{"id":2889,"date":"2020-07-09T16:15:04","date_gmt":"2020-07-09T14:15:04","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=2889"},"modified":"2023-04-16T16:19:26","modified_gmt":"2023-04-16T14:19:26","slug":"les-bases-en-python","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/les-bases-en-python\/","title":{"rendered":"Les bases en Python"},"content":{"rendered":"\n<p>Les bases en Python (binaire, hexad\u00e9cimale et d\u00e9cimale) ne font que traduire le syst\u00e8me de num\u00e9ration que nous utilisons de nos jours, qui n&#8217;est que la r\u00e9sultante d&#8217;une histoire. Nous comptons en base 10 car nous avons 10 doigts, mais en M\u00e9sopotamie, berceau des civilisations et des math\u00e9matiques, on comptait en base <em>sexag\u00e9simale<\/em>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">La base sexag\u00e9simale<\/h2>\n\n\n\n<p>En M\u00e9sopotamie, il fallait compter jusqu&#8217;\u00e0 plus de 10 pour savoir combien de b\u00eates on avait ou on voulait vendre. Dix doigts, ce n&#8217;\u00e9tait pas tr\u00e8s pratique&#8230; Aussi, on utilisait les phalanges.<\/p>\n\n\n\n<p>Prenons la main droite et servons-nous du pouce pour compter les phalanges des quatre autres doigts:<\/p>\n\n\n\n<figure class=\"wp-block-video aligncenter\"><video height=\"1146\" style=\"aspect-ratio: 1080 \/ 1146;\" width=\"1080\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/compter.mp4\"><\/video><figcaption>Compter jusqu&#8217;\u00e0 12 avec une seule main<\/figcaption><\/figure>\n\n\n\n<p>Nous arrivons jusqu&#8217;\u00e0 12. On peut alors lever le pouce de la main gauche et repartir pour compter \u00e0 nouveau jusqu&#8217;\u00e0 12 avec la main droite. Une fois arriv\u00e9s \u00e0 la deuxi\u00e8me douzaine, on peut lever l&#8217;index gauche pour d\u00e9signer le fait que l&#8217;on a compt\u00e9 2 douzaines. Comme la main gauche a 5 doigts (enfin, en g\u00e9n\u00e9ral&#8230;), on peut alors compter jusqu&#8217;\u00e0 \\(5 \\times 12=60\\). &#8220;60&#8221; est donc la base de la num\u00e9ration m\u00e9sopotamienne, la base <em>sexag\u00e9simale<\/em>. D&#8217;ailleurs, ne vous \u00eates-vous jamais demand\u00e9s pourquoi il y a 60 minutes dans une heure et 60 secondes dans une minute ? Pourquoi l&#8217;angle plein mesure-t-il 360\u00b0, c&#8217;est-\u00e0-dire 6 fois 60 ? Cela vient de cette base.<\/p>\n\n\n\n<p>La num\u00e9ration m\u00e9sopotamienne est bas\u00e9e sur le syst\u00e8me de position. Nous allons expliquer cela.<\/p>\n\n\n\n<p>Un clou repr\u00e9sente une unit\u00e9 (1).<\/p>\n\n\n\n<p>Un chevron repr\u00e9sente 10 unit\u00e9s (10). Bien que raisonnant en base sexag\u00e9simale, il semblerait que le dizaine ait une importance d\u00e8s cette \u00e8re.<\/p>\n\n\n\n<p>Ainsi, pour \u00e9crire &#8220;39&#8221;, ils utilisaient ces symboles en adoptant la convention que ce qui est le plus \u00e0 gauche est le plus important (comme dans notre syst\u00e8me de num\u00e9ration) :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"298\" height=\"133\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/39.png\" alt=\"\" class=\"wp-image-2893\"\/><figcaption>&#8220;39&#8221; \u00e9crit en m\u00e9sopotamien<\/figcaption><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">La base d\u00e9cimale<\/h2>\n\n\n\n<p>Si on analyse ce qui pr\u00e9c\u00e8de, on voit que les symboles repr\u00e9sentent:$$3\\times10+9\\times1$$que l&#8217;on peut aussi voir comme \u00e9tant:$$3\\times10^1 + 9\\times10^0.$$Et par extension, n&#8217;importe quel nombre s&#8217;\u00e9crivant \\(a_na_{n-1}a_{n-2}\\cdots a_0\\) peut s&#8217;\u00e9crire:$$a_n\\times10^n + a_{n-1}\\times10^{n-1} + a_{n-2}\\times10^{n-2}+\\cdots+a_1\\times10^1 + a_0\\times10^0=\\sum_{p=0}^n a_p\\times10^p.$$On \u00e9crira alors:$$\\overline{a_n\\cdots a_0}^{10}=\\sum_{p=0}^n a_p\\times10^p.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">La base hexad\u00e9cimale<\/h2>\n\n\n\n<p>Ce que l&#8217;on a fait pour 10, on peut le faire pour d&#8217;autres nombres. On peut donc imaginer une base 16, appel\u00e9e <em>hexad\u00e9cimale<\/em> (<em>hexa<\/em> = &#8220;six&#8221;, <em>d\u00e9cimale<\/em> = &#8220;dix&#8221;, <em>hexad\u00e9cimale<\/em> = 6 + 10 = 16).<\/p>\n\n\n\n<p>Comme nous n&#8217;avons que 10 symboles pour repr\u00e9senter des chiffres, il faudra utiliser des lettres pour les 6 manquants : A, B, C, D, E et F repr\u00e9senteront alors respectivement 10, 11, 12, 13, 14 et 15. Sur le m\u00eame principe que dans le syst\u00e8me d\u00e9cimal, les symboles les plus \u00e0 gauche repr\u00e9sentent les plus grandes puissances de 16.<\/p>\n\n\n\n<p>Par exemple,$$\\overline{A58E}^{16} = 10\\times16^3 + 5\\times16^2 + 8\\times16^1 + 14\\times16^0=42382.$$Cette base est utilis\u00e9e pour repr\u00e9senter les couleurs en informatique.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Le binaire<\/h2>\n\n\n\n<p>C&#8217;est la base qui ne comporte que deux symboles : &#8220;0&#8221; et &#8220;1&#8221;. <\/p>\n\n\n\n<p>Si nous voulons savoir quel nombre (d\u00e9cimal) repr\u00e9sente &#8220;1100101&#8221;, il suffit de l&#8217;imaginer dans un tableau comme celui-ci par exemple:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(2^6\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^5\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^4\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^3\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^2\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^1\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2^0\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>ce qui donne:$$\\overline{1100101}^2 = 2^6 + 2^5 + 2^2 + 2^0 = 101.$$On n&#8217;additionne ici que les puissances de 2 o\u00f9 il y a un &#8220;1&#8221;.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conversion de la base d\u00e9cimale \u00e0 l&#8217;hexad\u00e9cimal en Python<\/h2>\n\n\n\n<p>C&#8217;est le cas le plus simple car il y a une fonction d\u00e9di\u00e9e \u00e0 cela:<\/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=\"\">hex(42382)<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'0xa58a'<\/pre>\n\n\n\n<p>Vous constaterez que le r\u00e9sultat (a58e) est pr\u00e9c\u00e9d\u00e9 de &#8220;0x&#8221; pour sp\u00e9cifier que cela repr\u00e9sente un hexad\u00e9cimal.<\/p>\n\n\n\n<p>On peut aussi utiliser la fonction <em>format<\/em>:<\/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=\"\">format(42382,'#x')\nformat(42382,'x')<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'0xa58e'\n'a58e'<\/pre>\n\n\n\n<p>Notez que la deuxi\u00e8me solution supprime le pr\u00e9fixe &#8220;0x&#8221;.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conversion de la base d\u00e9cimale au binaire en Python<\/h2>\n\n\n\n<p>L\u00e0 encore, il y a une fonction :<\/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=\"\">bin(101)<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'0b1100101'<\/pre>\n\n\n\n<p>Le r\u00e9sultat (1100101) est cette fois-ci pr\u00e9c\u00e9d\u00e9e de &#8220;0b&#8221; pour sp\u00e9cifier que c&#8217;est un binaire.<\/p>\n\n\n\n<p>On peut aussi utiliser la fonction <em>format<\/em>:<\/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=\"\">format(101,'#b')\nformat(101,'b')<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'0b1100101'\n'1100101'<\/pre>\n\n\n\n<p>Constatez que l&#8217;on peut directement avoir le binaire sans le pr\u00e9fixe &#8220;0b&#8221;.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conversion de la base d\u00e9cimale en binaire et en hexad\u00e9cimal avec Python<\/h2>\n\n\n\n<p>Nous pouvons encore utiliser la fonction <em>format<\/em>:<\/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=\"\">format(0b1011001,'d')<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'89'<\/pre>\n\n\n\n<p>Ici, nous avons convertit le binaire \\(\\overline{1011001}^2\\) en d\u00e9cimal.<\/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=\"\">format(0xa45bc,'d')<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\">'673212'<\/pre>\n\n\n\n<p>Nous avons ici convertit l&#8217;hexad\u00e9cimal \\(\\overline{a45bc}^{16}\\) en d\u00e9cimal.<\/p>\n\n\n\n<p>Dans mon <a aria-label=\"undefined (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/www.mathweb.fr\/euclide\/numerique-et-sciences-informatiques-nsi\/\" target=\"_blank\" rel=\"noreferrer noopener\">livre de NSI niveau 1\u00e8re<\/a>, je pr\u00e9sente plusieurs programmes Python pour les conversions. C&#8217;est un bon entra\u00eenement pour celles et ceux souhaitant s&#8217;initier s\u00e9rieusement \u00e0 ce langage, mais \u00e0 l&#8217;avenir, il est pr\u00e9f\u00e9rable d&#8217;utiliser la fonction <em>format<\/em> pour toute conversion.<\/p>\n\n\n\n<p>Attention toutefois : les r\u00e9sultats sont de type <em>str<\/em> (cha\u00eene de caract\u00e8res); ce ne sont pas des nombres (et pour cause !).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les bases en Python (binaire, hexad\u00e9cimale et d\u00e9cimale) ne font que traduire le syst\u00e8me de num\u00e9ration que nous utilisons de nos jours, qui n&#8217;est que la r\u00e9sultante d&#8217;une histoire. Nous comptons en base 10 car nous avons 10 doigts, mais en M\u00e9sopotamie, berceau des civilisations et des math\u00e9matiques, on comptait [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2889","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les bases en Python - Mathweb.fr - Conversion en Python<\/title>\n<meta name=\"description\" content=\"Les bases en informatique : signification math\u00e9matique et conversion en Python. 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