{"id":1225,"date":"2016-03-25T14:58:34","date_gmt":"2016-03-25T14:58:34","guid":{"rendered":"http:\/\/pageperso.math.univ-toulouse.fr\/hippocampe\/?p=1225"},"modified":"2016-03-25T14:59:26","modified_gmt":"2016-03-25T14:59:26","slug":"lycee-saint-sernin-toulouse-nombres-irrationnels-et-fractions-continues","status":"publish","type":"post","link":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2016\/03\/25\/lycee-saint-sernin-toulouse-nombres-irrationnels-et-fractions-continues\/","title":{"rendered":"Lyc\u00e9e Saint-Sernin (Toulouse) &#8211; Nombres (ir)rationnels et fractions continues"},"content":{"rendered":"<p style=\"text-align: justify\">Les \u00e9l\u00e8ves de premi\u00e8re S du lyc\u00e9e Saint-Sernin sont venus \u00e0 l&#8217;institut de math\u00e9matiques de Toulouse les 9, 10 et 11 mars pour participer \u00e0 un stage hippocampe, d&#8217;initiation \u00e0 la recherche. \u00a0Le sujet de la semaine \u00e9tait bien myst\u00e9rieux : les nombres (ir)rationnels et les fractions continues.<\/p>\n<p style=\"text-align: justify\"><a href=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3802.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-medium wp-image-1226\" src=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3802-300x225.jpg\" alt=\"IMG_3802\" width=\"300\" height=\"225\" srcset=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3802-300x225.jpg 300w, https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3802-1024x768.jpg 1024w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Un nombre rationnel est une fraction, par exemple 3\/4. Par opposition, un nombre irrationnel est un nombre qui ne peut pas se mettre sous la forme d&#8217;une fraction de deux nombres entiers. Les premiers exemples de nombres irrationnels auxquels on peut penser sont\u00a0\u221a2 ou\u00a0\u03c0. Il n&#8217;est pas forc\u00e9ment \u00e9vident de montrer qu&#8217;ils sont irrationnels mais c&#8217;est un bon exercice de s&#8217;essayer sur\u00a0\u221a2.<\/p>\n<p style=\"text-align: justify\">Une mani\u00e8re de d\u00e9crire un nombre est d&#8217;utiliser sa d\u00e9composition d\u00e9cimale. Cependant, il arrive que cette d\u00e9composition soit infinie. C&#8217;est le cas pour les nombre irrationnel mais aussi pour certains nombres rationnels comme 1\/3 ou 2\/7. Dans ce stage, les \u00e9l\u00e8ves du lyc\u00e9e Saint-Sernin ont d\u00e9couvert une autre fa\u00e7on de d\u00e9crire les nombres : \u00e0 l&#8217;aide des fractions continues.<\/p>\n<p style=\"text-align: justify\"><a href=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3801.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\" size-medium wp-image-1228 alignright\" src=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3801-300x225.jpg\" alt=\"IMG_3801\" width=\"300\" height=\"225\" srcset=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3801-300x225.jpg 300w, https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/IMG_3801-1024x768.jpg 1024w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Il est facile de comprendre comment fonctionnent les fractions continue\u00a0sur les nombres rationnels. Si vous cherchez la fraction continue\u00a0associ\u00e9e au nombre 9\/5, il vous suffit de dessiner un carr\u00e9 de 9 par 5. Retirer ensuite le plus grand carr\u00e9 possible : ici, ce sera un carr\u00e9 de 5*5. R\u00e9it\u00e9rer cette op\u00e9ration : le prochain carr\u00e9 sera de 4*4. Pour les quatre prochaines \u00e9tapes, vous trouverez des carr\u00e9s de taille 1*1. La fraction continue associ\u00e9e \u00e0 9\/5 est donc 1+1\/(1+4). \u00c0 vous : pouvez-vous donner\u00a0la fraction continue associ\u00e9e \u00e0 la fraction 8\/5 ?<\/p>\n<p style=\"text-align: justify\">Passons maintenant aux nombres irrationnels et basons-nous sur la m\u00eame id\u00e9e. Consid\u00e9rons un rectangle de c\u00f4t\u00e9s 1 et\u00a0\u03c0. Le plus grand carr\u00e9 que l&#8217;on peut retirer de ce rectangle est un carr\u00e9 de taille 1*1, et on pourra en retirer 3 de cette taille. En continuant comme pr\u00e9c\u00e9demment, on trouve alors la fraction continue associ\u00e9e \u00e0\u00a0\u03c0. Les fractions continues permettent d&#8217;approcher tr\u00e8s efficacement les nombres par des fractions.<\/p>\n<p style=\"text-align: justify\">Pendant trois jours, les premi\u00e8res S de Saint-Sernin ont formalis\u00e9 et \u00e9tudi\u00e9 plus en d\u00e9tail la notion de fraction continue. Ils ont ensuite pr\u00e9par\u00e9 des posters pour les pr\u00e9senter aux membres de l&#8217;institut de math\u00e9matiques sur les sujets suivants :<\/p>\n<ul>\n<li style=\"text-align: justify\">convergence des fractions continues ;<\/li>\n<li style=\"text-align: justify\">monotonie de cette convergence ;<\/li>\n<li style=\"text-align: justify\">qualit\u00e9 de l&#8217;approximation donn\u00e9e par une fraction continue ;<\/li>\n<li style=\"text-align: justify\">l&#8217;incommensurabilit\u00e9 de la diagonale ;<\/li>\n<li style=\"text-align: justify\">la p\u00e9riodicit\u00e9 dans les d\u00e9veloppements d\u00e9cimaux ou en fraction continue.<\/li>\n<\/ul>\n\n\t\t<style>\n\t\t\t#gallery-1 {\n\t\t\t\tmargin: auto;\n\t\t\t}\n\t\t\t#gallery-1 .gallery-item {\n\t\t\t\tfloat: left;\n\t\t\t\tmargin-top: 10px;\n\t\t\t\ttext-align: center;\n\t\t\t\twidth: 33%;\n\t\t\t}\n\t\t\t#gallery-1 img {\n\t\t\t\tborder: 2px solid #cfcfcf;\n\t\t\t}\n\t\t\t#gallery-1 .gallery-caption {\n\t\t\t\tmargin-left: 0;\n\t\t\t}\n\t\t\t\/* see gallery_shortcode() in wp-includes\/media.php *\/\n\t\t<\/style>\n\t\t<div id='gallery-1' class='gallery galleryid-1225 gallery-columns-3 gallery-size-thumbnail'><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4796\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4796-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1180\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1180'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4797\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4797-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1181\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1181'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4798\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4798-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1182\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1182'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4799\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4799-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1183\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1183'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4800\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4800-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1184\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1184'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4801\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4801-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1185\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1185'>\n\t\t\t\tConvergence\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4802\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4802-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1186\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1186'>\n\t\t\t\tMonotonie\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4803\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4803-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1187\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1187'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4804\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4804-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1188\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1188'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4805\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4805-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1189\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1189'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4806\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4806-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1190\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1190'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4815\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4815-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1191\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1191'>\n\t\t\t\tIncommensurabilit\u00e9\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4816\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4816-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1192\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1192'>\n\t\t\t\tIncommensurabilit\u00e9\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4817\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4817-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1193\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1193'>\n\t\t\t\tIncommensurabilit\u00e9\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4818\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4818-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1194\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1194'>\n\t\t\t\tD\u00e9veloppement d\u00e9cimal\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4821\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4821-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1196\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1196'>\n\t\t\t\tP\u00e9riodicit\u00e9\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4822\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4822-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1197\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1197'>\n\t\t\t\tP\u00e9riodicit\u00e9\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4823\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4823-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1198\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1198'>\n\t\t\t\tP\u00e9riodicit\u00e9\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4824\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4824-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1199\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1199'>\n\t\t\t\tP\u00e9riodicit\u00e9\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4830\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4830-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1200\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1200'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4831\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4831-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1201\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1201'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4832\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4832-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1202\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1202'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/cimg4833-copie\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2016\/03\/CIMG4833-copie-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1203\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1203'>\n\t\t\t\tApproximation\n\t\t\t\t<\/dd><\/dl>\n\t\t\t<br style='clear: both' \/>\n\t\t<\/div>\n\n<p style=\"text-align: justify\">Pour aller plus loin : supposer acquis\u00a0le r\u00e9sultat de John Wallis suivant<\/p>\n<p style=\"text-align: center\">\u03c0\/2 = 2*\u220f(1-1\/(2i+1)\u00b2),<\/p>\n<p style=\"text-align: justify\">o\u00f9 le produit\u00a0\u220f est sur tous les i\u22651. En d\u00e9duire la fraction continue associ\u00e9e \u00e0 4\/\u03c0. Vous serez alors surpris de d\u00e9couvrir une formule tr\u00e8s simple donnant une tr\u00e8s bonne approximation du nombre irrationnel 4\/\u03c0.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les \u00e9l\u00e8ves de premi\u00e8re S du lyc\u00e9e Saint-Sernin sont venus \u00e0 l&#8217;institut de math\u00e9matiques de Toulouse les 9, 10 et 11 mars pour participer \u00e0 un stage hippocampe, d&#8217;initiation \u00e0 la recherche. \u00a0Le sujet de la semaine \u00e9tait bien myst\u00e9rieux : les nombres (ir)rationnels et les fractions continues. Un nombre rationnel est une fraction, par [&hellip;]<\/p>\n","protected":false},"author":131,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1225","post","type-post","status-publish","format-standard","hentry","category-stages"],"_links":{"self":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1225","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/users\/131"}],"replies":[{"embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/comments?post=1225"}],"version-history":[{"count":9,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1225\/revisions"}],"predecessor-version":[{"id":1236,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1225\/revisions\/1236"}],"wp:attachment":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/media?parent=1225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/categories?post=1225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/tags?post=1225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}