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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construction_de_nombres_calculables"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Construction de nombres calculables</span> </div> </a> <ul id="toc-Construction_de_nombres_calculables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statut_de_l'ensemble_des_réels_calculables" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statut_de_l'ensemble_des_réels_calculables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Statut de l'ensemble des réels calculables</span> </div> </a> <ul id="toc-Statut_de_l'ensemble_des_réels_calculables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prolongements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Prolongements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Prolongements</span> </div> </a> <button aria-controls="toc-Prolongements-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Prolongements</span> </button> <ul id="toc-Prolongements-sublist" class="vector-toc-list"> <li id="toc-Nombre_complexe_calculable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nombre_complexe_calculable"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Nombre complexe calculable</span> </div> </a> <ul id="toc-Nombre_complexe_calculable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suite_calculable_de_réels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suite_calculable_de_réels"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Suite calculable de réels</span> </div> </a> <ul id="toc-Suite_calculable_de_réels-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"> <span class="vector-toc-numb">5</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Nombre réel calculable</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 18 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-18" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">18 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF_%D9%82%D8%A7%D8%A8%D9%84%D8%A9_%D9%84%D9%84%D8%AD%D8%B3%D8%A7%D8%A8" title="أعداد قابلة للحساب – arabe" lang="ar" hreflang="ar" data-title="أعداد قابلة للحساب" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prora%C4%8Dunljiv_broj" title="Proračunljiv broj – bosniaque" lang="bs" hreflang="bs" data-title="Proračunljiv broj" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_computable" title="Nombre computable – catalan" lang="ca" hreflang="ca" data-title="Nombre computable" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Beregnelige_tal" title="Beregnelige tal – danois" lang="da" hreflang="da" data-title="Beregnelige tal" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Berechenbare_Zahl" title="Berechenbare Zahl – allemand" lang="de" hreflang="de" data-title="Berechenbare Zahl" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Computable_number" title="Computable number – anglais" lang="en" hreflang="en" data-title="Computable number" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_computable" title="Número computable – espagnol" lang="es" hreflang="es" data-title="Número computable" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%97%D7%A9%D7%99%D7%91" title="מספר חשיב – hébreu" lang="he" hreflang="he" data-title="מספר חשיב" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B7%D5%BE%D5%A1%D6%80%D5%AF%D5%A5%D5%AC%D5%AB_%D5%A9%D5%BE%D5%A5%D6%80" title="Հաշվարկելի թվեր – arménien" lang="hy" hreflang="hy" data-title="Հաշվարկելի թվեր" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A8%88%E7%AE%97%E5%8F%AF%E8%83%BD%E6%95%B0" title="計算可能数 – japonais" lang="ja" hreflang="ja" data-title="計算可能数" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%84%EC%82%B0_%EA%B0%80%EB%8A%A5%ED%95%9C_%EC%88%98" title="계산 가능한 수 – coréen" lang="ko" hreflang="ko" data-title="계산 가능한 수" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_computabel" title="Numer computabel – lombard" lang="lmo" hreflang="lmo" data-title="Numer computabel" data-language-autonym="Lombard" data-language-local-name="lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_comput%C3%A1vel" title="Número computável – portugais" lang="pt" hreflang="pt" data-title="Número computável" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8B%D1%87%D0%B8%D1%81%D0%BB%D0%B8%D0%BC%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вычислимое число – russe" lang="ru" hreflang="ru" data-title="Вычислимое число" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Izra%C4%8Dunljivo_%C5%A1tevilo" title="Izračunljivo število – slovène" lang="sl" hreflang="sl" data-title="Izračunljivo število" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ber%C3%A4kningsbart_tal" title="Beräkningsbart tal – suédois" lang="sv" hreflang="sv" data-title="Beräkningsbart tal" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hesaplanabilir_say%C4%B1" title="Hesaplanabilir sayı – turc" lang="tr" hreflang="tr" data-title="Hesaplanabilir sayı" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%AF%E8%A8%88%E7%AE%97%E6%95%B8" title="可計算數 – chinois" lang="zh" hreflang="zh" data-title="可計算數" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:10,000_digits_of_pi_-_poster.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/10%2C000_digits_of_pi_-_poster.svg/220px-10%2C000_digits_of_pi_-_poster.svg.png" decoding="async" width="220" height="311" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/10%2C000_digits_of_pi_-_poster.svg/330px-10%2C000_digits_of_pi_-_poster.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/10%2C000_digits_of_pi_-_poster.svg/440px-10%2C000_digits_of_pi_-_poster.svg.png 2x" data-file-width="744" data-file-height="1052" /></a><figcaption><a href="/wiki/%CE%A0_(nombre)" class="mw-redirect" title="Π (nombre)">π</a> est calculable avec un précision arbitraire, mais presque tous les nombres réels sont non calculables.</figcaption></figure> <p>En <a href="/wiki/Informatique" title="Informatique">informatique</a> et <a href="/wiki/Algorithmique" title="Algorithmique">algorithmique</a>, un <b>nombre réel calculable</b> est un <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">réel</a> pour lequel il existe un <a href="/wiki/Algorithmique" title="Algorithmique">algorithme</a> ou une <a href="/wiki/Machine_de_Turing" title="Machine de Turing">machine de Turing</a> permettant d'énumérer la suite de ses chiffres (éventuellement infinie), ou plus généralement des <a href="/wiki/Th%C3%A9orie_des_automates#Alphabet" title="Théorie des automates">symboles</a> de son écriture sous forme de <a href="/wiki/Cha%C3%AEne_de_caract%C3%A8res" title="Chaîne de caractères">chaîne de caractères</a>. De manière plus générale, et équivalente, un nombre réel est calculable si on peut en calculer une approximation aussi précise que l'on veut, avec une précision connue. </p><p>Cette notion a été mise en place par <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> en 1936<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. Elle a ensuite été développée dans différentes branches des <a href="/wiki/Math%C3%A9matiques_constructives" class="mw-redirect" title="Mathématiques constructives">mathématiques constructives</a>, et plus particulièrement l'<a href="/wiki/Analyse_constructive" title="Analyse constructive">analyse constructive</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p><p>L'ensemble des réels <a href="/wiki/Calculabilit%C3%A9" class="mw-redirect" title="Calculabilité">calculables</a> est un <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps</a> <a href="/wiki/Ensemble_d%C3%A9nombrable" title="Ensemble dénombrable">dénombrable</a>. Il contient, par exemple, tous les <a href="/wiki/Nombre_alg%C3%A9brique" title="Nombre algébrique">nombres algébriques</a> réels, ou des constantes célèbres comme <a href="/wiki/Pi" title="Pi"><span class="texhtml">π</span></a> ou <a href="/wiki/Constante_d%27Euler-Mascheroni" title="Constante d'Euler-Mascheroni"><span class="texhtml">γ</span></a>. </p><p>Les réels <i>non </i>calculables sont donc bien plus nombreux, bien qu'il soit généralement difficile de les définir, et sont en grande partie des nombres <a href="/wiki/Suite_al%C3%A9atoire" title="Suite aléatoire">aléatoires</a>. On parvient toutefois à en caractériser certains, comme la constante <a href="/wiki/Om%C3%A9ga_de_Chaitin" title="Oméga de Chaitin">Oméga de Chaitin</a> ou des nombres définis à partir du <a href="/wiki/Castor_affair%C3%A9" title="Castor affairé">castor affairé</a> ou des <a href="/wiki/Suite_de_Specker" title="Suite de Specker">suites de Specker</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définitions_principales"><span id="D.C3.A9finitions_principales"></span>Définitions principales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=1" title="Modifier la section : Définitions principales" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=1" title="Modifier le code source de la section : Définitions principales"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tout réel <i><span class="texhtml">x</span></i> est <a href="/wiki/Limite_d%27une_suite" title="Limite d'une suite">limite</a> de nombreuses <a href="/wiki/Suite_(math%C3%A9matiques)" title="Suite (mathématiques)">suites</a> de <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">nombres rationnels</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. Il existe en particulier des suites de <a href="/wiki/Couple_(math%C3%A9matiques)" title="Couple (mathématiques)">couples</a> d'<a href="/wiki/Entier_relatif" title="Entier relatif">entiers</a> <span class="texhtml">(<i>p<sub>n</sub></i>, <i>q<sub>n</sub></i>)</span>, avec <i><span class="texhtml">q<sub>n</sub></span> </i>≠ 0, telles que : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall n\in \mathbb {N} \quad \left|x-{\frac {p_{n}}{q_{n}}}\right|\leq {\frac {1}{2^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall n\in \mathbb {N} \quad \left|x-{\frac {p_{n}}{q_{n}}}\right|\leq {\frac {1}{2^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c64cf56e4779220e9beefd4e9e8214941d79030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.566ex; height:5.843ex;" alt="{\displaystyle \forall n\in \mathbb {N} \quad \left|x-{\frac {p_{n}}{q_{n}}}\right|\leq {\frac {1}{2^{n}}}.}"></span></center> <p>Le nombre <i><span class="texhtml">x</span></i> est dit calculable si, parmi ces suites <span class="texhtml">(<i>p<sub>n</sub></i>, <i>q<sub>n</sub></i>)</span>, il en existe qui sont <a href="/wiki/Fonction_calculable" class="mw-redirect" title="Fonction calculable">calculables</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>. (Il ne suffit pas pour cela que <i><span class="texhtml">x</span> </i>soit limite d'une suite calculable de rationnels, comme le montre l'exemple des <a href="/wiki/Suite_de_Specker" title="Suite de Specker">suites de Specker</a> : il faut de plus que pour au moins une telle suite, le <a href="/wiki/Module_de_convergence" title="Module de convergence">module de convergence</a> soit, lui aussi, calculable<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup>.) </p><p>Une définition équivalente est : </p> <div style="display:table; margin-top:1.5em; margin-bottom:1.5em; overflow:hidden; border:1px solid #aaa; padding:1em; background:none; margin-right:auto; margin-left:auto;;">Un réel est calculable si la suite des chiffres dans une <a href="/wiki/Base_(arithm%C3%A9tique)" title="Base (arithmétique)">base</a> quelconque est calculable<sup id="cite_ref-PDG4_6-0" class="reference"><a href="#cite_note-PDG4-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>.</div> <p>Cette définition est vraie si on autorise chaque "chiffre", pour une base quelconque, à être éventuellement négatif, et c'est vrai particulièrement pour la <a href="/wiki/Base_10" class="mw-redirect" title="Base 10">base 10</a><sup id="cite_ref-PDG4_6-1" class="reference"><a href="#cite_note-PDG4-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. En revanche, en <a href="/wiki/Syst%C3%A8me_binaire" title="Système binaire">système binaire</a>, les <a href="/wiki/Bit" title="Bit">bits</a> n'ont pas à être négatifs, et c'est la base généralement utilisée pour définir la calculabilité ainsi<sup id="cite_ref-Rice_7-0" class="reference"><a href="#cite_note-Rice-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup>. </p><p>Un nombre réel peut être calculable même si ses chiffres ne sont pas déterminés directement. Une troisième définition, toujours démontrée équivalente, est : </p> <div style="display:table; margin-top:1.5em; margin-bottom:1.5em; overflow:hidden; border:1px solid #aaa; padding:1em; background:none; margin-right:auto; margin-left:auto;;">Un réel <i>x</i> est calculable s'il existe un programme pour énumérer l'ensemble des <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">rationnels</a> supérieurs à x, et un autre pour énumérer l'ensemble des rationnels inférieurs à <i>x</i><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>.</div> <div class="mw-heading mw-heading2"><h2 id="Construction_de_nombres_calculables">Construction de nombres calculables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=2" title="Modifier la section : Construction de nombres calculables" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=2" title="Modifier le code source de la section : Construction de nombres calculables"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit <i><span class="texhtml">A</span> </i>un ensemble d'entiers naturels, le réel </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k\in A}2^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k\in A}2^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73467e4aee2f40a4cf5359c8a2b6a2431d760315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.272ex; height:5.676ex;" alt="{\displaystyle \sum _{k\in A}2^{-k}}"></span></center> <p>est calculable si et seulement si l'ensemble <i><span class="texhtml">A</span> </i>est <a href="/wiki/Ensemble_r%C3%A9cursif" title="Ensemble récursif">récursif</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>. </p><p>Plus concrètement, on sait par exemple que : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>4</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d1f9374822e94bfe20f8fc562773b9facd91f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.935ex; height:7.176ex;" alt="{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}}"></span> (<a href="/wiki/Formule_de_Leibniz#Série_alternée" title="Formule de Leibniz">formule de Leibniz</a>).</center> <p>Il est donc possible de déterminer des rationnels approchant <a href="/wiki/Pi" title="Pi"><span class="texhtml">π</span></a> avec une précision arbitraire (la théorie sur les <a href="/wiki/S%C3%A9rie_altern%C3%A9e" title="Série alternée">séries alternées</a> permet même de savoir pour quel entier <i><span class="texhtml">m</span> </i>il faut calculer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\sum _{k=0}^{m}{\frac {(-1)^{k}}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\sum _{k=0}^{m}{\frac {(-1)^{k}}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d3bd76fe6668a1c34a011796c36c935cfaee9c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.504ex; height:7.176ex;" alt="{\displaystyle 4\sum _{k=0}^{m}{\frac {(-1)^{k}}{2k+1}}}"></span> pour avoir un nombre donné de décimales exactes). </p><p>Mieux, tout nombre donné par une suite explicite à partir de nombres dont on a déjà montré qu'ils sont calculables l'est également. Par exemple non seulement <span class="texhtml">e</span> est calculable car </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} =\sum _{n=0}^{+\infty }{1 \over n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} =\sum _{n=0}^{+\infty }{1 \over n!}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0880d9de5c340cf0818e874013dca45e3c2f9930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.751ex; height:7.176ex;" alt="{\displaystyle \mathrm {e} =\sum _{n=0}^{+\infty }{1 \over n!}}"></span></center> <p>mais pour tout nombre calculable <i><span class="texhtml">x</span> </i>(par exemple <span class="texhtml"><i>x </i>= π</span>), le nombre <span class="texhtml">e<sup><i>x</i></sup></span> l'est également car </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{x}=\sum _{n=0}^{+\infty }{x^{n} \over n!}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{x}=\sum _{n=0}^{+\infty }{x^{n} \over n!}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaf19b4403e965d3ca7aba67e35e495d4d15f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.077ex; height:7.176ex;" alt="{\displaystyle \mathrm {e} ^{x}=\sum _{n=0}^{+\infty }{x^{n} \over n!}.}"></span></center> <p>Donc pour toute <a href="/wiki/Fonction_calculable" class="mw-redirect" title="Fonction calculable">fonction calculable</a>, l'image d'un nombre calculable est un nombre calculable ; par exemple le <a href="/wiki/Cosinus" title="Cosinus">cosinus</a> d'un rationnel donné est calculable (réciproquement, si un réel <i><span class="texhtml">x</span> </i>n'est pas calculable alors <span class="texhtml">e<sup><i>x</i></sup></span>, par exemple, ne l'est pas non plus, sinon <span class="texhtml"><i>x </i>= ln(e<sup><i>x</i></sup>)</span> le serait). </p> <div class="mw-heading mw-heading2"><h2 id="Statut_de_l'ensemble_des_réels_calculables"><span id="Statut_de_l.27ensemble_des_r.C3.A9els_calculables"></span>Statut de l'ensemble des réels calculables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=3" title="Modifier la section : Statut de l'ensemble des réels calculables" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=3" title="Modifier le code source de la section : Statut de l'ensemble des réels calculables"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>L'ensemble des réels calculables est un <a href="/wiki/Sous-corps" class="mw-redirect" title="Sous-corps">sous-corps</a> de ℝ.</li> <li>Il contient l'<a href="/wiki/Alg%C3%A8bre_des_p%C3%A9riodes" title="Algèbre des périodes">algèbre des périodes</a> (donc tous les <a href="/wiki/Nombre_alg%C3%A9brique" title="Nombre algébrique">nombres algébriques</a> réels et certains <a href="/wiki/Nombre_transcendant" title="Nombre transcendant">nombres transcendants</a> comme <span class="texhtml">π</span>), mais aussi la <a href="/wiki/Constante_d%27Euler-Mascheroni" title="Constante d'Euler-Mascheroni">constante d'Euler-Mascheroni</a> <span class="texhtml">γ</span> (dont on ignore si elle est rationnelle).</li> <li>C'est un <a href="/wiki/Corps_r%C3%A9el_clos" title="Corps réel clos">corps réel clos</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>.</li> <li>C'est un <a href="/wiki/Ensemble_d%C3%A9nombrable" title="Ensemble dénombrable">ensemble dénombrable</a> (un algorithme étant une suite finie de lettres d'un alphabet fini, l'ensemble des algorithmes, et donc <i>a fortiori</i> des nombres calculables, est dénombrable). La quasi-totalité des réels est donc <i>non</i> calculable (complémentaire d'un ensemble dénombrable). Ce sont en grande partie des nombres <a href="/wiki/Suite_al%C3%A9atoire" title="Suite aléatoire">aléatoires</a>. Bien qu'ils soient <a href="/wiki/Puissance_du_continu" title="Puissance du continu">très nombreux</a>, il est difficile d'en exhiber « explicitement ». On peut cependant citer la constante <a href="/wiki/Om%C3%A9ga_de_Chaitin" title="Oméga de Chaitin">Oméga de Chaitin</a> ou les nombres définis par le <a href="/wiki/Castor_affair%C3%A9" title="Castor affairé">castor affairé</a>.</li> <li>Puisque l'ensemble des réels calculables est dénombrable, on pourrait être tenté de dire que l'application du <a href="/wiki/Argument_de_la_diagonale_de_Cantor" title="Argument de la diagonale de Cantor">procédé diagonal de Cantor</a> à cet ensemble fournirait un algorithme pour calculer un nouveau nombre, ce qui conduirait à une contradiction. La réponse de Turing<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup> est que l'on ignore comment attribuer un numéro à chaque nombre calculable (plus précisément, on peut facilement démontrer, justement, qu'une telle attribution n'est pas calculable), or ceci doit être fait préalablement à la diagonalisation.</li> <li>L'égalité entre réels calculables n'est pas <a href="/wiki/Ensemble_r%C3%A9cursif" title="Ensemble récursif">décidable</a><sup id="cite_ref-Rice_7-1" class="reference"><a href="#cite_note-Rice-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Prolongements">Prolongements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=4" title="Modifier la section : Prolongements" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=4" title="Modifier le code source de la section : Prolongements"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Nombre_complexe_calculable">Nombre complexe calculable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=5" title="Modifier la section : Nombre complexe calculable" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=5" title="Modifier le code source de la section : Nombre complexe calculable"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Par extension, on appelle <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombre complexe</a> calculable un nombre complexe dont la <a href="/wiki/Partie_r%C3%A9elle" title="Partie réelle">partie réelle</a> et la <a href="/wiki/Partie_imaginaire" title="Partie imaginaire">partie imaginaire</a> sont calculables. </p> <div class="mw-heading mw-heading3"><h3 id="Suite_calculable_de_réels"><span id="Suite_calculable_de_r.C3.A9els"></span>Suite calculable de réels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=6" title="Modifier la section : Suite calculable de réels" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=6" title="Modifier le code source de la section : Suite calculable de réels"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une suite de réels <span class="texhtml">(<i>x<sub>m</sub></i>)</span> est dite calculable<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup> s'il existe une suite calculable (doublement indexée) de couples d'entiers <span class="texhtml">(<i>p<sub>m, n</sub></i>, <i>q<sub>m, n</sub></i>)</span>, avec <i><span class="texhtml">q<sub>m, n</sub></span> </i>≠ 0, telle que : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall m,n\in \mathbb {N} \quad \left|x_{m}-{\frac {p_{m,n}}{q_{m,n}}}\right|\leq {\frac {1}{2^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="1em" /> <mrow> <mo>|</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall m,n\in \mathbb {N} \quad \left|x_{m}-{\frac {p_{m,n}}{q_{m,n}}}\right|\leq {\frac {1}{2^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d0ed98921f31e317199c72f7956d634b6a867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.215ex; height:6.176ex;" alt="{\displaystyle \forall m,n\in \mathbb {N} \quad \left|x_{m}-{\frac {p_{m,n}}{q_{m,n}}}\right|\leq {\frac {1}{2^{n}}}.}"></span></center> <p>Chacun des réels <span class="texhtml"><i>x<sub>m</sub></i></span> est alors clairement calculable. </p><p>Si la suite (doublement indexée) des décimales des <span class="texhtml"><i>x<sub>m</sub></i></span> est calculable, alors <span class="texhtml">(<i>x<sub>m</sub></i>)</span> est une suite calculable de réels, mais <i>la réciproque est fausse</i>, et de même en remplaçant 10 par n'importe quelle <a href="/wiki/Base_(arithm%C3%A9tique)" title="Base (arithmétique)">base</a> > 1<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=7" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=7" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=8" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=8" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><a href="#Turing1937">Turing 1937</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><a href="#Di_Gianantonio1996">Di Gianantonio 1996</a>, <abbr class="abbr" title="page(s)">p.</abbr> 1</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text">Voir le <a href="/wiki/Construction_des_nombres_r%C3%A9els#Construction_via_les_suites_de_Cauchy" title="Construction des nombres réels">§ « Construction via les suites de Cauchy » de l'article « Construction des nombres réels »</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text"> Pour des définitions équivalentes et des références, cf. par exemple <a href="#Mostowski1979">Mostowski 1979</a>, <abbr class="abbr" title="page(s)">p.</abbr> 96, <a href="#Rice1954">Rice 1954</a> et <a href="#Pour-ElRichards1989">Pour-El et Richards 1989</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink noprint"><a href="#cite_ref-5">↑</a> </span><span class="reference-text">Il existe toute une hiérarchie de classes de réels définies de façon analogue. Par exemple en remplaçant les fonctions calculables par les <a href="/wiki/ELEMENTARY_(complexit%C3%A9)" title="ELEMENTARY (complexité)">fonctions récursives primaires</a> (dans le numérateur, le dénominateur, et le module de convergence de la suite de rationnels), on définit la notion (plus restrictive) de <i>nombre réel élémentaire</i> : cf. bibliographie de <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Katrin Tent et Martin Ziegler, <i>Low functions of reals. </i><span class="ouvrage">« <a rel="nofollow" class="external text" href="https://arxiv.org/abs/0903.1384"><cite style="font-style:normal;">0903.1384</cite></a> », texte en accès libre, sur <span class="italique"><a href="/wiki/ArXiv" title="ArXiv">arXiv</a></span></span>.</span> </li> <li id="cite_note-PDG4-6"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-PDG4_6-0">a</a> et <a href="#cite_ref-PDG4_6-1">b</a></sup> </span><span class="reference-text"><a href="#Di_Gianantonio1996">Di Gianantonio 1996</a>, def. d), <abbr class="abbr" title="page(s)">p.</abbr> 4</span> </li> <li id="cite_note-Rice-7"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Rice_7-0">a</a> et <a href="#cite_ref-Rice_7-1">b</a></sup> </span><span class="reference-text"><a href="#Rice1954">Rice 1954</a>, def. B, <abbr class="abbr" title="page(s)">p.</abbr> 784</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><a href="#Mostowski1957">Mostowski 1957</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text">J.P. Delayahe <i>Omega Numbers</i> in <i>Randomness and Complexity. From Leibniz to Chaitin</i> World Scientific, 2007, p. 355</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink noprint"><a href="#cite_ref-10">↑</a> </span><span class="reference-text"><a href="#Weihrauch2000">Weihrauch 2000</a>, <abbr class="abbr" title="page(s)">p.</abbr> 5, Example 1.3.2 ou <a href="#Weihrauch1995">Weihrauch 1995</a>, <abbr class="abbr" title="page(s)">p.</abbr> 21 example 1 (4)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink noprint"><a href="#cite_ref-11">↑</a> </span><span class="reference-text"><a href="#Mostowski1979">Mostowski 1979</a>, <abbr class="abbr" title="page(s)">p.</abbr> 96</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><a href="#Turing1937">Turing 1937</a>, § 8</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text">Cf. <a href="#Mostowski1979">Mostowski 1979</a>, <abbr class="abbr" title="page(s)">p.</abbr> 96 et quelques explications dans <span class="ouvrage"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> « <a rel="nofollow" class="external text" href="https://planetmath.org/Computablesequence"><cite style="font-style:normal;" lang="en"><span class="lang-en" lang="en">Computable sequence</span></cite></a> », sur <span class="italique"><a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a></span></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink noprint"><a href="#cite_ref-14">↑</a> </span><span class="reference-text"><a href="#Mostowski1957">Mostowski 1957</a> établit des inclusions strictes entre diverses variantes, qui correspondent pourtant à des définitions équivalentes d'un réel calculable, et fait remarquer l'analogie inexpliquée avec la non-équivalence, déjà remarquée par <a href="/wiki/Ernst_Specker" title="Ernst Specker">Specker</a>, de ces mêmes variantes dans la définition d'un réel « semi-calculable ».</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&veaction=edit&section=9" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Nombre_r%C3%A9el_calculable&action=edit&section=9" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Di_Gianantonio1996"><span class="ouvrage" id="Pietro_Di_Gianantonio1996"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Pietro <span class="nom_auteur">Di Gianantonio</span>, « <cite style="font-style:normal" lang="en">Real Number Computability and Domain Theory</cite> », <i><span class="lang-en" lang="en"><i><a href="/wiki/Information_and_Computation" title="Information and Computation">Information and Computation</a></i></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 127, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1,‎ <time>1996</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">12-25</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://users.dimi.uniud.it/~pietro.digianantonio/papers/copy_pdf/reali.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Real+Number+Computability+and+Domain+Theory&rft.jtitle=%27%27Information+and+Computation%27%27&rft.issue=1&rft.aulast=Di+Gianantonio&rft.aufirst=Pietro&rft.date=1996&rft.volume=127&rft.pages=12-25&rft_id=http%3A%2F%2Fusers.dimi.uniud.it%2F~pietro.digianantonio%2Fpapers%2Fcopy_pdf%2Freali.pdf&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Mostowski1979"><span class="ouvrage" id="Andrzej_Mostowski1979"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Andrzej_Mostowski" title="Andrzej Mostowski">Andrzej <span class="nom_auteur">Mostowski</span></a>, <cite class="italique" lang="en">Foundational Studies : Selected Works</cite>, <abbr class="abbr" title="volume">vol.</abbr> 1, Amsterdam/New York, <a href="/wiki/Elsevier_(%C3%A9diteur)" class="mw-redirect" title="Elsevier (éditeur)">Elsevier</a>, <time>1979</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-444-85102-4" title="Spécial:Ouvrages de référence/978-0-444-85102-4"><span class="nowrap">978-0-444-85102-4</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=60mICbpCca4C">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundational+Studies&rft.place=Amsterdam%2FNew+York&rft.pub=Elsevier&rft.stitle=Selected+Works&rft.aulast=Mostowski&rft.aufirst=Andrzej&rft.date=1979&rft.volume=1&rft.isbn=978-0-444-85102-4&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Mostowski1957"><span class="ouvrage" id="A._Mostowski1957"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> A. <span class="nom_auteur">Mostowski</span>, « <cite style="font-style:normal" lang="en">On computable sequences</cite> », <i><a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae"><span class="lang-en" lang="en">Fund. Math.</span></a></i>, <abbr class="abbr" title="volume">vol.</abbr> 44,‎ <time>1957</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">37-51</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm44/fm4413.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+computable+sequences&rft.jtitle=Fund.+Math.&rft.aulast=Mostowski&rft.aufirst=A.&rft.date=1957&rft.volume=44&rft.pages=37-51&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm44%2Ffm4413.pdf&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Pour-ElRichards1989"><span class="ouvrage" id="Marian_B._Pour-ElJ._Ian_Richards1989"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Marian B. <span class="nom_auteur">Pour-El</span> et J. Ian <span class="nom_auteur">Richards</span>, <cite class="italique" lang="en">Computability in Analysis and Physics</cite>, <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer</a>, <time>1989</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235422916">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+in+Analysis+and+Physics&rft.pub=Springer&rft.aulast=Pour-El&rft.aufirst=Marian+B.&rft.au=Richards%2C+J.+Ian&rft.date=1989&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Rice1954"><span class="ouvrage" id="H._G._Rice1954"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Henry_Gordon_Rice" title="Henry Gordon Rice">H. G. <span class="nom_auteur">Rice</span></a>, « <cite style="font-style:normal" lang="en">Recursive real numbers</cite> », <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society"><span class="lang-en" lang="en">Proc. Amer. Math. Soc.</span></a></i>, <abbr class="abbr" title="volume">vol.</abbr> 5,‎ <time>1954</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">784-791</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.ams.org/journals/proc/1954-005-05/S0002-9939-1954-0063328-5/">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Recursive+real+numbers&rft.jtitle=Proc.+Amer.+Math.+Soc.&rft.aulast=Rice&rft.aufirst=H.+G.&rft.date=1954&rft.volume=5&rft.pages=784-791&rft_id=http%3A%2F%2Fwww.ams.org%2Fjournals%2Fproc%2F1954-005-05%2FS0002-9939-1954-0063328-5%2F&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Turing1937"><span class="ouvrage" id="Alan_Turing1937"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Alan Turing, <cite class="italique" lang="en">On Computable Numbers, with an Application to the Entscheidungsproblem : Proceedings of the London Mathematical Society</cite>, London Mathematical Society, <time>1937</time> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1112/PLMS/S2-42.1.230">10.1112/PLMS/S2-42.1.230</a></span>, <a rel="nofollow" class="external text" href="https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+Computable+Numbers%2C+with+an+Application+to+the+Entscheidungsproblem&rft.pub=London+Mathematical+Society&rft.stitle=Proceedings+of+the+London+Mathematical+Society&rft.aulast=Turing&rft.aufirst=Alan&rft.date=1937&rft_id=info%3Adoi%2F10.1112%2FPLMS%2FS2-42.1.230&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span> et <span class="ouvrage" id="1938">« <cite style="font-style:normal">[<i>idem</i>] : A Correction</cite> », <i>Proc. London Math. Soc.</i>, <abbr class="abbr" title="deuxième">2<sup>e</sup></abbr> série, <abbr class="abbr" title="volume">vol.</abbr> 43,‎ <time>1938</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">544-546</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1112/plms/s2-43.6.544">10.1112/plms/s2-43.6.544</a></span>, <a rel="nofollow" class="external text" href="http://www.thocp.net/biographies/papers/turing_oncomputablenumbers_1936.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%5B%27%27idem%27%27%5D&rft.jtitle=Proc.+London+Math.+Soc.&rft.stitle=A+Correction&rft.date=1938&rft.volume=43&rft.pages=544-546&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-43.6.544&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></li> <li><span class="ouvrage" id="Weihrauch1995"><span class="ouvrage" id="Klaus_Weihrauch1995"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Klaus <span class="nom_auteur">Weihrauch</span>, <cite class="italique" lang="en">A Simple Introduction to Computable Analysis</cite>, <a href="/wiki/Hagen_(Rh%C3%A9nanie-du-Nord-Westphalie)#Enseignement_supérieur_et_recherche" title="Hagen (Rhénanie-du-Nord-Westphalie)">FernUniversität Hagen</a>, <abbr class="abbr" title="collection">coll.</abbr> « Informatik Berichte » (<abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 171), <time class="nowrap" datetime="1995-07" data-sort-value="1995-07">juillet 1995</time>, <abbr class="abbr" title="deuxième">2<sup>e</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr>, 79 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://eccc.hpi-web.de/static/books/A_Simple_Introduction_to_Computable_Analysis_Fragments_of_a_Book/">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Simple+Introduction+to+Computable+Analysis&rft.pub=FernUniversit%C3%A4t+Hagen&rft.edition=2&rft.aulast=Weihrauch&rft.aufirst=Klaus&rft.date=1995-07&rft.tpages=79&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li> <li><span class="ouvrage" id="Weihrauch2000"><span class="ouvrage" id="Klaus_Weihrauch2000"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Klaus <span class="nom_auteur">Weihrauch</span>, <cite class="italique" lang="en">Computable Analysis : An Introduction</cite>, Springer, <abbr class="abbr" title="collection">coll.</abbr> « Texts in Theoretical Computer Science », <time>2000</time>, 285 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-3-540-66817-6" title="Spécial:Ouvrages de référence/978-3-540-66817-6"><span class="nowrap">978-3-540-66817-6</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=OPolVWVFDJYC&pg=PA5">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computable+Analysis&rft.pub=Springer&rft.stitle=An+Introduction&rft.aulast=Weihrauch&rft.aufirst=Klaus&rft.date=2000&rft.tpages=285&rft.isbn=978-3-540-66817-6&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ANombre+r%C3%A9el+calculable"></span></span></span></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint collapsed" style=""> <tbody><tr><th class="navbox-title" colspan="3" 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_Notion_de_nombre" title="Modèle:Palette Notion de nombre"><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_Notion_de_nombre&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%">Notion de <a href="/wiki/Nombre" title="Nombre">nombre</a></div></th> </tr> <tr> <th class="navbox-group" style="">Ensembles usuels</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Entier_naturel" title="Entier naturel">Entier naturel</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40eac26c488d3257e3fbe63619729673145d228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Entier_relatif" title="Entier relatif">Entier relatif</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c672518c0350ca035befd41c26633a2d399431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.096ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Nombre_d%C3%A9cimal" title="Nombre décimal">Nombre décimal</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f303c0b908a47cc0dfb5e7a7293a94a22dd0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {D} }"></span>)</li> <li><a href="/wiki/Nombre_rationnel" title="Nombre rationnel">Nombre rationnel</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaa5ab94a056a5a25944ddf0c52c92a404715ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">Nombre réel</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7df6838b44979c6531f6a0306206fbdb0477ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Nombre_complexe" title="Nombre complexe">Nombre complexe</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe3a54bb4e56c039e18c3af24ba70ab377f7a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {C} }"></span>)</li></ul> </div></td> <td class="navbox-image" rowspan="5" style="vertical-align:middle;padding-left:7px"><span typeof="mw:File"><a href="/wiki/Fichier:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description" title="Mathématiques"><img alt="Mathématiques" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/70px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/105px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/140px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></td> </tr> <tr> <th class="navbox-group" style="">Extensions</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Quaternion" title="Quaternion">Quaternion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d00daea5df233d805f1ec5d5ae84845bac2ad06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5cf3960cf7ba384648447c15581d5d4589a6d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {O} }"></span>)</li> <li><a href="/wiki/S%C3%A9d%C3%A9nion" title="Sédénion">Sédénion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef48a593f4503abeab608e8781ba478b7d1b304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.914ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Nombre_complexe_d%C3%A9ploy%C3%A9" title="Nombre complexe déployé">Nombre complexe déployé</a></li> <li><a href="/wiki/Tessarine" title="Tessarine">Tessarine</a></li> <li><a href="/wiki/Nombre_bicomplexe" title="Nombre bicomplexe">Nombre bicomplexe</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {C} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {C} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9ab8af8ff8f4437f3c72de4e78738374ee4954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.018ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {C} _{2}}"></span>)</li> <li>Nombre multicomplexe (<a href="/wiki/Nombre_multicomplexe_(Segre)" title="Nombre multicomplexe (Segre)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {C} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {C} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4170ef11ece45d8b66656c919e34eefdb51e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {C} _{n}}"></span></a></li> <li><a href="/wiki/Nombre_multicomplexe_(Fleury)" title="Nombre multicomplexe (Fleury)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\mathcal {M}}\mathbb {C} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\mathcal {M}}\mathbb {C} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be2600f9a2a7a3fdbdb7b08d21dec751dfde32e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.125ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\mathcal {M}}\mathbb {C} _{n}}"></span></a>)</li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternion</a></li> <li><a href="/wiki/Coquaternion" title="Coquaternion">Coquaternion</a></li> <li><a href="/wiki/Quaternion_hyperbolique" title="Quaternion hyperbolique">Quaternion hyperbolique</a></li> <li><a href="/wiki/Octonion_d%C3%A9ploy%C3%A9" title="Octonion déployé">Octonion déployé</a></li> <li><a href="/wiki/Nombre_hypercomplexe" title="Nombre hypercomplexe">Nombre hypercomplexe</a></li> <li><a href="/wiki/Nombre_p-adique" title="Nombre p-adique">Nombre p-adique</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Q} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425658dc852cec8a2d4b6f7d513e60977710d7f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.114ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \mathbb {Q} _{p}}"></span>)</li> <li><a href="/wiki/Nombre_hyperr%C3%A9el" title="Nombre hyperréel">Nombre hyperréel</a></li> <li><a href="/wiki/Nombre_superr%C3%A9el" title="Nombre superréel">Nombre superréel</a></li> <li><a href="/wiki/Nombre_dual" title="Nombre dual">Nombre dual</a></li> <li><a href="/wiki/Droite_r%C3%A9elle_achev%C3%A9e" title="Droite réelle achevée">Droite réelle achevée</a></li> <li><a href="/wiki/Nombre_cardinal" title="Nombre cardinal">Nombre cardinal</a></li> <li><a href="/wiki/Nombre_ordinal" title="Nombre ordinal">Nombre ordinal</a></li> <li><a href="/wiki/Nombre_surr%C3%A9el" title="Nombre surréel">Nombre surréel</a></li> <li><a href="/wiki/Nombre_pseudo-r%C3%A9el" title="Nombre pseudo-réel">Nombre pseudo-réel</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Propriétés particulières</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Parit%C3%A9_(arithm%C3%A9tique)" title="Parité (arithmétique)">Parité</a></li> <li><a href="/wiki/Nombre_premier" title="Nombre premier">Nombre premier</a></li> <li><a href="/wiki/Nombre_compos%C3%A9" title="Nombre composé">Nombre composé</a></li> <li><a href="/wiki/Nombre_figur%C3%A9" title="Nombre figuré">Nombre figuré</a></li> <li><a href="/wiki/Nombre_parfait" title="Nombre parfait">Nombre parfait</a></li> <li><a href="/wiki/Nombre_positif" title="Nombre positif">Nombre positif</a></li> <li><a href="/wiki/Nombre_n%C3%A9gatif" title="Nombre négatif">Nombre négatif</a></li> <li><a href="/wiki/Fraction_dyadique" title="Fraction dyadique">Fraction dyadique</a></li> <li><span typeof="mw:File"><span title="Bon article"><img alt="Bon article" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/14px-Bon_article.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/21px-Bon_article.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/28px-Bon_article.svg.png 2x" data-file-width="20" data-file-height="20" /></span></span> <a href="/wiki/Nombre_irrationnel" title="Nombre irrationnel">Nombre irrationnel</a></li> <li><a href="/wiki/Nombre_alg%C3%A9brique" title="Nombre algébrique">Nombre algébrique</a></li> <li><a href="/wiki/Nombre_transcendant" title="Nombre transcendant">Nombre transcendant</a></li> <li><a href="/wiki/Nombre_imaginaire_pur" title="Nombre imaginaire pur">Nombre imaginaire pur</a></li> <li><a href="/wiki/Nombre_de_Liouville" title="Nombre de Liouville">Nombre de Liouville</a></li> <li><a href="/wiki/Alg%C3%A8bre_des_p%C3%A9riodes" title="Algèbre des périodes">Période</a></li> <li><a href="/wiki/Nombre_normal" title="Nombre normal">Nombre normal</a></li> <li><a href="/wiki/Nombre_univers" title="Nombre univers">Nombre univers</a></li> <li><a href="/wiki/Nombre_constructible" title="Nombre constructible">Nombre constructible</a></li> <li><a class="mw-selflink selflink">Nombre réel calculable</a></li> <li><a href="/wiki/Nombre_transfini" title="Nombre transfini">Nombre transfini</a></li> <li><a href="/wiki/Infiniment_petit" title="Infiniment petit">Infiniment petit</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Exemples</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Pi" title="Pi">Pi</a> (<span class="texhtml">π</span>)</li> <li><span typeof="mw:File"><span title="Bon article"><img alt="Bon article" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/14px-Bon_article.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/21px-Bon_article.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/28px-Bon_article.svg.png 2x" data-file-width="20" data-file-height="20" /></span></span> <a href="/wiki/Racine_carr%C3%A9e_de_deux" title="Racine carrée de deux">Racine carrée de deux</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6ac02637a190523aa10dde1b52ae41964dfff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.191ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\sqrt {2}}}"></span>)</li> <li><span typeof="mw:File"><span title="Bon article"><img alt="Bon article" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/14px-Bon_article.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/21px-Bon_article.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bon_article.svg/28px-Bon_article.svg.png 2x" data-file-width="20" data-file-height="20" /></span></span> <a href="/wiki/Nombre_d%27or" title="Nombre d'or">Nombre d’or</a> (φ)</li> <li><a href="/wiki/Z%C3%A9ro" title="Zéro">Zéro</a> (0)</li> <li><a href="/wiki/Unit%C3%A9_imaginaire" title="Unité imaginaire">Unité imaginaire</a> (<span class="texhtml">i</span>)</li> <li><a href="/wiki/E_(nombre)" title="E (nombre)">Constante de Neper</a> (<span class="texhtml">e</span>)</li> <li><a href="/wiki/Aleph-z%C3%A9ro" title="Aleph-zéro">Aleph-zéro</a> (ℵ<sub>0</sub>)</li> <li><a href="/wiki/Table_de_constantes_math%C3%A9matiques" title="Table de constantes mathématiques">Table de constantes mathématiques</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Articles liés</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Chiffre" title="Chiffre">Chiffre</a></li> <li><a href="/wiki/Num%C3%A9ration" title="Numération">Numération</a></li> <li><a href="/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques)">Fraction</a></li> <li><a href="/wiki/Op%C3%A9ration_(math%C3%A9matiques)" title="Opération (mathématiques)">Opération</a></li> <li><a href="/wiki/Calcul_(math%C3%A9matiques)" title="Calcul (mathématiques)">Calcul</a></li> <li><a href="/wiki/Alg%C3%A8bre" title="Algèbre">Algèbre</a></li> <li><a href="/wiki/Arithm%C3%A9tique" title="Arithmétique">Arithmétique</a></li> <li><a href="/wiki/Suite_d%27entiers" title="Suite d'entiers">Suite d'entiers</a></li> <li><a href="/wiki/Infini" title="Infini">Infini</a> (<span class="texhtml">∞</span>)</li> <li><a href="/wiki/Chiffre_significatif" title="Chiffre significatif">Chiffre significatif</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Informatique_th%C3%A9orique" title="Portail de l'informatique théorique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Max-cut.svg/30px-Max-cut.svg.png" decoding="async" width="30" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Max-cut.svg/45px-Max-cut.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Max-cut.svg/60px-Max-cut.svg.png 2x" data-file-width="200" data-file-height="160" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Informatique_th%C3%A9orique" title="Portail:Informatique théorique">Portail de l'informatique théorique</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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