{"id":1082,"date":"2019-03-15T14:22:51","date_gmt":"2019-03-15T13:22:51","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1082"},"modified":"2020-04-19T16:28:57","modified_gmt":"2020-04-19T14:28:57","slug":"python-et-le-nombre-dor","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/","title":{"rendered":"Python et le nombre d&rsquo;or"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>Voici un article qui est abordable d\u00e8s le lyc\u00e9e.<\/em><\/p>\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\/2019\/03\/15\/python-et-le-nombre-dor\/#La_suite_de_Fibonacci\" >La suite de Fibonacci<\/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\/2019\/03\/15\/python-et-le-nombre-dor\/#Le_nombre_dor\" >Le nombre d&rsquo;or<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/#Aller_plus_loin%E2%80%A6\" >Aller plus loin&#8230;<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_suite_de_Fibonacci\"><\/span>La suite de Fibonacci <span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Imaginons une suite de nombre qui commence par \u00ab\u00a01\u00a0\u00bb et \u00ab\u00a01\u00a0\u00bb.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On souhaite que le nombre qui vient juste apr\u00e8s soit \u00e9gal \u00e0 la somme des deux derniers nombres. Ainsi, le 3\u00e8me nombre est \u00e9gal \u00e0 1+1, soit \u00ab\u00a02\u00a0\u00bb. Apr\u00e8s \u00ab\u00a02\u00a0\u00bb, il y a 2+1=3, puis apr\u00e8s ce \u00ab\u00a03\u00a0\u00bb, il y a 3+2=5.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">Comprenez-vous maintenant comment on calcul les termes de cette suite de nombres ? On prend toujours les deux derniers, on les ajoute et \u00e7a nous donne le suivant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Comme ce proc\u00e9d\u00e9 est r\u00e9p\u00e9titif, on va pouvoir utiliser un programme pour trouver tous les nombres de cette suite. En Python, cela donne :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">F = [1,1]\nfor n in range(30):\n    F.extend([F[n+1]+F[n]])\n\nprint(F)<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">La premi\u00e8re ligne d\u00e9finie une liste (de nombres) que l&rsquo;on initialise avec les deux nombres desquels on part (donc ici, \u00ab\u00a01\u00a0\u00bb et \u00ab\u00a01\u00a0\u00bb). On a ainsi F[0]=1 et F[1]=1 (le premier item d&rsquo;une liste est toujours indic\u00e9 \u00e0 0).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, on cr\u00e9\u00e9 une boucle \u00ab\u00a0Pour\u00a0\u00bb afin de calculer ici 30 termes de plus : quand on \u00e9crit \u00ab\u00a0<strong><em>for n in range(30)<\/em><\/strong>\u00ab\u00a0, cela signifie que la variable n va prendre 30 valeurs enti\u00e8res en partant de 0.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dans cette boucle, on calcule la somme des deux derniers termes de la liste L, puis on ajoute le r\u00e9sultat \u00e0 la liste (c&rsquo;est la m\u00e9thode <em><strong>extend <\/strong><\/em>: on <em>\u00e9tend<\/em> la liste avec la valeur trouv\u00e9e).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Une fois sorti.e.s de la boucle, on affiche la liste, ce qui nous donne:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309]<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_nombre_dor\"><\/span>Le nombre d&rsquo;or<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, comme je suis quelqu&rsquo;un de tr\u00e8s bizarre (no comment, thanks!), j&rsquo;ai envie de calculer les quotients successifs de deux termes  cons\u00e9cutifs de cette suite. Je vais utiliser le code suivant:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">for n in range(31):\n    print(F[n+1]\/F[n])<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">ce qui m&rsquo;affiche:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\"> 1.0<br> 2.0<br> 1.5<br> 1.6666666666666667<br> 1.6<br> 1.625<br> 1.6153846153846154<br> 1.619047619047619<br> 1.6176470588235294<br> 1.6181818181818182<br> 1.6179775280898876<br> 1.6180555555555556<br> 1.6180257510729614<br> 1.6180371352785146<br> 1.618032786885246<br> 1.618034447821682<br> 1.6180338134001253<br> 1.618034055727554<br> 1.6180339631667064<br> 1.6180339985218033<br> 1.618033985017358<br> 1.6180339901755971<br> 1.618033988205325<br> 1.618033988957902<br> 1.6180339886704431<br> 1.6180339887802426<br> 1.618033988738303<br> 1.6180339887543225<br> 1.6180339887482036<br> 1.6180339887505408<br> 1.6180339887496482<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ne remarquez-vous pas quelque chose ? Ces quotients semblent se rapprocher d&rsquo;un nombre, dont la valeur approch\u00e9e au milli\u00e8me est 1,618. C&rsquo;est ce nombre que l&rsquo;on appelle le <em><strong>nombre d&rsquo;or<\/strong><\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dans la mesure o\u00f9 Python affiche 16 d\u00e9cimales, je me demande \u00e0 partir de quel rang j&rsquo;obtiendrai une valeur approch\u00e9e du nombre d&rsquo;or avec une pr\u00e9cision de \\(10^{-16}\\). Je vais utiliser ce code:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">F = [1,1,2,3]\nn = 3\n\nwhile (abs(F[n-2]\/F[n-3]-F[n]\/F[n-1])>10**(-16)):\n    u = F[n]+F[n-1]\n    F.extend([u])\n    n +=1\n    \nprint(F[n]\/F[n-1])<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Je demande ici \u00e0 calculer les termes successifs de la suite de Fibonacci <em>tant que<\/em> la valeur absolue de la diff\u00e9rence de deux quotients cons\u00e9cutifs est sup\u00e9rieure \u00e0 \\(10^{-16}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, la valeur affich\u00e9e sera une valeur approch\u00e9e du dernier quotient calcul\u00e9, qui sera aussi une valeur approch\u00e9e du nombre d&rsquo;or.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On obtient ici:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">1.618033988749895<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ce nombre d&rsquo;or est not\u00e9 par la lettre grecque: \\[ \\varphi \\approx 1,618033988749895.\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aller_plus_loin%E2%80%A6\"><\/span>Aller plus loin&#8230;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On peut dire beaucoup de choses sur le nombre d&rsquo;or, mais on peut aussi pr\u00e9ciser qu&rsquo;il existe un nombre d&rsquo;argent (appel\u00e9 ainsi par Gilles HAINRY, professeur \u00e0 l&rsquo;universit\u00e9 du Mans \u00e0 l&rsquo;\u00e9poque des faits, donc en 1996). Dans un cas g\u00e9n\u00e9ral, on parle de <em>nombres de m\u00e9tal<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour en savoir plus, je vous invite \u00e0 regarder l&rsquo;excellent ouvrage \u00ab\u00a0Ainsi de suites\u00a0\u00bb, \u00e9crit par&#8230;. Oh tiens ! Ecrit par moi \ud83d\ude42 ! Il est t\u00e9l\u00e9chargeable gratuitement sur ce site sur la page suivante :<a rel=\"noreferrer noopener\" aria-label=\" (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/www.mathweb.fr\/euclide\/ouvrages-personnels-de-mathematiques\/\" target=\"_blank\">https:\/\/www.mathweb.fr\/euclide\/ouvrages-personnels-de-mathematiques\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Voici un article qui est abordable d\u00e8s le lyc\u00e9e. La suite de Fibonacci Imaginons une suite de nombre qui commence par \u00ab\u00a01\u00a0\u00bb et \u00ab\u00a01\u00a0\u00bb. On souhaite que le nombre qui vient juste apr\u00e8s soit \u00e9gal \u00e0 la somme des deux derniers nombres. Ainsi, le 3\u00e8me nombre est \u00e9gal \u00e0 1+1, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,4,6,5],"tags":[],"class_list":["post-1082","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-informatique","category-mathematiques","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Python et le nombre d&#039;or - 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\/2019\/03\/15\/python-et-le-nombre-dor\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Python et le nombre d&#039;or - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Voici un article qui est abordable d\u00e8s le lyc\u00e9e. La suite de Fibonacci Imaginons une suite de nombre qui commence par \u00ab\u00a01\u00a0\u00bb et \u00ab\u00a01\u00a0\u00bb. On souhaite que le nombre qui vient juste apr\u00e8s soit \u00e9gal \u00e0 la somme des deux derniers nombres. 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La suite de Fibonacci Imaginons une suite de nombre qui commence par \u00ab\u00a01\u00a0\u00bb et \u00ab\u00a01\u00a0\u00bb. On souhaite que le nombre qui vient juste apr\u00e8s soit \u00e9gal \u00e0 la somme des deux derniers nombres. Ainsi, le 3\u00e8me nombre est \u00e9gal \u00e0 1+1, [&hellip;]","og_url":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/","og_site_name":"Mathweb.fr","article_published_time":"2019-03-15T13:22:51+00:00","article_modified_time":"2020-04-19T14:28:57+00:00","author":"St\u00e9phane Pasquet","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"St\u00e9phane Pasquet","Dur\u00e9e de lecture estim\u00e9e":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/#article","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/"},"author":{"name":"St\u00e9phane Pasquet","@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"headline":"Python et le nombre d&rsquo;or","datePublished":"2019-03-15T13:22:51+00:00","dateModified":"2020-04-19T14:28:57+00:00","mainEntityOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/"},"wordCount":527,"commentCount":0,"publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"articleSection":["Enseignement","Informatique","Math\u00e9matiques","Python"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/","url":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/","name":"Python et le nombre d'or - Mathweb.fr","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#website"},"datePublished":"2019-03-15T13:22:51+00:00","dateModified":"2020-04-19T14:28:57+00:00","breadcrumb":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/15\/python-et-le-nombre-dor\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.mathweb.fr\/euclide\/"},{"@type":"ListItem","position":2,"name":"Python et le nombre d&rsquo;or"}]},{"@type":"WebSite","@id":"https:\/\/www.mathweb.fr\/euclide\/#website","url":"https:\/\/www.mathweb.fr\/euclide\/","name":"Mathweb.fr","description":"Math\u00e9matiques, LaTeX et Python","publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.mathweb.fr\/euclide\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":["Person","Organization"],"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69","name":"St\u00e9phane Pasquet","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","width":74,"height":77,"caption":"St\u00e9phane Pasquet"},"logo":{"@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp"}}]}},"_links":{"self":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/1082","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/comments?post=1082"}],"version-history":[{"count":0,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/1082\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media?parent=1082"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/categories?post=1082"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/tags?post=1082"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}