{"id":382,"date":"2018-08-21T14:42:13","date_gmt":"2018-08-21T12:42:13","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=382"},"modified":"2020-09-10T11:59:26","modified_gmt":"2020-09-10T09:59:26","slug":"demontrer-une-absurdite-mathematique","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/","title":{"rendered":"Absurdit\u00e9 math\u00e9matique : d\u00e9montrer l&rsquo;impossible"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Absurdit\u00e9 math\u00e9matique : \u00e7a, c&rsquo;est fun ! Il y a forc\u00e9ment une erreur, mais o\u00f9 ? C&rsquo;est un jeu amusant auquel on peut participer pour am\u00e9liorer notre r\u00e9flexion. C&rsquo;est comme les <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Liste_de_paradoxes\" target=\"_blank\" rel=\"noreferrer noopener\">paradoxes math\u00e9matiques<\/a> !<\/p>\n\n\n\n<!--more-->\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"770\" height=\"365\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/absurdite.jpg\" alt=\"absurdit\u00e9 math\u00e9matiques\" class=\"wp-image-3370\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/absurdite.jpg 770w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/absurdite-300x142.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/absurdite-600x284.jpg 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/absurdite-768x364.jpg 768w\" sizes=\"auto, (max-width: 770px) 100vw, 770px\" \/><\/figure><\/div>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 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\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Demontrer_que_1_2_en_fin_de_college_absurdite_mathematique\" >D\u00e9montrer que 1 = 2 (en fin de coll\u00e8ge) : absurdit\u00e9 math\u00e9matique<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#La_%C2%AB_preuve_%C2%BB_de_labsurdite_mathematique\" >La \u00ab\u00a0preuve\u00a0\u00bb de l&rsquo;absurdit\u00e9 math\u00e9matique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Lexplication_de_la_preuve\" >L&rsquo;explication de la preuve<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Epilogue\" >\u00c9pilogue<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Demontrer_que_1_3_en_fin_de_Lycee_absurdite_mathematique\" >D\u00e9montrer que 1 = 3 (en fin de Lyc\u00e9e) : absurdit\u00e9 math\u00e9matique<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#La_preuve\" >La preuve<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Prerequis\" >Pr\u00e9requis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Le_raisonnement\" >Le raisonnement<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Ou_est_lerreur\" >O\u00f9 est l&rsquo;erreur ?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/21\/demontrer-une-absurdite-mathematique\/#Epilogue-2\" >\u00c9pilogue<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demontrer_que_1_2_en_fin_de_college_absurdite_mathematique\"><\/span>D\u00e9montrer que 1 = 2 (en fin de coll\u00e8ge) : absurdit\u00e9 math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est sans doute l&rsquo;une des fa\u00e7ons les plus c\u00e9l\u00e8bres, mais il faut tout de m\u00eame en parler.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_%C2%AB_preuve_%C2%BB_de_labsurdite_mathematique\"><\/span>La \u00ab\u00a0preuve\u00a0\u00bb de l&rsquo;absurdit\u00e9 math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons deux nombres quelconques non nuls, que nous allons noter\u00a0<em>a<\/em> et\u00a0<em>b<\/em>. S&rsquo;ils sont \u00e9gaux, on peut \u00e9crire : $$a = b.$$ Et donc, en multipliant les deux nombres par\u00a0<em>a<\/em>, on obtient l&rsquo;\u00e9galit\u00e9 : $$a \\times a = a \\times b,$$ que l&rsquo;on peut aussi \u00e9crire : $$a^2 = ab.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, enlevons \\(b^2\\) \u00e0 droite et \u00e0 gauche du signe \u00ab\u00a0=\u00a0\u00bb :$$a^2 &#8211; b^2 = ab -b^2,$$que l&rsquo;on peut aussi \u00e9crire sous la forme factoris\u00e9e :$$(a-b)(a+b) = b(a-b).$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, divisons par\u00a0<em>a &#8211; b<\/em> les deux nombres \u00e0 droite et \u00e0 gauche du signe \u00ab\u00a0=\u00a0\u00bb :$$\\frac{\\pmb{(a-b)}(a+b)}{\\pmb{a-b}} = \\frac{b\\pmb{(a-b)}}{\\pmb{a-b}},$$qui se simplifie ainsi :$$ a+b=b.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Comme\u00a0<em>a<\/em> =\u00a0<em>b<\/em>, on peut remplacer\u00a0<em>b<\/em>\u00a0par\u00a0<em>a<\/em> dans cette derni\u00e8re \u00e9galit\u00e9, et on obtient :$$a+a = a, $$c&rsquo;est-\u00e0-dire :$$2a = a. $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En divisant par\u00a0<em>a<\/em> \u00e0 droite et \u00e0 gauche du signe \u00ab\u00a0=\u00a0\u00bb, on obtient :$$ 2 = 1. $$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Lexplication_de_la_preuve\"><\/span>L&rsquo;explication de la preuve<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si ce \u00e0 quoi on aboutit est faux, c&rsquo;est que nous avons fait une erreur dans la preuve. Et ici, il y a une erreur en effet&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Elle se trouve lorsque l&rsquo;on divise par&nbsp;<em>a<\/em> &#8211;&nbsp;<em>b<\/em>; en effet, nous avons suppos\u00e9 que&nbsp;<em>a<\/em> =&nbsp;<em>b<\/em> et donc&nbsp;<em>a<\/em> &#8211;&nbsp;<em>b<\/em> = 0. Or, diviser par 0 est impossible. Donc quand nous divisons par&nbsp; &nbsp; &nbsp; &nbsp;<em>a<\/em> &#8211;&nbsp;<em>b<\/em>, nous faisons une erreur math\u00e9matique et par cons\u00e9quent, la preuve n&rsquo;est plus valable.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Epilogue\"><\/span>\u00c9pilogue<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Rien de bien passionnant \u00e0 ce niveau l\u00e0, mais cela a le m\u00e9rite de montrer aux jeunes gens qu&rsquo;il faut toujours faire attention quand on r\u00e9dige une d\u00e9monstration. Il est facile de faire une telle erreur et la plupart du temps, on ne s&rsquo;en aper\u00e7oit m\u00eame pas dans la mesure o\u00f9 le r\u00e9sultat obtenu peut \u00eatre coh\u00e9rent.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demontrer_que_1_3_en_fin_de_Lycee_absurdite_mathematique\"><\/span>D\u00e9montrer que 1 = 3 (en fin de Lyc\u00e9e) : absurdit\u00e9 math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Ici, nous avons besoin de savoir ce que sont les nombres complexes.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_preuve\"><\/span>La preuve<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Prerequis\"><\/span>Pr\u00e9requis<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Avant tout, un petit rappel :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\begin{cases}\\forall\\ a&gt;0,\\ \\forall\\ x\\in\\mathbb{R}, &amp; a^x = \\text{e}^{x\\ln(a)}\\\\ \\forall\\ y\\in\\mathbb{R}^*, &amp; y^0 = 1 \\end{cases}&nbsp;\\]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_raisonnement\"><\/span>Le raisonnement<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;apr\u00e8s les pr\u00e9requis, on peut \u00e9crire :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;1^{\\frac{1}{3}}=\\text{e}^{\\frac{1}{3}\\ln(1)}=\\left(\\text{e}^{\\frac{1}{3}}\\right)^{\\ln(1)}=\\left(\\text{e}^{\\frac{1}{3}}\\right)^{0}=1.\\tag{1}&nbsp;\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, pour tout \\(k\\in\\mathbb{Z}\\) :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;1=\\text{e}^{2k\\text{i}\\pi} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">donc :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;1^{\\frac{1}{3}}=\\big(\\text{e}^{2k\\text{i}\\pi}\\big)^{\\frac{1}{3}}=\\text{e}^{\\frac{2k\\text{i}\\pi}{3}}.\\tag{2}&nbsp;\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, des \u00e9galit\u00e9s (1) et (2), par transitivit\u00e9, on en d\u00e9duit que :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;\\text{e}^{\\frac{2k\\text{i}\\pi}{3}}=1.&nbsp;\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors que :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;\\cos\\left(\\frac{2k\\pi}{3}\\right)+\\text{i}\\sin\\left(\\frac{2k\\pi}{3}\\right)=1&nbsp;\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">et donc que :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;\\begin{cases}&nbsp;\\cos\\left(\\frac{2k\\pi}{3}\\right)=1\\\\[1em]\\sin\\left(\\frac{2k\\pi}{3}\\right)=0\\end{cases}\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">ou encore :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;\\frac{2k\\pi}{3}= 2k\\pi \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d&rsquo;o\u00f9, en simplifiant par \\(2k\\pi\\), \\(k\\neq0\\) :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;\\frac{1}{3}=1 \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">soit finalement :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[&nbsp;1=3. \\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ou_est_lerreur\"><\/span>O\u00f9 est l&rsquo;erreur ?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Premi\u00e8re chose : la formule vue au coll\u00e8ge sur les exposants :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ x^{pq} = \\left(x^p\\right)^q,\\ p\\in\\mathbb{Z},\\ q\\in\\mathbb{Z} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">est aussi vraie si les exposants sont r\u00e9els.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, on a (pour \\( x &gt; 0\\)) d&rsquo;une part :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ x^{ab} = \\text{e}^{ab\\ln(x)},\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d&rsquo;autre part :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\left(x^a\\right)^b=\\left(\\text{e}^{a\\ln(x)}\\right)^b = \\text{e}^{b\\text{e}^{\\ln\\left(a\\ln(x)\\right)}}= \\text{e}^{b \\times a\\ln(x)}=\\text{e}^{ab\\ln (x)}. \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On obtient bien les deux m\u00eames expressions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De plus, l&rsquo;\u00e9galit\u00e9 (2) est vraie aussi. Il faut juste avoir \u00e0 l&rsquo;esprit qu&rsquo;elle est vraie pour tout&nbsp;<em>k<\/em> entier.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Alors, o\u00f9 est le bug ? Et bien, il est dans le fait de consid\u00e9rer l&rsquo;\u00e9galit\u00e9 (1) dans le corps des r\u00e9els.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, l&rsquo;\u00e9galit\u00e9 (1) est trait\u00e9e dans le corps des r\u00e9els alors que l&rsquo;\u00e9galit\u00e9 (2) est trait\u00e9e dans le corps des complexes. On ne peut pas tenir un raisonnement en passant d&rsquo;un corps \u00e0 l&rsquo;autre. C&rsquo;est le m\u00eame probl\u00e8me quand on dit que 1 + 1 = 0&#8230; mais dans l&rsquo;anneau \\(\\mathbb{Z}\/2\\mathbb{Z}\\). Quand on raisonne, il faut toujours se fixer un \u00ab\u00a0lieu\u00a0\u00bb&#8230; et ici, on prend le corps des nombres complexes (de toute \u00e9vidence).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;\u00e9galit\u00e9 (1) nous dit que seule \u00ab\u00a01\u00a0\u00bb est une racine cubique de \u00ab\u00a01\u00a0\u00bb : c&rsquo;est vrai dans \\(\\mathbb{R}\\) mais pas dans \\(\\mathbb{C}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le raisonnement ne tient donc pas d\u00e8s cette \u00e9galit\u00e9.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Epilogue-2\"><\/span>\u00c9pilogue<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">N&rsquo;oubliez pas que ce qui est tentant n&rsquo;est pas toujours bon&#8230; (et c&rsquo;est aussi valable pour le Nutella&#8230;)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si cet article vous a plus, vous aimeriez peut-\u00eatre <a href=\"https:\/\/www.mathweb.fr\/euclide\/les-differents-types-de-raisonnements-en-mathematiques\/\" target=\"_blank\" rel=\"noreferrer noopener\">cette page sur les diff\u00e9rents raisonnements math\u00e9matiques<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Absurdit\u00e9 math\u00e9matique : \u00e7a, c&rsquo;est fun ! Il y a forc\u00e9ment une erreur, mais o\u00f9 ? C&rsquo;est un jeu amusant auquel on peut participer pour am\u00e9liorer notre r\u00e9flexion. C&rsquo;est comme les paradoxes math\u00e9matiques !<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[24,25,26,23],"class_list":["post-382","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-absurdite","tag-demonstration","tag-identites-remarquables","tag-incoherence"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Absurdit\u00e9 math\u00e9matique : d\u00e9montrer l&#039;impossible - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"D\u00e9montrer une absurdit\u00e9 math\u00e9matique : voil\u00e0 une chose \u00e9trange. 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