{"id":14237,"date":"2025-06-29T03:14:11","date_gmt":"2025-06-29T01:14:11","guid":{"rendered":"https:\/\/www.gastonballiot.fr\/?p=14237"},"modified":"2025-06-22T16:05:31","modified_gmt":"2025-06-22T14:05:31","slug":"les-bases-de-numeration","status":"publish","type":"post","link":"https:\/\/www.gastonballiot.fr\/?p=14237","title":{"rendered":"Les bases de num\u00e9ration. Comment \u00e9crire un nombre."},"content":{"rendered":"

**   r\u00e9cr\u00e9ation scientifique n\u00b05   **<\/em><\/p>\n

Nous \u00e9crivons les nombres dans le syst\u00e8me d\u00e9cimal – base dix –
\nCertainement parce que nous avons dix doigts. Si nous avions eu douze doigts cela aurait peut \u00eatre mieux pour jouer du \u00eeano et nous aurions un syst\u00e8me \u00e0 base douze.<\/p>\n

Chaque chiffre correspond suivabt sa position \u00e0 une puissance de la base. Ce choix de notation a \u00e9t\u00e9 un grand progr\u00e8s dans l’\u00e9criture des nombres.
\nA comparer \u00e0 l’\u00e9criture en chiffres romains..<\/p>\n

Dans le syst\u00e8me d\u00e9cimal nous avons dix chiffres de 0 \u00e0 9
\net 257 repr\u00e9sente 7 unit\u00e9s + 5 dizaines + 2 centaines – dizaines de dizaines –
\nDans une base douze cette m\u00eame \u00e9criture repr\u00e9senterait le nombre 7+5×12+2×12\u00b2 soit l\u00e9entier que nous notons en d\u00e9cimal  355
\nDans la base huit on a huit chiffres 0,1,2,3,4,5,6,7 et le nombre entier not\u00e9 257 repr\u00e9sente 7+5×8+2×8\u00b2 soit 175 en d\u00e9cimal.<\/p>\n

En informatique on utilise les syst\u00e8me binaires (base deux) et hexad\u00e9cimal (base seize)<\/p>\n

En binaire on a deux chiffres 0 et 1
\n1010 –> le nombre dix
\nEn hexad\u00e9cimal on a seize chiffres not\u00e9s 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
\n5A –> le nombre quatre vingt dix
\n28 –> le nombre quarante<\/p>\n

Vous avez certainement vu des codes \u00e9crits en hexad\u00e9cimal sur certains appareils, par exemple Livebox-51F3<\/p>\n\n\n\n\n\n\n\n\n\n\n
d\u00e9cimal<\/td>\nbinaire<\/td>\nhexad\u00e9cimal<\/td>\n<\/tr>\n
1<\/td>\n1<\/td>\n1<\/td>\n<\/tr>\n
2<\/td>\n10<\/td>\n2<\/td>\n<\/tr>\n
5<\/td>\n101<\/td>\n5<\/td>\n<\/tr>\n
10<\/td>\n1010<\/td>\nA<\/td>\n<\/tr>\n
16<\/td>\n10000<\/td>\n10<\/td>\n<\/tr>\n
65<\/td>\n1000001<\/td>\n41<\/td>\n<\/tr>\n
43<\/td>\n101011<\/td>\n2B<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

La conversion entre binaire et hexad\u00e9cimal est tr\u00e8s simple et l’informaticien qui veut \u00e9crire un octet (huit chiffres binaires) l’\u00e9crit en hexa avec deux chiffres seulement.<\/p>\n

\n

\u00c9criture d\u2019un octet en hexad\u00e9cimal<\/h4>\n

\u00c9crire un octet en utilisant le binaire n\u2019est pas pratique (8 chiffres)
\n10001110 ou 01110101 ou 11111111 pour \u00ab le plus grand \u00bb
\nEn l\u2019\u00e9crivant en d\u00e9cimal ce n\u2019est pas pratique non plus, 0 pour le plus petit ou 255 pour le plus
\ngrand, la conversion d\u2019un syst\u00e8me \u00e0 l\u2019autre \u00e9tant fastidieuse.
\nLes informaticiens utilisent donc le syst\u00e8me hexad\u00e9cimal (\u00e0 base 16) car 16 \u00e9tant une puissance de
\n2 les conversions avec le binaire sont tr\u00e8s simples.
\nTout octet en hexad\u00e9cimal s\u2019\u00e9crit avec deux chiffres seulement au lieu de huit en binaire.
\nLes seize chiffres h\u00e9xa sont 0 1 2 3 4 5 6 7 8 9 A B C D E F
\nExemple 11111111 s\u2019\u00e9crit FF. On remarquera que le premier chiffre hexa est obtenu avec les quatre
\npremiers chiffres binaires et le second avec les quatre derniers, donc conversion tr\u00e8s facile.
\nExemple l\u2019octet 01011011 s\u2019\u00e9crira 5B  <\/p>\n","protected":false},"excerpt":{"rendered":"

**   r\u00e9cr\u00e9ation scientifique n\u00b05   ** Nous \u00e9crivons les nombres dans le syst\u00e8me d\u00e9cimal – base dix – Certainement parce que nous avons dix doigts. Si nous avions eu douze doigts cela aurait peut \u00eatre mieux pour jouer du \u00eeano et nous aurions un syst\u00e8me \u00e0 base douze. Chaque chiffre correspond suivabt sa position \u00e0 une puissance de la base. Ce choix de notation a \u00e9t\u00e9 un grand progr\u00e8s dans l’\u00e9criture des nombres. A comparer \u00e0 l’\u00e9criture en chiffres romains.. Dans le … Lire la suite…<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[662],"tags":[654,663],"class_list":["post-14237","post","type-post","status-publish","format-standard","hentry","category-recreation-scientifique","tag-mathematiques","tag-nombres","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"jetpack_publicize_connections":[],"yoast_head":"\nLes bases de num\u00e9ration. 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