{"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&#8217;or"},"content":{"rendered":"\n<p><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_82_2 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&#8217;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>Imaginons une suite de nombre qui commence par &#8220;1&#8221; et &#8220;1&#8221;.<\/p>\n\n\n\n<p>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 &#8220;2&#8221;. Apr\u00e8s &#8220;2&#8221;, il y a 2+1=3, puis apr\u00e8s ce &#8220;3&#8221;, il y a 3+2=5.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>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>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>La premi\u00e8re ligne d\u00e9finie une liste (de nombres) que l&#8217;on initialise avec les deux nombres desquels on part (donc ici, &#8220;1&#8221; et &#8220;1&#8221;). On a ainsi F[0]=1 et F[1]=1 (le premier item d&#8217;une liste est toujours indic\u00e9 \u00e0 0).<\/p>\n\n\n\n<p>Ensuite, on cr\u00e9\u00e9 une boucle &#8220;Pour&#8221; afin de calculer ici 30 termes de plus : quand on \u00e9crit &#8220;<strong><em>for n in range(30)<\/em><\/strong>&#8220;, cela signifie que la variable n va prendre 30 valeurs enti\u00e8res en partant de 0.<\/p>\n\n\n\n<p>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&#8217;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>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&#8217;or<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Maintenant, comme je suis quelqu&#8217;un de tr\u00e8s bizarre (no comment, thanks!), j&#8217;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>ce qui m&#8217;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>Ne remarquez-vous pas quelque chose ? Ces quotients semblent se rapprocher d&#8217;un nombre, dont la valeur approch\u00e9e au milli\u00e8me est 1,618. C&#8217;est ce nombre que l&#8217;on appelle le <em><strong>nombre d&#8217;or<\/strong><\/em>.<\/p>\n\n\n\n<p>Dans la mesure o\u00f9 Python affiche 16 d\u00e9cimales, je me demande \u00e0 partir de quel rang j&#8217;obtiendrai une valeur approch\u00e9e du nombre d&#8217;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>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>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&#8217;or.<\/p>\n\n\n\n<p>On obtient ici:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">1.618033988749895<\/pre>\n\n\n\n<p>Ce nombre d&#8217;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>On peut dire beaucoup de choses sur le nombre d&#8217;or, mais on peut aussi pr\u00e9ciser qu&#8217;il existe un nombre d&#8217;argent (appel\u00e9 ainsi par Gilles HAINRY, professeur \u00e0 l&#8217;universit\u00e9 du Mans \u00e0 l&#8217;\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>Pour en savoir plus, je vous invite \u00e0 regarder l&#8217;excellent ouvrage &#8220;Ainsi de suites&#8221;, \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 &#8220;1&#8221; et &#8220;1&#8221;. 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 v27.5 - 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 &#8220;1&#8221; et &#8220;1&#8221;. 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 &#8220;1&#8221; et &#8220;1&#8221;. 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&#8217;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}]}}