{"id":643,"date":"2018-09-06T10:02:32","date_gmt":"2018-09-06T08:02:32","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=643"},"modified":"2021-10-26T17:14:41","modified_gmt":"2021-10-26T15:14:41","slug":"zero-exposant-zero","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/","title":{"rendered":"Z\u00e9ro exposant z\u00e9ro"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Au m\u00eame titre que \\(\\displaystyle\\frac{0}{0}\\), l&rsquo;\u00e9criture \\(0^0\\) reste un myst\u00e8re pour beaucoup de personnes novices qui manipulent les math\u00e9matiques. Comme pour la premi\u00e8re \u00e9criture, nous allons voir qu&rsquo;il est impossible d&rsquo;imposer une \u00e9galit\u00e9 fixe pour la seconde.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"662\" height=\"373\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/0-exposant-0.png\" alt=\"\" class=\"wp-image-3220\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/0-exposant-0.png 662w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/0-exposant-0-300x169.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/0-exposant-0-600x338.png 600w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/figure><\/div>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/#Premiere_approche_les_puissances_au_college\" >Premi\u00e8re approche : les puissances au coll\u00e8ge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/#Une_approche_fonctionnelle\" >Une approche fonctionnelle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/#Une_approche_ensembliste\" >Une approche ensembliste<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/#Conclusion\" >Conclusion<\/a><\/li><\/ul><\/nav><\/div>\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Premiere_approche_les_puissances_au_college\"><\/span>Premi\u00e8re approche : les puissances au coll\u00e8ge<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Au coll\u00e8ge, nous voyons que pour \\(n\\in\\mathbb{N}\\),&nbsp;\\[x^n=\\underbrace{x\\times x\\times x\\times\\cdots\\times x}_{n~\\text{facteurs}} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">o\u00f9 <em>x<\/em>&nbsp;repr\u00e9sente un nombre non diff\u00e9rent de z\u00e9ro.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On voit aussi la relation :&nbsp; \\[ x^n\\times x^m=x^{n+m}.\\qquad(1)\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous voyons aussi la relation :&nbsp;\\[ x^{-n}=\\frac{1}{x^n},\\quad x\\neq 0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Donc, si on prend <em>m&nbsp;<\/em>= &#8211;<em>n<\/em>, la relation (1) devient :&nbsp; \\[&nbsp;x^n\\times x^{-n}=x^0\\qquad (2) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, \\[&nbsp;x^n\\times x^{-n}=\\frac{x^n}{x^n}=1,\\quad x\\neq 0. \\qquad (3) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On d\u00e9duit alors des \u00e9galit\u00e9s (2) et (3) :&nbsp;\\[ x^0=1,\\quad x\\neq 0.\\]<\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_approche_fonctionnelle\"><\/span>Une approche fonctionnelle<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons la fonction \\(f(x)=x^x=\\text{e}^{x\\ln x}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous savons que \\(\\lim\\limits_{x\\to 0} x\\ln x=0\\) donc \\(\\lim\\limits_{x\\to 0} f(x)=\\text{e}^0=1\\). On aurait alors montr\u00e9 que :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ 0^0=1.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 ce stade, on est soulag\u00e9 : \u00ab\u00a0Ouf ! On a trouv\u00e9 la valeur de \\(0^0 \\) ! \u00ab\u00a0. Mais comme je suis un rabat-joie, j&rsquo;aimerais parler d&rsquo;une autre fonction :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ g(x)=x^{\\frac{1}{\\ln x}}=\\text{e}^{\\frac{1}{\\ln x}\\times\\ln x}=\\text{e}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On a : \\[ \\lim\\limits_{x\\to 0} \\ln x=-\\infty \\] donc \\[\\lim\\limits_{x\\to 0} \\frac{1}{\\ln x}=0.\\] Ainsi, \\[\\lim\\limits_{x\\to 0} g(x)=0^0.\\] Mais \\[\\lim\\limits_{x\\to 0} g(x)=\\text{e}\\] d&rsquo;apr\u00e8s l&rsquo;expression trouv\u00e9e pr\u00e9c\u00e9demment. D&rsquo;o\u00f9 :&nbsp;\\[ 0^0=\\text{e}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On pourrait m\u00eame consid\u00e9rer la fonction :&nbsp;\\[ h(x)=\\left(\\text{e}^{-\\frac{1}{x}}\\right)^x.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, on a d&rsquo;une part :\\[ h(x)=\\text{e}^{-1}. \\] D&rsquo;autre part, \\[ \\lim\\limits_{x\\to 0^+}\\left(-\\frac{1}{x}\\right)=-\\infty\\] et \\[\\lim\\limits_{X\\to -\\infty} \\text{e}^X=0\\;,\\] donc : \\[ \\lim\\limits_{x\\to 0^+} h(x)=0^0=\\text{e}^{-1}.\\] Ainsi,&nbsp;&nbsp;\\[ 0^0=\\text{e}^{-1}.\\]<\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_approche_ensembliste\"><\/span>Une approche ensembliste<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">En th\u00e9orie des ensembles, on voit que si deux ensembles M et N ont respectivement un cardinal \u00e9gal \u00e0 <em>n<\/em>&nbsp;et <em>m<\/em>, alors le nombre d&rsquo;applications de N vers M est \u00e9gal \u00e0 \\(m^n\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, \\(0^0\\) repr\u00e9sente, dans ce contexte pr\u00e9cis, le nombre d&rsquo;applications de l&rsquo;ensemble vide vers l&rsquo;ensemble vide, c&rsquo;est-\u00e0-dire que : \\[ 0^0=1.\\]<\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Nous pouvons, \u00e0 la vue de ce qui a \u00e9t\u00e9 \u00e9crit pr\u00e9c\u00e9demment, conclure que l&rsquo;\u00e9criture \u00a0\u00bb \\(0^0\\) \u00a0\u00bb n&rsquo;a aucune signification si elle est sortie d&rsquo;un certain contexte, et selon le contexte dans lequel on la place, elle n&rsquo;a pas la m\u00eame signification. Ce n&rsquo;est qu&rsquo;une \u00e9criture qui symbolise un ph\u00e9nom\u00e8ne math\u00e9matique particulier selon le contexte dans lequel on la place. Ainsi, parler de \\(0^0\\) seul n&rsquo;a pas de sens.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Article fortement inspir\u00e9 de l&rsquo;article de Jean Jacquelin, \u00ab\u00a00 puissance 0\u00a0\u00bb, paru dans le magazine Quadrature n\u00b066.<\/em><\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Zero-exposant-zero.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Zero exposant zero<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Zero-exposant-zero.pdf\" class=\"wp-block-file__button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Obtenir le fichier source \\(\\LaTeX\\) du document PDF:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Z\u00e9ro-exposant-z\u00e9ro.zip\">Z\u00e9ro exposant z\u00e9ro<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Z\u00e9ro-exposant-z\u00e9ro.zip\" class=\"wp-block-file__button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Au m\u00eame titre que \\(\\displaystyle\\frac{0}{0}\\), l&rsquo;\u00e9criture \\(0^0\\) reste un myst\u00e8re pour beaucoup de personnes novices qui manipulent les math\u00e9matiques. Comme pour la premi\u00e8re \u00e9criture, nous allons voir qu&rsquo;il est impossible d&rsquo;imposer une \u00e9galit\u00e9 fixe pour la seconde.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[62,63],"class_list":["post-643","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-puissance","tag-zero"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Z\u00e9ro exposant z\u00e9ro - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/06\/zero-exposant-zero\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Z\u00e9ro exposant z\u00e9ro - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Au m\u00eame titre que (displaystylefrac{0}{0}), l&rsquo;\u00e9criture (0^0) reste un myst\u00e8re pour beaucoup de personnes novices qui manipulent les math\u00e9matiques. 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