Division — Wikipédia

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Historique </div> </a> <ul id="toc-Vocabulaire_et_notations._Historique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathématiques_et_langue_française" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathématiques_et_langue_française"> <div class="vector-toc-text"> 3 Mathématiques et langue française </div> </a> <ul id="toc-Mathématiques_et_langue_française-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Définition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Définition"> <div class="vector-toc-text"> 4 Définition </div> </a> <button aria-controls="toc-Définition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Définition </button> <ul id="toc-Définition-sublist" class="vector-toc-list"> <li id="toc-Approche_élémentaire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approche_élémentaire"> <div class="vector-toc-text"> 4.1 Approche élémentaire </div> </a> <ul id="toc-Approche_élémentaire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Définition_complète" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_complète"> <div class="vector-toc-text"> 4.2 Définition complète </div> </a> <ul id="toc-Définition_complète-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> 5 Propriétés </div> </a> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithmes_de_la_division_de_nombres_décimaux" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithmes_de_la_division_de_nombres_décimaux"> <div class="vector-toc-text"> 6 Algorithmes de la division de nombres décimaux </div> </a> <button aria-controls="toc-Algorithmes_de_la_division_de_nombres_décimaux-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Algorithmes de la division de nombres décimaux </button> <ul id="toc-Algorithmes_de_la_division_de_nombres_décimaux-sublist" class="vector-toc-list"> <li id="toc-Division_non_abrégée" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_non_abrégée"> <div class="vector-toc-text"> 6.1 Division non abrégée </div> </a> <ul id="toc-Division_non_abrégée-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division_de_nombres_codés_en_binaire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_de_nombres_codés_en_binaire"> <div class="vector-toc-text"> 6.2 Division de nombres codés en binaire </div> </a> <ul id="toc-Division_de_nombres_codés_en_binaire-sublist" class="vector-toc-list"> <li id="toc-Division_euclidienne_binaire" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Division_euclidienne_binaire"> <div class="vector-toc-text"> 6.2.1 Division euclidienne binaire </div> </a> <ul id="toc-Division_euclidienne_binaire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Méthodes_lentes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Méthodes_lentes"> <div class="vector-toc-text"> 6.2.2 Méthodes lentes </div> </a> <ul id="toc-Méthodes_lentes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Méthodes_rapides" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Méthodes_rapides"> <div class="vector-toc-text"> 6.2.3 Méthodes rapides </div> </a> <ul id="toc-Méthodes_rapides-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Division_d'objets_autres_que_des_nombres_réels" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Division_d'objets_autres_que_des_nombres_réels"> <div class="vector-toc-text"> 7 Division d'objets autres que des nombres réels </div> </a> <button aria-controls="toc-Division_d'objets_autres_que_des_nombres_réels-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Division d'objets autres que des nombres réels </button> <ul id="toc-Division_d'objets_autres_que_des_nombres_réels-sublist" class="vector-toc-list"> <li id="toc-Division_complexe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_complexe"> <div class="vector-toc-text"> 7.1 Division complexe </div> </a> <ul id="toc-Division_complexe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division_matricielle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_matricielle"> <div class="vector-toc-text"> 7.2 Division matricielle </div> </a> <ul id="toc-Division_matricielle-sublist" class="vector-toc-list"> <li id="toc-Division_à_droite" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Division_à_droite"> <div class="vector-toc-text"> 7.2.1 Division à droite </div> </a> <ul id="toc-Division_à_droite-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division_à_gauche" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Division_à_gauche"> <div class="vector-toc-text"> 7.2.2 Division à gauche </div> </a> <ul id="toc-Division_à_gauche-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Division_de_polynômes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_de_polynômes"> <div class="vector-toc-text"> 7.3 Division de polynômes </div> </a> <ul id="toc-Division_de_polynômes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> 8 Notes et références </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> 9 Voir aussi </div> </a> <button aria-controls="toc-Voir_aussi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Voir aussi </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> 9.1 Articles connexes </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <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" > Basculer la table des matières </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">Division</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="Aller à un article dans une autre langue. Disponible en 120 langues." > <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-120" aria-hidden="true" > 120 langues </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 – afrikaans" lang="af" hreflang="af" data-title="Deling" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Division_(Mathematik)" title="Division (Mathematik) – alémanique" lang="gsw" hreflang="gsw" data-title="Division (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="alémanique" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Divisi%C3%B3n" title="División – aragonais" lang="an" hreflang="an" data-title="División" data-language-autonym="Aragonés" data-language-local-name="aragonais" class="interlanguage-link-target">Aragonés</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) – arabe" lang="ar" hreflang="ar" data-title="قسمة (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target">العربية</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 – arabe marocain" lang="ary" hreflang="ary" data-title="قسيم" data-language-autonym="الدارجة" data-language-local-name="arabe marocain" class="interlanguage-link-target">الدارجة</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 – arabe égyptien" lang="arz" hreflang="arz" data-title="قسمه" data-language-autonym="مصرى" data-language-local-name="arabe égyptien" class="interlanguage-link-target">مصرى</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) – assamais" lang="as" hreflang="as" data-title="হৰণ (গণিত)" data-language-autonym="অসমীয়া" data-language-local-name="assamais" class="interlanguage-link-target">অসমীয়া</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) – asturien" lang="ast" hreflang="ast" data-title="División (matemátiques)" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jaljaya%C3%B1a" title="Jaljayaña – aymara" lang="ay" hreflang="ay" data-title="Jaljayaña" data-language-autonym="Aymar aru" data-language-local-name="aymara" class="interlanguage-link-target">Aymar aru</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) – azerbaïdjanais" lang="az" hreflang="az" data-title="Bölmə (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target">Azərbaycanca</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">تۆرکجه</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 – bachkir" lang="ba" hreflang="ba" data-title="Бүлеү" data-language-autonym="Башҡортса" data-language-local-name="bachkir" class="interlanguage-link-target">Башҡортса</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">Bikol Central</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 – biélorusse" lang="be" hreflang="be" data-title="Дзяленне" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%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">Беларуская (тарашкевіца)</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 – bulgare" lang="bg" hreflang="bg" data-title="Деление" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target">Български</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 – bengali" lang="bn" hreflang="bn" data-title="ভাগ" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target">বাংলা</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། – tibétain" lang="bo" hreflang="bo" data-title="བགོ་རྩིས།" data-language-autonym="བོད་ཡིག" data-language-local-name="tibétain" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Rannadur" title="Rannadur – breton" lang="br" hreflang="br" data-title="Rannadur" data-language-autonym="Brezhoneg" data-language-local-name="breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Dijeljenje_(matematika)" title="Dijeljenje (matematika) – bosniaque" lang="bs" hreflang="bs" data-title="Dijeljenje (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target">Bosanski</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">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Divisi%C3%B3" title="Divisió – catalan" lang="ca" hreflang="ca" data-title="Divisió" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target">Català</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">閩東語 / Mìng-dĕ̤ng-ngṳ̄</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 – sorani" lang="ckb" hreflang="ckb" data-title="دابەشکردن" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target">کوردی</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 – tchèque" lang="cs" hreflang="cs" data-title="Dělení" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target">Čeština</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 – tchouvache" lang="cv" hreflang="cv" data-title="Пайласси" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhannu_(mathemateg)" title="Rhannu (mathemateg) – gallois" lang="cy" hreflang="cy" data-title="Rhannu (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="gallois" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Division_(matematik)" title="Division (matematik) – danois" lang="da" hreflang="da" data-title="Division (matematik)" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Division_(Mathematik)" title="Division (Mathematik) – allemand" lang="de" hreflang="de" data-title="Division (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target">Deutsch</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 – grec" lang="el" hreflang="el" data-title="Διαίρεση" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Division_(mathematics)" title="Division (mathematics) – anglais" lang="en" hreflang="en" data-title="Division (mathematics)" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Divido" title="Divido – espéranto" lang="eo" hreflang="eo" data-title="Divido" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target">Esperanto</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) – espagnol" lang="es" hreflang="es" data-title="División (matemática)" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Jagamine" title="Jagamine – estonien" lang="et" hreflang="et" data-title="Jagamine" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zatiketa_(matematika)" title="Zatiketa (matematika) – basque" lang="eu" hreflang="eu" data-title="Zatiketa (matematika)" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="تقسیم – persan" lang="fa" hreflang="fa" data-title="تقسیم" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Jakolasku" title="Jakolasku – finnois" lang="fi" hreflang="fi" data-title="Jakolasku" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tabana" title="Tabana – fidjien" lang="fj" hreflang="fj" data-title="Tabana" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fidjien" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Brot" title="Brot – féroïen" lang="fo" hreflang="fo" data-title="Brot" data-language-autonym="Føroyskt" data-language-local-name="féroïen" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Dialen_(Matematiik)" title="Dialen (Matematiik) – frison septentrional" lang="frr" hreflang="frr" data-title="Dialen (Matematiik)" data-language-autonym="Nordfriisk" data-language-local-name="frison septentrional" class="interlanguage-link-target">Nordfriisk</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">贛語</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Divizyon" title="Divizyon – créole guyanais" lang="gcr" hreflang="gcr" data-title="Divizyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="créole guyanais" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Roinn_(matamataig)" title="Roinn (matamataig) – gaélique écossais" lang="gd" hreflang="gd" data-title="Roinn (matamataig)" data-language-autonym="Gàidhlig" data-language-local-name="gaélique écossais" class="interlanguage-link-target">Gàidhlig</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) – galicien" lang="gl" hreflang="gl" data-title="División (matemáticas)" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target">Galego</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 – hakka" lang="hak" hreflang="hak" data-title="Chhù-fap" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="hakka" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</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 – hébreu" lang="he" hreflang="he" data-title="חילוק" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target">עברית</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">हिन्दी</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Baatna" title="Baatna – hindi fidjien" lang="hif" hreflang="hif" data-title="Baatna" data-language-autonym="Fiji Hindi" data-language-local-name="hindi fidjien" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dijeljenje" title="Dijeljenje – croate" lang="hr" hreflang="hr" data-title="Dijeljenje" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Oszt%C3%A1s" title="Osztás – hongrois" lang="hu" hreflang="hu" data-title="Osztás" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target">Magyar</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) – arménien" lang="hy" hreflang="hy" data-title="Բաժանում (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Division_(mathematica)" title="Division (mathematica) – interlingua" lang="ia" hreflang="ia" data-title="Division (mathematica)" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target">Interlingua</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">Jaku Iban</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pembagian" title="Pembagian – indonésien" lang="id" hreflang="id" data-title="Pembagian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target">Bahasa Indonesia</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">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Deiling" title="Deiling – islandais" lang="is" hreflang="is" data-title="Deiling" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Divisione_(matematica)" title="Divisione (matematica) – italien" lang="it" hreflang="it" data-title="Divisione (matematica)" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target">Italiano</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 – japonais" lang="ja" hreflang="ja" data-title="除法" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Divijan_(matimatix)" title="Divijan (matimatix) – créole jamaïcain" lang="jam" hreflang="jam" data-title="Divijan (matimatix)" data-language-autonym="Patois" data-language-local-name="créole jamaïcain" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Paran" title="Paran – javanais" lang="jv" hreflang="jv" data-title="Paran" data-language-autonym="Jawa" data-language-local-name="javanais" class="interlanguage-link-target">Jawa</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 – kazakh" lang="kk" hreflang="kk" data-title="Бөлу" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target">Қазақша</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 – kannada" lang="kn" hreflang="kn" data-title="ಭಾಗಾಕಾರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target">ಕನ್ನಡ</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 – coréen" lang="ko" hreflang="ko" data-title="나눗셈" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Parkirin" title="Parkirin – kurde" lang="ku" hreflang="ku" data-title="Parkirin" data-language-autonym="Kurdî" data-language-local-name="kurde" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D2%AF%D2%AF" title="Бөлүү – kirghize" lang="ky" hreflang="ky" data-title="Бөлүү" data-language-autonym="Кыргызча" data-language-local-name="kirghize" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Divisio_(mathematica)" title="Divisio (mathematica) – latin" lang="la" hreflang="la" data-title="Divisio (mathematica)" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target">Latina</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">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%9E%E0%BA%B2%E0%BA%81%E0%BA%A7%E0%BA%B4%E0%BA%8A%E0%BA%B2_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ພາກວິຊາ (ຄະນິດສາດ) – lao" lang="lo" hreflang="lo" data-title="ພາກວິຊາ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dalyba" title="Dalyba – lituanien" lang="lt" hreflang="lt" data-title="Dalyba" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target">Lietuvių</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 – letton" lang="lv" hreflang="lv" data-title="Dalīšana" data-language-autonym="Latviešu" data-language-local-name="letton" class="interlanguage-link-target">Latviešu</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 – macédonien" lang="mk" hreflang="mk" data-title="Делење" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target">Македонски</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 – malayalam" lang="ml" hreflang="ml" data-title="ഹരണം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembahagian" title="Pembahagian – malais" lang="ms" hreflang="ms" data-title="Pembahagian" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target">Bahasa Melayu</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) – maltais" lang="mt" hreflang="mt" data-title="Diviżjoni (matematika)" data-language-autonym="Malti" data-language-local-name="maltais" class="interlanguage-link-target">Malti</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) – mirandais" lang="mwl" hreflang="mwl" data-title="Debison (matemática)" data-language-autonym="Mirandés" data-language-local-name="mirandais" class="interlanguage-link-target">Mirandés</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) – népalais" lang="ne" hreflang="ne" data-title="भाग (गणित)" data-language-autonym="नेपाली" data-language-local-name="népalais" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Delen" title="Delen – néerlandais" lang="nl" hreflang="nl" data-title="Delen" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target">Nederlands</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 – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Divisjon i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Divisjon_(matematikk)" title="Divisjon (matematikk) – norvégien bokmål" lang="nb" hreflang="nb" data-title="Divisjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target">Norsk bokmål</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">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Division" title="Division – occitan" lang="oc" hreflang="oc" data-title="Division" data-language-autonym="Occitan" data-language-local-name="occitan" class="interlanguage-link-target">Occitan</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) – odia" lang="or" hreflang="or" data-title="ହରଣ (ଗଣିତ)" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="odia" class="interlanguage-link-target">ଓଡ଼ିଆ</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 – pendjabi" lang="pa" hreflang="pa" data-title="ਤਕਸੀਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dzielenie" title="Dzielenie – polonais" lang="pl" hreflang="pl" data-title="Dzielenie" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target">Polski</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 – pachto" lang="ps" hreflang="ps" data-title="تقسيم" data-language-autonym="پښتو" data-language-local-name="pachto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Divis%C3%A3o" title="Divisão – portugais" lang="pt" hreflang="pt" data-title="Divisão" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Rakiy" title="Rakiy – quechua" lang="qu" hreflang="qu" data-title="Rakiy" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target">Runa Simi</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) – roumain" lang="ro" hreflang="ro" data-title="Împărțire (matematică)" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target">Română</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) – russe" lang="ru" hreflang="ru" data-title="Деление (математика)" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target">Русский</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">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dijeljenje" title="Dijeljenje – serbo-croate" lang="sh" hreflang="sh" data-title="Dijeljenje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</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">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Delenie_(matematika)" title="Delenie (matematika) – slovaque" lang="sk" hreflang="sk" data-title="Delenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target">Slovenčina</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">سرائیکی</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Deljenje" title="Deljenje – slovène" lang="sl" hreflang="sl" data-title="Deljenje" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target">Slovenščina</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">ChiShona</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">Soomaaliga</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Pjes%C3%ABtimi" title="Pjesëtimi – albanais" lang="sq" hreflang="sq" data-title="Pjesëtimi" data-language-autonym="Shqip" data-language-local-name="albanais" class="interlanguage-link-target">Shqip</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 – serbe" lang="sr" hreflang="sr" data-title="Дељење" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Division_(matematik)" title="Division (matematik) – suédois" lang="sv" hreflang="sv" data-title="Division (matematik)" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ugawanyaji" title="Ugawanyaji – swahili" lang="sw" hreflang="sw" data-title="Ugawanyaji" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target">Kiswahili</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) – tamoul" lang="ta" hreflang="ta" data-title="வகுத்தல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AD%E0%B0%BE%E0%B0%97%E0%B0%B9%E0%B0%BE%E0%B0%B0%E0%B0%82" title="భాగహారం – télougou" lang="te" hreflang="te" data-title="భాగహారం" data-language-autonym="తెలుగు" data-language-local-name="télougou" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AB%E0%B8%B2%E0%B8%A3" title="การหาร – thaï" lang="th" hreflang="th" data-title="การหาร" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Paghahati" title="Paghahati – tagalog" lang="tl" hreflang="tl" data-title="Paghahati" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/B%C3%B6lme" title="Bölme – turc" lang="tr" hreflang="tr" data-title="Bölme" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target">Türkçe</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 – ukrainien" lang="uk" hreflang="uk" data-title="Ділення" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target">Українська</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) – ourdou" lang="ur" hreflang="ur" data-title="تقسیم (ریاضی)" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bo%CA%BBlish" title="Boʻlish – ouzbek" lang="uz" hreflang="uz" data-title="Boʻlish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Divizion" title="Divizion – vénitien" lang="vec" hreflang="vec" data-title="Divizion" data-language-autonym="Vèneto" data-language-local-name="vénitien" class="interlanguage-link-target">Vèneto</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 – vietnamien" lang="vi" hreflang="vi" data-title="Phép chia" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target">Tiếng Việt</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">Winaray</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">吴语</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 – kalmouk" lang="xal" hreflang="xal" data-title="Хувалһан" data-language-autonym="Хальмг" data-language-local-name="kalmouk" class="interlanguage-link-target">Хальмг</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 – xhosa" lang="xh" hreflang="xh" data-title="Ukwaba/ukwahlula-hlula" data-language-autonym="IsiXhosa" data-language-local-name="xhosa" class="interlanguage-link-target">IsiXhosa</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 – yiddish" lang="yi" hreflang="yi" data-title="צעטיילונג" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target">ייִדיש</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">Vahcuengh</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 – chinois" lang="zh" hreflang="zh" data-title="除法" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target">中文</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 – minnan" lang="nan" hreflang="nan" data-title="Tû-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="minnan" 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liées</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Division" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k">Suivi des pages liées</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u">Téléverser un fichier</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q">Pages spéciales</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Division&oldid=219840689" title="Adresse permanente de cette version de cette page">Lien permanent</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Division&action=info" title="Davantage d’informations sur cette page">Informations sur la page</a></li><li id="t-cite" class="mw-list-item"><a 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mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> Pour les articles homonymes, voir <a href="/wiki/Division_(homonymie)" class="mw-disambig" title="Division (homonymie)">Division (homonymie)</a>. </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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>Division en tant que partage. Illustration de 20÷4 : partage d'un ensemble de 20 pommes en 4 parts égales.</figcaption></figure> La division est une opération mathématique. Dans une première approche, la division consiste à calculer combien de fois on peut mettre un nombre dans un autre. Par exemple, si l'on dispose de 20 pommes à distribuer à 4 personnes, il faut donner 5 pommes à chacun (on peut mettre 5 fois 4 pommes pour avoir 20 pommes). Le nombre que l'on divise (20 dans l'exemple) s'appelle le dividende. Nous l'avons divisé par 4 : il s'agit du diviseur. Le résultat s'appelle le quotient ou rapport (5 dans l'exemple). En termes mathématiques, à deux nombres, le dividende <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><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}"> et le diviseur <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, la division associe un troisième nombre (<a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">loi de composition interne</a>), qui est le quotient. Selon la norme NF EN ISO 80000-2<a href="#cite_note-1">[1]</a>,<a href="#cite_note-2">[2]</a>, le quotient de <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><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}"> par <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> se note : <ul><li><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><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}"> ;</li> <li><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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b6c014398323cb45578581a536033fce1b28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.39ex; height:2.843ex;" alt="{\displaystyle a/b}"> (<a href="/wiki/Barre_oblique" title="Barre oblique">barre oblique</a>, <a href="/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques)">fraction</a> en ligne) ;</li> <li><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"> (<a href="/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques)">fraction</a>).</li></ul> Selon la même norme, il convient de ne pas utiliser la notation <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><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}"> (<a href="/wiki/Ob%C3%A9lus" title="Obélus">obélus</a>). Dans une première approche, on peut voir la quantité <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><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}"> comme une séparation de la quantité <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><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}"> en <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> parts égales. Mais cette approche est surtout adaptée à la division entre nombres entiers, où la notion de quantité est assez intuitive. On distingue couramment la division « exacte » (celle dont on parle ici) de la division « avec reste » (la <a href="/wiki/Division_euclidienne" title="Division euclidienne">division euclidienne</a>). D'un point de vue pratique, on peut voir la division comme le <a href="/wiki/Produit_math%C3%A9matique" class="mw-redirect" title="Produit mathématique">produit</a> du premier par l'<a href="/wiki/Inverse" title="Inverse">inverse</a> du second. Si un nombre est non nul, la fonction « division par ce nombre » est la réciproque de la fonction « <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> par ce nombre ». Cela permet de déterminer un certain nombre de propriétés de l'opération. De manière plus générale, on peut définir le quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=a/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9ca57e5cd22002590b2506c4d775e8b0595aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.644ex; height:2.843ex;" alt="{\displaystyle y=a/b}"> comme étant la solution de l'<a href="/wiki/%C3%89quation" title="Équation">équation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\times y=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>×</mo> <mi>y</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\times y=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830d83c6b9985acf0bb3dedbf446a6a875d04f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.322ex; height:2.509ex;" alt="{\displaystyle b\times y=a}">. Ceci permet d'étendre la définition à d'autres objets mathématiques que les nombres, à tous les éléments d'un <a href="/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques)">anneau</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Problématique">Problématique</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=1" title="Modifier la section : Problématique" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=1" title="Modifier le code source de la section : Problématique">modifier le code</a>]</div> La division sert : <ul><li>à faire un partage équitable entre un nombre de parts déterminé à l'avance, et donc à déterminer la taille d'une part. Par exemple :</li></ul> <dl><dd>Question : Si on répartit équitablement 500 grammes de poudre de perlimpimpin entre huit personnes, combien chacune d'elles obtiendra-t-elle ?</dd> <dd>Réponse : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {500}{8}}=62{,}5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>500</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>62</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {500}{8}}=62{,}5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097ef10809062d5931685fbb02a9393e52ec71be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.556ex; height:5.176ex;" alt="{\displaystyle {\dfrac {500}{8}}=62{,}5}">, chacun obtient 62,5 grammes de poudre de perlimpimpin</dd></dl> <ul><li>à déterminer le nombre de parts possible, d'une taille déterminée à l'avance. Par exemple :</li></ul> <dl><dd>Question : Si on répartit 500 grammes de poudre de perlimpimpin par tranche de 70 <abbr class="abbr" title="gramme">g</abbr>, combien de personnes pourra-t-on servir ?</dd> <dd>Réponse : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {500}{70}}=7{,}14\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>500</mn> <mn>70</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>14</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {500}{70}}=7{,}14\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d102e1d60836db6fd413b601a1cacd1a47163403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.667ex; height:5.343ex;" alt="{\displaystyle {\dfrac {500}{70}}=7{,}14\ldots }">, on pourra servir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee716ec61382a6b795092c0edd859d12e64cbba8" 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 7}"> personnes et il restera de quoi servir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4d58e3805ad7764adda51f644ece004afd557c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/7}"> de personne (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx 0,14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <mn>0</mn> <mo>,</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx 0,14}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54e6bed6523e3d75b346ab3e0c7612269741de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.975ex; height:2.509ex;" alt="{\displaystyle \approx 0,14}">).</dd></dl> La multiplication est également l'expression d'une <a href="/wiki/Loi_proportionnelle" class="mw-redirect" title="Loi proportionnelle">loi proportionnelle</a>. La division permet donc « d'inverser » cette loi. Par exemple, on sait qu'à un endroit donné, le <a href="/wiki/Poids" title="Poids">poids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> (force qui tire un objet vers le bas, exprimée en <a href="/wiki/Newton_(unit%C3%A9)" title="Newton (unité)">newtons</a>) est proportionnelle à la <a href="/wiki/Masse" title="Masse">masse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (quantité fixe, exprimée en kilogrammes), le coefficient de proportionnalité étant appelé « gravité » <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=m\times g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>m</mi> <mo>×</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=m\times g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/905ab050cd88a84849b0d8fdce278a699145d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.841ex; height:2.509ex;" alt="{\displaystyle P=m\times g}"></dd></dl> Un pèse-personne mesure la force qui lui est exercée, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> ; la gravité étant connue, on peut en déduire la masse d'une personne en inversant la loi par une division : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=P/g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=P/g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aea28707a27e981811b6281a1ed9af1d9713bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.163ex; height:2.843ex;" alt="{\displaystyle m=P/g}">.</dd></dl> Dans le même ordre d'idées, dans le cas d'un <a href="/wiki/Mouvement_rectiligne_uniforme" title="Mouvement rectiligne uniforme">mouvement rectiligne uniforme</a>, la distance parcourue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> (en kilomètres) est proportionnelle au temps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> (en heures), le coefficient étant la vitesse moyenne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> (en kilomètres par heure) : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=v\times t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>v</mi> <mo>×</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=v\times t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8279975ba8777ee9f90b92c4e61b39ed37408ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.122ex; height:2.176ex;" alt="{\displaystyle d=v\times t}">.</dd></dl> On peut inverser cette loi pour déterminer le temps qu'il faut pour parcourir une distance donnée à une vitesse moyenne donnée : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=d/v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=d/v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43660e92c8f7157d90eb0768a754903d673d0abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.444ex; height:2.843ex;" alt="{\displaystyle t=d/v}">.</dd></dl> De manière plus générale, la division intervient dans l'inversion d'une <a href="/wiki/Loi_affine" title="Loi affine">loi affine</a>, c'est-à-dire du type <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=ax+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ce6c0c441910ff30fb2bff13287f12f4f14a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle y=ax+b}"></dd></dl> ce qui donne si a est non nul <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(y-b)/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(y-b)/a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccca841fe1c9c2ae8c946b2861cb979ae86f856" 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 x=(y-b)/a}"></dd></dl> Par ailleurs, on peut approcher la plupart des lois par une loi affine en faisant un <a href="/wiki/D%C3%A9veloppement_limit%C3%A9" title="Développement limité">développement limité</a> à l'ordre 1. La division permet donc d'inverser une loi de manière approximative : si l'on connaît la valeur de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> et de sa <a href="/wiki/D%C3%A9riv%C3%A9e" title="Dérivée">dérivée</a> en un point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}">, on peut écrire « autour de ce point » <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)\approx f(x_{0})+f'(x_{0})\times (x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)\approx f(x_{0})+f'(x_{0})\times (x-x_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2072d3c616c6141624179082dde27fc9f77fc46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.484ex; height:3.009ex;" alt="{\displaystyle y=f(x)\approx f(x_{0})+f'(x_{0})\times (x-x_{0})}"></dd></dl> et ainsi inverser la loi : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\simeq x_{0}+{\frac {f(x)-f(x_{0})}{f'(x_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≃</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\simeq x_{0}+{\frac {f(x)-f(x_{0})}{f'(x_{0})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2992f6df6fb439a8a6cd5610fa42bd4bfc77c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.218ex; height:6.509ex;" alt="{\displaystyle x\simeq x_{0}+{\frac {f(x)-f(x_{0})}{f'(x_{0})}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)\simeq x_{0}+{\frac {y-f(x_{0})}{f'(x_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)\simeq x_{0}+{\frac {y-f(x_{0})}{f'(x_{0})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b44271ed1bd8130fef9628d77d64c7609d54138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.245ex; height:6.509ex;" alt="{\displaystyle f^{-1}(y)\simeq x_{0}+{\frac {y-f(x_{0})}{f'(x_{0})}}}"></dd></dl> Ceci est par exemple utilisé dans l'<a href="/wiki/Algorithme_de_Newton" class="mw-redirect" title="Algorithme de Newton">algorithme de Newton</a>, qui recherche les zéros d'une fonction : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(0)\simeq x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>≃</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(0)\simeq x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e96e1e4c55708060ccc391181865cec35b9874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.982ex; height:6.509ex;" alt="{\displaystyle f^{-1}(0)\simeq x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vocabulaire_et_notations._Historique">Vocabulaire et notations. Historique</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=2" title="Modifier la section : Vocabulaire et notations. Historique" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=2" title="Modifier le code source de la section : Vocabulaire et notations. Historique">modifier le code</a>]</div> Le symbole actuel de la division est un trait horizontal séparant le numérateur (dividende) du dénominateur (diviseur). Par exemple, <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><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}"> divisé par <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> se note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9599ed0e64c39080cb58e4823f9ef3842b7b617d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\dfrac {a}{b}}}">. Le dénominateur donne la dénomination et le numérateur énumère : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {3}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1eead031a21bbe7e23e950a2d275825be9dac30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\dfrac {3}{4}}}"> indique qu'il s'agit de quarts, et qu'il y en a trois → trois quarts. <a href="/wiki/Diophante" class="mw-redirect" title="Diophante">Diophante</a> et les <a href="/wiki/Romains" class="mw-redirect" title="Romains">Romains</a>, au <a href="/wiki/IVe_si%C3%A8cle" title="IVe siècle"><abbr class="abbr" title="4ᵉ siècle">IVe</abbr> siècle</a> écrivaient déjà des fractions sous une forme semblable, les <a href="/wiki/Inde" title="Inde">Indiens</a> également au <a href="/wiki/XIIe_si%C3%A8cle" title="XIIe siècle"><abbr class="abbr" title="12ᵉ siècle">XIIe</abbr> siècle</a> et la notation moderne fut adoptée par les <a href="/wiki/Arabes" title="Arabes">Arabes</a>. Le symbole « : » a été plus tard utilisé par <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a>. Les fabricants de calculatrices impriment l'obélus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }"> ou la barre oblique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle /}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle /}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0c4de1fba637d9799f6c64a6c77bf016d0ce1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle /}"> sur la touche « opérateur division ». L'utilisation de ces symboles est plus ambiguë que la barre de fraction, puisqu'elle demande de définir des priorités, mais elle est pratique pour l'écriture « en ligne » utilisée en imprimerie ou sur un écran. Dans les publications scientifiques, on utilise plus volontiers les notations fractionnelles. La notation avec les deux-points est souvent utilisée pour représenter un rapport de quantités entières ou de longueurs. Aujourd'hui en France, en classe de <abbr class="abbr" title="Sixième">6e</abbr> de collège, les notations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }">, : et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle /}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle /}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0c4de1fba637d9799f6c64a6c77bf016d0ce1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle /}"> sont utilisées, car la division a pour les élèves un statut d'opération. Une nuance de sens est communément admise : <ul><li><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><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}"> et <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><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}"> désignent une opération (non effectuée), et le vocabulaire approprié est dividende pour <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><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}"> et diviseur pour <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> ;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9599ed0e64c39080cb58e4823f9ef3842b7b617d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\dfrac {a}{b}}}"> et <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b6c014398323cb45578581a536033fce1b28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.39ex; height:2.843ex;" alt="{\displaystyle a/b}"> désignent l'<a href="/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques)">écriture fractionnaire</a> du résultat de cette opération, et le vocabulaire approprié est numérateur pour <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><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}"> et dénominateur pour <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Mathématiques_et_langue_française">Mathématiques et langue française</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=3" title="Modifier la section : Mathématiques et langue française" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=3" title="Modifier le code source de la section : Mathématiques et langue française">modifier le code</a>]</div> On peut diviser une entité en un nombre de parties dont l'addition donne cette entité, par un moyen implicite ou explicite. Ainsi, on peut : <ul><li>diviser un gâteau en deux parts, par un coup de couteau</li> <li>simplement diviser un gâteau en deux [parts, par un moyen quelconque]</li> <li>diviser 1 en 2 demis, par la représentation mentale mathématique que l'on s'en fait</li> <li>simplement diviser 1 en <abbr class="abbr" title="Trente-sixième">36e</abbr></li> <li>etc.</li></ul> On peut également diviser par dichotomie ou par malice, mais diviser par 2 est un concept mathématique : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{b}}=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{b}}=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d278b8e5365a74e0def6f5a58807adbb2b4c56e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.171ex; height:4.843ex;" alt="{\displaystyle {\dfrac {a}{b}}=c}"> : « <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><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}"> divisé par <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> est égal à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ».</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Définition">Définition</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=4" title="Modifier la section : Définition" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=4" title="Modifier le code source de la section : Définition">modifier le code</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Division.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Division.png/220px-Division.png" decoding="async" width="220" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Division.png/330px-Division.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Division.png/440px-Division.png 2x" data-file-width="498" data-file-height="243" /></a><figcaption>Technique de division avec des variables.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Approche_élémentaire">Approche élémentaire</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=5" title="Modifier la section : Approche élémentaire" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=5" title="Modifier le code source de la section : Approche élémentaire">modifier le code</a>]</div> On commence par définir la division « avec reste » entre <a href="/wiki/Entier_naturel" title="Entier naturel">nombres entiers naturels</a>. Cette division peut s'approcher de manière intuitive par la notion de « partage, distribution équitable », et donne une procédure de calcul. <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Division_euclidienne" title="Division euclidienne">Division euclidienne</a>.</div></div> Cette notion permet déjà de mettre en évidence le problème de la division par <a href="/wiki/Z%C3%A9ro" title="Zéro">zéro</a> : comment partager une quantité en 0 part ? Cela n'a pas de sens, il faut au moins une part. Puis, on définit la notion de <a href="/wiki/Nombre_d%C3%A9cimal" title="Nombre décimal">nombre décimal</a>, et l'on étend la procédure de calcul en l'appliquant de manière récursive au reste, voir la section ci-dessous <a href="#Division_non_abrégée">Division non abrégée</a>. Cela permet de définir la notion de <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">nombre rationnel</a>. La notion de partage convient encore mais est plus difficile à appréhender. On peut imaginer des portions de part, donc diviser par des nombres fractionnaires : diviser une quantité par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d7ca945b5a3fb32ceb9513350b49ae0a19a4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.359ex; height:2.509ex;" alt="{\displaystyle 0,1}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6769669820a5abd2668e74b135580cb2c338598b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle 1/10}">), c'est dire que la quantité initiale représente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6769669820a5abd2668e74b135580cb2c338598b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle 1/10}"> part, et donc trouver la taille d'une part complète. Diviser une quantité par un nombre négatif, cela revient à calculer la taille d'une part que l'on enlève. Les <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombres réels</a> sont construits à partir des rationnels. Un nombre irrationnel ne peut pas se concevoir comme une quantité ; par contre, il peut se voir comme une proportion : le rapport entre la diagonale d'un carré et son côté, le rapport entre le périmètre d'un cercle et son diamètre. Dès lors, la division ne peut plus être définie comme un partage, mais comme la réciproque de la multiplication. Avec cette définition, on ne peut toujours pas diviser par 0 : puisque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\times a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>×</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\times a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05eeef4d767aa40a68dcac221dab7564bf7f2e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.494ex; height:2.176ex;" alt="{\displaystyle 0\times a=0}"> pour tout nombre, il existe une infinité de réciproques. On peut aussi — puisque diviser par un nombre revient à multiplier par son inverse — voir ce problème comme celui de la <a href="/wiki/Limite_(math%C3%A9matiques_%C3%A9l%C3%A9mentaires)" title="Limite (mathématiques élémentaires)">limite</a> en 0 de la <a href="/wiki/Fonction_inverse" title="Fonction inverse">fonction inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049b4bda6d9e222e496f2670248d9ecfb75841d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.171ex; height:2.843ex;" alt="{\displaystyle f(x)=1/x}"> : elle a deux limites, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> à gauche et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"> à droite. Le fait « d'étendre » les nombres réels en incluant des « pseudo-nombres infinis », <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> (<a href="/wiki/Droite_r%C3%A9elle_achev%C3%A9e" title="Droite réelle achevée">droite réelle achevée</a>), ne règle pas le problème, puisque reste le problème du signe. <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Division_par_z%C3%A9ro" title="Division par zéro">Division par zéro</a>.</div></div> La notion de division est donc fondamentale en <a href="/wiki/Alg%C3%A8bre" title="Algèbre">algèbre</a> et en <a href="/wiki/Analyse_r%C3%A9elle" title="Analyse réelle">analyse</a>. <div class="mw-heading mw-heading3"><h3 id="Définition_complète">Définition complète</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=6" title="Modifier la section : Définition complète" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=6" title="Modifier le code source de la section : Définition complète">modifier le code</a>]</div> Étant donné un <a href="/wiki/Anneau_int%C3%A8gre" title="Anneau intègre">anneau intègre</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,+,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,+,\times )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8afe943ffa3da32884751e2dbeeb2a5702e4ce98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.237ex; height:2.843ex;" alt="{\displaystyle (A,+,\times )}">, la division sur <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><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}"> est la loi de composition : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} \times \mathrm {A} \to \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} \times \mathrm {A} \to \mathrm {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3ecf48738f9fc8ee35f0eba42fe749bd4f3b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.684ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} \times \mathrm {A} \to \mathrm {A} }">, notée par exemple « <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }"> », telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (a,b,c)\in \mathrm {A} \times \mathrm {A} \times \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (a,b,c)\in \mathrm {A} \times \mathrm {A} \times \mathrm {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63f1b4803b414a42fa717f60e67050a4e2f1957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.155ex; height:2.843ex;" alt="{\displaystyle \forall (a,b,c)\in \mathrm {A} \times \mathrm {A} \times \mathrm {A} }">, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:b=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8a501c531bbbec4e350caf2c7e9cb53b49d5f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.27ex; height:2.176ex;" alt="{\displaystyle a:b=c}"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\times c=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>×</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\times c=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ee958ff9b9ff375fdf8b8a348bfd7c1d0f2659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.173ex; height:2.176ex;" alt="{\displaystyle b\times c=a}">.</dd></dl> L'intégrité de l'anneau assure que la division a bien un résultat unique. Par contre, elle n'est définie que sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} \times (\mathrm {A} -\{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} \times (\mathrm {A} -\{0\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fac2b9d3506eda80444979c3df6a9fa22f2a213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.464ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} \times (\mathrm {A} -\{0\})}"> si et seulement si <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><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}"> est un <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps commutatif</a>, et en aucun cas définie pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}">. Si la division n'est pas définie partout, on peut étendre conjointement la division et l'ensemble <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><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}"> : dans le cas commutatif, on définit sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} \times \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} \times \mathrm {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7522a18f1fd6980a4f13267e0487e660a8e6521b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} \times \mathrm {A} }"> une <a href="/wiki/Relation_d%27%C3%A9quivalence" title="Relation d'équivalence">relation d'équivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> par <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\sim (a',b')\iff a\times b'=a'\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>×</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\sim (a',b')\iff a\times b'=a'\times b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822713dba4cdd14c31d0a5067dd8bc93ab17b8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.11ex; height:3.009ex;" alt="{\displaystyle (a,b)\sim (a',b')\iff a\times b'=a'\times b}"></dd></dl> et on écrit <dl><dd><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><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}"> la classe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"> dans l'<a href="/wiki/Anneau_quotient" title="Anneau quotient">anneau quotient</a>.</dd></dl> Cet anneau quotient est un corps dont le neutre est la classe 1 : 1. C'est ainsi que l'on <a href="/wiki/Construction_des_nombres_rationnels" title="Construction des nombres rationnels">construit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> en symétrisant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> pour la multiplication (ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> à partir de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> en symétrisant l'addition). Cette définition ne recouvre pas celle de <a href="/wiki/Division_euclidienne" title="Division euclidienne">division euclidienne</a>, qui se pose de manière analogue mais dont le sens est radicalement différent. Dans l'idée, elle sert aussi à inverser la multiplication (dans <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><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}">, combien de fois <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">). Le problème de définition ne se pose plus, puisque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (a,b)\in \mathbb {N} \times (\mathbb {N} -\{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (a,b)\in \mathbb {N} \times (\mathbb {N} -\{0\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60098090d27e8c5c3c27a0d0d0cac75b7ba10261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.537ex; height:2.843ex;" alt="{\displaystyle \forall (a,b)\in \mathbb {N} \times (\mathbb {N} -\{0\})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n\in \mathbb {N} \ |\ b\times n\leqslant a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>b</mi> <mo>×</mo> <mi>n</mi> <mo>⩽</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n\in \mathbb {N} \ |\ b\times n\leqslant a\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6940f0252d43c92babfb61e0b2ca4dbcaebe37ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.607ex; height:2.843ex;" alt="{\displaystyle \{n\in \mathbb {N} \ |\ b\times n\leqslant a\}}"> est une partie de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> non vide et majorée, qui admet donc un plus grand élément. Cette division, fondamentale en arithmétique, introduit la notion de reste. Néanmoins, comme pour toutes les divisions, le <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> de la définition ne peut être <a href="/wiki/Z%C3%A9ro" title="Zéro">zéro</a>. <div class="mw-heading mw-heading2"><h2 id="Propriétés">Propriétés</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=7" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=7" title="Modifier le code source de la section : Propriétés">modifier le code</a>]</div> La division n'était pas à proprement parler une opération (<a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">loi de composition interne</a>, définie partout), ses « propriétés » n'ont pas d'implications structurelles sur les ensembles de nombres, et doivent être comprises comme des propriétés des nombres en écriture fractionnaire. Non-propriétés <ul><li>non <a href="/wiki/Commutativit%C3%A9" class="mw-redirect" title="Commutativité">commutative</a> car <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\div 3\neq 3\div 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>÷</mo> <mn>3</mn> <mo>≠</mo> <mn>3</mn> <mo>÷</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\div 3\neq 3\div 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239c543575bd826dba5f94113b6cd40cf75f2866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.429ex; height:2.676ex;" alt="{\displaystyle 5\div 3\neq 3\div 5}"></li> <li>non <a href="/wiki/Associativit%C3%A9" title="Associativité">associative</a> car <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\div (4\div 3)\neq (12\div 4)\div 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>÷</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>÷</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>12</mn> <mo>÷</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>÷</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\div (4\div 3)\neq (12\div 4)\div 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a23a44ae1fcb0c5d316d8dbbfdb53258fc201a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.378ex; height:2.843ex;" alt="{\displaystyle 12\div (4\div 3)\neq (12\div 4)\div 3}"></li></ul> Remarques <ul><li>pseudo-<a href="/wiki/%C3%89l%C3%A9ment_neutre" title="Élément neutre">élément neutre</a> à droite : 1 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{1}}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{1}}=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcc2f6df2a2f99a3482318464109456439fa256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.394ex; height:4.676ex;" alt="{\displaystyle {\dfrac {a}{1}}=a}"></dd></dl></li> <li>pseudo-<a href="/wiki/%C3%89l%C3%A9ment_absorbant" title="Élément absorbant">élément absorbant</a> à gauche : 0 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {0}{b}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> si </mtext> </mstyle> </mrow> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>0</mn> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {0}{b}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed05fb57e9df9327519aff40952a8211233be5d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.276ex; height:5.343ex;" alt="{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {0}{b}}=0}"></dd></dl></li> <li>égalité de fractions <ul><li>de même dénominateur <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {a}{b}}={\dfrac {c}{b}}\iff a=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> si </mtext> </mstyle> </mrow> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {a}{b}}={\dfrac {c}{b}}\iff a=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2632d51d8b75d71dcbe59129a27b325baa77522a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.256ex; height:4.843ex;" alt="{\displaystyle {\mbox{ si }}b\neq 0,{\dfrac {a}{b}}={\dfrac {c}{b}}\iff a=c}"></dd></dl></li> <li>en général (qui découle de la construction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{ si }}b\neq 0\wedge d\neq 0,{\dfrac {a}{b}}={\dfrac {c}{d}}\iff ad=bc\,{\text{(produit en croix)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> si </mtext> </mstyle> </mrow> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>∧</mo> <mi>d</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mi>d</mi> <mo>=</mo> <mi>b</mi> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(produit en croix)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{ si }}b\neq 0\wedge d\neq 0,{\dfrac {a}{b}}={\dfrac {c}{d}}\iff ad=bc\,{\text{(produit en croix)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3290e54a1b7f89fc06f58e2d56bd65ebd7761718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.905ex; height:4.843ex;" alt="{\displaystyle {\mbox{ si }}b\neq 0\wedge d\neq 0,{\dfrac {a}{b}}={\dfrac {c}{d}}\iff ad=bc\,{\text{(produit en croix)}}}"></dd></dl></li></ul></li> <li>ordre <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{ si }}b>0,{\dfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> si </mtext> </mstyle> </mrow> <mi>b</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{ si }}b>0,{\dfrac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ca56b99331551c2df2f2f96ec783785d533f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.083ex; height:4.843ex;" alt="{\displaystyle {\mbox{ si }}b>0,{\dfrac {a}{b}}}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {c}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {c}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39301e5eed2e24d6b776873508d6e2178e9e536c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.843ex; height:4.843ex;" alt="{\displaystyle {\dfrac {c}{b}}}"> sont dans le même ordre que <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><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}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="Algorithmes_de_la_division_de_nombres_décimaux">Algorithmes de la division de nombres décimaux</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=8" title="Modifier la section : Algorithmes de la division de nombres décimaux" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=8" title="Modifier le code source de la section : Algorithmes de la division de nombres décimaux">modifier le code</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Division_non_abrégée">Division non abrégée</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=9" title="Modifier la section : Division non abrégée" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=9" title="Modifier le code source de la section : Division non abrégée">modifier le code</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:Division_non_abr%C3%A9g%C3%A9e_22_6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Division_non_abr%C3%A9g%C3%A9e_22_6.svg/200px-Division_non_abr%C3%A9g%C3%A9e_22_6.svg.png" decoding="async" width="200" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Division_non_abr%C3%A9g%C3%A9e_22_6.svg/300px-Division_non_abr%C3%A9g%C3%A9e_22_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Division_non_abr%C3%A9g%C3%A9e_22_6.svg/400px-Division_non_abr%C3%A9g%C3%A9e_22_6.svg.png 2x" data-file-width="69" data-file-height="53" /></a><figcaption>Division non abrégée de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e140641068b3a0763290941fb20531be45227748" 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 23}"> par <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><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}">.</figcaption></figure> L'algorithme de division non abrégée, encore appelé division longue, sert à déterminer une écriture décimale du quotient de deux <a href="/wiki/Entier_relatif" title="Entier relatif">nombres entiers</a>. C'est une extension de la division euclidienne (voir <a href="/wiki/Poser_une_division#En_arithmétique" class="mw-redirect" title="Poser une division">Poser une division > Généralisation en arithmétique</a>). D'un point de vue pratique, il consiste à continuer la procédure, en « descendant des zéros », les nouveaux chiffres calculés s'ajoutant après la virgule. Cela se justifie par <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}\times 10^{n}={\frac {a\times 10^{n}}{b}}\Longrightarrow {\frac {a}{b}}={\frac {a\times 10^{n}}{b}}\div 10^{n}}"> <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> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>b</mi> </mfrac> </mrow> <mo stretchy="false">⟹</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>÷</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\times 10^{n}={\frac {a\times 10^{n}}{b}}\Longrightarrow {\frac {a}{b}}={\frac {a\times 10^{n}}{b}}\div 10^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed897f8f23a4e591932a734376f04f4a726b993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:45.092ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}\times 10^{n}={\frac {a\times 10^{n}}{b}}\Longrightarrow {\frac {a}{b}}={\frac {a\times 10^{n}}{b}}\div 10^{n}}"></dd></dl> où n est le nombre de décimales que l'on veut. Ainsi, on ajoute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> zéros à droite du numérateur, on effectue une division euclidienne classique, puis sur le quotient obtenu, les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> derniers chiffres sont après la virgule. Notons que l'on a intérêt à calculer la <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1^{\mathrm {e} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1^{\mathrm {e} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a488af33063b2ba75a741cb20b7f5ab70f4f92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.36ex; height:2.509ex;" alt="{\displaystyle n+1^{\mathrm {e} }}"> décimale pour savoir dans quel sens faire l'<a href="/wiki/Arrondi_(math%C3%A9matiques)" title="Arrondi (mathématiques)">arrondi</a>. Par exemple, pour calculer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 23:6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>23</mn> <mo>:</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23:6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16bff4d530595ed4d95a7f394ab1020d7e0edc54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.425ex; height:2.176ex;" alt="{\displaystyle 23:6}"> avec deux décimales, la procédure revient à calculer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\ 300:6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mtext> </mtext> <mn>300</mn> <mo>:</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ 300:6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cacdf01b4b6189841684d8475e35eccf4889269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.33ex; height:2.176ex;" alt="{\displaystyle 2\ 300:6}"> ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =383}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>383</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =383}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d576a9e09f40f79540d3f96ed863fab8f0fbfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle =383}">, reste <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}">) puis à diviser le résultat par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"> (ce qui donne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3,83}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> <mn>83</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,83}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049661708ca4ab063ccdf8287eb24f8cfc9d49c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.521ex; height:2.509ex;" alt="{\displaystyle 3,83}">). Cette procédure se généralise au quotient de deux <a href="/wiki/Nombre_d%C3%A9cimal" title="Nombre décimal">nombres décimaux</a> ; cela se justifie par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}={\frac {a\times 10^{n}}{b\times 10^{n}}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>b</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}={\frac {a\times 10^{n}}{b\times 10^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83794c584422d8bf5cad13b04de5ed284a46832c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.614ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}={\frac {a\times 10^{n}}{b\times 10^{n}}}}"></dd></dl> où <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> est un entier positif tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times 10^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times 10^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b771df47bce6150b178f9ffc5471f2037cd6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.614ex; height:2.343ex;" alt="{\displaystyle a\times 10^{n}}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\times 10^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\times 10^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b634c9c643cbc0dc81f7ec8629bf0acfe90bee60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.381ex; height:2.343ex;" alt="{\displaystyle b\times 10^{n}}">n soient des entiers ; on peut par exemple prendre le plus grand nombre de décimales dans les nombres <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><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}"> et <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. Deux situations peuvent se présenter : <ul><li>on finit par avoir un reste nul : on obtient donc un nombre décimal ;</li> <li>on finit par retomber par un reste qui est déjà apparu auparavant : la division « ne se termine pas », elle boucle à l'infini ; dans ce cas, le quotient est un rationnel non décimal, et on peut prouver que son <a href="/wiki/D%C3%A9veloppement_d%C3%A9cimal" title="Développement décimal">développement décimal</a> admet une période, dont la longueur est strictement inférieure au diviseur.</li></ul> Dans une division non exacte <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><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}">, (<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><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}"> et <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> étant deux nombres entiers, <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> non nul), si on note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1a924af25b278a955bfab1e1b0a437aa3389a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.096ex; height:2.343ex;" alt="{\displaystyle q_{p}}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02ec163fac8837ac757005d783b375e52808b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.108ex; height:2.343ex;" alt="{\displaystyle r_{p}}"> respectivement le quotient et le reste obtenus en poussant les itérations jusqu'à obtenir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> chiffres après la virgule du quotient, on obtient un encadrement ou une égalité : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{b}}\approx q_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>≈</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{b}}\approx q_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a8d34517a4dddddd596c63bf8182c7b0f985e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.261ex; height:4.843ex;" alt="{\displaystyle {\dfrac {a}{b}}\approx q_{p}}"> à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7351ec4a044a74cde7a786259c7afd84eee0cf1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.663ex; height:2.509ex;" alt="{\displaystyle 10^{-p}}"> près ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{p}<{\dfrac {a}{b}}<q_{p}+10^{-p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo><</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{p}<{\dfrac {a}{b}}<q_{p}+10^{-p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243f6223c531083b221c3925f42c09b016c4c28e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.958ex; height:4.843ex;" alt="{\displaystyle q_{p}<{\dfrac {a}{b}}<q_{p}+10^{-p}}"></dd></dl> et <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {a}{b}}=q_{p}+{\dfrac {r_{p}\cdot 10^{-p}}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mrow> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {a}{b}}=q_{p}+{\dfrac {r_{p}\cdot 10^{-p}}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d858f975dbfdb51929aa4b6e36c5defeaf831a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.387ex; height:6.009ex;" alt="{\displaystyle {\dfrac {a}{b}}=q_{p}+{\dfrac {r_{p}\cdot 10^{-p}}{b}}}"></dd></dl> Un nombre irrationnel (réel, sans être rationnel) ne peut s'écrire sous forme de fraction, par définition. C'est par contre la limite d'une suite de nombres rationnels (voir <a href="/wiki/Construction_des_nombres_r%C3%A9els#Définition_en_tant_qu'ensemble_2" title="Construction des nombres réels">Construction des nombres réels</a>). <div class="mw-heading mw-heading3"><h3 id="Division_de_nombres_codés_en_binaire">Division de nombres codés en binaire</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=10" title="Modifier la section : Division de nombres codés en binaire" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=10" title="Modifier le code source de la section : Division de nombres codés en binaire">modifier le code</a>]</div> La division en <a href="/wiki/Syst%C3%A8me_binaire" title="Système binaire">système binaire</a> est une opération fondamentale pour l'informatique. <div class="mw-heading mw-heading4"><h4 id="Division_euclidienne_binaire">Division euclidienne binaire</h4>[<a href="/w/index.php?title=Division&veaction=edit&section=11" title="Modifier la section : Division euclidienne binaire" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=11" title="Modifier le code source de la section : Division euclidienne binaire">modifier le code</a>]</div> On considère dans un premier temps des entiers naturels (positifs). La division se fait de la même manière que la division euclidienne. Soient deux nombres <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><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}"> et <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> bits. La notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ec626563476d6ab3c5dd577ccd7da0c6a6b3ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.842ex; height:2.843ex;" alt="{\displaystyle a(i)}"> désigne le <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ième bit (en partant de la droite, notés de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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 0}"> à <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}">), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(i:j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>:</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(i:j)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d8fa3206d61494eac92359e177f19d4ded6e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.737ex; height:2.843ex;" alt="{\displaystyle a(i:j)}"> désigne les bits situés entre les positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">. La fonction suivante calcule le quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> et le reste <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}">, en <a href="/wiki/Pseudo-code" title="Pseudo-code">pseudo-code</a> : <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre>fonction [Q, R] = diviser(a, b) si b == 0 alors génère l'exception "division par zéro" ; Q := 0 ; R := a ; // initialisation pour i = n-1 → 0 si a(n:i) >= b alors Q(i) = 1 ; // i-ème bit du quotient R = a(n:i) - b ; // reste fin fin retourne [Q, R] ; fin </pre></div> On cherche une portion de « bits forts » (chiffres de gauche) de <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><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}"> qui est supérieure à <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> ; si l'on n'en trouve pas, alors le quotient garde la valeur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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 0}">, et le reste <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"> prend la valeur <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><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}"> (puisqu'au final il est constitué de tous les chiffres de <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><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}">). Si à un certain rang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n:i)\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>:</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n:i)\geq b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d98a68f2b64c3be076a0bcf4eef06f257f690e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.269ex; height:2.843ex;" alt="{\displaystyle a(n:i)\geq b}">, alors du fait du système binaire, la réponse à <dl><dd>en a(n:i) combien de fois <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></dd></dl> est nécessairement « une fois » : en base <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><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}">, la réponse est entre <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><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}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9=10-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>=</mo> <mn>10</mn> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9=10-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9475847660d2debecb20241e3414d88f07721a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.589ex; height:2.343ex;" alt="{\displaystyle 9=10-1}"> ; ici, elle est entre <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><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}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=2-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=2-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/164a7fa1e1b2a92a2f10a08fb93b75ca9e4b53f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1=2-1}">. On a donc le <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ème bit du quotient qui vaut <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><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}"> ; le reste <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"> à cette étape est la différence. Ce pseudo-programme n'est pas optimisé pour des raisons de clarté ; une version plus efficace serait : <div class="mw-highlight mw-highlight-lang-text mw-content-ltr mw-highlight-lines" dir="ltr"><pre>fonction [Q, R] = diviser(a, b) si b == 0 alors génère l'exception "division par zéro" ; Q := 0 ; R := 0 ; // initialisation pour i = n-1 → 0 R = décalage_à_gauche_de_bits(R, 1) ; // équivaut à rajouter un 0 à droite R(1) = a(i) ; // le bit de poids faible de R est le i-ème bit du numérateur si R >= b alors Q(i) = 1 ; // i-ème bit du quotient R = R - b ; // reste fin fin retourne [Q, R] ; fin </pre></div> Cet algorithme marche également avec les <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombres réels</a> codés en <a href="/wiki/Virgule_fixe" title="Virgule fixe">virgule fixe</a>. Pour les nombres réels codés en <a href="/wiki/Virgule_flottante" title="Virgule flottante">virgule flottante</a>, il suffit de voir que : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2^{m}\,a}{2^{n}\,b}}=2^{m-n}{\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>a</mi> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2^{m}\,a}{2^{n}\,b}}=2^{m-n}{\frac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048a682465e69c78202c9237bd5c167fe5f50fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.557ex; height:5.509ex;" alt="{\displaystyle {\frac {2^{m}\,a}{2^{n}\,b}}=2^{m-n}{\frac {a}{b}}}"></dd></dl> donc on procède à la différence des exposants (des <a href="/wiki/Entier_relatif" title="Entier relatif">entiers relatifs</a>, codés en virgule fixe) et à la division des <a href="/wiki/Mantisse" title="Mantisse">mantisses</a> (des <a href="/wiki/Entier_naturel" title="Entier naturel">entiers naturels</a>). On voit que l'on ne fait appel qu'à des fonctions élémentaires — comparaison, décalage, assignation, <a href="/wiki/Soustraction" title="Soustraction">soustraction</a> — ce qui permet de le mettre en œuvre de manière simple dans un microprocesseur. <div class="mw-heading mw-heading4"><h4 id="Méthodes_lentes">Méthodes lentes</h4>[<a href="/w/index.php?title=Division&veaction=edit&section=12" title="Modifier la section : Méthodes lentes" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=12" title="Modifier le code source de la section : Méthodes lentes">modifier le code</a>]</div> Dans la division euclidienne de <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><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}"> par <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, pour des nombres représentés en base <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><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}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a46dbc619c25f03bb05880f1c44bdcf0753d52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B=2}"> en binaire), à l'étape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> : <ul><li>on considère le reste de l'étape précédente, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55baa9cdea636d16e2c79b9fa9f294abb2fad5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.664ex; height:2.509ex;" alt="{\displaystyle R_{i-1}}"> ;</li> <li>on détermine le <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-ème chiffre du quotient en partant de la gauche, donc le chiffre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=n-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=n-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53922bf37f3097e6548e77d6f094f8a9263b874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.121ex; height:2.509ex;" alt="{\displaystyle j=n-i}"> en partant de la droite : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n-i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e806282e9c85d5f8cab61abe521b30a064666e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.903ex; height:2.509ex;" alt="{\displaystyle Q_{n-i}}"> ;</li> <li>le nouveau reste vaut</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}=B\times R_{i-1}-Q_{n-i}\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>B</mi> <mo>×</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=B\times R_{i-1}-Q_{n-i}\times b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4a82d7d948d54b3998d419209e906e079322da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.512ex; height:2.509ex;" alt="{\displaystyle R_{i}=B\times R_{i-1}-Q_{n-i}\times b}">.</dd></dl> Cette construction par récurrence constitue la base des méthodes dites lentes. Connaissant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55baa9cdea636d16e2c79b9fa9f294abb2fad5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.664ex; height:2.509ex;" alt="{\displaystyle R_{i-1}}">, on suppose que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n-i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n-i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3230ce61d99b0f9c2b7361a1d7d409030d372072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.164ex; height:2.509ex;" alt="{\displaystyle Q_{n-i}=1}">, on a alors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}=2\times R_{i-1}-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=2\times R_{i-1}-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55b0ca044f304fe818a8715adc8e7d6b3877e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.167ex; height:2.509ex;" alt="{\displaystyle R_{i}=2\times R_{i-1}-b}">.</dd></dl> Si la valeur trouvée est négative, c'est que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n-i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n-i}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed858df9c2c5f51f0bca73838c25cd5565ad6a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.164ex; height:2.509ex;" alt="{\displaystyle Q_{n-i}=0}">. Par rapport à la procédure euclidienne, plutôt que de prendre les chiffres de gauche du numérateur, on ajout des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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 0}"> à droite du dénominateur. À la première étape, on en ajoute autant que le nombre de bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> servant à coder les nombres (il faut donc un espace mémoire double), puis on multiplie le numérateur par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"> jusqu'à vérifier la condition. Cela donne en pseudo-code (on omet le test de <a href="/wiki/Division_par_z%C3%A9ro" title="Division par zéro">division par zéro</a>) : <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre>fonction [Q, R] = diviser(a, b) si b == 0 alors génère l'exception "division par zéro" ; R := a // valeur initiale du reste b := décalage_à_gauche_de_bits(b, n) pour i = n-1 → 0 R := décalage_à_gauche_de_bits(b, 1) si R >= b alors Q(i) := 1 // le i-ème bit de i est 1 R = R - b sinon Q(i) := 0 // le i-ème bit de i est 0 fin fin retourner[Q, R] fin </pre></div> Lorsque l'on optimise l'algorithme pour réduire le nombre d'opérations, on obtient le pseudo-code suivant. <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre>fonction [Q, R] = diviser(a, b) si b == 0 alors génère l'exception "division par zéro" ; R := a // valeur initiale du reste b := décalage_à_gauche_de_bits(b, n) pour i = n-1 → 0 R := 2*R - b // le reste décalé est-il supérieur à b ? si R >= 0 alors Q(i) := 1 // le i-ème bit de i est 1 sinon Q(i) := 0 // le i-ème bit de i est 0 R := R + b // on restaure la valeur du reste en gardant le décalage fin fin retourner[Q, R] fin </pre></div> Du fait de la dernière instruction de la boucle, on qualifie cette méthode de « division avec restauration ». La méthode peut être améliorée en générant un quotient utilisant les nombres +1 et −1. Par exemple, le nombre codé par les bits <dl><dd><code>11101010</code></dd></dl> correspond en fait au nombre <dl><dd><code>11111111</code></dd></dl> c'est-à- dire à 11101010 -00010101 --------- 11010101 La méthode est dite « sans restauration » et son pseudo-code est : <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre>fonction [Q, R] = diviser(a, b) si b == 0 alors génère l'exception "division par zéro" ; R[0] := a i := 0 tant que i < n si R[i] >= 0 alors Q[n-(i+1)] := 1 R[i+1] := 2*R[i] - b sinon Q[n-(i+1)] := -1 R[i+1] := 2*R[i] + b fin i := i + 1 fin Q = transforme(Q) retourner[Q, R] fin </pre></div> La méthode de division SRT — du nom de ses inventeurs, Sweeney, Robertson, et Tocher —, est une méthode sans restauration, mais la détermination des bits du quotient se fait en utilisant une <a href="/wiki/Table_de_correspondance" title="Table de correspondance">table de correspondance</a> ayant pour entrées <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><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}"> et <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. C'est un algorithme utilisé dans de nombreux <a href="/wiki/Microprocesseur" title="Microprocesseur">microprocesseurs</a>. Alors que la méthode sans restauration classique ne permet de générer qu'un bit par <a href="/wiki/Cycle_d%27horloge" class="mw-redirect" title="Cycle d'horloge">cycle d'horloge</a>, la méthode SRT permet de générer deux bits par cycle<a href="#cite_note-3">[3]</a>. L'<a href="/wiki/Bug_de_la_division_du_Pentium" title="Bug de la division du Pentium">erreur de division du Pentium</a> était due à une erreur dans l'établissement de la table de correspondance<a href="#cite_note-4">[4]</a>. <div class="mw-heading mw-heading4"><h4 id="Méthodes_rapides">Méthodes rapides</h4>[<a href="/w/index.php?title=Division&veaction=edit&section=13" title="Modifier la section : Méthodes rapides" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=13" title="Modifier le code source de la section : Méthodes rapides">modifier le code</a>]</div> Les méthodes rapides consistent à évaluer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e380214d9e8b81984d7a9d034696a3e19ea23ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.751ex; height:2.843ex;" alt="{\displaystyle x=1/b}">, puis à calculer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=a\times x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>a</mi> <mo>×</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=a\times x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1430002e4dfea34811221835cc5b8898481927a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle Q=a\times x}">. La méthode de Newton-Raphson consiste à déterminer <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ec010ae7264f61ef9bced0326003d1f0e376ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.322ex; height:2.843ex;" alt="{\displaystyle 1/b}"> par la <a href="/wiki/M%C3%A9thode_de_Newton" title="Méthode de Newton">méthode de Newton</a>. La méthode de Newton permet de trouver le zéro d'une fonction en connaissant sa valeur et la valeur de sa dérivée en chaque point. Il faut donc trouver une fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{b}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{b}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32585d77264efc75b75cc18177cc6a14225d9737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.216ex; height:2.843ex;" alt="{\displaystyle f_{b}(x)}"> qui vérifie <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{b}(x)=0\Longleftrightarrow x=1/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⟺</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{b}(x)=0\Longleftrightarrow x=1/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab57729df89077cea3d4dfb409322f49dd1945fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.834ex; height:2.843ex;" alt="{\displaystyle f_{b}(x)=0\Longleftrightarrow x=1/b}"></dd></dl> et telle que l'on puisse effectuer l'itération <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{i-1}-{\frac {f_{b}(x_{i-1})}{f'_{b}(x_{i-1})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{i-1}-{\frac {f_{b}(x_{i-1})}{f'_{b}(x_{i-1})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a982f9e9dc8d0b3e8b0a4e20ef264afe7b3359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.25ex; height:6.676ex;" alt="{\displaystyle x_{i}=x_{i-1}-{\frac {f_{b}(x_{i-1})}{f'_{b}(x_{i-1})}}}"></dd></dl> sans avoir à connaître <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ec010ae7264f61ef9bced0326003d1f0e376ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.322ex; height:2.843ex;" alt="{\displaystyle 1/b}">. On peut par exemple utiliser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{b}(x)=1/x-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{b}(x)=1/x-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18cbd5e1ff7a0d20799349565e8da51ac3705cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.807ex; height:2.843ex;" alt="{\displaystyle f_{b}(x)=1/x-b}">, ce qui donne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{i-1}+x_{i-1}(1-bx_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{i-1}+x_{i-1}(1-bx_{i-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30695f84813fad9aada201b9c237e3d949ad2f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.568ex; height:2.843ex;" alt="{\displaystyle x_{i}=x_{i-1}+x_{i-1}(1-bx_{i-1})}">. Pour initialiser la procédure, il faut trouver une approximation de <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ec010ae7264f61ef9bced0326003d1f0e376ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.322ex; height:2.843ex;" alt="{\displaystyle 1/b}">. Pour cela, on normalise <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><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}"> et <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> par des décalages de bits afin d'avoir <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> compris dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,5;1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>5</mn> <mo>;</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,5;1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86d66c4f2a0184a3d46b1cf1dbd54ca923c30eb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [0,5;1]}">. On peut ensuite prendre une valeur arbitraire dans cet intervalle — par exemple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\approx 0,75}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≈</mo> <mn>0</mn> <mo>,</mo> <mn>75</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\approx 0,75}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0993886cf70c9130a5a928dec7f8a46b9c6ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.617ex; height:2.509ex;" alt="{\displaystyle b\approx 0,75}"> donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/b\approx 1,3{\underline {3}}\ldots }"> <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> <mo>≈</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>3</mn> <mo>_</mo> </munder> </mrow> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/b\approx 1,3{\underline {3}}\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449308c530e0cc3ce2eabff772a78b13474f050e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:14.053ex; height:3.343ex;" alt="{\displaystyle 1/b\approx 1,3{\underline {3}}\ldots }">, ou encore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq 1/b\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq 1/b\leq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e906b7409c5fd634160756be850e8d4ecc0628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.844ex; height:2.843ex;" alt="{\displaystyle 1\leq 1/b\leq 2}"> donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/b\approx 1,5}"> <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> <mo>≈</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/b\approx 1,5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea0666d3f3ab475cfd64c497ee2d24c509103d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:2.843ex;" alt="{\displaystyle 1/b\approx 1,5}"> —, ou bien faire le développement limité de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/x}"> en un point — par exemple en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,75}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>75</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,75}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4dfde8802ef018a95570b04609b0623891afb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.521ex; height:2.509ex;" alt="{\displaystyle 0,75}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/b\approx 2,6{\underline {6}}\ldots }"> <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> <mo>≈</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>6</mn> <mo>_</mo> </munder> </mrow> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/b\approx 2,6{\underline {6}}\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ccac783c065305d33137d9b751965d48148cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:14.053ex; height:3.343ex;" alt="{\displaystyle 1/b\approx 2,6{\underline {6}}\ldots }"> − <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,7{\underline {7}}\ldots \times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>7</mn> <mo>_</mo> </munder> </mrow> <mo>…</mo> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,7{\underline {7}}\ldots \times b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6b2c5ff02dfad8c5b5592acd890221a782c9ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:11.47ex; height:3.176ex;" alt="{\displaystyle 1,7{\underline {7}}\ldots \times b}">) ou en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/1,5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mo>,</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/1,5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8978a58bab8e7be01267a82a47a60ac1da9d1a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle 1/1,5}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/b\approx 3}"> <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> <mo>≈</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/b\approx 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5ce94eb74415254670c0d4c257ff23c1af0d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.583ex; height:2.843ex;" alt="{\displaystyle 1/b\approx 3}"> − <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,25\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>25</mn> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,25\times b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d9f557aa6ef13d09ac7098528b79b84d2b1a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.359ex; height:2.509ex;" alt="{\displaystyle 2,25\times b}">). On retient souvent l'approximation affine <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}={\frac {48}{17}}-{\frac {32}{17}}b\ (\simeq 2{,}82-1,88b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>48</mn> <mn>17</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>17</mn> </mfrac> </mrow> <mi>b</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mo>≃</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>82</mn> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>88</mn> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}={\frac {48}{17}}-{\frac {32}{17}}b\ (\simeq 2{,}82-1,88b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51dfa247d2a814cfb46311e85e0ec66e3c31f639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.979ex; height:5.343ex;" alt="{\displaystyle x_{0}={\frac {48}{17}}-{\frac {32}{17}}b\ (\simeq 2{,}82-1,88b)}"></dd></dl> les valeurs de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 48/17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>48</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 48/17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e3b013e142352b67113fcbb4e688bed35d6647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.812ex; height:2.843ex;" alt="{\displaystyle 48/17}"> et de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 32/17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32/17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10670ebcf0c3bcb642ba89ba6261de8f518d260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.812ex; height:2.843ex;" alt="{\displaystyle 32/17}"> étant précalculées (stockées « en dur »). Cette méthode <a href="/wiki/Vitesse_de_convergence_des_suites#Convergence_q-quadratique" title="Vitesse de convergence des suites">converge de manière quadratique</a>. Pour une précision sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> bits, il faut donc un nombre d'étapes <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><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}"> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\log _{2}\left({\frac {p+1}{\log _{2}17}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>17</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\log _{2}\left({\frac {p+1}{\log _{2}17}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5d55d502ea2850180c2a616aa6867713212ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.21ex; height:6.176ex;" alt="{\displaystyle s=\log _{2}\left({\frac {p+1}{\log _{2}17}}\right)}"></dd></dl> (arrondi au supérieur), soit trois étapes pour un codage en simple précision et quatre étapes en double précision et double précision étendue (selon la norme <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>). Voici le pseudo-code de l'algorithme. <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre>fonction [Q] = diviser(a, b) e := exposant(b) // b = M*2^e (représentation en virgule flottante) b' := b/2^{e + 1} // normalisation ; peut se faire par un décalage de e+1 bits à droite a' := a/2^{e + 1} // 0.5 <= b <= 1 ; a'/b' = a/b X := 48/17 - 32/17*b' // initialisation de la méthode de Newton pour i = 1 → s // s précalculé en fonction du nombre p de bits de la mantisse X := X + X*(1 - b*X) fin Q := a'*X retourne[Q] fin </pre></div> La méthode de Goldschmidt<a href="#cite_note-5">[5]</a> est fondée, elle, sur la considération suivante : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {Q} }{1}}={\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {Q} }{1}}={\frac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4865c07209ed1c95fd029570d273c0af33b082e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.809ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {Q} }{1}}={\frac {a}{b}}}"></dd></dl> donc, il existe un facteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\times F=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>×</mo> <mi>F</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\times F=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9722c149a2a7f6c632b3ddf46761c8c99cd3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.84ex; height:2.176ex;" alt="{\displaystyle b\times F=1}">, et ainsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times F=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>F</mi> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times F=Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d296780adf8ca8847fd468adcd57677dd59d3fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle a\times F=Q}">. Notons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=1/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=1/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e25d189957f7932e7631eb20f7f49755462b856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.162ex; height:2.843ex;" alt="{\displaystyle F=1/b}">. Le facteur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> est évalué par une suite (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}">) : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}=f_{k}\times f_{k-1}\times \ldots f_{1}=\prod (f_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>…</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>∏</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}=f_{k}\times f_{k-1}\times \ldots f_{1}=\prod (f_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce3e03920610ab0f357bd9e4f59fda268beebe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.039ex; height:3.843ex;" alt="{\displaystyle F_{k}=f_{k}\times f_{k-1}\times \ldots f_{1}=\prod (f_{i})}"></dd></dl> telle que la suite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{k})=F_{k}\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{k})=F_{k}\times b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8326b19f07e9f1ae221b66950b7214e701e22ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.415ex; height:2.843ex;" alt="{\displaystyle (b_{k})=F_{k}\times b}"> converge vers <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><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}">. Pour cela, on normalise la fraction pour que <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> se trouve dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]0;1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>;</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ]0;1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce0afd2865e794abecc52498533cd63249990e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle ]0;1]}">, et l'on définit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i+1}=2-b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i+1}=2-b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5014191e8ac235fac86007e7dd4a2778f2deeeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.938ex; height:2.509ex;" alt="{\displaystyle f_{i+1}=2-b_{i}}">.</dd></dl> Le quotient final vaut <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=F_{k}\times a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>×</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=F_{k}\times a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd559968bae7da47ded70df8cb4a1ee0f3cabbd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.59ex; height:2.509ex;" alt="{\displaystyle Q=F_{k}\times a}">.</dd></dl> Notons que l'on a <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i+1}=f_{i+1}\times F_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i+1}=f_{i+1}\times F_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/946f6e274efddbb367970c9573889d7423192fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.667ex; height:2.509ex;" alt="{\displaystyle F_{i+1}=f_{i+1}\times F_{i}}"> ;</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i+1}=f_{i+1}\times b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i+1}=f_{i+1}\times b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a43b95bd3da1f6c16b5b245c926ba97c4936df61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.673ex; height:2.509ex;" alt="{\displaystyle b_{i+1}=f_{i+1}\times b_{i}}">.</dd></dl> Cette méthode est notamment utilisée sur les microprocesseurs <a href="/wiki/Advanced_Micro_Devices" title="Advanced Micro Devices">AMD</a> <a href="/wiki/Athlon" title="Athlon">Athlon</a> et suivants<a href="#cite_note-6">[6]</a>,<a href="#cite_note-7">[7]</a>. La méthode binomiale est similaire à la méthode de Goldschmidt, mais consiste à prendre pour suite de facteurs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}=1+x^{2i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}=1+x^{2i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e69da8f4b8d7cb0d4e8a025d36eff317116c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.992ex; height:3.009ex;" alt="{\displaystyle f_{i}=1+x^{2i}}"> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb144ec1780ecbc13348980f8309e6739d547db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.429ex; height:2.343ex;" alt="{\displaystyle x=1-b}">. En effet, on a : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Q} ={\frac {a}{1-x}}={\frac {a(1+x)}{(1-x)(1+x)}}={\frac {a(1+x)}{1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Q} ={\frac {a}{1-x}}={\frac {a(1+x)}{(1-x)(1+x)}}={\frac {a(1+x)}{1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2624118c62186b46c105ec3629b10a1feca8b911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.6ex; height:6.509ex;" alt="{\displaystyle \mathrm {Q} ={\frac {a}{1-x}}={\frac {a(1+x)}{(1-x)(1+x)}}={\frac {a(1+x)}{1-x^{2}}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Q} ={\frac {a(1+x)(1+x^{2})}{1-x^{4}}}={\frac {a(1+x)(1+x^{2})(1+x^{4})}{1-x^{8}}}=\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Q} ={\frac {a(1+x)(1+x^{2})}{1-x^{4}}}={\frac {a(1+x)(1+x^{2})(1+x^{4})}{1-x^{8}}}=\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e92b94c54e66f9bf42ea95535d03f1a319ebe78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.831ex; height:6.343ex;" alt="{\displaystyle \mathrm {Q} ={\frac {a(1+x)(1+x^{2})}{1-x^{4}}}={\frac {a(1+x)(1+x^{2})(1+x^{4})}{1-x^{8}}}=\cdots }"></dd></dl> suivant la formule du <a href="/wiki/Bin%C3%B4me_de_Newton" class="mw-redirect" title="Binôme de Newton">binôme de Newton</a>. Si l'on normalise <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> pour qu'il soit dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,5;1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>5</mn> <mo>;</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,5;1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86d66c4f2a0184a3d46b1cf1dbd54ca923c30eb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [0,5;1]}">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> est dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0;0,5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>;</mo> <mn>0</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0;0,5]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b6c4862722a6a8c653ec1e366bacf7a3789cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [0;0,5]}"> et à l'étape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, le dénominateur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-x^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-x^{2n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7aee339a086f9ffd8ddca13cae056a64504937d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.373ex; height:2.843ex;" alt="{\displaystyle 1-x^{2n}}"> est égal à <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><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}"> avec une erreur de inférieure à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0414b5d9e81c2eb5bd85a6ca4af24b69d5336dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.659ex; height:2.509ex;" alt="{\displaystyle 2^{-n}}">, ce qui garantit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" 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 2^{n}}"> chiffres significatifs. Cette méthode est parfois désignée comme « méthode IBM »<a href="#cite_note-8">[8]</a>. <div class="mw-heading mw-heading2"><h2 id="Division_d'objets_autres_que_des_nombres_réels">Division d'objets autres que des nombres réels</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=14" title="Modifier la section : Division d'objets autres que des nombres réels" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=14" title="Modifier le code source de la section : Division d'objets autres que des nombres réels">modifier le code</a>]</div> On peut donc définir la division <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca8d6f1e2b9f1adc1c6d0fca9da3bb685fecb46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.818ex; height:2.843ex;" alt="{\displaystyle x=a/b}"> pour tout ensemble muni d'une multiplication, comme étant la solution de l'équation. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times b=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times b=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3232311257faa3e6560cfbf1097fa4f1a6492730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.496ex; height:2.176ex;" alt="{\displaystyle x\times b=a}"></dd></dl> Nous allons voir l'exemple des nombres complexes, des polynômes et des matrices. <div class="mw-heading mw-heading3"><h3 id="Division_complexe">Division complexe</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=15" title="Modifier la section : Division complexe" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=15" title="Modifier le code source de la section : Division complexe">modifier le code</a>]</div> Commençons par la notation polaire. Soient deux <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}=r_{1}\mathrm {e} ^{i\theta _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}=r_{1}\mathrm {e} ^{i\theta _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c540a25ca3b0ab41027629b234e7f757f12d36d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.771ex; height:3.009ex;" alt="{\displaystyle z_{1}=r_{1}\mathrm {e} ^{i\theta _{1}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}=r_{2}\mathrm {e} ^{i\theta _{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}=r_{2}\mathrm {e} ^{i\theta _{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0752893e5ebfefbacb2b3d3f428d1fe642f488ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.771ex; height:3.009ex;" alt="{\displaystyle z_{2}=r_{2}\mathrm {e} ^{i\theta _{2}}}">. La division complexe <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=z_{1}/z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=z_{1}/z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35332b71d99195d05b50f0f5fe6c63aced81bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.861ex; height:2.843ex;" alt="{\displaystyle x=z_{1}/z_{2}}"></dd></dl> est donc définie par <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times z_{2}=z_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times z_{2}=z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca0ed3dce706e211e91bf99eb957948faab7b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.539ex; height:2.009ex;" alt="{\displaystyle x\times z_{2}=z_{1}}"></dd></dl> Si on note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r\mathrm {e} ^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\mathrm {e} ^{i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0f6003e075fd673bc1433be69b3a7baea7008a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.08ex; height:2.676ex;" alt="{\displaystyle x=r\mathrm {e} ^{i\theta }}">, alors l'équation devient <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\mathrm {e} ^{i\theta }\times r_{2}\mathrm {e} ^{i\theta _{2}}=r_{1}\mathrm {e} ^{i\theta _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>×</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\mathrm {e} ^{i\theta }\times r_{2}\mathrm {e} ^{i\theta _{2}}=r_{1}\mathrm {e} ^{i\theta _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3fe5ca941017c2df2235111d231c06509ec0538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.666ex; height:3.009ex;" alt="{\displaystyle r\mathrm {e} ^{i\theta }\times r_{2}\mathrm {e} ^{i\theta _{2}}=r_{1}\mathrm {e} ^{i\theta _{1}}}"></dd></dl> soit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rr_{2}\mathrm {e} ^{i(\theta +\theta _{2})}=r_{1}\mathrm {e} ^{i\theta _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rr_{2}\mathrm {e} ^{i(\theta +\theta _{2})}=r_{1}\mathrm {e} ^{i\theta _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b45f50b8c76888777abd3049cc05a3ff99d7005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.551ex; height:3.176ex;" alt="{\displaystyle rr_{2}\mathrm {e} ^{i(\theta +\theta _{2})}=r_{1}\mathrm {e} ^{i\theta _{1}}}"></dd></dl> donc <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{aligned}rr_{2}=\ &r_{1}\\\theta +\theta _{2}=\ &\theta _{1}\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}r=\ &{\frac {r_{1}}{r_{2}}}\\\theta =\ &\theta _{1}-\theta _{2}\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> <mo stretchy="false">⟹</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}rr_{2}=\ &r_{1}\\\theta +\theta _{2}=\ &\theta _{1}\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}r=\ &{\frac {r_{1}}{r_{2}}}\\\theta =\ &\theta _{1}-\theta _{2}\end{aligned}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbad9bd8d6398cc5cb5fc45550591b34679aa3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.014ex; height:8.176ex;" alt="{\displaystyle \left\{{\begin{aligned}rr_{2}=\ &r_{1}\\\theta +\theta _{2}=\ &\theta _{1}\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}r=\ &{\frac {r_{1}}{r_{2}}}\\\theta =\ &\theta _{1}-\theta _{2}\end{aligned}}\right.}"></dd></dl> Ceci n'est défini que si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3aa55ea004066cc9667ba4c350198358866399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.364ex; height:2.676ex;" alt="{\displaystyle r_{2}\neq 0}"> c'est-à-dire <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b6e97e9bb66c69d7c181365c7ce75103b5b058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.396ex; height:2.676ex;" alt="{\displaystyle z_{2}\neq 0}"> On peut donc écrire <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{1}\mathrm {e} ^{\mathrm {i} \theta _{1}}}{r_{2}\mathrm {e} ^{\mathrm {i} \theta _{2}}}}={\frac {r_{1}}{r_{2}}}\mathrm {e} ^{\mathrm {i} (\theta _{1}-\theta _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r_{1}\mathrm {e} ^{\mathrm {i} \theta _{1}}}{r_{2}\mathrm {e} ^{\mathrm {i} \theta _{2}}}}={\frac {r_{1}}{r_{2}}}\mathrm {e} ^{\mathrm {i} (\theta _{1}-\theta _{2})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7075114dd4b759d2b0d89bc14f818f3fcea36b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.786ex; height:6.343ex;" alt="{\displaystyle {\frac {r_{1}\mathrm {e} ^{\mathrm {i} \theta _{1}}}{r_{2}\mathrm {e} ^{\mathrm {i} \theta _{2}}}}={\frac {r_{1}}{r_{2}}}\mathrm {e} ^{\mathrm {i} (\theta _{1}-\theta _{2})}}"></dd></dl> Voyons maintenant la notation cartésienne. On a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}=a+b\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}=a+b\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f19034cbdddc5802e736a71b616d7a61efff95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.944ex; height:2.509ex;" alt="{\displaystyle z_{1}=a+b\times i}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}=c+d\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}=c+d\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c9531cb07e5f0d9598e8a78de97102b9e432be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.94ex; height:2.509ex;" alt="{\displaystyle z_{2}=c+d\times i}">, et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=e+f\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>e</mi> <mo>+</mo> <mi>f</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=e+f\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc41143b06e89c4cfa1768b7895c789079450d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.273ex; height:2.509ex;" alt="{\displaystyle x=e+f\times i}">. L'équation de définition <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times z_{2}=z_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times z_{2}=z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca0ed3dce706e211e91bf99eb957948faab7b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.539ex; height:2.009ex;" alt="{\displaystyle x\times z_{2}=z_{1}}"></dd></dl> devient <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e+f\times i)\times (c+d\times i)=a+b\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>e</mi> <mo>+</mo> <mi>f</mi> <mo>×</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>×</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e+f\times i)\times (c+d\times i)=a+b\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35dbc5a510f5eccba2237912044fd659975d4432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.819ex; height:2.843ex;" alt="{\displaystyle (e+f\times i)\times (c+d\times i)=a+b\times i}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ec-fd+(de+cf)i=a+b\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>c</mi> <mo>−</mo> <mi>f</mi> <mi>d</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>e</mi> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ec-fd+(de+cf)i=a+b\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61b583c92e6dbafcd81e67a65c6d49679a20f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.112ex; height:2.843ex;" alt="{\displaystyle ec-fd+(de+cf)i=a+b\times i}"></dd></dl> soit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{aligned}ec-fd=\ &a\\de+cf=\ &b\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}e=\ &{\frac {ac+bd}{c^{2}+d^{2}}}\\f=\ &{\frac {bc-ad}{c^{2}+d^{2}}}\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>e</mi> <mi>c</mi> <mo>−</mo> <mi>f</mi> <mi>d</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>e</mi> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> <mo stretchy="false">⟹</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>e</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}ec-fd=\ &a\\de+cf=\ &b\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}e=\ &{\frac {ac+bd}{c^{2}+d^{2}}}\\f=\ &{\frac {bc-ad}{c^{2}+d^{2}}}\end{aligned}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893bc40af37d4c3d546d6c671579b96e28d2a1c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:35.826ex; height:11.509ex;" alt="{\displaystyle \left\{{\begin{aligned}ec-fd=\ &a\\de+cf=\ &b\end{aligned}}\right.\Longrightarrow \left\{{\begin{aligned}e=\ &{\frac {ac+bd}{c^{2}+d^{2}}}\\f=\ &{\frac {bc-ad}{c^{2}+d^{2}}}\end{aligned}}\right.}"></dd></dl> qui est défini si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}+d^{2}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}+d^{2}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8a872d23d8b27da02f617f51473116e695c220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.435ex; height:3.176ex;" alt="{\displaystyle c^{2}+d^{2}\neq 0}">, c'est-à-dire si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z_{2}|\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z_{2}|\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ebed3d171b9f0279f0eaa557164dcb11b5991a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.69ex; height:2.843ex;" alt="{\displaystyle |z_{2}|\neq 0}">, ce qui équivaut à dire que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abf655fa14f7ea44ad0ca781b59ff59c5f49117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}"> est non-nul. On peut donc écrire <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}\mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}\mathrm {i} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca671e4efb54fd39fbfefb86f735fafb23d7a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.23ex; height:5.843ex;" alt="{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}\mathrm {i} }"></dd></dl> Une mise en informatique « brute » de la méthode de calcul peut mener à des résultats problématiques. Dans un ordinateur, la précision des nombres est limitée par le mode de représentation. Si l'on utilise la <a href="/wiki/Virgule_flottante" title="Virgule flottante">double précision</a> selon la norme <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>, la <a href="/wiki/Valeur_absolue" title="Valeur absolue">valeur absolue</a> des nombres est limitée à environ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [10^{-307};10^{308}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mo>;</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>308</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [10^{-307};10^{308}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ad8542437ccc07fe235bddda1cc14dfc2d837d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.652ex; height:3.176ex;" alt="{\displaystyle [10^{-307};10^{308}]}">. Si le calcul génère une valeur absolue supérieure à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{308}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>308</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{308}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc41a9b36b5ca6956ccf5ee2a1790af579510b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.023ex; height:2.676ex;" alt="{\displaystyle 10^{308}}">, le résultat est considéré comme « infini » (<code>Inf</code>, <a href="/wiki/D%C3%A9passement_d%27entier" title="Dépassement d'entier">erreur de dépassement</a>) ; et si la valeur absolue est inférieure à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-307}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-307}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76eca23b7e472ffe7ce5439789b84eb8da9c815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.676ex;" alt="{\displaystyle 10^{-307}}">, le résultat est considéré comme nul (<a href="/wiki/Soupassement_arithm%C3%A9tique" title="Soupassement arithmétique">soupassement</a>). Or, la formule ci-dessus faisant intervenir des produits et des carrés, on peut avoir un intermédiaire de calcul dépassant les capacités de représentation alors que le résultat final (les nombres <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">) peuvent être représentés. Notons que dans la norme IEEE 754, <code>1/0</code> donne <code>Inf</code> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }">) et <code>−1/0</code> donne <code>−Inf</code> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">), mais en mettant un drapeau indiquant l'erreur « division par zéro » ; le calcul de <code>1/Inf</code> donne <code>0</code>. Si par exemple l'on met en œuvre cette formule pour calculer<a href="#cite_note-9">[9]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+i)/(1+10^{-307}i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+i)/(1+10^{-307}i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a758dbf279752c70c3ea76010963a23b0affaa00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.693ex; height:3.176ex;" alt="{\displaystyle (1+i)/(1+10^{-307}i)}">, on obtient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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 0}"> alors que le résultat est environ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-307}-10^{-307}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-307}-10^{-307}i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9daf11f9103d430f07945a224ccc254a96e0c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.246ex; height:2.843ex;" alt="{\displaystyle 10^{-307}-10^{-307}i}"> (ce qui est représentable) ;</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+i)/(10^{-307}+10^{-307}i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+i)/(10^{-307}+10^{-307}i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e8b08c4de7c8cc4425a078243f1c4e6162882e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.833ex; height:3.176ex;" alt="{\displaystyle (1+i)/(10^{-307}+10^{-307}i)}">, on obtient <code>NaN</code> (résultat d'une opération <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0/0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0/0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1260a93f6fb76e30f25d5633b42e39e2c2fda79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 0/0}">), alors que le résultat est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{307}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>307</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{307}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc56e65c8d9943c4bc4ee7df09bad0a255f9b806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.023ex; height:2.676ex;" alt="{\displaystyle 10^{307}}"> (représentable) ;</dd></dl> Ces erreurs sont dues au calcul de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (10^{307})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>307</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (10^{307})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1840d205515211040ce66bb5a21b1c368d0ae524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.887ex; height:3.176ex;" alt="{\displaystyle (10^{307})^{2}}"> et de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (10^{-307})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>307</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (10^{-307})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c84c2bb42e65b0e2e106916a1203253117ad585" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.165ex; height:3.176ex;" alt="{\displaystyle (10^{-307})^{2}}">. Pour limiter ces erreurs, on peut utiliser la méthode de Smith<a href="#cite_note-10">[10]</a>, qui consiste à factoriser de manière pertinente : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}=\left\{{\begin{aligned}{\text{si }}|d|\ll |c|{\text{ : }}&{\frac {a+b(d/c)}{c+d(d/c)}}+{\frac {b-a(d/c)}{c+d(d/c)}}\mathrm {i} \\{\text{si }}|d|\gg |c|{\text{ : }}&{\frac {a(c/d)+b}{c(c/d)+d}}+{\frac {b(c/d)-a}{c(c/d)+d}}\mathrm {i} \\\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> : </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> : </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}=\left\{{\begin{aligned}{\text{si }}|d|\ll |c|{\text{ : }}&{\frac {a+b(d/c)}{c+d(d/c)}}+{\frac {b-a(d/c)}{c+d(d/c)}}\mathrm {i} \\{\text{si }}|d|\gg |c|{\text{ : }}&{\frac {a(c/d)+b}{c(c/d)+d}}+{\frac {b(c/d)-a}{c(c/d)+d}}\mathrm {i} \\\end{aligned}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5be10431dace4a8331b473f9b7cfe6522a5f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:53.367ex; height:13.176ex;" alt="{\displaystyle {\frac {a+b\times \mathrm {i} }{c+d\times \mathrm {i} }}=\left\{{\begin{aligned}{\text{si }}|d|\ll |c|{\text{ : }}&{\frac {a+b(d/c)}{c+d(d/c)}}+{\frac {b-a(d/c)}{c+d(d/c)}}\mathrm {i} \\{\text{si }}|d|\gg |c|{\text{ : }}&{\frac {a(c/d)+b}{c(c/d)+d}}+{\frac {b(c/d)-a}{c(c/d)+d}}\mathrm {i} \\\end{aligned}}\right.}"></dd></dl> En effet, dans le premier cas, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |d/c|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |d/c|<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5980146461e51ca7d54974f31829a6897ae65e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.94ex; height:2.843ex;" alt="{\displaystyle |d/c|<1}"> donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |xd/c|<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |xd/c|<x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aa127ac4ff2d21ee8c76b0f3c8e9da3dc12235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.437ex; height:2.843ex;" alt="{\displaystyle |xd/c|<x}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a,b,d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a,b,d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab18ef45d446a4940b452575a1d175f0f87a4013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.939ex; height:2.509ex;" alt="{\displaystyle x=a,b,d}">), donc la méthode est moins sensible aux dépassements. <div class="mw-heading mw-heading3"><h3 id="Division_matricielle">Division matricielle</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=16" title="Modifier la section : Division matricielle" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=16" title="Modifier le code source de la section : Division matricielle">modifier le code</a>]</div> La <a href="/wiki/Multiplication_matricielle" class="mw-redirect" title="Multiplication matricielle">multiplication matricielle</a> n'étant pas commutative, on définit deux divisions matricielles. <div class="mw-heading mw-heading4"><h4 id="Division_à_droite">Division à droite</h4>[<a href="/w/index.php?title=Division&veaction=edit&section=17" title="Modifier la section : Division à droite" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=17" title="Modifier le code source de la section : Division à droite">modifier le code</a>]</div> Soit <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><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}"> une matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"> et <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><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}"> une matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}">, alors on appelle « division à droite de <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><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}"> par <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><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}"> » et on note <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=A/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A/B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379c494f7b6727d46e3481ad4bbd74cdc75f8329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.748ex; height:2.843ex;" alt="{\displaystyle X=A/B}"></dd></dl> la matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e459f087ef822dc6fba54b953c60de61be69c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.05ex; height:2.009ex;" alt="{\displaystyle m\times p}"> vérifiant l'équation : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XB=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XB=A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8f9572c3216d07089f4926176059d7c87df569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.586ex; height:2.176ex;" alt="{\displaystyle XB=A}"></dd></dl> <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid #aaa; text-align: justify;"> Théorème — Si <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><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}"> est une <a href="/wiki/Matrice_r%C3%A9guli%C3%A8re" class="mw-redirect" title="Matrice régulière">matrice régulière</a> (matrice carrée inversible), alors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=A/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A/B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379c494f7b6727d46e3481ad4bbd74cdc75f8329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.748ex; height:2.843ex;" alt="{\displaystyle X=A/B}"></dd></dl> est équivalent à <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=AB^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <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 X=AB^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bd055b520a185b311c6994b6056809b8149bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.918ex; height:2.676ex;" alt="{\displaystyle X=AB^{-1}}"></dd></dl> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration</div><div class="NavContent" style="padding-bottom:0.4em"> Il vient immédiatement de <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XB=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XB=A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8f9572c3216d07089f4926176059d7c87df569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.586ex; height:2.176ex;" alt="{\displaystyle XB=A}"></dd></dl> que <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (XB)^{-1}=AB^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <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 (XB)^{-1}=AB^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68170da515c7bdb5dc20e2c169eaa15096b16fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.824ex; height:3.176ex;" alt="{\displaystyle (XB)^{-1}=AB^{-1}}"></dd></dl> ce qui donne le résultat par associativité. </div><div class="clear" style="clear:both;"></div> </div> <div class="mw-heading mw-heading4"><h4 id="Division_à_gauche">Division à gauche</h4>[<a href="/w/index.php?title=Division&veaction=edit&section=18" title="Modifier la section : Division à gauche" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=18" title="Modifier le code source de la section : Division à gauche">modifier le code</a>]</div> Soit <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><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}"> une matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"> et <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><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}"> une matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e459f087ef822dc6fba54b953c60de61be69c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.05ex; height:2.009ex;" alt="{\displaystyle m\times p}">, alors on appelle « division à gauche de <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><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}"> par <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><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}"> » et on note <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=A\backslash B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>A</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A\backslash B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d75218911eaf9e8d02e0d89564e399eaf54e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.748ex; height:2.843ex;" alt="{\displaystyle X=A\backslash B}"></dd></dl> la matrice de dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"> vérifiant l'équation : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AX=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AX=B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51b626e9793be3e577dc5157905281ec0af0d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.586ex; height:2.176ex;" alt="{\displaystyle AX=B}"></dd></dl> Si la matrice <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><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}"> n'est pas de <a href="/wiki/Rang_(math%C3%A9matiques)" class="mw-redirect" title="Rang (mathématiques)">rang</a> maximal, la solution n'est pas unique. La matrice <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> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66bb9c954299f9ac5d2fd5c5cf5cd347ed14a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.018ex; height:2.509ex;" alt="{\displaystyle A^{+}B}">, où <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.509ex;" alt="{\displaystyle A^{+}}"> désigne la matrice <a href="/wiki/Pseudo-inverse" title="Pseudo-inverse">pseudo-inverse</a>, est la solution de <a href="/wiki/Norme_(math%C3%A9matiques)" title="Norme (mathématiques)">norme</a> minimale. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid #aaa; text-align: justify;"> Théorème — Si <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><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}"> est une <a href="/wiki/Matrice_r%C3%A9guli%C3%A8re" class="mw-redirect" title="Matrice régulière">matrice régulière</a> (matrice carrée inversible), alors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=A\backslash B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>A</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A\backslash B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d75218911eaf9e8d02e0d89564e399eaf54e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.748ex; height:2.843ex;" alt="{\displaystyle X=A\backslash B}"></dd></dl> est équivalent à <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=A^{-1}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A^{-1}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe53e2f9ce5d9abae9614f9811d922b5f961a84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.918ex; height:2.676ex;" alt="{\displaystyle X=A^{-1}B}"></dd></dl> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration</div><div class="NavContent" style="padding-bottom:0.4em"> Il vient immédiatement de <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AX=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AX=B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51b626e9793be3e577dc5157905281ec0af0d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.586ex; height:2.176ex;" alt="{\displaystyle AX=B}"></dd></dl> que <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}(AX)=A^{-1}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}(AX)=A^{-1}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02bdd92e6e472747c4c216cfe11835d7577843e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.547ex; height:3.176ex;" alt="{\displaystyle A^{-1}(AX)=A^{-1}B}"></dd></dl> ce qui donne le résultat par associativité. </div><div class="clear" style="clear:both;"></div> </div> On a par ailleurs : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B/A={^{t}({^{t}A}\backslash {^{t}B})}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mi>A</mi> </mrow> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mi>B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/A={^{t}({^{t}A}\backslash {^{t}B})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bb8f90b8881c743b7f7f35fa8295310e18c4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.725ex; height:3.009ex;" alt="{\displaystyle B/A={^{t}({^{t}A}\backslash {^{t}B})}}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Division_de_polynômes">Division de polynômes</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=19" title="Modifier la section : Division de polynômes" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=19" title="Modifier le code source de la section : Division de polynômes">modifier le code</a>]</div> On peut définir la division de polynômes à la manière de la division euclidienne : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} =\mathrm {B} \times \mathrm {Q} +\mathrm {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} =\mathrm {B} \times \mathrm {Q} +\mathrm {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed18f4c1babd667cc7be5b085a8f1cc742f051a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.687ex; height:2.509ex;" alt="{\displaystyle \mathrm {A} =\mathrm {B} \times \mathrm {Q} +\mathrm {R} }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {deg} (R)<\mathrm {deg} (B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">g</mi> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">g</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {deg} (R)<\mathrm {deg} (B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f42e60e63c0b77a73538b16089b0f5e0f6149b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.22ex; height:2.843ex;" alt="{\displaystyle \mathrm {deg} (R)<\mathrm {deg} (B)}"></dd></dl> où <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"> sont des polynômes ; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> est le quotient et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"> est le reste. <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Division_d%27un_polyn%C3%B4me" title="Division d'un polynôme">Division d'un polynôme</a>.</div></div> On peut également appliquer la méthode de <a href="/wiki/Construction_des_nombres_rationnels" title="Construction des nombres rationnels">construction des nombres rationnels</a> aux polynômes, ce qui permet de définir de manière formelle une fraction de polynômes. Mais il ne s'agit pas là d'un « calcul » de division, il s'agit de définir un nouvel objet mathématique, un nouvel outil. <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Fraction_rationnelle" title="Fraction rationnelle">Fraction rationnelle</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références">Notes et références</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=20" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=20" title="Modifier le code source de la section : Notes et références">modifier le code</a>]</div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> 14:00-17:00, « <a rel="nofollow" class="external text" href="https://www.iso.org/fr/standard/64973.html"><cite style="font-style:normal;">ISO 80000-2:2019</cite></a> », sur ISO, <time class="nowrap" datetime="2020-05-19" data-sort-value="2020-05-19">19 mai 2020</time> (consulté le <time class="nowrap" datetime="2023-12-07" data-sort-value="2023-12-07">7 décembre 2023</time>) </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> « <a rel="nofollow" class="external text" href="https://www.boutique.afnor.org/fr-fr/norme/nf-en-iso-800002/grandeurs-et-unites-partie-2-signes-et-symboles-mathematiques-a-employer-da/fa180173/1349"><cite style="font-style:normal;">NF EN ISO 80000-2</cite></a> », sur Afnor EDITIONS (consulté le <time class="nowrap" datetime="2023-12-07" data-sort-value="2023-12-07">7 décembre 2023</time>) </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> [<a rel="nofollow" class="external text" href="http://www.intel.com/support/processors/pentium/sb/CS-013008.htm">Pentium® Processor Divide Algorithm</a>] </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> [<a rel="nofollow" class="external text" href="http://www.intel.com/support/processors/pentium/fdiv/wp/">Statistical Analysis of Floating Point Flaw</a>], Intel Corporation, 1994 </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Robert Elliot Goldschmidt, « <cite style="font-style:normal">Applications of Division by Convergence</cite> », MSc dissertation, M.I.T.,‎ <time>1964</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Applications+of+Division+by+Convergence&rft.jtitle=MSc+dissertation&rft.aulast=Goldschmidt&rft.aufirst=Robert+Elliot&rft.date=1964&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADivision"> </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Stuart F. Oberman, <cite style="font-style:normal" lang="en">« Floating Point Division and Square Root Algorithms and Implementation in the AMD-K7 Microprocessor »</cite>, dans <cite class="italique" lang="en">Proc. IEEE Symposium on Computer Arithmetic</cite>, <time>1999</time>, <abbr class="abbr" title="pages">p.</abbr> 106–115<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Proc.+IEEE+Symposium+on+Computer+Arithmetic&rft.atitle=Floating+Point+Division+and+Square+Root+Algorithms+and+Implementation+in+the+AMD-K7+Microprocessor&rft.aulast=Oberman&rft.aufirst=Stuart+F.&rft.date=1999&rft.pages=106%E2%80%93115&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADivision"> </li> <li id="cite_note-7"><a href="#cite_ref-7">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Peter Soderquist et Miriam Leeser, « <cite style="font-style:normal" lang="en">Division and Square Root: Choosing the Right Implementation</cite> », IEEE Micro, <abbr class="abbr" title="volume">vol.</abbr> 17, <abbr class="abbr" title="numéro">no</abbr> 4,‎ <time class="nowrap" datetime="1997" data-sort-value="1997">juillet/août 1997</time>, <abbr class="abbr" title="pages">p.</abbr> 56–66<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Division+and+Square+Root%3A+Choosing+the+Right+Implementation&rft.jtitle=IEEE+Micro&rft.issue=4&rft.aulast=Soderquist&rft.aufirst=Peter&rft.au=Leeser%2C+Miriam&rft.date=1997&rft.volume=17&rft.pages=56%E2%80%9366&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADivision"> </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> Paul Molitor, [<abbr class="abbr indicateur-langue" title="Langue : allemand">(de)</abbr> <a rel="nofollow" class="external text" href="https://mephisto.informatik.uni-halle.de/aufzeichnungen/vhdl/folien/vhdl06_sw.pdf">Entwurf digitaler Systeme mit VHDL</a>] </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> [<abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a rel="nofollow" class="external text" href="http://www.scilab.org/content/download/249/1710/file/scilabisnotnaive.pdf">Scilab is not naive</a>] </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Robert L. Smith, « <cite style="font-style:normal" lang="en">Algorithm 116: Complex division</cite> », Communications of the ACM, New-York, Association for Computing Machinery, <abbr class="abbr" title="volume">vol.</abbr> 5, <abbr class="abbr" title="numéro">no</abbr> 8,‎ <time class="nowrap" datetime="1962-08" data-sort-value="1962-08">août 1962</time>, <abbr class="abbr" title="page">p.</abbr> 435 (<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1145/368637.368661">10.1145/368637.368661</a>, <a rel="nofollow" class="external text" href="http://dl2.acm.org/citation.cfm?id=368661">lire en ligne</a>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Algorithm+116%3A+Complex+division&rft.jtitle=Communications+of+the+ACM&rft.issue=8&rft.aulast=Smith&rft.aufirst=Robert+L.&rft.date=1962-08&rft.volume=5&rft.pages=435&rft_id=info%3Adoi%2F10.1145%2F368637.368661&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADivision"> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2>[<a href="/w/index.php?title=Division&veaction=edit&section=21" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=21" title="Modifier le code source de la section : Voir aussi">modifier le code</a>]</div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> Sur les autres projets Wikimedia : <ul class="noarchive plainlinks"> <li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/Op%C3%A9rations_%C3%A9l%C3%A9mentaires" class="extiw" title="v:Opérations élémentaires">Opérations élémentaires</a>, sur Wikiversity</li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3>[<a href="/w/index.php?title=Division&veaction=edit&section=22" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Division&action=edit&section=22" title="Modifier le code source de la section : Articles connexes">modifier le code</a>]</div> <ul><li><a href="/wiki/Divisibilit%C3%A9" title="Divisibilité">Divisibilité</a></li> <li><a href="/wiki/Division_par_deux" title="Division par deux">Division par deux</a></li> <li><a href="/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques)">Fraction</a></li> <li><a href="/wiki/Congruence_sur_les_entiers" title="Congruence sur les entiers">Congruence sur les entiers</a></li> <li><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint collapsed" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Op%C3%A9rations_binaires" title="Modèle:Palette Opérations binaires"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Op%C3%A9rations_binaires&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Op%C3%A9ration_binaire" title="Opération binaire">Opérations binaires</a></div></th> </tr> <tr> <td class="navbox-list" style="" colspan="2"><table class="navbox-columns-table" style="text-align:left;margin:0px;width:100%;"><tbody><tr><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;">Numériques</td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;">En ensemble ordonné</td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;">Structurelles</td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;">Autres</td></tr><tr style="vertical-align:top;"><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> Élémentaires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> <a href="/wiki/Addition" title="Addition">Addition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"> <a href="/wiki/Soustraction" title="Soustraction">Soustraction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"> <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }"> <a class="mw-selflink selflink">Division</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97daf05d8be90237eb4d2e4a46d3733ca829bd1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:2.009ex;" alt="{\displaystyle {\hat {}}}"> <a href="/wiki/Puissance_d%27un_nombre" title="Puissance d'un nombre">Puissance</a> Arithmétiques <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {div} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2d000acb2863a923fa83e5bee0635410b909bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.167ex; height:2.176ex;" alt="{\displaystyle \mathrm {div} }"> <a href="/wiki/Division_euclidienne" title="Division euclidienne">Quotient euclidien</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {mod} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {mod} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a50b3797e46642fd900093e002af888607a78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.391ex; height:2.176ex;" alt="{\displaystyle \mathrm {mod} }"> <a href="/wiki/Reste" title="Reste">Reste euclidien</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {pgcd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pgcd} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a99af40437989ee3ff61087ef6784068cc7d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.78ex; height:2.509ex;" alt="{\displaystyle \mathrm {pgcd} }"> <a href="/wiki/Plus_grand_commun_diviseur" title="Plus grand commun diviseur">PGCD</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ppcm} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ppcm} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2414692d872334199729ef7ec1ee384b8a57f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.553ex; height:2.009ex;" alt="{\displaystyle \mathrm {ppcm} }"> <a href="/wiki/Plus_petit_commun_multiple" title="Plus petit commun multiple">PPCM</a> Combinatoires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ()}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ()}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bc8aa05e1302397bb3e7877e842784991351df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.843ex;" alt="{\displaystyle ()}"> <a href="/wiki/Coefficient_binomial" title="Coefficient binomial">Coefficient binomial</a> <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><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}"> <a href="/wiki/Arrangement" title="Arrangement">Arrangement</a> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> Ensembles de parties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"> <a href="/wiki/Union_(math%C3%A9matiques)" title="Union (mathématiques)">Union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \backslash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant" mathvariant="normal">∖</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \backslash }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e865e5b244a7a5d4fdf7a95fe4cea6d25e581ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \backslash }"> <a href="/wiki/Alg%C3%A8bre_des_parties_d%27un_ensemble#Difference" title="Algèbre des parties d'un ensemble">Différence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }"> <a href="/wiki/Intersection_(math%C3%A9matiques)" title="Intersection (mathématiques)">Intersection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> <a href="/wiki/Alg%C3%A8bre_des_parties_d%27un_ensemble#Difference_symetrique" title="Algèbre des parties d'un ensemble">Différence symétrique</a> Ordre total <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695d28931288a686335c3969dfd15bb76ea873db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \min }"> <a href="/wiki/Extremum" title="Extremum">Minimum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e49fca3e322708b32d21eaa8b095dc05f09538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }"> <a href="/wiki/Extremum" title="Extremum">Maximum</a> Treillis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"> <a href="/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné)">Borne inférieure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"> <a href="/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné)">Borne supérieure</a> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <div style="column-count:2;column-gap:1em;text-align:center" class="colonnes"> Ensembles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"> <a href="/wiki/Produit_cart%C3%A9sien" title="Produit cartésien">Produit cartésien</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\cup }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∪</mo> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\cup }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca2f2880f761544ce268abf4057fcc1f50c0e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.509ex;" alt="{\displaystyle {\dot {\cup }}}"> <a href="/wiki/Op%C3%A9ration_ensembliste#Somme_disjointe" title="Opération ensembliste">Somme disjointe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97daf05d8be90237eb4d2e4a46d3733ca829bd1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:2.009ex;" alt="{\displaystyle {\hat {}}}"> <a href="/wiki/Op%C3%A9ration_ensembliste#Exponentiation" title="Opération ensembliste">Puissance ensembliste</a> Groupes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"> <a href="/wiki/Somme_directe" title="Somme directe">Somme directe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"> <a href="/wiki/Produit_libre" title="Produit libre">Produit libre</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wr }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≀</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wr }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4cda60813dda84640dc58dc142774abfa5a07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \wr }"> <a href="/wiki/Produit_en_couronne" title="Produit en couronne">Produit en couronne</a> Modules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"> <a href="/wiki/Produit_tensoriel" title="Produit tensoriel">Produit tensoriel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea3aafc91ebbd147d45c3c69e88431c48cbe9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle \mathrm {Hom} }"> <a href="/wiki/Foncteur_Hom" title="Foncteur Hom">Homomorphisme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Tor} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Tor} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab99a129c5d7d0c5726c68a87fff94e73a0d57fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.752ex; height:2.176ex;" alt="{\displaystyle \mathrm {Tor} }"> <a href="/wiki/Foncteur_Tor" title="Foncteur Tor">Torsion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ext} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ext} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ext} }"> <a href="/wiki/Foncteur_Ext" title="Foncteur Ext">Extension</a> Arbres <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"> <a href="/w/index.php?title=Enracinement_(math%C3%A9matiques)&action=edit&redlink=1" class="new" title="Enracinement (mathématiques) (page inexistante)">Enracinement</a> Variétés connexes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72e8254fd59fa4060c66c9310acbaf6df2ce894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.936ex; height:2.509ex;" alt="{\displaystyle \#}"> <a href="/wiki/Somme_connexe" title="Somme connexe">Somme connexe</a> Espaces pointés <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"> <a href="/wiki/Bouquet_(math%C3%A9matiques)" title="Bouquet (mathématiques)">Bouquet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"> <a href="/wiki/Smash-produit" title="Smash-produit">Smash-produit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"> <a href="/wiki/Joint_(math%C3%A9matiques)" title="Joint (mathématiques)">Joint</a> </div> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <div style="column-count:2;column-gap:1em;text-align:center" class="colonnes"> Fonctionnelles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> <a href="/wiki/Composition_de_fonctions" title="Composition de fonctions">Composition de fonctions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"> <a href="/wiki/Produit_de_convolution" title="Produit de convolution">Produit de convolution</a> Vectorielles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"> <a href="/wiki/Produit_scalaire" title="Produit scalaire">Produit scalaire</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"> <a href="/wiki/Produit_vectoriel" title="Produit vectoriel">Produit vectoriel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59549550bdbbf3ee3c3e699ef776a2fb75d925b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.509ex;" alt="{\displaystyle \times \,}"> <a href="/wiki/Produit_vectoriel_en_dimension_7" title="Produit vectoriel en dimension 7">Produit vectoriel généralisé</a> Algébriques <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [,]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>,</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [,]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04349be7aa458edb6f1ab423189a8455c6e21f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.328ex; height:2.843ex;" alt="{\displaystyle [,]}"> <a href="/wiki/Crochet_de_Lie" title="Crochet de Lie">Crochet de Lie</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{,\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>,</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{,\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6e62f0d087cac84c136e76f9c5e8c9f6af5f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.359ex; height:2.843ex;" alt="{\displaystyle \{,\}}"> <a href="/wiki/Crochet_de_Poisson" title="Crochet de Poisson">Crochet de Poisson</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"> <a href="/wiki/Produit_ext%C3%A9rieur" title="Produit extérieur">Produit extérieur</a> Homologiques <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \smile }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⌣</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \smile }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3885eca63e224e40912ea2b44c5fe85f3d4f8be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.309ex; margin-bottom: -0.48ex; width:2.324ex; height:1.343ex;" alt="{\displaystyle \smile }"> <a href="/wiki/Cup-produit" title="Cup-produit">Cup-produit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"> <a href="/w/index.php?title=Produit_d%27intersection&action=edit&redlink=1" class="new" title="Produit d'intersection (page inexistante)">Produit d'intersection</a> Séquentielles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> <a href="/wiki/Concat%C3%A9nation" title="Concaténation">Concaténation</a> </div> </div></td></tr></tbody></table></td> </tr> <tr> <td class="navbox-banner" style="background-color:transparent; color:inherit;" colspan="2"><div class="liste-horizontale">Logique booléenne : <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"> <a href="/wiki/Fonction_ET" title="Fonction ET">ET (conjonction)</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"> <a href="/wiki/Fonction_OU" title="Fonction OU">OU (disjonction)</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"> <a href="/wiki/Fonction_OU_exclusif" title="Fonction OU exclusif">OU exclusif</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> <a href="/wiki/Implication_(logique)" title="Implication (logique)">IMP (implication)</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"> <a href="/wiki/%C3%89quivalence_logique" title="Équivalence logique">EQV (équivalence)</a></li></ul> </div></td></tr></tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><a href="/wiki/Portail:Arithm%C3%A9tique_et_th%C3%A9orie_des_nombres" title="Arithmétique et théorie des nombres"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/34px-Pascal%27s_triangle_5.svg.png" decoding="async" width="34" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/50px-Pascal%27s_triangle_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/67px-Pascal%27s_triangle_5.svg.png 2x" data-file-width="540" data-file-height="389" /></a> <a href="/wiki/Portail:Arithm%C3%A9tique_et_th%C3%A9orie_des_nombres" title="Portail:Arithmétique et théorie des nombres">Arithmétique et théorie des nombres</a> </li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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