{"id":569,"date":"2018-11-26T10:31:13","date_gmt":"2018-11-26T09:31:13","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=569"},"modified":"2019-02-22T08:10:59","modified_gmt":"2019-02-22T07:10:59","slug":"la-plus-belle-formule-des-mathematiques","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/11\/26\/la-plus-belle-formule-des-mathematiques\/","title":{"rendered":"La plus belle formule des math\u00e9matiques"},"content":{"rendered":"<p>Quand on leur pose la question &#8220;quelle est la plus belle formule des math\u00e9matiques ?&#8221;, la plupart des math\u00e9maticiens r\u00e9pondent :<\/p>\n<p style=\"text-align: center\"><em><span style=\"color: #000000\"><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><i>e <sup>i \u03c0<\/sup><\/i><\/span><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\">\u00a0+\u00a0<\/span><\/span><\/em><span style=\"color: #000000\"><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\">1\u00a0=\u00a00<\/span><\/span><\/p>\n<p>Cette formule est due \u00e0 Leonhard Euler (1707 &#8211; 1783), auteur \u00e9galement de la formule plus utile mais moins belle :<\/p>\n<p style=\"text-align: center\"><span style=\"color: #000000\"><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><i>e <sup>i x<\/sup><\/i><\/span><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\">\u00a0=\u00a0cos\u00a0<i>x<\/i>\u00a0+\u00a0<i>i<\/i>\u00a0sin\u00a0<i>x<\/i> <\/span><\/span><\/p>\n<p><strong>Remarque :<\/strong> Cette formule est utile en particulier en trigonom\u00e9trie.<\/p>\n<blockquote>\n<h2 style=\"text-align: left\"><em>Beaut\u00e9 d&#8217;une formule<\/em><\/h2>\n<p><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">\u00c0 quoi tient la beaut\u00e9 de cette formule\u00a0? Sans doute dans la r\u00e9union des cinq constantes les plus importantes des math\u00e9matiques\u00a0: 0 et 1, les neutres de l\u2019addition et de la multiplication, le nombre complexe <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">i<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">, racine carr\u00e9e de \u20131 et les deux principales constantes transcendantes\u00a0: <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">e<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"> et <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">\u03c0<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">. Nous y voyons appara\u00eetre aussi les lois les plus usuelles\u00a0: addition, multiplication et exponentiation tandis que le cercle se devine sous la pr\u00e9sence du nombre d\u2019Archim\u00e8de\u00a0: <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">\u03c0<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">. De plus, cette formule lie l&#8217;arithm\u00e9tique (0 et 1), l&#8217;alg\u00e8bre (le nombre<\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"> i<\/span><\/span><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">),<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"> la g\u00e9om\u00e9trie (le nombre \u03c0) et l&#8217;analyse (le nombre\u00a0<\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">e<em> et l&#8217;exponentielle).<\/em><\/span><\/span><\/p>\n<h2>Beaut\u00e9 d&#8217;une preuve<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-572 aligncenter\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_-300x300.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_-300x300.png 300w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_-150x150.png 150w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_-768x768.png 768w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_-1024x1024.png 1024w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/euler.ai_.png 1074w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">Cette beaut\u00e9 se retrouve dans une d\u00e9monstration. D\u2019apr\u00e8s la formule d\u2019Euler ci dessus, <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">e <\/span><\/span><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"><sup>i x<\/sup><\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"> est repr\u00e9sent\u00e9 dans le plan par le point du cercle trigonom\u00e9trique (centre 0, rayon 1) \u00e0 l\u2019extr\u00e9mit\u00e9 du rayon d\u2019angle au centre x (avec l\u2019horizontale). En faisant varier x de 0 \u00e0 <\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">\u03c0<\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">, ce point passe de 1 \u00e0 \u20131. En ajoutant 1 \u00e0 e<sup>i\u03c0<\/sup>, on atteint alors 0. La formule\u00a0:<\/span><\/span><\/em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">e <\/span><\/span><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\"><sup>i \u03c0<\/sup><\/span><\/span><em><span style=\"margin: 0px;font-family: 'Arial',sans-serif;font-size: 11pt\"><span style=\"color: #000000\">\u00a0+\u00a01\u00a0=\u00a00 est ainsi d\u00e9montr\u00e9e par le mouvement d\u2019un point sur un cercle.<\/span><\/span><\/em><\/p>\n<h2>Beaut\u00e9 d&#8217;un objet<\/h2>\n<p><i><span style=\"font-family: Arial\">Lors du tricentenaire d&#8217;Euler, cette formule nous a inspir\u00e9 un bel objet : une lampe en verre que nous vous laissons admirer.\u00a0<\/span><\/i><\/p>\n<figure id=\"attachment_575\" aria-describedby=\"caption-attachment-575\" style=\"width: 225px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-575 size-medium\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/LampeEuler112007-225x300.jpg\" alt=\"\" width=\"225\" height=\"300\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/LampeEuler112007-225x300.jpg 225w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/LampeEuler112007-768x1024.jpg 768w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2018\/11\/LampeEuler112007.jpg 1678w\" sizes=\"auto, (max-width: 225px) 100vw, 225px\" \/><figcaption id=\"caption-attachment-575\" class=\"wp-caption-text\">Lampe en hommage \u00e0 Euler. \u00a9 Herv\u00e9 Lehning<\/figcaption><\/figure><\/blockquote>\n<!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"<p>Quand on leur pose la question &#8220;quelle est la plus belle formule des math\u00e9matiques ?&#8221;, la plupart des math\u00e9maticiens r\u00e9pondent : e i \u03c0\u00a0+\u00a01\u00a0=\u00a00 Cette formule est due \u00e0 Leonhard Euler (1707 &#8211; 1783), auteur \u00e9galement de la formule plus utile mais moins belle : e i x\u00a0=\u00a0cos\u00a0x\u00a0+\u00a0i\u00a0sin\u00a0x Remarque : Cette formule est utile en &hellip; <a href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/11\/26\/la-plus-belle-formule-des-mathematiques\/\" class=\"more-link\">Continuer la lecture de <span class=\"screen-reader-text\">La plus belle formule des math\u00e9matiques<\/span>  <span class=\"meta-nav\">&rarr;<\/span><\/a><!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt --><\/p>\n","protected":false},"author":12,"featured_media":577,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[5,29],"tags":[224,222,223],"class_list":["post-569","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-arts","category-maths-2","tag-beaute","tag-euler","tag-formule"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>La plus belle formule des math\u00e9matiques, par Herv\u00e9 Lehning<\/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:\/\/blogs.futura-sciences.com\/lehning\/2018\/11\/26\/la-plus-belle-formule-des-mathematiques\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La plus belle formule des math\u00e9matiques, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Quand on leur pose la question &#8220;quelle est la plus belle formule des math\u00e9matiques ?&#8221;, la plupart des math\u00e9maticiens r\u00e9pondent : e i \u03c0\u00a0+\u00a01\u00a0=\u00a00 Cette formule est due \u00e0 Leonhard Euler (1707 &#8211; 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