_{3},\\\vdots \\r_{n-2}=r_{n-1}q_{n}+r_{n}\,,\,0\leq r_{3}<r_{3},\\r_{n-1}=r_{n}q_{n+1}\end{matrix}}}"></span> اب اگر آخری عدد بچا ہے جو صفر نہیں ہے، تو <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_{n}=\gcd(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ r_{n}=\gcd(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6df5091bd08ff11c90fcde964013c021edb4927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.504ex; height:2.843ex;" alt="{\displaystyle \ r_{n}=\gcd(a,b)}"></span> ۔ </p><p>مثال: ہم 198اور 1050 کا عادِ اعظم نکالتے ہیں: <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}1050&=198&\times 5&+&60\\&\swarrow &&&\swarrow \\198&=60&\times 3&+&18\\&\swarrow &&&\swarrow \\60&=18&\times 3&+&6\\&\swarrow &&&\swarrow \\18&=6&\times 3&+&0\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>1050</mn> </mtd> <mtd> <mo>=</mo> <mn>198</mn> </mtd> <mtd> <mo>×</mo> <mn>5</mn> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>60</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> <mtd /> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> </mtr> <mtr> <mtd> <mn>198</mn> </mtd> <mtd> <mo>=</mo> <mn>60</mn> </mtd> <mtd> <mo>×</mo> <mn>3</mn> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>18</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> <mtd /> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> </mtr> <mtr> <mtd> <mn>60</mn> </mtd> <mtd> <mo>=</mo> <mn>18</mn> </mtd> <mtd> <mo>×</mo> <mn>3</mn> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> <mtd /> <mtd /> <mtd> <mo stretchy="false">↙</mo> </mtd> </mtr> <mtr> <mtd> <mn>18</mn> </mtd> <mtd> <mo>=</mo> <mn>6</mn> </mtd> <mtd> <mo>×</mo> <mn>3</mn> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}1050&=198&\times 5&+&60\\&\swarrow &&&\swarrow \\198&=60&\times 3&+&18\\&\swarrow &&&\swarrow \\60&=18&\times 3&+&6\\&\swarrow &&&\swarrow \\18&=6&\times 3&+&0\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a136712d21647e9f0a7f8c0ece21990cf2ef3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:27.736ex; height:22.509ex;" alt="{\displaystyle {\begin{matrix}1050&=198&\times 5&+&60\\&\swarrow &&&\swarrow \\198&=60&\times 3&+&18\\&\swarrow &&&\swarrow \\60&=18&\times 3&+&6\\&\swarrow &&&\swarrow \\18&=6&\times 3&+&0\end{matrix}}}"></span> اس لیے <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 \ \gcd(198,1050)=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>198</mn> <mo>,</mo> <mn>1050</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \gcd(198,1050)=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141e3e3fe2f0af85af02400c8c493c1dbb9df617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.696ex; height:2.843ex;" alt="{\displaystyle \ \gcd(198,1050)=6}"></span> ۔ </p> <div class="mw-heading mw-heading3"><h3 id="نظریہ"><span id=".D9.86.D8.B8.D8.B1.DB.8C.DB.81"></span>نظریہ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=9" title="تدوینِ قطعہ: نظریہ"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>اگر <i>a</i> اور <i>b</i> کا عاد اعظم <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 \ \gcd(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \gcd(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2434005e392a9ac4514766fc7ca5be6089bbea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.526ex; height:2.843ex;" alt="{\displaystyle \ \gcd(a,b)}"></span> ہو، تو اس عاد اعظم کو <i>a</i> اور <i>b</i> کے لکیری جوڑ کے طور پر لکھا جا سکتا ہے۔ یعنی ایسے صحیح اعداد <i>x</i> اور <i>y</i> موجود ہوں گے کہ <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 \ \gcd(a,b)=xa+yb}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>a</mi> <mo>+</mo> <mi>y</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \gcd(a,b)=xa+yb}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db989089b4618fe6d8cc4af3946b6d7dba4fbb24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.177ex; height:2.843ex;" alt="{\displaystyle \ \gcd(a,b)=xa+yb}"></span> </p><p>اعداد <i>x</i> اور <i>y</i> کو نکالنے کے لیے عاد اعظم الخوارزم کو اُلٹی طرف سے پڑھا جا سکتا ہے۔ اوپر کی مثال ہم الٹی جانب لکھتے ہیں: <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}6&=&60-18\times 3&&\\&=&60-(198-60\times 3)\times 3&=&-198\times 3+60\times 10\\&=&-198\times 3+(1050-198\times 5)\times 10&=&1050\times 10-198\times 53\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>6</mn> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>60</mn> <mo>−</mo> <mn>18</mn> <mo>×</mo> <mn>3</mn> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>60</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>198</mn> <mo>−</mo> <mn>60</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>3</mn> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo>−</mo> <mn>198</mn> <mo>×</mo> <mn>3</mn> <mo>+</mo> <mn>60</mn> <mo>×</mo> <mn>10</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo>−</mo> <mn>198</mn> <mo>×</mo> <mn>3</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1050</mn> <mo>−</mo> <mn>198</mn> <mo>×</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>10</mn> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>1050</mn> <mo>×</mo> <mn>10</mn> <mo>−</mo> <mn>198</mn> <mo>×</mo> <mn>53</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}6&=&60-18\times 3&&\\&=&60-(198-60\times 3)\times 3&=&-198\times 3+60\times 10\\&=&-198\times 3+(1050-198\times 5)\times 10&=&1050\times 10-198\times 53\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f85a26f8ca0bf41bfdfc5364140668c3876b6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:70.223ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}6&=&60-18\times 3&&\\&=&60-(198-60\times 3)\times 3&=&-198\times 3+60\times 10\\&=&-198\times 3+(1050-198\times 5)\times 10&=&1050\times 10-198\times 53\end{matrix}}}"></span> <br /> گویا <i>x=10</i> اور <i>y=-53</i> </p><p><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 \gcd(1050,198)=10\times 1050-53\times 198}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>1050</mn> <mo>,</mo> <mn>198</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> <mo>×</mo> <mn>1050</mn> <mo>−</mo> <mn>53</mn> <mo>×</mo> <mn>198</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(1050,198)=10\times 1050-53\times 198}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebaa91d2a360fde7ee1963040c626ed2651b3b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.874ex; height:2.843ex;" alt="{\displaystyle \gcd(1050,198)=10\times 1050-53\times 198}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="ذواضعاف_اقل"><span id=".D8.B0.D9.88.D8.A7.D8.B6.D8.B9.D8.A7.D9.81_.D8.A7.D9.82.D9.84"></span>ذواضعاف اقل</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=10" title="تدوینِ قطعہ: ذواضعاف اقل"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%B0%D9%88%D8%A7%D8%B6%D8%B9%D8%A7%D9%81_%D8%A7%D9%82%D9%84" title="ذواضعاف اقل">ذواضعاف اقل</a> </p><p>دو اعداد <i>a</i> اور <i>b</i> کا ذواضعاف اقل ایسے عدد <i>m</i> کو کہا جاتا ہے، جبکہ <i>m</i> ایسا عدد ہو جو <i>a</i> اور <i>b</i> کے مثبت ضربیات میں سب سے چھوٹا ہو۔ انگریزی میں اسے least common multiple (lcm) کہتے ہیں۔ <br /> مثال: چلو <i>a=4</i>, <i>b=6</i>، پھر <br /> <br /> 4 کے مثبت ضربیات: 4, 8, 12, 16, 20, ..... <br /> 6 کے مثبت ضربیات: 6, 12, 18, 24, 30, ..... <br /> <br /> اس لیے <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 \ lcm(4,6)=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>l</mi> <mi>c</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ lcm(4,6)=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367bc7d0865535b864bf52f4ebf68737d3833b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.913ex; height:2.843ex;" alt="{\displaystyle \ lcm(4,6)=12}"></span> ۔ </p> <div class="mw-heading mw-heading3"><h3 id="نظریہ_2"><span id=".D9.86.D8.B8.D8.B1.DB.8C.DB.81_2"></span>نظریہ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=11" title="تدوینِ قطعہ: نظریہ"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>دو مثبت صحیح اعداد کا ضربی جوڑ ان اعداد کے عاد اعظم اور دواضعاف اقل کے ضربی جوڑ کے برابر ہوتا ہے: <br /> <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}\ a\times b=\gcd(a,b)\times lcm(a,b)\,\,,\\a>0,\,b>0\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> <mtext> </mtext> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>l</mi> <mi>c</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\ a\times b=\gcd(a,b)\times lcm(a,b)\,\,,\\a>0,\,b>0\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a2711bf40c3c1a1a35a68726347fae9d9c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.129ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}\ a\times b=\gcd(a,b)\times lcm(a,b)\,\,,\\a>0,\,b>0\end{matrix}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="مرکب_عدد"><span id=".D9.85.D8.B1.DA.A9.D8.A8_.D8.B9.D8.AF.D8.AF"></span>مرکب عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=12" title="تدوینِ قطعہ: مرکب عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D9%85%D8%B1%DA%A9%D8%A8_%D8%B9%D8%AF%D8%AF" title="مرکب عدد">مرکب عدد</a> </p><p>ایک مثبت عدد جس کے دو سے زیادہ ضربی اجزا ہوں، اسے مرکب عدد کہا جاتا ہے۔ انگریزی میں اسے کمپوزٹ (composite) کہتے ہیں۔ مثال: </p><p><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 \ 116=2\times 2\times 29}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>116</mn> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>29</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 116=2\times 2\times 29}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6014dea25a6d8de6f47b78d74277193802f69a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.497ex; height:2.176ex;" alt="{\displaystyle \ 116=2\times 2\times 29}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="مفرد_عدد"><span id=".D9.85.D9.81.D8.B1.D8.AF_.D8.B9.D8.AF.D8.AF"></span>مفرد عدد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=13" title="تدوینِ قطعہ: مفرد عدد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون</i>: <a href="/wiki/%D8%A7%D9%88%D9%84%DB%8C_%D8%B9%D8%AF%D8%AF" class="mw-redirect" title="اولی عدد">مفرد عدد</a> </p><p>ایک مثبت عدد کو مفرد کہا جاتا ہے اگر اس عدد کے صرف دو ضربی اجزا ہوں (ایک یہ خود اور دوسرا 1)۔ مثلاً 25 سے چھوٹے مفرد اعداد یہ ہیں: <br /> 2, 3, 5, 7, 11, 13, 17, 19, 23 <br /> انگریزی میں <a href="/wiki/%D9%85%D9%81%D8%B1%D8%AF_%D8%B9%D8%AF%D8%AF" title="مفرد عدد">مفرد عدد</a> کو پرائم (prime) کہا جاتا ہے۔ </p><p>عدد<i> 1</i> نہ مفرد ہے نہ مرکب۔ </p> <div class="mw-heading mw-heading3"><h3 id="حساب_کا_بنیادی_نظریہ"><span id=".D8.AD.D8.B3.D8.A7.D8.A8_.DA.A9.D8.A7_.D8.A8.D9.86.DB.8C.D8.A7.D8.AF.DB.8C_.D9.86.D8.B8.D8.B1.DB.8C.DB.81"></span>حساب کا بنیادی نظریہ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=14" title="تدوینِ قطعہ: حساب کا بنیادی نظریہ"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>فرض کرو <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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span>۔ اب <i>n</i> کو مفرد اعداد کے جز ضربی کے طور پر لکھا جا سکتا ہے۔ اور یہ جُزِ ضربی منفرد ہوں گے، صرف ترتیب مختلف ہو سکتی ہے۔ مثال: <br /> <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 \ 299376=2^{4}\times 3^{5}\times 7\times 11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>299376</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>×</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 299376=2^{4}\times 3^{5}\times 7\times 11}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70fea97eab71116fb059871fd8d37053efa5284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.096ex; height:2.676ex;" alt="{\displaystyle \ 299376=2^{4}\times 3^{5}\times 7\times 11}"></span> <br /> جہاں 2, 3, 7, 11, مفرد اعداد ہیں۔ ان مفرد اعداد کے علاوہ کوئی دوسرا مفرد اعداد کا مجموعہ نہیں، جو 299376 کے ضربی جز بن سکیں، صرف ترتیب مختلف ہو سکتی ہے، مثلاً <br /> <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 \ 299376=7\times 3\times 2\times 2\times 3\times 3\times 2\times 3\times 11\times 3\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>299376</mn> <mo>=</mo> <mn>7</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>11</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 299376=7\times 3\times 2\times 2\times 3\times 3\times 2\times 3\times 11\times 3\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4d55c1e5e12a05cf9addde89e92c1909347df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:53.007ex; height:2.176ex;" alt="{\displaystyle \ 299376=7\times 3\times 2\times 2\times 3\times 3\times 2\times 3\times 11\times 3\times 2}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="نظریہ_3"><span id=".D9.86.D8.B8.D8.B1.DB.8C.DB.81_3"></span>نظریہ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=15" title="تدوینِ قطعہ: نظریہ"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>مفرد اعداد کی تعداد لامحدود ہے۔ <br /> ثبوت: <br /> ثبوت نفی طریقہ سے دیتے ہیں۔ فرض کرو کہ مفرد اعداد کا مجموعہ محدود ہے۔ تو اس مجموعہ کو یوں لکھ لیتے ہیں: <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 \{p_{1},p_{2},p_{3},\cdots ,p_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p_{1},p_{2},p_{3},\cdots ,p_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c74a02542029d506fea2a68c2375073e20f6090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.5ex; height:2.843ex;" alt="{\displaystyle \{p_{1},p_{2},p_{3},\cdots ,p_{k}\}}"></span> اب اس عدد کو دیکھو: <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=p_{1}p_{2}p_{3}\cdots p_{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=p_{1}p_{2}p_{3}\cdots p_{k}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e42f1955e53820e75a2d18fd20e4f9d4269c7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.366ex; height:2.509ex;" alt="{\displaystyle Q=p_{1}p_{2}p_{3}\cdots p_{k}+1}"></span> اب یا تو<i>Q</i> مفرد ہے یا پھر اس کے مفرد جز ضربی موجود ہیں۔ اگر مفرد ہے تو مفروضے کی نفی ہو گئی۔ دوسری صورت میں دیکھو کہ اوپر دیے مفرد اعداد میں سے کوئی بھی <i>Q</i> کو تقسیم نہیں کرتا جو بنیادی نظریہ کے خلاف ہے۔ اس لیے یہ صورت بھی مفروضے کی نفی کرتی ہے۔ پس ہم یہ نتیجہ اخذ کرتے ہیں کہ یہ مفروضہ کہ "مفرد اعداد کی تعداد محدود ہے" ہی غلط تھا۔ </p> <div class="mw-heading mw-heading3"><h3 id="نظریہ_4"><span id=".D9.86.D8.B8.D8.B1.DB.8C.DB.81_4"></span>نظریہ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=16" title="تدوینِ قطعہ: نظریہ"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>اگر صحیح عدد <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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span> کے کوئی جز ضربی ایسے نہیں جو <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 {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}"></span> سے چھوٹے ہوں (<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 \leq {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq {\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3e52a44ef4a08ec479a3a024bb0c4e403c8422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.784ex; height:3.009ex;" alt="{\displaystyle \leq {\sqrt {n}}}"></span>)، تو عدد <i>n</i> مفرد ہے۔ </p> <div class="mw-heading mw-heading3"><h3 id="مفرد_عدد_کی_چھاننی"><span id=".D9.85.D9.81.D8.B1.D8.AF_.D8.B9.D8.AF.D8.AF_.DA.A9.DB.8C_.DA.86.DA.BE.D8.A7.D9.86.D9.86.DB.8C"></span>مفرد عدد کی چھاننی</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=17" title="تدوینِ قطعہ: مفرد عدد کی چھاننی"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>مفرد اعداد ڈھونڈنے کے لیے چھاننی کا طریقہ مفید ہے۔ فرض کرو کہ ہمیں 300 سے کم اعداد میں سے مفرد عدد تلاش کرنے ہیں، تو 300 تک کے اعداد لکھ لو <br /> <u>2</u> <u>3</u> <s>4</s> 5 <s>6</s> 7 <s>8</s> <s>9</s> <s>10</s> 11 <s>12</s> 13 <s>14</s> <s>15</s> <s>16</s> 17 ........ <br /> اب 2 سے شروع کرتے ہیں۔ اس کے نیچے لکیر لگا دو۔ اب 2 کے ضربیات کاٹ دو۔ اس کے بعد 3 کے نیچے لکیر لگاؤ۔ اب 3 کے ضربیات کاٹ دو۔ اس ظرح نہ کٹے اعداد کے نیچے لکیر لگا کر اس کے ضربیات کاٹنے (چھاننے) کا عمل جاری رکھو۔ جو بھی سب سے چھوٹا عدد جس کے نیچے لکیر نہیں لگی یا کٹا ہوا نہیں، تو وہ عدد مفرد ہے۔ چونکہ <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 \lfloor {\sqrt {300}}\rfloor =17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>300</mn> </msqrt> </mrow> <mo fence="false" stretchy="false">⌋</mo> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor {\sqrt {300}}\rfloor =17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/083026875833f0bf67ec402599580cf121813b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.911ex; height:3.009ex;" alt="{\displaystyle \lfloor {\sqrt {300}}\rfloor =17}"></span>، اس لیے ہمیں 17 تک کے اعداد کے نیچے لکیر لگانے کا عمل جاری رکھنا ہے۔ </p> <div class="mw-heading mw-heading3"><h3 id="مفرد_عدد_کی_پہچان"><span id=".D9.85.D9.81.D8.B1.D8.AF_.D8.B9.D8.AF.D8.AF_.DA.A9.DB.8C_.D9.BE.DB.81.DA.86.D8.A7.D9.86"></span>مفرد عدد کی پہچان</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=18" title="تدوینِ قطعہ: مفرد عدد کی پہچان"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>مفرد عدد کی یہ ایک کسوٹی ہے: اگر عدد <i>p</i> مفرد ہے تو لازم ہے کہ وہ اس امتحان میں پورا اترے <br /> <i>p-1</i> کو 2 کی طاقت علاحدہ کر کے لکھو <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 p-1=2^{\zeta }\times \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msup> <mo>×</mo> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-1=2^{\zeta }\times \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62522cd683f799d82d30350a3908d14a3d25f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.539ex; height:3.176ex;" alt="{\displaystyle p-1=2^{\zeta }\times \eta }"></span> <br /> تو <i>p</i> کے مفرد ہونے کے لیے لازم ہے کہ نیچے دی دو مساوات میں سے ایک کی تسکین ہو: <br /> <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 \beta ^{\eta }\equiv \pm 1\mod p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msup> <mo>≡</mo> <mo>±</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\eta }\equiv \pm 1\mod p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720a1069bbf589cbcf4295fc16b31b10d049c5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.896ex; height:2.676ex;" alt="{\displaystyle \beta ^{\eta }\equiv \pm 1\mod p}"></span> </p><p><b>یا</b> <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 \beta ^{2^{j}\eta }\equiv -1\mod p\,,\,j=1,2,\cdots ,\zeta -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mi>η</mi> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>p</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>ζ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2^{j}\eta }\equiv -1\mod p\,,\,j=1,2,\cdots ,\zeta -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35625ae07e3ac39c2a8286f37da74e6935dae7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.932ex; height:3.343ex;" alt="{\displaystyle \beta ^{2^{j}\eta }\equiv -1\mod p\,,\,j=1,2,\cdots ,\zeta -1}"></span> <br /> ہر نیچے دیے <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> کے لیے <br /> <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 \beta =2,3,\cdots ,p-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =2,3,\cdots ,p-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106969c6c36300f05c9aee70fe556c59e1131257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.14ex; height:2.509ex;" alt="{\displaystyle \beta =2,3,\cdots ,p-2}"></span> </p><p>مثال: عدد 511 مفرد نہیں کیونکہ 7 سے تقسیم ہوتا ہے۔ مگر <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 \beta =81}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>81</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =81}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4af2ef1bb73400e4ba1b82fa94a5056ba4bd715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.755ex; height:2.509ex;" alt="{\displaystyle \beta =81}"></span> کے لیے کسوٹی پر پورا اترتا ہے <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}p-1=510=2^{1}\times 255\\\beta =81\\\beta ^{\eta }=81^{255}\equiv 1\mod 511\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> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>510</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mn>255</mn> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> <mo>=</mo> <mn>81</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>255</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mspace width="0.667em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>511</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p-1=510=2^{1}\times 255\\\beta =81\\\beta ^{\eta }=81^{255}\equiv 1\mod 511\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938627fb343d84bea4ca6dda6c07d125b1e536f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.505ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}p-1=510=2^{1}\times 255\\\beta =81\\\beta ^{\eta }=81^{255}\equiv 1\mod 511\end{matrix}}}"></span> <br /> جس سے پتہ چلتا ہے کہ تمام <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> کے لیے تسلی کرنی چاہیے۔ </p><p>عملی طور پر یہ کسوٹی مفرد عدد ڈھونڈنے کے لیے استعمال ہوتی ہے۔ بہت بڑے اعداد کی تجزی کرنا ممکن نہیں ہوتا۔ کچھ عملیات میں یہ کرتے ہیں کہ کسی عدد کے بمطابق بہت سے رینڈم (بے ترتیب) <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> لے کر (مگر سارے نہیں) تجربہ کیا جاتا ہے، اگر کسوٹی پر کوئی عدد پورا اترے تو اسے مفرد تصور کر لیا جاتا ہے۔ </p> <div class="mw-heading mw-heading3"><h3 id="توزیع_مفرد_اعداد"><span id=".D8.AA.D9.88.D8.B2.DB.8C.D8.B9_.D9.85.D9.81.D8.B1.D8.AF_.D8.A7.D8.B9.D8.AF.D8.A7.D8.AF"></span>توزیع مفرد اعداد</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=19" title="تدوینِ قطعہ: توزیع مفرد اعداد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>اگر <i>x</i> سے کم مفرد اعداد کی تعداد کو <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 \pi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a62d9bf7b2ba03b30a95e7059362cc4b3865123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle \pi (x)}"></span> لکھا جائے تو <br /> <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 {\frac {\pi (x)}{x/\ln(x)}}\to 1\,\,as\,\,x\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>a</mi> <mi>s</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi (x)}{x/\ln(x)}}\to 1\,\,as\,\,x\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f80a67434cbb29aeed9e0b0597f3d5e87c0527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.706ex; height:6.509ex;" alt="{\displaystyle {\frac {\pi (x)}{x/\ln(x)}}\to 1\,\,as\,\,x\to \infty }"></span> </p> <pre>اسے توزیع مفرد اعداد (Distribution of Prime Number) کہتے ہیں۔ <a href="/wiki/1896%D8%A1" title="1896ء">1896ء</a> میں دو ریاضی دانوں، فرانسیسی ریاضی دان <a href="/w/index.php?title=%D8%AC%DB%8C%DA%A9_%DB%81%DB%8C%DA%88%D8%A7%D9%85%D8%B1%DA%88&action=edit&redlink=1" class="new" title="جیک ہیڈامرڈ (صفحہ موجود نہیں)">جیک ہیڈامرڈ</a> اور بیلجین ریاضی دان <a href="/w/index.php?title=%DA%86%D8%A7%D8%B1%D9%84%D8%B3_%DA%98%D8%A7%DA%BA_%DA%AF%D9%88%D8%B3%D8%AA%D8%A7%D9%88_%D9%86%DB%8C%DA%A9%D9%88%D9%84%D8%A7_%D8%A8%D8%A7%D8%B1%D9%88%D9%86_%DA%88%D9%88%D9%84%DB%81_%D9%88%D8%A7%D9%84%DB%81%DB%92_%D9%BE%D9%88%D8%B3%D8%A7&action=edit&redlink=1" class="new" title="چارلس ژاں گوستاو نیکولا بارون ڈولہ والہے پوسا (صفحہ موجود نہیں)">چارلس ژاں گوستاو نیکولا بارون ڈولہ والہے پوسا</a> نے الگ الگ ثابت کیا۔ دونوں نے اس مقصد کے لیے <a href="/wiki/%D9%85%D8%AE%D8%AA%D9%84%D8%B7_%D8%AA%D8%AD%D9%84%DB%8C%D9%84" title="مختلط تحلیل">مختلط تحلیل</a> (Complex Analysis) کا استعمال کیا۔ بیسوں صدی میں اس کو اور بہت سے ریاضی دانوں نے بغیر مختلط تحلیل کے ذریعے ثابت کیا۔ سب سے آسان حل امریکی ریاضی دان <a href="/w/index.php?title=%DA%88%D9%88%D9%86%D9%84%DA%88_%D9%86%DB%8C%D9%88%D9%85%DB%8C%D9%86&action=edit&redlink=1" class="new" title="ڈونلڈ نیومین (صفحہ موجود نہیں)">ڈونلڈ نیومین</a> نے دیا حالانکہ اس میں بھی مختلط تحلیل استعمال ہوئی ہے۔ <a href="/wiki/2005%D8%A1" title="2005ء">2005ء</a> میں اسے پہلی دفعہ <a href="/wiki/%DA%A9%D9%85%D9%BE%DB%8C%D9%88%D9%B9%D8%B1" title="کمپیوٹر">کمپیوٹر</a> (Computer) کی مدد سے ثابت کیا گیا۔ </pre> <p><i>تفصیلی مضمون</i>: <a href="/w/index.php?title=%D9%86%D8%B8%D8%B1%DB%8C%DB%81_%D9%85%D9%81%D8%B1%D8%AF_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF&action=edit&redlink=1" class="new" title="نظریہ مفرد اعداد (صفحہ موجود نہیں)">نظریہ مفرد اعداد</a> </p> <div class="mw-heading mw-heading2"><h2 id="مزید_دیکھیے"><span id=".D9.85.D8.B2.DB.8C.D8.AF_.D8.AF.DB.8C.DA.A9.DA.BE.DB.8C.DB.92"></span>مزید دیکھیے</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%B9%D8%AF%D8%AF&action=edit&section=20" title="تدوینِ قطعہ: مزید دیکھیے"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D8%B5%D8%AD%DB%8C%D8%AD_%D8%B9%D8%AF%D8%AF" title="صحیح عدد">صحیح عدد</a></li> <li><a href="/wiki/%D9%85%DA%A9%D9%85%D9%84_%D8%B9%D8%AF%D8%AF" class="mw-redirect" title="مکمل عدد">مکمل عدد</a></li> <li><a href="/wiki/%D9%82%D8%AF%D8%B1%D8%AA%DB%8C_%D8%B9%D8%AF%D8%AF" title="قدرتی عدد">قدرتی عدد</a></li> <li><a href="/wiki/%D8%AC%D9%81%D8%AA_%D8%B9%D8%AF%D8%AF" title="جفت عدد">جفت عدد</a></li> <li><a href="/wiki/%D8%B7%D8%A7%D9%82_%D8%B9%D8%AF%D8%AF" title="طاق عدد">طاق عدد</a></li> <li><a href="/wiki/%D8%B9%D8%A7%D8%AF_%D8%A7%D8%B9%D8%B8%D9%85" title="عاد اعظم">عاد اعظم</a></li> <li><a href="/wiki/%D8%B0%D9%88%D8%A7%D8%B6%D8%B9%D8%A7%D9%81_%D8%A7%D9%82%D9%84" title="ذواضعاف اقل">ذواضعاف اقل</a></li> <li><a href="/wiki/%D9%85%D8%B1%DA%A9%D8%A8_%D8%B9%D8%AF%D8%AF" title="مرکب عدد">مرکب عدد</a></li> <li><a href="/wiki/%D8%A7%D9%88%D9%84%DB%8C_%D8%B9%D8%AF%D8%AF" class="mw-redirect" title="اولی عدد">مفرد عدد</a></li> <li><a href="/w/index.php?title=%D9%86%D8%B8%D8%B1%DB%8C%DB%81_%D9%85%D9%81%D8%B1%D8%AF_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF&action=edit&redlink=1" class="new" title="نظریہ مفرد اعداد (صفحہ موجود نہیں)">نظریہ مفرد اعداد</a></li></ul> <div style="background-color: #f9f9f9; border: 1px dashed #aaaaaa; padding: .2em; text-align: center; margin: .5em 5em .5em 5em; font-size: large;"><small> </small><p><small><code>E=mc<sup>2</sup></code>     اردو ویکیپیڈیا پر ریاضی مساوات کو بائیں سے دائیں <small>LTR</small> پڑھیٔے     <a href="https://en.wikipedia.org/wiki/Table_of_mathematical_symbols" class="extiw" title="en:Table of mathematical symbols">ریاضی علامات </a></small> 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