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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">Início</div> </a> </li> <li id="toc-Propriedades_importantes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriedades_importantes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Propriedades importantes</span> </div> </a> <button aria-controls="toc-Propriedades_importantes-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>Alternar a subsecção Propriedades importantes</span> </button> <ul id="toc-Propriedades_importantes-sublist" class="vector-toc-list"> <li id="toc-Nos_números_inteiros" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nos_números_inteiros"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Nos números inteiros</span> </div> </a> <ul id="toc-Nos_números_inteiros-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nos_números_racionais,_reais_e_em_outros_corpos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nos_números_racionais,_reais_e_em_outros_corpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Nos números racionais, reais e em outros corpos</span> </div> </a> <ul id="toc-Nos_números_racionais,_reais_e_em_outros_corpos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divisão_de_polinômios" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisão_de_polinômios"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Divisão de polinômios</span> </div> </a> <ul id="toc-Divisão_de_polinômios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Em_estruturas_mais_gerais" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Em_estruturas_mais_gerais"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Em estruturas mais gerais</span> </div> </a> <ul id="toc-Em_estruturas_mais_gerais-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representação" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representação"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Representação</span> </div> </a> <button aria-controls="toc-Representação-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>Alternar a subsecção Representação</span> </button> <ul id="toc-Representação-sublist" class="vector-toc-list"> <li id="toc-Divisão_entre_números_consecutivos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisão_entre_números_consecutivos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Divisão entre números consecutivos</span> </div> </a> <ul id="toc-Divisão_entre_números_consecutivos-sublist" class="vector-toc-list"> <li id="toc-Caso_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Caso_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Caso 1</span> </div> </a> <ul id="toc-Caso_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caso_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Caso_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Caso 2</span> </div> </a> <ul id="toc-Caso_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas_e_referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas_e_referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notas e referências</span> </div> </a> <ul id="toc-Notas_e_referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <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="Alternar o índice" > <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">Alternar o índice</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">Divisão</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="Ir para um artigo noutra língua. Disponível em 119 línguas" > <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-119" 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">119 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Deling" title="Deling — africanês" lang="af" hreflang="af" data-title="Deling" data-language-autonym="Afrikaans" data-language-local-name="africanês" 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/Division_(Mathematik)" title="Division (Mathematik) — alemão suíço" lang="gsw" hreflang="gsw" data-title="Division (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Divisi%C3%B3n" title="División — aragonês" lang="an" hreflang="an" data-title="División" data-language-autonym="Aragonés" data-language-local-name="aragonês" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%B3%D9%85%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="قسمة (رياضيات) — árabe" lang="ar" hreflang="ar" data-title="قسمة (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%82%D8%B3%D9%8A%D9%85" title="قسيم — Moroccan Arabic" lang="ary" hreflang="ary" data-title="قسيم" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%82%D8%B3%D9%85%D9%87" title="قسمه — Egyptian Arabic" lang="arz" hreflang="arz" data-title="قسمه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B9%E0%A7%B0%E0%A6%A3_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="হৰণ (গণিত) — assamês" lang="as" hreflang="as" data-title="হৰণ (গণিত)" data-language-autonym="অসমীয়া" data-language-local-name="assamês" 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/Divisi%C3%B3n_(matem%C3%A1tiques)" title="División (matemátiques) — asturiano" lang="ast" hreflang="ast" data-title="División (matemátiques)" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jaljaya%C3%B1a" title="Jaljayaña — aimará" lang="ay" hreflang="ay" data-title="Jaljayaña" data-language-autonym="Aymar aru" data-language-local-name="aimará" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/B%C3%B6lm%C9%99_(riyaziyyat)" title="Bölmə (riyaziyyat) — azerbaijano" lang="az" hreflang="az" data-title="Bölmə (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" 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%A8%D8%A4%D9%84%D9%85%D9%87" title="بؤلمه — South Azerbaijani" lang="azb" hreflang="azb" data-title="بؤلمه" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%91%D2%AF%D0%BB%D0%B5%D2%AF" title="Бүлеү — bashkir" lang="ba" hreflang="ba" data-title="Бүлеү" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Pagbanga_(matematika)" title="Pagbanga (matematika) — Central Bikol" lang="bcl" hreflang="bcl" data-title="Pagbanga (matematika)" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B7%D1%8F%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5" title="Дзяленне — bielorrusso" lang="be" hreflang="be" data-title="Дзяленне" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" 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%94%D0%B7%D1%8F%D0%BB%D0%B5%D0%BD%D1%8C%D0%BD%D0%B5" title="Дзяленьне — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дзяленьне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Деление — búlgaro" lang="bg" hreflang="bg" data-title="Деление" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A6%BE%E0%A6%97" title="ভাগ — bengalês" lang="bn" hreflang="bn" data-title="ভাগ" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%96%E0%BD%82%E0%BD%BC%E0%BC%8B%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8D" title="བགོ་རྩིས། — tibetano" lang="bo" hreflang="bo" data-title="བགོ་རྩིས།" data-language-autonym="བོད་ཡིག" data-language-local-name="tibetano" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Rannadur" title="Rannadur — bretão" lang="br" hreflang="br" data-title="Rannadur" data-language-autonym="Brezhoneg" data-language-local-name="bretão" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Dijeljenje_(matematika)" title="Dijeljenje (matematika) — bósnio" lang="bs" hreflang="bs" data-title="Dijeljenje (matematika)" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A5%D1%83%D0%B1%D0%B0%D0%B0%D0%BB%D1%82%D0%B0" title="Хубаалта — Russia Buriat" lang="bxr" hreflang="bxr" data-title="Хубаалта" 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 mw-list-item"><a href="https://ca.wikipedia.org/wiki/Divisi%C3%B3" title="Divisió — catalão" lang="ca" hreflang="ca" data-title="Divisió" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/D%E1%B9%B3%CC%80-hu%C3%A1k" title="Dṳ̀-huák — Mindong" lang="cdo" hreflang="cdo" data-title="Dṳ̀-huák" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A8%DB%95%D8%B4%DA%A9%D8%B1%D8%AF%D9%86" title="دابەشکردن — curdo central" lang="ckb" hreflang="ckb" data-title="دابەشکردن" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/D%C4%9Blen%C3%AD" title="Dělení — checo" lang="cs" hreflang="cs" data-title="Dělení" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B9%D0%BB%D0%B0%D1%81%D1%81%D0%B8" title="Пайласси — chuvash" lang="cv" hreflang="cv" data-title="Пайласси" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhannu_(mathemateg)" title="Rhannu (mathemateg) — galês" lang="cy" hreflang="cy" data-title="Rhannu (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="galês" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Division_(matematik)" title="Division (matematik) — dinamarquês" lang="da" hreflang="da" data-title="Division (matematik)" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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/Division_(Mathematik)" title="Division (Mathematik) — alemão" lang="de" hreflang="de" data-title="Division (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CE%AF%CF%81%CE%B5%CF%83%CE%B7" title="Διαίρεση — grego" lang="el" hreflang="el" data-title="Διαίρεση" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Division_(mathematics)" title="Division (mathematics) — inglês" lang="en" hreflang="en" data-title="Division (mathematics)" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Divido" title="Divido — esperanto" lang="eo" hreflang="eo" data-title="Divido" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Divisi%C3%B3n_(matem%C3%A1tica)" title="División (matemática) — espanhol" lang="es" hreflang="es" data-title="División (matemática)" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Jagamine" title="Jagamine — estónio" lang="et" hreflang="et" data-title="Jagamine" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zatiketa_(matematika)" title="Zatiketa (matematika) — basco" lang="eu" hreflang="eu" data-title="Zatiketa (matematika)" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="تقسیم — persa" lang="fa" hreflang="fa" data-title="تقسیم" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Jakolasku" title="Jakolasku — finlandês" lang="fi" hreflang="fi" data-title="Jakolasku" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tabana" title="Tabana — fijiano" lang="fj" hreflang="fj" data-title="Tabana" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijiano" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Brot" title="Brot — feroês" lang="fo" hreflang="fo" data-title="Brot" data-language-autonym="Føroyskt" data-language-local-name="feroês" 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/Division" title="Division — francês" lang="fr" hreflang="fr" data-title="Division" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Dialen_(Matematiik)" title="Dialen (Matematiik) — frísio setentrional" lang="frr" hreflang="frr" data-title="Dialen (Matematiik)" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%99%A4%E6%B3%95" title="除法 — gan" lang="gan" hreflang="gan" data-title="除法" 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-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Divizyon" title="Divizyon — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Divizyon" 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-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Roinn_(matamataig)" title="Roinn (matamataig) — gaélico escocês" lang="gd" hreflang="gd" data-title="Roinn (matamataig)" data-language-autonym="Gàidhlig" data-language-local-name="gaélico escocês" 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/Divisi%C3%B3n_(matem%C3%A1ticas)" title="División (matemáticas) — galego" lang="gl" hreflang="gl" data-title="División (matemáticas)" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Chh%C3%B9-fap" title="Chhù-fap — hacá" lang="hak" hreflang="hak" data-title="Chhù-fap" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="hacá" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%99%D7%9C%D7%95%D7%A7" title="חילוק — hebraico" lang="he" hreflang="he" data-title="חילוק" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AD%E0%A4%BE%E0%A4%9C%E0%A4%A8_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="विभाजन (गणित) — hindi" lang="hi" hreflang="hi" data-title="विभाजन (गणित)" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Baatna" title="Baatna — Fiji Hindi" lang="hif" hreflang="hif" data-title="Baatna" 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-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dijeljenje" title="Dijeljenje — croata" lang="hr" hreflang="hr" data-title="Dijeljenje" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Oszt%C3%A1s" title="Osztás — húngaro" lang="hu" hreflang="hu" data-title="Osztás" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%AA%D5%A1%D5%B6%D5%B8%D6%82%D5%B4_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Բաժանում (մաթեմատիկա) — arménio" lang="hy" hreflang="hy" data-title="Բաժանում (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Division_(mathematica)" title="Division (mathematica) — interlíngua" lang="ia" hreflang="ia" data-title="Division (mathematica)" data-language-autonym="Interlingua" data-language-local-name="interlíngua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Pemagi_(matematik)" title="Pemagi (matematik) — iban" lang="iba" hreflang="iba" data-title="Pemagi (matematik)" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pembagian" title="Pembagian — indonésio" lang="id" hreflang="id" data-title="Pembagian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Divido_(matematiko)" title="Divido (matematiko) — ido" lang="io" hreflang="io" data-title="Divido (matematiko)" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Deiling" title="Deiling — islandês" lang="is" hreflang="is" data-title="Deiling" data-language-autonym="Íslenska" data-language-local-name="islandês" 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/Divisione_(matematica)" title="Divisione (matematica) — italiano" lang="it" hreflang="it" data-title="Divisione (matematica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%99%A4%E6%B3%95" title="除法 — japonês" lang="ja" hreflang="ja" data-title="除法" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Divijan_(matimatix)" title="Divijan (matimatix) — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Divijan (matimatix)" 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-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Paran" title="Paran — javanês" lang="jv" hreflang="jv" data-title="Paran" data-language-autonym="Jawa" data-language-local-name="javanês" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D1%83" title="Бөлу — cazaque" lang="kk" hreflang="kk" data-title="Бөлу" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AD%E0%B2%BE%E0%B2%97%E0%B2%BE%E0%B2%95%E0%B2%BE%E0%B2%B0" title="ಭಾಗಾಕಾರ — canarim" lang="kn" hreflang="kn" data-title="ಭಾಗಾಕಾರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarim" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%98%EB%88%97%EC%85%88" title="나눗셈 — coreano" lang="ko" hreflang="ko" data-title="나눗셈" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Parkirin" title="Parkirin — curdo" lang="ku" hreflang="ku" data-title="Parkirin" data-language-autonym="Kurdî" data-language-local-name="curdo" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D2%AF%D2%AF" title="Бөлүү — quirguiz" lang="ky" hreflang="ky" data-title="Бөлүү" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Divisio_(mathematica)" title="Divisio (mathematica) — latim" lang="la" hreflang="la" data-title="Divisio (mathematica)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Divide_(matematica)" title="Divide (matematica) — Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Divide (matematica)" 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-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dalyba" title="Dalyba — lituano" lang="lt" hreflang="lt" data-title="Dalyba" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dal%C4%AB%C5%A1ana" title="Dalīšana — letão" lang="lv" hreflang="lv" data-title="Dalīšana" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D0%B5%D1%9A%D0%B5" title="Делење — macedónio" lang="mk" hreflang="mk" data-title="Делење" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B9%E0%B4%B0%E0%B4%A3%E0%B4%82" title="ഹരണം — malaiala" lang="ml" hreflang="ml" data-title="ഹരണം" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembahagian" title="Pembahagian — malaio" lang="ms" hreflang="ms" data-title="Pembahagian" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Divi%C5%BCjoni_(matematika)" title="Diviżjoni (matematika) — maltês" lang="mt" hreflang="mt" data-title="Diviżjoni (matematika)" data-language-autonym="Malti" data-language-local-name="maltês" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Debison_(matem%C3%A1tica)" title="Debison (matemática) — mirandês" lang="mwl" hreflang="mwl" data-title="Debison (matemática)" data-language-autonym="Mirandés" data-language-local-name="mirandês" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%97_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="भाग (गणित) — nepalês" lang="ne" hreflang="ne" data-title="भाग (गणित)" data-language-autonym="नेपाली" data-language-local-name="nepalês" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Delen" title="Delen — neerlandês" lang="nl" hreflang="nl" data-title="Delen" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Divisjon_i_matematikk" title="Divisjon i matematikk — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Divisjon i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Divisjon_(matematikk)" title="Divisjon (matematikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Divisjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Divisione" title="Divisione — Novial" lang="nov" hreflang="nov" data-title="Divisione" 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/Division" title="Division — occitano" lang="oc" hreflang="oc" data-title="Division" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%B9%E0%AC%B0%E0%AC%A3_(%E0%AC%97%E0%AC%A3%E0%AC%BF%E0%AC%A4)" title="ହରଣ (ଗଣିତ) — oriá" lang="or" hreflang="or" data-title="ହରଣ (ଗଣିତ)" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="oriá" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%95%E0%A8%B8%E0%A9%80%E0%A8%AE" title="ਤਕਸੀਮ — panjabi" lang="pa" hreflang="pa" data-title="ਤਕਸੀਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dzielenie" title="Dzielenie — polaco" lang="pl" hreflang="pl" data-title="Dzielenie" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AA%D9%82%D8%B3%D9%8A%D9%85" title="تقسيم — pastó" lang="ps" hreflang="ps" data-title="تقسيم" data-language-autonym="پښتو" data-language-local-name="pastó" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Rakiy" title="Rakiy — quíchua" lang="qu" hreflang="qu" data-title="Rakiy" data-language-autonym="Runa Simi" data-language-local-name="quíchua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/%C3%8Emp%C4%83r%C8%9Bire_(matematic%C4%83)" title="Împărțire (matematică) — romeno" lang="ro" hreflang="ro" data-title="Împărțire (matematică)" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Деление (математика) — russo" lang="ru" hreflang="ru" data-title="Деление (математика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%A6%E1%B1%9F%E1%B1%B4%E1%B1%9F%E1%B1%AC" title="ᱦᱟᱴᱟᱬ — santali" lang="sat" hreflang="sat" data-title="ᱦᱟᱴᱟᱬ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dijeljenje" title="Dijeljenje — servo-croata" lang="sh" hreflang="sh" data-title="Dijeljenje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Division_(mathematics)" title="Division (mathematics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Division (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Delenie_(matematika)" title="Delenie (matematika) — eslovaco" lang="sk" hreflang="sk" data-title="Delenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="تقسیم — Saraiki" lang="skr" hreflang="skr" data-title="تقسیم" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target"><span>سرائیکی</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Deljenje" title="Deljenje — esloveno" lang="sl" hreflang="sl" data-title="Deljenje" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kugovanisa" title="Kugovanisa — shona" lang="sn" hreflang="sn" data-title="Kugovanisa" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/U_qeybin" title="U qeybin — somali" lang="so" hreflang="so" data-title="U qeybin" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Pjes%C3%ABtimi" title="Pjesëtimi — albanês" lang="sq" hreflang="sq" data-title="Pjesëtimi" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D1%99%D0%B5%D1%9A%D0%B5" title="Дељење — sérvio" lang="sr" hreflang="sr" data-title="Дељење" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Division_(matematik)" title="Division (matematik) — sueco" lang="sv" hreflang="sv" data-title="Division (matematik)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ugawanyaji" title="Ugawanyaji — suaíli" lang="sw" hreflang="sw" data-title="Ugawanyaji" data-language-autonym="Kiswahili" data-language-local-name="suaíli" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%95%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வகுத்தல் (கணிதம்) — tâmil" lang="ta" hreflang="ta" data-title="வகுத்தல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AD%E0%B0%BE%E0%B0%97%E0%B0%B9%E0%B0%BE%E0%B0%B0%E0%B0%82" title="భాగహారం — telugu" lang="te" hreflang="te" data-title="భాగహారం" data-language-autonym="తెలుగు" data-language-local-name="telugu" 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%81%E0%B8%B2%E0%B8%A3%E0%B8%AB%E0%B8%B2%E0%B8%A3" title="การหาร — tailandês" lang="th" hreflang="th" data-title="การหาร" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Paghahati" title="Paghahati — tagalo" lang="tl" hreflang="tl" data-title="Paghahati" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/B%C3%B6lme" title="Bölme — turco" lang="tr" hreflang="tr" data-title="Bölme" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Ділення — ucraniano" lang="uk" hreflang="uk" data-title="Ділення" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%82%D8%B3%DB%8C%D9%85_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="تقسیم (ریاضی) — urdu" lang="ur" hreflang="ur" data-title="تقسیم (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bo%CA%BBlish" title="Boʻlish — usbeque" lang="uz" hreflang="uz" data-title="Boʻlish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Divizion" title="Divizion — véneto" lang="vec" hreflang="vec" data-title="Divizion" data-language-autonym="Vèneto" data-language-local-name="véneto" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_chia" title="Phép chia — vietnamita" lang="vi" hreflang="vi" data-title="Phép chia" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pagtunga-tunga" title="Pagtunga-tunga — waray" lang="war" hreflang="war" data-title="Pagtunga-tunga" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%99%A4%E6%B3%95" title="除法 — wu" lang="wuu" hreflang="wuu" data-title="除法" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A5%D1%83%D0%B2%D0%B0%D0%BB%D2%BB%D0%B0%D0%BD" title="Хувалһан — kalmyk" lang="xal" hreflang="xal" data-title="Хувалһан" data-language-autonym="Хальмг" data-language-local-name="kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Ukwaba/ukwahlula-hlula" title="Ukwaba/ukwahlula-hlula — xosa" lang="xh" hreflang="xh" data-title="Ukwaba/ukwahlula-hlula" data-language-autonym="IsiXhosa" data-language-local-name="xosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A6%D7%A2%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A0%D7%92" title="צעטיילונג — iídiche" lang="yi" hreflang="yi" data-title="צעטיילונג" data-language-autonym="ייִדיש" data-language-local-name="iídiche" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Cawzfap" title="Cawzfap — zhuang" lang="za" hreflang="za" data-title="Cawzfap" data-language-autonym="Vahcuengh" data-language-local-name="zhuang" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%99%A4%E6%B3%95" title="除法 — chinês" lang="zh" hreflang="zh" data-title="除法" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/T%C3%BB-hoat" title="Tû-hoat — min nan" lang="nan" hreflang="nan" data-title="Tû-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%99%A4" title="除 — cantonês" lang="yue" hreflang="yue" data-title="除" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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Para o álbum do cantor Ed Sheeran, veja <a href="/wiki/%C3%B7_(%C3%A1lbum)" title="÷ (álbum)">÷ (álbum)</a>.</div> <table class="vertical-navbox nowraplinks hlist" cellspacing="5" cellpadding="0" style="float:right;clear:right;width:20.8em;margin:0 0 1.0em 1.0em;background-color: var(--background-color-neutral-subtle,#f9f9f9); color:inherit; border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;min-width:20.8em;width: 22em"><tbody><tr><td class="" style="padding-top:0.4em;line-height:1.2em;">Parte da série sobre</td> </tr><tr><th class="" style="padding:0.2em 0.4em 0.2em;padding-top:0;font-size:145%;line-height:1.2em;"><a href="/wiki/Matem%C3%A1tica" title="Matemática">Matemática</a></th> </tr><tr><td class="{{{classe-imagem}}}" style="padding:0.2em 0 0.4em;"><span typeof="mw:File"><a href="/wiki/Ficheiro:8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg/170px-8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg.png" decoding="async" width="170" height="41" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg/255px-8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg/340px-8%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA%D8%A7%D8%AA.svg.png 2x" data-file-width="183" data-file-height="44" /></a></span></td> </tr><tr><td class="" style="padding:0.3em 0.4em 0.3em;font-weight:bold; padding-bottom:0.35em;"> <ul><li><a href="/wiki/Hist%C3%B3ria_da_matem%C3%A1tica" title="História da matemática">História</a></li></ul></td> </tr><tr> <td class="" style="padding:0 0.1em 0.4em; ;"> <table class="vertical-navbox-subgroup" style="width:100%;margin:0px;border-spacing:0px"><tbody><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;border-top:1px solid #aaa;background:#ddddff;text-align:center;"><a href="/wiki/%C3%81reas_da_matem%C3%A1tica" title="Áreas da matemática">Áreas</a></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Teoria_dos_n%C3%BAmeros" title="Teoria dos números">Teoria dos números</a></li> <li><a href="/wiki/Geometria" title="Geometria">Geometria</a></li> <li><a href="/wiki/%C3%81lgebra" title="Álgebra">Álgebra</a></li> <li><a href="/wiki/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal">Cálculo</a> e <a href="/wiki/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática">Análise</a></li> <li><a href="/wiki/Matem%C3%A1tica_discreta" title="Matemática discreta">Matemática discreta</a></li> <li><a href="/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática">Lógica matemática</a> e <a href="/wiki/Teoria_dos_conjuntos" title="Teoria dos conjuntos">Teoria dos conjuntos</a></li> <li><a href="/wiki/Probabilidade" title="Probabilidade">Probabilidade</a></li> <li><a href="/wiki/Estat%C3%ADstica" title="Estatística">Estatística</a> e <a href="/wiki/Teoria_da_decis%C3%A3o" title="Teoria da decisão">Teoria da decisão</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;border-top:1px solid #aaa;background:#ddddff;text-align:center;">Relação com as ciências</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/F%C3%ADsica_matem%C3%A1tica" title="Física matemática">Física</a></li> <li><a href="/wiki/Qu%C3%ADmica_matem%C3%A1tica" title="Química matemática">Química</a></li> <li><a href="/wiki/Matem%C3%A1tica_computacional" title="Matemática computacional">Computação</a></li> <li><a href="/wiki/Biologia_matem%C3%A1tica_e_te%C3%B3rica" title="Biologia matemática e teórica">Biologia</a></li> <li><a href="/wiki/Lingu%C3%ADstica_computacional" title="Linguística computacional">Linguística</a></li> <li><a href="/wiki/Economia_matem%C3%A1tica" title="Economia matemática">Economia</a></li> <li><a href="/wiki/Filosofia_da_matem%C3%A1tica" title="Filosofia da matemática">Filosofia</a></li> <li><a href="/wiki/Educa%C3%A7%C3%A3o_matem%C3%A1tica" title="Educação matemática">Educação</a></li></ul></div></div></td> </tr></tbody></table></td> </tr><tr> <th class="" style="padding:0.1em; {{{estilo-rótulo}}};"><span typeof="mw:File"><a href="/wiki/Ficheiro:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/20px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/30px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Matem%C3%A1tica" title="Portal:Matemática">Portal da Matemática</a></th> </tr><tr><td style="text-align:right;font-size:115%;"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Matem%C3%A1tica" title="Predefinição:Matemática"><abbr title="Ver esta predefinição">v</abbr></a></li><li class="nv-discutir"><a href="/wiki/Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Matem%C3%A1tica" title="Predefinição Discussão:Matemática"><abbr title="Discutir esta predefinição">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:Matem%C3%A1tica&action=edit"><abbr title="Editar esta predefinição">e</abbr></a></li></ul></div></td> </tr> </tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Divide20by4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Divide20by4.svg/220px-Divide20by4.svg.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Divide20by4.svg/330px-Divide20by4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Divide20by4.svg/440px-Divide20by4.svg.png 2x" data-file-width="500" data-file-height="600" /></a><figcaption>Ilustração de 20 maçãs dividas igualmente em 4 grupos, totalizando 5 maças em cada grupo.</figcaption></figure> <p><b>Divisão</b> é a <a href="/wiki/Matem%C3%A1tica" title="Matemática">operação matemática</a> inversa da <a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">multiplicação</a>. O ato de dividir por algum elemento de um <a href="/wiki/Conjunto" title="Conjunto">conjunto</a> só faz sentido quando a multiplicação por aquele elemento for uma <a href="/wiki/Fun%C3%A7%C3%A3o_bijetora" class="mw-redirect" title="Função bijetora">função bijetora</a>. </p><p>No <a href="/wiki/Anel_(matem%C3%A1tica)" title="Anel (matemática)">anel</a> dos <a href="/wiki/N%C3%BAmeros_inteiros" class="mw-redirect" title="Números inteiros">números inteiros</a> a hipótese da bijetividade não é satisfeita para o zero, assim, não se define <a href="/wiki/Divis%C3%A3o_por_zero" title="Divisão por zero">divisão por zero</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Propriedades_importantes">Propriedades importantes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=1" title="Editar secção: Propriedades importantes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=1" title="Editar código-fonte da secção: Propriedades importantes"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As propriedades da divisão são herdadas, via inversão, da multiplicação. Não existe, entretanto, a propriedade de fechamento no conjunto dos números reais, uma vez que a <a href="/wiki/Divis%C3%A3o_por_zero" title="Divisão por zero">divisão por zero</a> não produz como resultado um número real. </p> <div class="mw-heading mw-heading3"><h3 id="Nos_números_inteiros"><span id="Nos_n.C3.BAmeros_inteiros"></span>Nos números inteiros</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=2" title="Editar secção: Nos números inteiros" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=2" title="Editar código-fonte da secção: Nos números inteiros"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os números inteiros não formam um <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a>, portanto a divisão (como foi definido) só faz sentido quando o número que vai ser dividido (<i>dividendo</i><sup id="cite_ref-Vianna-nomenc_1-0" class="reference"><a href="#cite_note-Vianna-nomenc-1"><span>[</span>1<span>]</span></a></sup>) é um múltiplo inteiro do número pelo qual se vai dividir (<i>divisor</i><sup id="cite_ref-Vianna-nomenc_1-1" class="reference"><a href="#cite_note-Vianna-nomenc-1"><span>[</span>1<span>]</span></a></sup>). Para tratar dos casos em que o <i>dividendo</i> não é um múltiplo do <i>divisor</i> é necessário definir <i>quociente</i> e <i>resto</i>. </p><p>Se a e b são dois números inteiros positivos (com <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\geq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2db6581643a6d21f1efd54d284cfcc99b850531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle b\geq a}"></span>), o <b>quociente</b><sup id="cite_ref-Vianna-nomenc_1-2" class="reference"><a href="#cite_note-Vianna-nomenc-1"><span>[</span>1<span>]</span></a></sup> da divisão de a por b é o <i>maior número inteiro q tal que </i><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 bq\leq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>q</mi> <mo>≤</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bq\leq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf048a963e04c952145248303103dd464d59f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.395ex; height:2.509ex;" alt="{\displaystyle bq\leq a}"></span>. O <b>resto</b><sup id="cite_ref-Vianna-nomenc2_2-0" class="reference"><a href="#cite_note-Vianna-nomenc2-2"><span>[</span>2<span>]</span></a></sup> da divisão de a por b com quociente q é o <i>número inteiro r tal que </i><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=a-bq.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-bq.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027057e7f6f40c712988baab39b6434bcc22d860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.931ex; height:2.509ex;" alt="{\displaystyle r=a-bq.}"></span> </p><p>A noção de resto no anel dos números inteiros está intimamente conectada com a noção de <a href="/wiki/Aritm%C3%A9tica_modular" title="Aritmética modular">congruência</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nos_números_racionais,_reais_e_em_outros_corpos"><span id="Nos_n.C3.BAmeros_racionais.2C_reais_e_em_outros_corpos"></span>Nos <a href="/wiki/N%C3%BAmeros_racionais" class="mw-redirect" title="Números racionais">números racionais</a>, <a href="/wiki/N%C3%BAmeros_reais" class="mw-redirect" title="Números reais">reais</a> e em outros <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpos</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=3" title="Editar secção: Nos números racionais, reais e em outros corpos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=3" title="Editar código-fonte da secção: Nos números racionais, reais e em outros corpos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Por se tratarem de corpos, a divisão nesse caso fica reduzida a multiplicação pelo inverso. </p><p>Por um exemplo, para dividirmos um número racional <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_{1}={\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}={\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77e62758fa637c91d0154db8a893ac91dc2f04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.256ex; height:4.843ex;" alt="{\displaystyle q_{1}={\frac {a}{b}}}"></span> por <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_{2}={\frac {c}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}={\frac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94344e404d37f9e0bc936ba615325fd255997069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.242ex; height:4.843ex;" alt="{\displaystyle q_{2}={\frac {c}{d}}}"></span> (com as hipóteses de que a,b,c e d sejam inteiros e que b,c e d sejam diferentes de zero) devemos prosseguir da seguinte forma </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 {\frac {q_{1}}{q_{2}}}=q_{1}\ q_{2}^{-1}={\frac {a}{b}}\ {\frac {d}{c}}={\frac {ad}{bc}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</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> </mfrac> </mrow> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{1}}{q_{2}}}=q_{1}\ q_{2}^{-1}={\frac {a}{b}}\ {\frac {d}{c}}={\frac {ad}{bc}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b161d204e0798f733a33689f78723f6481817362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.287ex; height:5.843ex;" alt="{\displaystyle {\frac {q_{1}}{q_{2}}}=q_{1}\ q_{2}^{-1}={\frac {a}{b}}\ {\frac {d}{c}}={\frac {ad}{bc}}}"></span> </p><p>Em <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} _{13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{13}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5287b3dae710d1bd805f8bd3b924290b46edf06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.427ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{13}}"></span> (grupo multiplicativo dos inteiros módulo 13), que também é um corpo, a divisão de 7 por 5 se daria da seguinte forma: </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 {\frac {7}{5}}=7.5^{-1}=7.8=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mn>7.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>7.8</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{5}}=7.5^{-1}=7.8=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909320a589726d9ed4552921e5da2e45eb48896f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.733ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{5}}=7.5^{-1}=7.8=4}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Divisão_de_polinômios"><span id="Divis.C3.A3o_de_polin.C3.B4mios"></span>Divisão de polinômios</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=4" title="Editar secção: Divisão de polinômios" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=4" title="Editar código-fonte da secção: Divisão de polinômios"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pode-se definir a operação de divisão para <a href="/wiki/Polin%C3%B4mio" class="mw-redirect" title="Polinômio">polinômios</a>. Então, como no caso dos inteiros, tem-se um <i>resto</i>.<sup id="cite_ref-Serrasq-div-pol_3-0" class="reference"><a href="#cite_note-Serrasq-div-pol-3"><span>[</span>3<span>]</span></a></sup> Veja <a href="/wiki/Divis%C3%A3o_polinomial" title="Divisão polinomial">divisão polinomial</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Em_estruturas_mais_gerais">Em estruturas mais gerais</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=5" title="Editar secção: Em estruturas mais gerais" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=5" title="Editar código-fonte da secção: Em estruturas mais gerais"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A divisão é possível em estruturas que não são dotadas dos axiomas de corpo. Em <a href="/wiki/Analogia" title="Analogia">analogia</a> ao caso dos números inteiros, tenta-se encontrar um quociente e um resto. Isso nem sempre pode ser feito com o auxílio da <a href="/wiki/Rela%C3%A7%C3%A3o_de_ordem" title="Relação de ordem">relação de ordem</a>, pois a mesma nem sempre está presente. Quando pode-se definir uma função conveniente, trabalhamos com <a href="/w/index.php?title=Dom%C3%ADnio_Euclidiano&action=edit&redlink=1" class="new" title="Domínio Euclidiano (página não existe)">domínios euclidianos</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Representação"><span id="Representa.C3.A7.C3.A3o"></span>Representação</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=6" title="Editar secção: Representação" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=6" title="Editar código-fonte da secção: Representação"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sejam <b>a</b> e <b>b</b> elementos do conjunto dos números inteiros, e <b>b</b> diferente de zero. Podemos representar uma divisão da seguinte forma: </p> <ul><li>Como uma <a href="/wiki/Fra%C3%A7%C3%A3o" title="Fração">fração</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101faa653872fb89cf12cbea8c0a99b1c507943a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}},}"></span> (utilizando uma barra horizontal entre os dois números);</li> <li>Através de uma <a href="/wiki/Barra_(caractere)" title="Barra (caractere)">barra</a> inclinada: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{a}\!{/}\!{}_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mrow> <mspace width="negativethinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{a}\!{/}\!{}_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94984169876110520db56d48ed028f2cc66ef36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.428ex; height:2.843ex;" alt="{\displaystyle {}^{a}\!{/}\!{}_{b}}"></span>. (É utilizado para fazer operações em computadores);</li> <li>Com a simbologia usual da divisão, utilizando dois pontos e uma barra horizontal entre eles: <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\div b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>÷</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\div b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021958e4ad3e896d02f70c725544ad3fac40727c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.068ex; height:2.176ex;" alt="{\displaystyle a\div b}"></span>;</li> <li>Utilizando dois pontos entre os dois números na horizontal: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3149b4f815ad9e8b3e8cdd29adcd02a42c22e5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.165ex; height:2.176ex;" alt="{\displaystyle a:b}"></span>;</li> <li>Usando a notação do <a href="/wiki/Inverso" class="mw-redirect" title="Inverso">inverso multiplicativo</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f564d5f29302ecae14d03ea091a413e7f39ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.56ex; height:2.676ex;" alt="{\displaystyle ab^{-1}}"></span>.</li></ul> <p><b>Divisão de números consecutivos</b> </p><p>Seja o número impar <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="{\textstyle a>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c788c56e25e030d4380b011d633e078583afac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\textstyle a>1}"></span> e o seu consecutivo <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="{\textstyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/548efa927166ec8ec116d1951e10a116d46bbf97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\textstyle a+1}"></span> . </p><p>Seja a divisão <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/(a+1)=s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/(a+1)=s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed917087f3634f2d15bf5baf7c9d0ba36cb154ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.623ex; height:2.843ex;" alt="{\displaystyle a/(a+1)=s}"></span>. Esta divisão apresenta as duas peculiaridade a seguir : </p><p>a - O quociente <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> é menor do que <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> e tende para <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> com o aumento de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, então <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 \lim _{a\to \infty }\rightarrow s=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">→</mo> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{a\to \infty }\rightarrow s=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a847aeee775c8f06a99fb48a197c95d3d6841e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.121ex; height:3.676ex;" alt="{\displaystyle \lim _{a\to \infty }\rightarrow s=1}"></span> </p><p>b - Na imensa maioria das proposições o quociente <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> apresenta infinitos algarismos após a virgula decimal. </p><p>Seja a divisão <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+1)/a=s'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo>=</mo> <msup> <mi>s</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)/a=s'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343b068273d2b07916254b8488833affcf9e0b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.009ex;" alt="{\displaystyle (a+1)/a=s'}"></span>. Esta divisão apresenta as duas peculiaridade a seguir: </p><p>a - O quociente <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 s'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5136680c63706cfd17ceddb4acddbfdd0ba5ef2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle s'}"></span> é maior do que <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> e tende para <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> com o aumento de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, então <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 \lim _{a\to \infty }\rightarrow s'=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">→</mo> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{a\to \infty }\rightarrow s'=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d94a3db8b4d3e9915f7c87df7009d6aca6ae169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.806ex; height:4.009ex;" alt="{\displaystyle \lim _{a\to \infty }\rightarrow s'=1}"></span> </p><p>b - Na imensa maioria das proposições o quociente <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 s'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5136680c63706cfd17ceddb4acddbfdd0ba5ef2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle s'}"></span> apresenta infinitos algarismos após a virgula decimal. </p><p><b>Entretanto</b> </p><p>Em nenhuma das proposições para <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 s,s'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> <msup> <mi>s</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,s'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8f1d4c78e318aaea8bb89df4e8bca94277880f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.9ex; height:2.843ex;" alt="{\displaystyle s,s'}"></span> ocorre com estes números consecutivos, de o quociente apresentar em ambas proposições os dois quocientes com números finitos de algarismos após a virgula decimal. </p> <div class="mw-heading mw-heading3"><h3 id="Divisão_entre_números_consecutivos"><span id="Divis.C3.A3o_entre_n.C3.BAmeros_consecutivos"></span>Divisão entre números consecutivos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=7" title="Editar secção: Divisão entre números consecutivos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=7" title="Editar código-fonte da secção: Divisão entre números consecutivos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Na divisão entre dois números consecutivos temos dois casos a considerar: </p><p>Caso 1 - Caso em que o número maior tem paridade par </p><p>Caso 2 - Caso em que o número maior tem paridade impar </p> <div class="mw-heading mw-heading4"><h4 id="Caso_1">Caso 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=8" title="Editar secção: Caso 1" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=8" title="Editar código-fonte da secção: Caso 1"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>sejam dois números consecutivos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span> com <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A>B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>></mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A>B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5791db2db54217e5290e070826a0bacc33e0388d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A>B}"></span> e de paridade par. </p><p>A divisão <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A/B=1+1/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/B=1+1/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0beb620bbd9864923ee42f90eeb81c4f0151ff1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.86ex; height:2.843ex;" alt="{\displaystyle A/B=1+1/B}"></span> , e a outra divisão <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/A=1-1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/A=1-1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e22096692753c1563c693470b90ef1131bf9d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.839ex; height:2.843ex;" alt="{\displaystyle B/A=1-1/A}"></span> . </p><p>Na imensa maioria dos casos cada uma dessas expressões tem como resultados números com infinitos algarismos após o ponto decimal. </p><p>Em absolutamente todos os casos ao menos uma das duas expressões acima apresenta infinitos algarismos após o ponto decimal. </p><p>No sistema decimal a decomposição única do número <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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span> é dada por <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 10=2.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo>=</mo> <mn>2.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10=2.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9906b1ec59bba8a341c61a686fc118a2aaee050f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.395ex; height:2.176ex;" alt="{\displaystyle 10=2.5}"></span> , então a fração <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 1/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61137d1df14e664dcb4a34328ed4c24e992bfa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle 1/B}"></span> só não será uma dizima infinita quando <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=5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87783de26b1f99ac56633266cbad593b1d007f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.243ex; height:2.343ex;" alt="{\displaystyle B=5^{n}}"></span> pois <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> é um número de paridade impar. </p><p>A fração <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 1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd62775834c956066187d9cd8b6af20170fc509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 1/A}"></span> só não será uma dizima infinita quando <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=2^{m}.10^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mn>.10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2^{m}.10^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edacb045b8f634f2fd9ce4aabac1dd8e80dfc4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.869ex; height:2.343ex;" alt="{\displaystyle A=2^{m}.10^{n}}"></span> . </p><p>A expressão <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 5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e4e56efc40e9f35f396645f596b6aee8e586e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 5^{n}}"></span> termina sempre com o número <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 25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8acf6937f13156a1301ebb614c5364d16597e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 25}"></span> exceto para <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/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" 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>. </p><p>Para termos dois números consecutivos nas condições acima o número <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> tem que terminar com o número <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 26}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>26</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 26}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95afbbb0014a068ce8ff3db0c90da36a5b1b2fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 26}"></span> exceto para o primeiro caso onde <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=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed4054d120e910b9de1e55cb2d42d9bc3abb3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A=6}"></span> , e o número <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> , terá que ser da forma <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 2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667d0154f26e56e3f7979803f08afac16b4dcb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}"></span> onde a expressão <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 1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd62775834c956066187d9cd8b6af20170fc509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 1/A}"></span> não será uma dizima infinita. </p><p>Como os números da forma <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 2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667d0154f26e56e3f7979803f08afac16b4dcb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}"></span> com algarismo <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 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"></span> na na última posição são sempre terminados em <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 16,56,96,36,76}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mo>,</mo> <mn>56</mn> <mo>,</mo> <mn>96</mn> <mo>,</mo> <mn>36</mn> <mo>,</mo> <mn>76</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16,56,96,36,76}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/218e7fab6df6289a4bba9507c5775af2f3bff67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.76ex; height:2.509ex;" alt="{\displaystyle 16,56,96,36,76}"></span> jamais teremos o par consecutivo com os dois últimos algarismos sendo <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 25,26}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>,</mo> <mn>26</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25,26}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b195ddd0a115e5e78d72d7fde87799237cc00f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.684ex; height:2.509ex;" alt="{\displaystyle 25,26}"></span> e com a propriedade de serem da forma <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 5^{n},2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{n},2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe22d5aac893ed570107b382184e9fdbfcb9675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.252ex; height:2.676ex;" alt="{\displaystyle 5^{n},2^{m}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Caso_2">Caso 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=9" title="Editar secção: Caso 2" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=9" title="Editar código-fonte da secção: Caso 2"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sejam dois números consecutivos <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,A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>,</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B,A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e237f4d9fa5728edbb0115527cc4263dad68d79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle B,A}"></span> com <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>A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>></mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B>A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970e9f4fd851be4786963b9603ab9cb5dd27ae3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle B>A}"></span> e de paridade impar. </p><p>A divisão <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/A=1+1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/A=1+1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb36390f69ab8b1578f98c4ce3a41943fc87cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.839ex; height:2.843ex;" alt="{\displaystyle B/A=1+1/A}"></span> e a outra divisão <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A/B=1-1/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/B=1-1/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65691d8cce25b8b1a8b6b72335e6f2acf28be658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.86ex; height:2.843ex;" alt="{\displaystyle A/B=1-1/B}"></span> </p><p>Na imensa maioria dos casos, cada uma destas duas expressões tem como resultado números com infinitos algarismos após o ponto decimal. </p><p>Em absolutamente todos os casos ao menos uma destas duas expressões apresenta como resultado números com infinitos algarismos após o ponto decimal. </p><p>No sistema decimal a decomposição única do número <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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span> é <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 10=2.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo>=</mo> <mn>2.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10=2.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9906b1ec59bba8a341c61a686fc118a2aaee050f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.395ex; height:2.176ex;" alt="{\displaystyle 10=2.5}"></span>, então a fração <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 1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd62775834c956066187d9cd8b6af20170fc509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 1/A}"></span> só não uma dizima infinita quando <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=2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253af40f5d4afa715bdcb79f9385e25842f736b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.679ex; height:2.343ex;" alt="{\displaystyle A=2^{m}}"></span> . </p><p>A fração <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 1/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61137d1df14e664dcb4a34328ed4c24e992bfa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle 1/B}"></span> só não será uma dizima infinita quando <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=5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87783de26b1f99ac56633266cbad593b1d007f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.243ex; height:2.343ex;" alt="{\displaystyle B=5^{n}}"></span> . </p><p>A expressão <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 5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e4e56efc40e9f35f396645f596b6aee8e586e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 5^{n}}"></span> termina sempre no número <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 25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8acf6937f13156a1301ebb614c5364d16597e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 25}"></span> exceto para <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/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" 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> . </p><p>Para termos dois números consecutivos nas condições acima o número <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> tem que terminar em <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 24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be92101c8b0277e66fdefeef1ccdd7788e88ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 24}"></span> , exceto para o primeiro caso onde <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=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ff5d5843d4361c204185dd3f7c827f5ac1662c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A=4}"></span> , e número <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> terá que ser da forma <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 2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667d0154f26e56e3f7979803f08afac16b4dcb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}"></span> , onde a expressão <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 1/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd62775834c956066187d9cd8b6af20170fc509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 1/A}"></span> não será uma dizima infinita. </p><p>O valor de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> só termina em <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 24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be92101c8b0277e66fdefeef1ccdd7788e88ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 24}"></span> , para <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 m=10,30,50,70,90,110...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mn>30</mn> <mo>,</mo> <mn>50</mn> <mo>,</mo> <mn>70</mn> <mo>,</mo> <mn>90</mn> <mo>,</mo> <mn>110...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=10,30,50,70,90,110...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f43d5c7cca71c52fd28bd1452b9bcfca0b9f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.361ex; height:2.509ex;" alt="{\displaystyle m=10,30,50,70,90,110...}"></span> e para nenhum destes casos o número sucessivo terminado em <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 25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8acf6937f13156a1301ebb614c5364d16597e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 25}"></span> é da forma <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 5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e4e56efc40e9f35f396645f596b6aee8e586e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 5^{n}}"></span>, impedindo que tenhamos números consecutivos terminados em <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 24,25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> <mo>,</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24,25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38500682c6d735bf40ea0fafb84bb96ecb18b5d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.684ex; height:2.509ex;" alt="{\displaystyle 24,25}"></span> que sejam da forma <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 2^{m},5^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m},5^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62687546fe79bb4b5fd19c2eb5d00760ec8f3b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.252ex; height:2.676ex;" alt="{\displaystyle 2^{m},5^{n}}"></span> . </p><p>Estas divisões são aplicadas nas soluções para o Último Teorema de Fermat e para a <a href="/wiki/Conjectura" title="Conjectura">Conjectura</a> de Beal a partir das equações do Terno Pitagórico obtidas por Geometria , pois qualquer raiz de um número racional com dizima infinita não terá como resposta um número inteiro. </p><p>Todas as outras fórmulas para a determinação do Terno Pitagórico inclusive as Fórmulas de Euclides não se aplicam, porque são fórmulas incompletas. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=10" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=10" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Crit%C3%A9rios_de_divisibilidade" title="Critérios de divisibilidade">Critérios de divisibilidade</a></li> <li><a href="/wiki/Divis%C3%A3o_por_zero" title="Divisão por zero">Divisão por zero</a></li> <li><a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">Multiplicação</a></li> <li><a href="/wiki/Anel_(matem%C3%A1tica)" title="Anel (matemática)">Anel (matemática)</a></li> <li><a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">Corpo (matemática)</a></li> <li><a href="/wiki/Algoritmo_de_Euclides" title="Algoritmo de Euclides">Algoritmo de Euclides</a></li></ul> <h2 id="Notas_e_referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Notas_e_refer.C3.AAncias"></span>Notas e referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Vianna-nomenc-1"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-Vianna-nomenc_1-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-Vianna-nomenc_1-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-Vianna-nomenc_1-2">c</a></b></i></sup></span> <span class="reference-text">Essa nomenclatura é utilizada por Vianna (1914), <a href="https://pt.wikisource.org/wiki/P%C3%A1gina:Elementos_de_Arithmetica.djvu/43" class="extiw" title="s:Página:Elementos de Arithmetica.djvu/43">p. 39</a></span> </li> <li id="cite_note-Vianna-nomenc2-2"><span class="mw-cite-backlink"><a href="#cite_ref-Vianna-nomenc2_2-0">↑</a></span> <span class="reference-text">Essa nomenclatura é utilizada por Vianna (1914), <a href="https://pt.wikisource.org/wiki/P%C3%A1gina:Elementos_de_Arithmetica.djvu/44" class="extiw" title="s:Página:Elementos de Arithmetica.djvu/44">p. 40</a></span> </li> <li id="cite_note-Serrasq-div-pol-3"><span class="mw-cite-backlink"><a href="#cite_ref-Serrasq-div-pol_3-0">↑</a></span> <span class="reference-text">Serrasqueiro (1906), <a href="https://pt.wikisource.org/wiki/P%C3%A1gina:Tratado_de_Algebra_Elementar.djvu/37" class="extiw" title="s:Página:Tratado de Algebra Elementar.djvu/37">p. 35-37</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Referências"><span id="Refer.C3.AAncias"></span>Referências</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=11" title="Editar secção: Referências" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=11" title="Editar código-fonte da secção: Referências"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Vianna, João José Luiz (1914). <i>Elementos de Arithmetica</i> 15 ed. Rio de Janeiro: Francisco Alves</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ADivis%C3%A3o&rft.aufirst=Jo%C3%A3o+Jos%C3%A9+Luiz&rft.aulast=Vianna&rft.btitle=Elementos+de+Arithmetica&rft.date=1914&rft.edition=15&rft.genre=book&rft.place=Rio+de+Janeiro&rft.pub=Francisco+Alves&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Serrasqueiro, José Adelino (1906). <i>Tratado de Algebra Elementar</i> 9 ed. Largo da Sé Velha: Livraria Central de J. Diogo Pires</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ADivis%C3%A3o&rft.aufirst=Jos%C3%A9+Adelino&rft.aulast=Serrasqueiro&rft.btitle=Tratado+de+Algebra+Elementar&rft.date=1906&rft.edition=9&rft.genre=book&rft.place=Largo+da+S%C3%A9+Velha&rft.pub=Livraria+Central+de+J.+Diogo+Pires&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>i</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divis%C3%A3o&veaction=edit&section=12" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divis%C3%A3o&action=edit&section=12" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r69325977">@media(max-width:768px){.mw-parser-output .mobile-stack{width:100%!important;float:none!important;margin:10px auto!important;text-align:center!important;box-sizing:border-box!important}.mw-parser-output .mobile-stack .image-container{float:none!important;display:inline-block;padding-bottom:10px}.mw-parser-output .mobile-stack .text-container{margin-left:0!important}}</style><div class="noprint mobile-stack" style="clear: right; border: solid #aaa 1px; margin: 0 0 1em 1em; font-size: 90%; background: var(--background-color-neutral-subtle, #f9f9f9); color: inherit; width: 238px; padding: 5px; spacing: 0; text-align: left; float: right;"> <div class="image-container" style="float: left; vertical-align:middle; margin: 0 10px 0 5px;"><figure class="mw-halign-none" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/38px-Wikibooks-logo.svg.png" decoding="async" width="38" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/57px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/76px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span><figcaption></figcaption></figure></div> <div class="text-container" style="margin-left: 30px; line-height:normal; vertical-align:middle;">O <a href="/wiki/Wikilivros" title="Wikilivros">Wikilivro</a> <i><a href="https://pt.wikibooks.org/wiki/Special:Search/Teoria_de_n%C3%BAmeros" class="extiw" title="b:Special:Search/Teoria de números">Teoria de números</a></i> tem uma página sobre <i><b><a href="https://pt.wikibooks.org/wiki/Special:Search/Teoria_de_n%C3%BAmeros/M%C3%A1ximo_divisor_comum#Algoritmo_da_divisão_(de_Euclides)" class="extiw" title="b:Special:Search/Teoria de números/Máximo divisor comum">Algoritmo da divisão de Euclides</a></b></i> </div> </div> <table class="infobox noprint" style="width:250px; line-height:2.2em; font-size:90%"> <tbody><tr style="line-height:1.3em"> <td colspan="2" style="text-align: center;">Outros projetos <a href="/wiki/Wikimedia" class="mw-redirect" title="Wikimedia">Wikimedia</a> também contêm material sobre este tema: </td></tr> <tr> <th style="width: 37px;"><span typeof="mw:File"><a href="https://pt.wiktionary.org/wiki/Special:Search/Divis%C3%A3o" title="Wikcionário"><img alt="Wikcionário" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Wiktionary-logo-pt.png/25px-Wiktionary-logo-pt.png" decoding="async" width="25" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Wiktionary-logo-pt.png/38px-Wiktionary-logo-pt.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Wiktionary-logo-pt.png/50px-Wiktionary-logo-pt.png 2x" data-file-width="140" data-file-height="135" /></a></span> </th> <td><a href="https://pt.wiktionary.org/wiki/Special:Search/Divis%C3%A3o" class="extiw" title="wikt:Special:Search/Divisão"><span title="Procurar por definições no Wikicionário"><b>Definições</b></span></a> no <a href="https://pt.wiktionary.org/wiki/P%C3%A1gina_principal" class="extiw" title="wikt:Página principal"><span title="Wikcionário">Wikcionário</span></a> </td></tr> </tbody></table><div id="interProject" style="display:none;"> <ul><li><a href="https://pt.wiktionary.org/wiki/Special:Search/Divis%C3%A3o" class="extiw" title="wikt:Special:Search/Divisão"><span title="Wikcionário">Wikcionário</span></a></li></ul> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=oHpKzOXgTyQ">Explicando o Algoritmo da Divisão</a></li></ul> <table class="noprint" style="border-top:none; border-bottom:none; background:transparent; color: inherit"><tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Ficheiro:E-to-the-i-pi.svg" class="mw-file-description"><img alt="Ícone de esboço" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/34px-E-to-the-i-pi.svg.png" decoding="async" width="34" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/51px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/68px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span></td> <td><i>Este artigo sobre <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a> é um <a href="/wiki/Wikip%C3%A9dia:Esbo%C3%A7o" title="Wikipédia:Esboço">esboço</a>. Você pode ajudar a Wikipédia <b><span class="plainlinks" style="white-space:nowrap"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Divis%C3%A3o&action=edit">expandindo-o</a></span></b>.</i><div class="plainlinks hlist navbar mini" style="position:absolute; right:15px; font-size:smaller; display:none;"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Esbo%C3%A7o-matem%C3%A1tica" title="Predefinição:Esboço-matemática"><abbr title="Ver esta predefinição">v</abbr></a></li><li class="nv-discutir"><a href="/wiki/Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Esbo%C3%A7o-matem%C3%A1tica" title="Predefinição Discussão:Esboço-matemática"><abbr title="Discutir esta predefinição">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:Esbo%C3%A7o-matem%C3%A1tica&action=edit"><abbr title="Editar esta predefinição">e</abbr></a></li></ul></div></td> </tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r69328899">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:0 auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><div role="navigation" class="navbox" aria-labelledby="Aritmética_elementar" style="padding:3px"><table class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Aritm%C3%A9tica_elementar" title="Predefinição:Aritmética elementar"><abbr title="Ver esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">v</abbr></a></li><li class="nv-discutir"><a href="/w/index.php?title=Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Aritm%C3%A9tica_elementar&action=edit&redlink=1" class="new" title="Predefinição Discussão:Aritmética elementar (página não existe)"><abbr title="Discutir esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:Aritm%C3%A9tica_elementar&action=edit"><abbr title="Editar esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">e</abbr></a></li></ul></div><div id="Aritmética_elementar" style="font-size:114%;margin:0 4em"><a href="/wiki/Aritm%C3%A9tica_elementar" class="mw-redirect" title="Aritmética elementar">Aritmética elementar</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px;background:transparent;color:inherit;"><div style="padding:0px;"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:auto; margin-left:auto; margin-right:auto;"><tbody><tr style="vertical-align:top;"><td style="width:5em;">   </td><td class="navbox-list" style="padding:0px;text-align: center;;;width:25%;"><div> <span typeof="mw:File"><a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/50px-Symbol_support_vote.svg.png" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/75px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/100px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span><br /><a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição">Adição</a><div style="font-size: x-large;"><a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição">+</a></div> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align: center;;;width:25%;"><div> <span typeof="mw:File"><a href="/wiki/Subtra%C3%A7%C3%A3o" title="Subtração"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Symbol_oppose_vote.svg/50px-Symbol_oppose_vote.svg.png" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Symbol_oppose_vote.svg/75px-Symbol_oppose_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Symbol_oppose_vote.svg/100px-Symbol_oppose_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span><br /><a href="/wiki/Subtra%C3%A7%C3%A3o" title="Subtração">Subtração</a><div style="font-size: x-large;"><a href="/wiki/Subtra%C3%A7%C3%A3o" title="Subtração">−</a></div> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align: center;;;width:25%;"><div> <span typeof="mw:File"><a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Symbol_multiplication_vote.svg/50px-Symbol_multiplication_vote.svg.png" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Symbol_multiplication_vote.svg/75px-Symbol_multiplication_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Symbol_multiplication_vote.svg/100px-Symbol_multiplication_vote.svg.png 2x" data-file-width="1800" data-file-height="1850" /></a></span><br /><a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">Multiplicação</a><div style="font-size: x-large;"><a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">×</a></div> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align: center;;;width:25%;"><div> <span typeof="mw:File"><a href="/wiki/Divis%C3%A3o" title="Divisão"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Symbol_divide_vote.svg/50px-Symbol_divide_vote.svg.png" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Symbol_divide_vote.svg/75px-Symbol_divide_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Symbol_divide_vote.svg/100px-Symbol_divide_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span><br /><a class="mw-selflink selflink">Divisão</a><div style="font-size: x-large;"><a class="mw-selflink selflink">÷</a></div> </div></td></tr></tbody></table></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="Controle_de_autoridade" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Controle_de_autoridade" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Ajuda:Controle_de_autoridade" title="Ajuda:Controle de autoridade">Controle de autoridade</a></th><td class="navbox-list navbox-odd plainlinks" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Wikidata:Main_Page" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span>: <span class="uid"><a href="https://www.wikidata.org/wiki/Q1226939" class="extiw" title="wikidata:Q1226939">Q1226939</a></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_Central_de_Floren%C3%A7a" title="Biblioteca Nacional Central de Florença">BNCF</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=1131">1131</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">EBID</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/division-mathematics">ID</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4150319-3">4150319-3</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/N%C3%BAmero_de_controle_da_Biblioteca_do_Congresso" title="Número de controle da Biblioteca do Congresso">LCCN</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85038610">sh85038610</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_da_Dieta" title="Biblioteca Nacional da Dieta">NDL</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00575006">00575006</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclop%C3%A9dia_Treccani" title="Enciclopédia Treccani">Treccani</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/divisione">divisione</a></span></span></span></li></ul> </div></td></tr></tbody></table></div> <ul class="noprint navigation-box" style="border-top: solid silver 1px; border-right: solid silver 1px; border-bottom:1px solid silver; border-left: solid silver 1px; padding:3px; background-color: #F9F9F9; text-align: center; margin-top:10px; margin-left: 0; clear: both;"><li style="display: inline;"><span style="white-space: nowrap; margin: auto 1.5em"><span style="margin-right: 0.5em"><span typeof="mw:File"><a href="/wiki/Ficheiro:Nuvola_apps_edu_mathematics-p.svg" title="Portal da matemática"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/25px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/38px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/50px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span style="font-weight:bold"><a href="/wiki/Portal:Matem%C3%A1tica" title="Portal:Matemática">Portal da matemática</a></span></span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Divisão&oldid=69360535">https://pt.wikipedia.org/w/index.php?title=Divisão&oldid=69360535</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:Aritm%C3%A9tica" title="Categoria:Aritmética">Aritmética</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!P%C3%A1ginas_que_usam_sidebar_com_o_par%C3%A2metro_filha" title="Categoria:!Páginas que usam sidebar com o parâmetro filha">!Páginas que usam sidebar com o parâmetro filha</a></li><li><a href="/wiki/Categoria:!Esbo%C3%A7os_sobre_matem%C3%A1tica" title="Categoria:!Esboços sobre matemática">!Esboços sobre matemática</a></li><li><a href="/wiki/Categoria:!Esbo%C3%A7os_maiores_que_10000_bytes" title="Categoria:!Esboços maiores que 10000 bytes">!Esboços maiores que 10000 bytes</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 19h52min de 14 de janeiro de 2025.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; 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