{"id":8322,"date":"2023-04-17T13:16:17","date_gmt":"2023-04-17T11:16:17","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=8322"},"modified":"2023-04-17T14:01:36","modified_gmt":"2023-04-17T12:01:36","slug":"developpement-decimal-periodique-en-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/17\/developpement-decimal-periodique-en-python\/","title":{"rendered":"D\u00e9veloppement d\u00e9cimal p\u00e9riodique en Python"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Obtenir le d\u00e9veloppement d\u00e9cimal p\u00e9riodique d&rsquo;un nombre rationnel en Python peut para\u00eetre au prime abord simple&#8230; Et pourtant, quand on y r\u00e9fl\u00e9chir un peu, on s&rsquo;aper\u00e7oit que ce n&rsquo;est pas si trivial que \u00e7a.<\/p>\n\n\n\n<!--more-->\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\/2023\/04\/17\/developpement-decimal-periodique-en-python\/#Developpement_decimal_periodique_en_Python_un_exemple\" >D\u00e9veloppement d\u00e9cimal p\u00e9riodique en Python: un exemple<\/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\/2023\/04\/17\/developpement-decimal-periodique-en-python\/#Developpement_decimal_periodique_en_Python_implementation\" >D\u00e9veloppement d\u00e9cimal p\u00e9riodique en Python: impl\u00e9mentation<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/04\/17\/developpement-decimal-periodique-en-python\/#Une_premiere_implementation\" >Une premi\u00e8re impl\u00e9mentation<\/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\/2023\/04\/17\/developpement-decimal-periodique-en-python\/#Une_implementation_avec_le_module_richconsole\" >Une impl\u00e9mentation avec le module rich.console<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Developpement_decimal_periodique_en_Python_un_exemple\"><\/span>D\u00e9veloppement d\u00e9cimal p\u00e9riodique en Python: un exemple<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Consid\u00e9rons le nombre rationnel \\(\\displaystyle\\frac{12233}{99000}\\). Son \u00e9criture d\u00e9cimale est:$$\\frac{12233}{99000}=0,123\\underline{56}$$c&rsquo;est-\u00e0-dire:$$\\frac{12233}{99000}=0,12356565656\\ldots$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il est de coutume de souligner la s\u00e9quence des nombres qui se r\u00e9p\u00e8tent ind\u00e9finiment. Cette s\u00e9quence est appel\u00e9e la <em>p\u00e9riode<\/em> de l&rsquo;\u00e9criture d\u00e9cimale.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Quand on pose la division, on a ceci:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"204\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-1-300x204.png\" alt=\"d\u00e9veloppement d\u00e9cimal p\u00e9riodique python\" class=\"wp-image-8323\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-1-300x204.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-1-600x407.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-1.png 617w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On voit que d\u00e8s que l&rsquo;on obtient un reste d\u00e9j\u00e0 trouv\u00e9, cela signifie que la p\u00e9riode \u00e0 \u00e9t\u00e9 trouv\u00e9e. C&rsquo;est ce constat qui va nous permettre d&rsquo;impl\u00e9menter cela en Python.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Developpement_decimal_periodique_en_Python_implementation\"><\/span>D\u00e9veloppement d\u00e9cimal p\u00e9riodique en Python: impl\u00e9mentation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Comme nous allons nous servir des restes et des quotients de divisions euclidiennes, il nous faudra cr\u00e9er deux listes (par exemple <strong>rList<\/strong> et <strong>qList<\/strong> repr\u00e9sentant respectivement la liste des restes et des quotients). <\/p>\n\n\n\n<figure class=\"wp-block-pullquote is-style-Paragraph-paragraph\"><blockquote><p>Notre fonction devdec(&lsquo;a\/b&rsquo;) devra retourner une cha\u00eene de caract\u00e8res indiquant l&rsquo;\u00e9criture d\u00e9cimale de la fraction \\(\\frac{a}{b}\\) en mettant en relief la p\u00e9riode.<\/p><\/blockquote><\/figure>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On cr\u00e9e une boucle tant que le reste n&rsquo;est pas dans <em>rList<\/em>. Mais qu&rsquo;allons-nous faire dans cette boucle? Comme dans la division pos\u00e9e ci-dessus, nous allons regarder le quotient et le reste de la division euclidienne de 10<em>r<\/em> par <em>b<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">D\u00e8s lors que nous savons quoi faire, l&rsquo;impl\u00e9mentation est quasi-imm\u00e9diate.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_premiere_implementation\"><\/span>Une premi\u00e8re impl\u00e9mentation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from fractions import Fraction\n\ndef devdec(frac):\n    a,b = Fraction(frac).numerator, Fraction(frac).denominator\n    q, r = a\/\/b, a%b\n    qList = [str(q) , '.']\n    rList = []\n    while r not in rList:\n        rList.append( r )\n        q, r = (r*10)\/\/b, (r*10)%b\n        qList.append( str(q) )\n        \n    i = rList.index( r ) + 2\n    result = ''\n    \n    for n in range( len( qList ) ):\n        if n != i:\n            result += qList[n]\n        else:\n            result += '\\033[1;33m'+qList[n]\n            \n    return result<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Pour plus de pratique dans l&rsquo;\u00e9criture de l&rsquo;appel \u00e0 la fonction, j&rsquo;ai opt\u00e9 pour l&rsquo;utilisation du module <em>fractions<\/em> (plut\u00f4t que de taper \u00ab\u00a0devdec(a,b)\u00a0\u00bb, j&rsquo;ai pr\u00e9f\u00e9r\u00e9 \u00ab\u00a0devdec(&lsquo;a\/b&rsquo;)\u00a0\u00bb, plus explicite \u00e0 mes yeux).<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"794\" height=\"82\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3.png\" alt=\"\" class=\"wp-image-8327\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3.png 794w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3-600x62.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3-300x31.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-3-768x79.png 768w\" sizes=\"auto, (max-width: 794px) 100vw, 794px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_implementation_avec_le_module_richconsole\"><\/span>Une impl\u00e9mentation avec le module rich.console<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from fractions import Fraction\nfrom rich.console import Console\n\ndef devdec(frac):\n    result_console = Console()\n    a,b = Fraction(frac).numerator, Fraction(frac).denominator\n    q, r = a\/\/b, a%b\n    qList = [str(q) , '.']\n    rList = []\n    while r not in rList:\n        rList.append( r )\n        q, r = (r*10)\/\/b, (r*10)%b\n        qList.append( str(q) )\n        \n    i = rList.index( r ) + 2\n    result = '[white]'\n    \n    for n in range( len( qList ) ):\n        if n != i:\n            result += qList[n]\n        else:\n            result += '[u]'+qList[n]\n            \n    result + '[\/u][\/white]'\n    result_console.print( result , style='white' )<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">>>> devdec('12233\/99000')<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"801\" height=\"84\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4.png\" alt=\"\" class=\"wp-image-8328\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4.png 801w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4-600x63.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4-300x31.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/04\/image-4-768x81.png 768w\" sizes=\"auto, (max-width: 801px) 100vw, 801px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Vous pouvez formater le texte comme vous le souhaitez (voir <a href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/02\/11\/mettre-des-couleurs-en-mode-console-a-un-texte-sous-python\/\" target=\"_blank\" rel=\"noreferrer noopener\">cet article<\/a> pour mettre des couleurs ou autre).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Obtenir le d\u00e9veloppement d\u00e9cimal p\u00e9riodique d&rsquo;un nombre rationnel en Python peut para\u00eetre au prime abord simple&#8230; Et pourtant, quand on y r\u00e9fl\u00e9chir un peu, on s&rsquo;aper\u00e7oit que ce n&rsquo;est pas si trivial que \u00e7a.<\/p>\n","protected":false},"author":1,"featured_media":8329,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[384,221,383],"class_list":["post-8322","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-couleurs","tag-fractions","tag-soulignement"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ 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