{"id":2118,"date":"2020-03-05T11:30:08","date_gmt":"2020-03-05T10:30:08","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2118"},"modified":"2023-08-30T15:53:07","modified_gmt":"2023-08-30T13:53:07","slug":"une-preuve-que-2-4","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/05\/une-preuve-que-2-4\/","title":{"rendered":"Une preuve que 2 = 4"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Nous allons voir dans cet article une preuve (bien entendu erron\u00e9e) que 2 = 4. Ce que nous allons voir est compr\u00e9hensible par des \u00e9l\u00e8ves de Terminale ayant vu la notion de continuit\u00e9 de fonctions.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_87_1 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-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/05\/une-preuve-que-2-4\/#Dabord_la_pseudo-preuve\" >D&rsquo;abord, la pseudo-preuve<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/05\/une-preuve-que-2-4\/#Ensuite_la_demolition_de_la_pseudo-preuve\" >Ensuite, la d\u00e9molition de la pseudo-preuve<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Dabord_la_pseudo-preuve\"><\/span>D&rsquo;abord, la pseudo-preuve<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons l&rsquo;\u00e9quation:$$x^{x^{x^{\\ldots}}}=2.$$Elle admet \\(\\sqrt2\\) comme solution. \u00c9vident non ? Non ? Bon, je m&rsquo;explique: d&rsquo;abord, si l&rsquo;\u00e9quation admet une solution alors elle est n\u00e9cessairement positive. De plus, $$\\begin{align}  { \\color{blue} x^{ \\color{ blue } x^{ \\color{ blue } x^{ \\color{ blue } \\ldots}}}}=2  &amp; \\iff  x^{\\color{ blue }x^{ \\color{ blue } x^{ \\color{ blue } \\ldots}}}=2  \\\\ &amp; \\iff x^{\\color{blue}2} = 2\\\\ &amp; \\iff x=\\sqrt2. \\end{align} $$On peut alors \u00e9crire:$$\\sqrt2^{\\sqrt2^{\\sqrt2^{\\ldots}}}=2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons maintenant l&rsquo;\u00e9quation: $$x^{x^{x^{\\ldots}}}=4.$$ De mani\u00e8re analogue, on a:$$\\begin{align} x^{x^{x^{\\ldots}}}=4  &amp; \\iff x^4 = 4\\\\&amp; \\iff (x^2)^2 &#8211; 2^2 = 0\\\\&amp;\\iff(x^2-2)(x^2+2)=0\\\\&amp;\\iff x^2-2=0\\\\&amp;\\iff x^2=2\\\\&amp;\\iff x=\\sqrt2\\text{ car }x&gt;0. \\end{align}$$Ainsi,$$\\sqrt{2}^{\\sqrt2^{\\sqrt2^{\\ldots}}}=4.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Des deux \u00e9galit\u00e9s trouv\u00e9es, on en d\u00e9duit que 2 = 4.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ensuite_la_demolition_de_la_pseudo-preuve\"><\/span>Ensuite, la d\u00e9molition de la pseudo-preuve<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons la fonction:$$f(x)=\\sqrt[x]{x}=\\text{e}^{\\frac{\\ln(x)}{x}}.$$Une \u00e9tude de fonction classique montre que sur \\([1;+\\infty[\\), <em>f<\/em>(<em>x<\/em>) prend ses valeurs dans \\([1;\\sqrt[\\text{e}]{\\text{e}}]\\), qu&rsquo;elle est strictement croissante sur [1;e], prenant ses valeurs de 1 \u00e0  \\( \\sqrt[\\text{e}]{\\text{e}} \\), puis strictement d\u00e9croissante sur [e;+\\(\\infty\\)[ prenant ses valeurs de  \\( \\sqrt[\\text{e}]{\\text{e}} \\)  \u00e0 1. Ainsi, d&rsquo;apr\u00e8s le th\u00e9or\u00e8me de la bijection (corollaire du th\u00e9or\u00e8me des valeurs interm\u00e9diaires), l&rsquo;\u00e9quation <em>f(x)=a<\/em> admet deux solutions si 1 &lt; <em>a<\/em> &lt;  \\( \\sqrt[\\text{e}]{\\text{e}} \\), la premi\u00e8re sur [1;e] et la seconde sur [e;+\\(\\infty\\)[.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour 1 &lt; <em>a<\/em> &lt; \\(\\sqrt[\\text{e}]{\\text{e}}\\), l&rsquo;\u00e9quation d&rsquo;inconnue <em>x<\/em> &gt; 0:$$a^x = x$$admet donc deux solutions. En effet,$$\\begin{align}a^x = x &amp; \\iff \\left(a^x\\right)^{\\frac{1}{x}} = x^{\\frac{1}{x}}\\\\&amp;\\iff a = \\sqrt[x]{x}\\end{align}$$Notons \\(1&lt;x_1&lt;\\text{e}\\) et \\(\\text{e}&lt;x_2\\) ces deux solutions. Alors, <em>a<\/em> peut s&rsquo;\u00e9crire de deux fa\u00e7ons:$$a = \\sqrt[x_1]{x_1} = \\sqrt[x_2]{x_2}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soit <em>a<\/em> \\(\\geq\\) 1. Consid\u00e9rons maintenant la suite \\((u_n(a)\\) d\u00e9finie par:$$\\begin{cases}u_1(a)=a\\\\u_{n+1}(a)=a^{u_n(a)}\\end{cases}$$pour \\(n\\geq1\\). Si la suite admet une limite, notons-la <em>u(a)<\/em>. Alors,$$u(a)=a^{a^{a^{\\ldots}}}.$$On peut aussi \u00e9crire:$$\\lim\\limits_{n\\to+\\infty} u_{n+1}(a)= \\lim\\limits_{n\\to+\\infty}  a^{u_n(a)} \\iff u(a)=a^{u(a)} \\iff a = \\sqrt[u(a)]{u(a)}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;apr\u00e8s ce qui a \u00e9t\u00e9 dit en amont, $$a &lt; \\sqrt[\\text{e}]{\\text{e}}.$$Cela signifie donc que la suite ne peut converger que si on a cette condition.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, \\(  \\sqrt[\\text{e}]{\\text{e}}  \\approx 1,44466 \\). En prenant \\(a = \\sqrt2\\approx1,414\\), la suite admet une limite \u00e9gale \u00e0 2 donc l&rsquo;\u00e9quation:$$x^{x^{x^{\\ldots}}}=2$$admet bien une solution qui est \\(\\sqrt2\\). En revanche, l&rsquo;\u00e9quation: $$x^{x^{x^{\\ldots}}}=4$$ne peut admettre de solution car 4 n&rsquo;est pas compris entre 1\/e et e. C&rsquo;est la raison pour laquelle la pseudo-preuve n&rsquo;est pas une preuve.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si vous souhaitez en savoir plus, je vous encourage \u00e0 lire <a rel=\"noreferrer noopener\" aria-label=\"cet article (s\u2019ouvre dans un nouvel onglet)\"  href=\"http:\/\/www.nntdm.net\/papers\/nntdm-16\/NNTDM-16-3-18-24.pdf\" target=\"_blank\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">cet article<\/a> (cliquez dessus pour \u00eatre redirig\u00e9s).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons voir dans cet article une preuve (bien entendu erron\u00e9e) que 2 = 4. Ce que nous allons voir est compr\u00e9hensible par des \u00e9l\u00e8ves de Terminale ayant vu la notion de continuit\u00e9 de fonctions.<\/p>\n","protected":false},"author":1,"featured_media":8793,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[185,25,176,186,108],"class_list":["post-2118","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-enseignement","category-mathematiques","tag-continuite","tag-demonstration","tag-exponentielle","tag-fonction","tag-suites"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Une preuve que 2 = 4 - 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\/2020\/03\/05\/une-preuve-que-2-4\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Une preuve que 2 = 4 - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Nous allons voir dans cet article une preuve (bien entendu erron\u00e9e) que 2 = 4. 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