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cloud","\u002Fsnt-2nde\u002Fsd-donnees-structurees\u002Fcloud","0.snt-2nde\u002Fsd-donnees-structurees\u002F5.cloud",{"title":119,"path":120,"stem":121,"children":122},"Localisation, cartographie et mobilité","\u002Fsnt-2nde\u002Fse-localisation-cartographie","0.snt-2nde\u002Fse-localisation-cartographie\u002F0.index",[123,124,128,132,136,140],{"title":119,"path":120,"stem":121},{"title":125,"path":126,"stem":127},"GPS et Galileo — se situer grâce aux satellites","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fgps-galileo","0.snt-2nde\u002Fse-localisation-cartographie\u002F1.gps-galileo",{"title":129,"path":130,"stem":131},"Décoder une trame NMEA","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Ftrame-nmea","0.snt-2nde\u002Fse-localisation-cartographie\u002F2.trame-nmea",{"title":133,"path":134,"stem":135},"Cartes numériques — GeoPortail et 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l'appareil photo","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Falgorithmes-photo","0.snt-2nde\u002Fsg-photo-numerique\u002F2.algorithmes-photo",{"title":177,"path":178,"stem":179},"Les métadonnées EXIF","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fmetadonnees-exif","0.snt-2nde\u002Fsg-photo-numerique\u002F3.metadonnees-exif",{"title":181,"path":182,"stem":183},"Traiter une image en Python","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Ftraitement-rgb","0.snt-2nde\u002Fsg-photo-numerique\u002F4.traitement-rgb",{"title":185,"path":186,"stem":187,"children":188},"Notions transversales de programmation","\u002Fsnt-2nde\u002Fsh-prog-notions","0.snt-2nde\u002Fsh-prog-notions\u002F0.index",[189,190,194,198,202],{"title":185,"path":186,"stem":187},{"title":191,"path":192,"stem":193},"Séquences d'instructions et variables","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fsequences-affectations","0.snt-2nde\u002Fsh-prog-notions\u002F1.sequences-affectations",{"title":195,"path":196,"stem":197},"Instructions conditionnelles","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fconditionnelles","0.snt-2nde\u002Fsh-prog-notions\u002F2.conditionnelles",{"title":199,"path":200,"stem":201},"Boucles bornées et non bornées","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fboucles","0.snt-2nde\u002Fsh-prog-notions\u002F3.boucles",{"title":203,"path":204,"stem":205},"Définitions et appels de fonctions","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Ffonctions","0.snt-2nde\u002Fsh-prog-notions\u002F4.fonctions",{"title":207,"path":208,"stem":209,"children":210},"Démo des widgets MDC","\u002Fsnt-2nde\u002Fzz-demo-widgets","0.snt-2nde\u002Fzz-demo-widgets\u002F0.index",[211,212,216,220],{"title":207,"path":208,"stem":209},{"title":213,"path":214,"stem":215},"Démo statique — callouts, code, BO, illustration","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fstatique","0.snt-2nde\u002Fzz-demo-widgets\u002F1.statique",{"title":217,"path":218,"stem":219},"Démo interactive — self-eval et python-snippet","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Finteractif","0.snt-2nde\u002Fzz-demo-widgets\u002F2.interactif",{"title":221,"path":222,"stem":223},"Démo · 14 widgets de manipulation NSI","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fwidgets-manipulation","0.snt-2nde\u002Fzz-demo-widgets\u002F3.widgets-manipulation",{"title":225,"path":226,"stem":227,"children":228},"NSI — Première","\u002Fnsi-1ere","1.nsi-1ere\u002F0.index",[229,230,234,256,282,304,328,354,380,410],{"title":225,"path":226,"stem":227},{"title":231,"path":232,"stem":233},"Relecture pédagogique — 8 chapitres NSI 1ère (PA → PH)","\u002Fnsi-1ere\u002Freview","1.nsi-1ere\u002FREVIEW",{"title":235,"path":236,"stem":237,"children":238},"Histoire de 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aujourd'hui)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Freseaux-et-web","1.nsi-1ere\u002Fpa-histoire-info\u002F4.reseaux-et-web",{"title":257,"path":258,"stem":259,"children":260},"Représentation des données — types et valeurs","\u002Fnsi-1ere\u002Fpb-types-valeurs","1.nsi-1ere\u002Fpb-types-valeurs\u002F0.index",[261,262,266,270,274,278],{"title":257,"path":258,"stem":259},{"title":263,"path":264,"stem":265},"Bases de numération et entiers positifs","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbases-entiers-positifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs",{"title":267,"path":268,"stem":269},"Entiers relatifs et complément à deux","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs",{"title":271,"path":272,"stem":273},"Nombres flottants et virgule 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filtrage","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Frecherche","1.nsi-1ere\u002Fpd-tables-donnees\u002F2.recherche",{"title":319,"path":320,"stem":321},"Trier une table","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftri","1.nsi-1ere\u002Fpd-tables-donnees\u002F3.tri",{"title":323,"path":324,"stem":325,"children":326},"Fusion et indexation","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ffusion-indexation","1.nsi-1ere\u002Fpd-tables-donnees\u002F4.fusion-indexation",[327],{"title":323,"path":324,"stem":325},{"title":329,"path":330,"stem":331,"children":332},"IHM sur le Web","\u002Fnsi-1ere\u002Fpe-ihm-web","1.nsi-1ere\u002Fpe-ihm-web\u002F0.index",[333,334,338,342,346,350],{"title":329,"path":330,"stem":331},{"title":335,"path":336,"stem":337},"HTML et CSS — rappel de structure","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fhtml-css-bases","1.nsi-1ere\u002Fpe-ihm-web\u002F1.html-css-bases",{"title":339,"path":340,"stem":341},"Formulaires HTML","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fformulaires","1.nsi-1ere\u002Fpe-ihm-web\u002F2.formulaires",{"title":343,"path":344,"stem":345},"Événements JavaScript","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fevenements-js","1.nsi-1ere\u002Fpe-ihm-web\u002F3.evenements-js",{"title":347,"path":348,"stem":349},"Manipulation du DOM","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fdom-interaction","1.nsi-1ere\u002Fpe-ihm-web\u002F4.dom-interaction",{"title":351,"path":352,"stem":353},"Client, serveur et HTTP","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fclient-serveur-http","1.nsi-1ere\u002Fpe-ihm-web\u002F5.client-serveur-http",{"title":355,"path":356,"stem":357,"children":358},"Architectures matérielles et systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os","1.nsi-1ere\u002Fpf-archi-os\u002F0.index",[359,360,364,368,372,376],{"title":355,"path":356,"stem":357},{"title":361,"path":362,"stem":363},"Le modèle de von Neumann","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fvon-neumann","1.nsi-1ere\u002Fpf-archi-os\u002F1.von-neumann",{"title":365,"path":366,"stem":367},"Périphériques et interface homme-machine","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fperipheriques-ihm","1.nsi-1ere\u002Fpf-archi-os\u002F2.peripheriques-ihm",{"title":369,"path":370,"stem":371},"Les systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fsystemes-exploitation","1.nsi-1ere\u002Fpf-archi-os\u002F3.systemes-exploitation",{"title":373,"path":374,"stem":375},"Linux et le shell","\u002Fnsi-1ere\u002Fpf-archi-os\u002Flinux-shell","1.nsi-1ere\u002Fpf-archi-os\u002F4.linux-shell",{"title":377,"path":378,"stem":379},"Réseaux et protocoles","\u002Fnsi-1ere\u002Fpf-archi-os\u002Freseaux","1.nsi-1ere\u002Fpf-archi-os\u002F5.reseaux",{"title":381,"path":382,"stem":383,"children":384},"Langages et programmation","\u002Fnsi-1ere\u002Fpg-langages-prog","1.nsi-1ere\u002Fpg-langages-prog\u002F0.index",[385,386,390,394,398,402,406],{"title":381,"path":382,"stem":383},{"title":387,"path":388,"stem":389},"Diversité et unité des langages de programmation","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fdiversite-langages","1.nsi-1ere\u002Fpg-langages-prog\u002F1.diversite-langages",{"title":391,"path":392,"stem":393},"Constructions élémentaires","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fconstructions-elementaires","1.nsi-1ere\u002Fpg-langages-prog\u002F2.constructions-elementaires",{"title":395,"path":396,"stem":397},"Fonctions","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Ffonctions","1.nsi-1ere\u002Fpg-langages-prog\u002F3.fonctions",{"title":399,"path":400,"stem":401},"Spécification d'une fonction","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fspecification","1.nsi-1ere\u002Fpg-langages-prog\u002F4.specification",{"title":403,"path":404,"stem":405},"Mise au point des programmes","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fmise-au-point","1.nsi-1ere\u002Fpg-langages-prog\u002F5.mise-au-point",{"title":407,"path":408,"stem":409},"Utilisation de bibliothèques","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fbibliotheques","1.nsi-1ere\u002Fpg-langages-prog\u002F6.bibliotheques",{"title":411,"path":412,"stem":413,"children":414},"Algorithmique","\u002Fnsi-1ere\u002Fph-algorithmique","1.nsi-1ere\u002Fph-algorithmique\u002F0.index",[415,416,420,424,428,432,436],{"title":411,"path":412,"stem":413},{"title":417,"path":418,"stem":419},"Parcours séquentiel d'un tableau","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fparcours-sequentiel","1.nsi-1ere\u002Fph-algorithmique\u002F1.parcours-sequentiel",{"title":421,"path":422,"stem":423},"Bases de la complexité","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fcomplexite-bases","1.nsi-1ere\u002Fph-algorithmique\u002F2.complexite-bases",{"title":425,"path":426,"stem":427},"Tris par sélection et par insertion","\u002Fnsi-1ere\u002Fph-algorithmique\u002Ftris","1.nsi-1ere\u002Fph-algorithmique\u002F3.tris",{"title":429,"path":430,"stem":431},"Recherche dichotomique","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fdichotomie","1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie",{"title":433,"path":434,"stem":435},"Algorithmes gloutons","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fgloutons","1.nsi-1ere\u002Fph-algorithmique\u002F5.gloutons",{"title":437,"path":438,"stem":439},"Les k plus proches voisins (k-NN)","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fknn","1.nsi-1ere\u002Fph-algorithmique\u002F6.knn",{"title":441,"path":442,"stem":443,"children":444},"NSI — Terminale","\u002Fnsi-tale","2.nsi-tale\u002F0.index",[445,446,450,476,498,536,560],{"title":441,"path":442,"stem":443},{"title":447,"path":448,"stem":449},"Review pédagogique — chapitres NSI Terminale (TA, TB, TC, TD, TE)","\u002Fnsi-tale\u002Freview","2.nsi-tale\u002FREVIEW",{"title":451,"path":452,"stem":453,"children":454},"Structures de données","\u002Fnsi-tale\u002Fta-structures-donnees","2.nsi-tale\u002Fta-structures-donnees\u002F0.index",[455,456,460,464,468,472],{"title":451,"path":452,"stem":453},{"title":457,"path":458,"stem":459},"Objet, classe, interface","\u002Fnsi-tale\u002Fta-structures-donnees\u002Finterface-implementation","2.nsi-tale\u002Fta-structures-donnees\u002F1.interface-implementation",{"title":461,"path":462,"stem":463},"Implémenter une structure — l'exemple de la file FIFO","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fimplementer-une-structure","2.nsi-tale\u002Fta-structures-donnees\u002F2.implementer-une-structure",{"title":465,"path":466,"stem":467},"Listes, piles, files, dictionnaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Flistes-piles-files-dict","2.nsi-tale\u002Fta-structures-donnees\u002F3.listes-piles-files-dict",{"title":469,"path":470,"stem":471},"Arbres binaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Farbres-binaires","2.nsi-tale\u002Fta-structures-donnees\u002F4.arbres-binaires",{"title":473,"path":474,"stem":475},"Graphes","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fgraphes","2.nsi-tale\u002Fta-structures-donnees\u002F5.graphes",{"title":477,"path":478,"stem":479,"children":480},"Bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees","2.nsi-tale\u002Ftb-bases-de-donnees\u002F0.index",[481,482,486,490,494],{"title":477,"path":478,"stem":479},{"title":483,"path":484,"stem":485},"Le modèle relationnel","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fmodele-relationnel","2.nsi-tale\u002Ftb-bases-de-donnees\u002F1.modele-relationnel",{"title":487,"path":488,"stem":489},"Bases de données relationnelles et anomalies de schéma","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fbase-relationnelle","2.nsi-tale\u002Ftb-bases-de-donnees\u002F2.base-relationnelle",{"title":491,"path":492,"stem":493},"Le langage SQL","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsql","2.nsi-tale\u002Ftb-bases-de-donnees\u002F3.sql",{"title":495,"path":496,"stem":497},"Les systèmes de gestion de bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsgbd","2.nsi-tale\u002Ftb-bases-de-donnees\u002F4.sgbd",{"title":499,"path":500,"stem":501,"children":502},"Architectures matérielles, systèmes d'exploitation et réseaux","\u002Fnsi-tale\u002Ftc-archi-os-reseaux","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F0.index",[503,504,508,512,516,520,524,528,532],{"title":499,"path":500,"stem":501},{"title":505,"path":506,"stem":507},"Le système sur puce (SoC)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fsysteme-sur-puce","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F1.systeme-sur-puce",{"title":509,"path":510,"stem":511},"Processus et ordonnancement par l'OS","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprocessus-os","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F2.processus-os",{"title":513,"path":514,"stem":515},"L'interblocage (deadlock)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Finterblocage","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F3.interblocage",{"title":517,"path":518,"stem":519},"Principes du routage","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Froutage-principes","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F4.routage-principes",{"title":521,"path":522,"stem":523},"Protocoles de routage RIP et OSPF","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprotocoles-rip-ospf","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F5.protocoles-rip-ospf",{"title":525,"path":526,"stem":527},"Chiffrement symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-symetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F6.chiffrement-symetrique",{"title":529,"path":530,"stem":531},"Chiffrement asymétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-asymetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F7.chiffrement-asymetrique",{"title":533,"path":534,"stem":535},"HTTPS — l'échange de clé symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fhttps-tls","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F8.https-tls",{"title":381,"path":537,"stem":538,"children":539},"\u002Fnsi-tale\u002Ftd-langages-prog","2.nsi-tale\u002Ftd-langages-prog\u002F0.index",[540,541,544,548,552,556],{"title":381,"path":537,"stem":538},{"title":403,"path":542,"stem":543},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmise-au-point","2.nsi-tale\u002Ftd-langages-prog\u002F1.mise-au-point",{"title":545,"path":546,"stem":547},"Modularité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmodularite","2.nsi-tale\u002Ftd-langages-prog\u002F2.modularite",{"title":549,"path":550,"stem":551},"Récursivité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Frecursivite","2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite",{"title":553,"path":554,"stem":555},"Paradigmes de programmation","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fparadigmes","2.nsi-tale\u002Ftd-langages-prog\u002F4.paradigmes",{"title":557,"path":558,"stem":559},"Programme comme donnée — calculabilité et indécidabilité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fprogramme-comme-donnee","2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee",{"title":411,"path":561,"stem":562,"children":563},"\u002Fnsi-tale\u002Fte-algorithmique","2.nsi-tale\u002Fte-algorithmique\u002F0.index",[564,565,569,573,577,581],{"title":411,"path":561,"stem":562},{"title":566,"path":567,"stem":568},"Algorithmes sur les arbres binaires","\u002Fnsi-tale\u002Fte-algorithmique\u002Farbres-binaires-algos","2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos",{"title":570,"path":571,"stem":572},"Parcours de graphes — DFS, BFS, cycles, chemins","\u002Fnsi-tale\u002Fte-algorithmique\u002Fgraphes-parcours","2.nsi-tale\u002Fte-algorithmique\u002F2.graphes-parcours",{"title":574,"path":575,"stem":576},"Diviser pour régner","\u002Fnsi-tale\u002Fte-algorithmique\u002Fdiviser-pour-regner","2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner",{"title":578,"path":579,"stem":580},"Programmation dynamique","\u002Fnsi-tale\u002Fte-algorithmique\u002Fprogrammation-dynamique","2.nsi-tale\u002Fte-algorithmique\u002F4.programmation-dynamique",{"title":582,"path":583,"stem":584},"Recherche textuelle — l'algorithme de Boyer-Moore","\u002Fnsi-tale\u002Fte-algorithmique\u002Frecherche-textuelle","2.nsi-tale\u002Fte-algorithmique\u002F5.recherche-textuelle",[586,5651,8824,11701,12958],{"id":587,"title":263,"bo":588,"body":590,"description":5635,"eval":5636,"extension":5637,"labo":5636,"meta":5638,"navigation":5639,"path":264,"quizzes":5640,"readingTime":2889,"seo":5649,"stem":265,"tp":5636,"__hash__":5650},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs.md",[589],"PB01",{"type":591,"value":592,"toc":5624},"minimark",[593,598,832,859,895,1133,1593,1858,1967,2098,2283,2287,2294,2376,2870,2877,2965,3073,3077,3088,3119,3602,3707,3823,3826,3834,3838,3845,4026,4226,4233,4342,4346,4550,5112,5280,5337,5341,5546,5550,5620],[594,595,597],"h2",{"id":596},"introduction","Introduction",[599,600,601,602,606,607,700,701,763,764,827,828,831],"p",{},"Quand vous écrivez ",[603,604,605],"strong",{},"2026",", vous utilisez sans y penser une convention :\nchaque chiffre occupe une position qui correspond à une puissance de 10. La\nposition des unités vaut ",[608,609,612,641],"span",{"className":610},[611],"katex",[608,613,616],{"className":614},[615],"katex-mathml",[617,618,620],"math",{"xmlns":619},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[621,622,623,636],"semantics",{},[624,625,626],"mrow",{},[627,628,629,633],"msup",{},[630,631,632],"mn",{},"10",[630,634,635],{},"0",[637,638,640],"annotation",{"encoding":639},"application\u002Fx-tex","10^0",[608,642,646],{"className":643,"ariaHidden":645},[644],"katex-html","true",[608,647,650,655,660],{"className":648},[649],"base",[608,651],{"className":652,"style":654},[653],"strut","height:0.8141em;",[608,656,659],{"className":657},[658],"mord","1",[608,661,663,666],{"className":662},[658],[608,664,635],{"className":665},[658],[608,667,670],{"className":668},[669],"msupsub",[608,671,674],{"className":672},[673],"vlist-t",[608,675,678],{"className":676},[677],"vlist-r",[608,679,682],{"className":680,"style":654},[681],"vlist",[608,683,685,690],{"style":684},"top:-3.063em;margin-right:0.05em;",[608,686],{"className":687,"style":689},[688],"pstrut","height:2.7em;",[608,691,697],{"className":692},[693,694,695,696],"sizing","reset-size6","size3","mtight",[608,698,635],{"className":699},[658,696],", celle des dizaines ",[608,702,704,722],{"className":703},[611],[608,705,707],{"className":706},[615],[617,708,709],{"xmlns":619},[621,710,711,719],{},[624,712,713],{},[627,714,715,717],{},[630,716,632],{},[630,718,659],{},[637,720,721],{"encoding":639},"10^1",[608,723,725],{"className":724,"ariaHidden":645},[644],[608,726,728,731,734],{"className":727},[649],[608,729],{"className":730,"style":654},[653],[608,732,659],{"className":733},[658],[608,735,737,740],{"className":736},[658],[608,738,635],{"className":739},[658],[608,741,743],{"className":742},[669],[608,744,746],{"className":745},[673],[608,747,749],{"className":748},[677],[608,750,752],{"className":751,"style":654},[681],[608,753,754,757],{"style":684},[608,755],{"className":756,"style":689},[688],[608,758,760],{"className":759},[693,694,695,696],[608,761,659],{"className":762},[658,696],", celle des centaines\n",[608,765,767,786],{"className":766},[611],[608,768,770],{"className":769},[615],[617,771,772],{"xmlns":619},[621,773,774,783],{},[624,775,776],{},[627,777,778,780],{},[630,779,632],{},[630,781,782],{},"2",[637,784,785],{"encoding":639},"10^2",[608,787,789],{"className":788,"ariaHidden":645},[644],[608,790,792,795,798],{"className":791},[649],[608,793],{"className":794,"style":654},[653],[608,796,659],{"className":797},[658],[608,799,801,804],{"className":800},[658],[608,802,635],{"className":803},[658],[608,805,807],{"className":806},[669],[608,808,810],{"className":809},[673],[608,811,813],{"className":812},[677],[608,814,816],{"className":815,"style":654},[681],[608,817,818,821],{"style":684},[608,819],{"className":820,"style":689},[688],[608,822,824],{"className":823},[693,694,695,696],[608,825,782],{"className":826},[658,696],", et ainsi de suite. C'est la ",[603,829,830],{},"base 10",", ou notation décimale.",[599,833,834,835,838,839,842,843,847,848,851,852,855,856],{},"Un ordinateur, lui, ne distingue physiquement que deux états : courant qui\npasse ou non, aimantation positive ou négative, tension haute ou basse. Il est\ndonc naturel qu'il travaille en ",[603,836,837],{},"base 2"," : il n'utilise que deux chiffres, 0\net 1, appelés ",[603,840,841],{},"bits"," (",[844,845,846],"em",{},"binary digit","). Pour faciliter la lecture humaine de\nces longues suites de bits, on utilise aussi la ",[603,849,850],{},"base 16"," (hexadécimal) et,\nplus rarement, la ",[603,853,854],{},"base 8"," (octal). ",[857,858],"bo-ref",{"code":589},[594,860,862,863],{"id":861},"le-principe-dune-base-bbb","Le principe d'une base ",[608,864,866,881],{"className":865},[611],[608,867,869],{"className":868},[615],[617,870,871],{"xmlns":619},[621,872,873,879],{},[624,874,875],{},[876,877,878],"mi",{},"b",[637,880,878],{"encoding":639},[608,882,884],{"className":883,"ariaHidden":645},[644],[608,885,887,891],{"className":886},[649],[608,888],{"className":889,"style":890},[653],"height:0.6944em;",[608,892,878],{"className":893},[658,894],"mathnormal",[599,896,897,898,926,927,985,986,1014,1015,1043,1044,1099,1100,1103,1104,1132],{},"Dans une base ",[608,899,901,914],{"className":900},[611],[608,902,904],{"className":903},[615],[617,905,906],{"xmlns":619},[621,907,908,912],{},[624,909,910],{},[876,911,878],{},[637,913,878],{"encoding":639},[608,915,917],{"className":916,"ariaHidden":645},[644],[608,918,920,923],{"className":919},[649],[608,921],{"className":922,"style":890},[653],[608,924,878],{"className":925},[658,894]," (avec ",[608,928,930,950],{"className":929},[611],[608,931,933],{"className":932},[615],[617,934,935],{"xmlns":619},[621,936,937,947],{},[624,938,939,941,945],{},[876,940,878],{},[942,943,944],"mo",{},"≥",[630,946,782],{},[637,948,949],{"encoding":639},"b \\geq 2",[608,951,953,975],{"className":952,"ariaHidden":645},[644],[608,954,956,960,963,968,972],{"className":955},[649],[608,957],{"className":958,"style":959},[653],"height:0.8304em;vertical-align:-0.136em;",[608,961,878],{"className":962},[658,894],[608,964],{"className":965,"style":967},[966],"mspace","margin-right:0.2778em;",[608,969,944],{"className":970},[971],"mrel",[608,973],{"className":974,"style":967},[966],[608,976,978,982],{"className":977},[649],[608,979],{"className":980,"style":981},[653],"height:0.6444em;",[608,983,782],{"className":984},[658],"), on dispose de ",[608,987,989,1002],{"className":988},[611],[608,990,992],{"className":991},[615],[617,993,994],{"xmlns":619},[621,995,996,1000],{},[624,997,998],{},[876,999,878],{},[637,1001,878],{"encoding":639},[608,1003,1005],{"className":1004,"ariaHidden":645},[644],[608,1006,1008,1011],{"className":1007},[649],[608,1009],{"className":1010,"style":890},[653],[608,1012,878],{"className":1013},[658,894]," chiffres distincts, allant\nde ",[608,1016,1018,1031],{"className":1017},[611],[608,1019,1021],{"className":1020},[615],[617,1022,1023],{"xmlns":619},[621,1024,1025,1029],{},[624,1026,1027],{},[630,1028,635],{},[637,1030,635],{"encoding":639},[608,1032,1034],{"className":1033,"ariaHidden":645},[644],[608,1035,1037,1040],{"className":1036},[649],[608,1038],{"className":1039,"style":981},[653],[608,1041,635],{"className":1042},[658]," à ",[608,1045,1047,1066],{"className":1046},[611],[608,1048,1050],{"className":1049},[615],[617,1051,1052],{"xmlns":619},[621,1053,1054,1063],{},[624,1055,1056,1058,1061],{},[876,1057,878],{},[942,1059,1060],{},"−",[630,1062,659],{},[637,1064,1065],{"encoding":639},"b-1",[608,1067,1069,1090],{"className":1068,"ariaHidden":645},[644],[608,1070,1072,1076,1079,1083,1087],{"className":1071},[649],[608,1073],{"className":1074,"style":1075},[653],"height:0.7778em;vertical-align:-0.0833em;",[608,1077,878],{"className":1078},[658,894],[608,1080],{"className":1081,"style":1082},[966],"margin-right:0.2222em;",[608,1084,1060],{"className":1085},[1086],"mbin",[608,1088],{"className":1089,"style":1082},[966],[608,1091,1093,1096],{"className":1092},[649],[608,1094],{"className":1095,"style":981},[653],[608,1097,659],{"className":1098},[658],". Un nombre s'écrit comme une suite de ces chiffres, et la\n",[603,1101,1102],{},"valeur"," du nombre se calcule en sommant chaque chiffre multiplié par la\npuissance de ",[608,1105,1107,1120],{"className":1106},[611],[608,1108,1110],{"className":1109},[615],[617,1111,1112],{"xmlns":619},[621,1113,1114,1118],{},[624,1115,1116],{},[876,1117,878],{},[637,1119,878],{"encoding":639},[608,1121,1123],{"className":1122,"ariaHidden":645},[644],[608,1124,1126,1129],{"className":1125},[649],[608,1127],{"className":1128,"style":890},[653],[608,1130,878],{"className":1131},[658,894]," correspondant à sa position.",[599,1134,1135,1136,1456,1457,1485,1486,1559,1560,1563,1564,1592],{},"Pour un nombre noté ",[608,1137,1139,1204],{"className":1138},[611],[608,1140,1142],{"className":1141},[615],[617,1143,1144],{"xmlns":619},[621,1145,1146,1201],{},[624,1147,1148,1152,1167,1179,1182,1188,1194],{},[942,1149,1151],{"stretchy":1150},"false","(",[1153,1154,1155,1158],"msub",{},[876,1156,1157],{},"c",[624,1159,1160,1163,1165],{},[876,1161,1162],{},"n",[942,1164,1060],{},[630,1166,659],{},[1153,1168,1169,1171],{},[876,1170,1157],{},[624,1172,1173,1175,1177],{},[876,1174,1162],{},[942,1176,1060],{},[630,1178,782],{},[942,1180,1181],{},"…",[1153,1183,1184,1186],{},[876,1185,1157],{},[630,1187,659],{},[1153,1189,1190,1192],{},[876,1191,1157],{},[630,1193,635],{},[1153,1195,1196,1199],{},[942,1197,1198],{"stretchy":1150},")",[876,1200,878],{},[637,1202,1203],{"encoding":639},"(c_{n-1} c_{n-2} \\ldots c_1 c_0)_b",[608,1205,1207],{"className":1206,"ariaHidden":645},[644],[608,1208,1210,1214,1218,1273,1322,1326,1330,1333,1374,1414],{"className":1209},[649],[608,1211],{"className":1212,"style":1213},[653],"height:1em;vertical-align:-0.25em;",[608,1215,1151],{"className":1216},[1217],"mopen",[608,1219,1221,1224],{"className":1220},[658],[608,1222,1157],{"className":1223},[658,894],[608,1225,1227],{"className":1226},[669],[608,1228,1231,1264],{"className":1229},[673,1230],"vlist-t2",[608,1232,1234,1259],{"className":1233},[677],[608,1235,1238],{"className":1236,"style":1237},[681],"height:0.3011em;",[608,1239,1241,1244],{"style":1240},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[608,1242],{"className":1243,"style":689},[688],[608,1245,1247],{"className":1246},[693,694,695,696],[608,1248,1250,1253,1256],{"className":1249},[658,696],[608,1251,1162],{"className":1252},[658,894,696],[608,1254,1060],{"className":1255},[1086,696],[608,1257,659],{"className":1258},[658,696],[608,1260,1263],{"className":1261},[1262],"vlist-s","​",[608,1265,1267],{"className":1266},[677],[608,1268,1271],{"className":1269,"style":1270},[681],"height:0.2083em;",[608,1272],{},[608,1274,1276,1279],{"className":1275},[658],[608,1277,1157],{"className":1278},[658,894],[608,1280,1282],{"className":1281},[669],[608,1283,1285,1314],{"className":1284},[673,1230],[608,1286,1288,1311],{"className":1287},[677],[608,1289,1291],{"className":1290,"style":1237},[681],[608,1292,1293,1296],{"style":1240},[608,1294],{"className":1295,"style":689},[688],[608,1297,1299],{"className":1298},[693,694,695,696],[608,1300,1302,1305,1308],{"className":1301},[658,696],[608,1303,1162],{"className":1304},[658,894,696],[608,1306,1060],{"className":1307},[1086,696],[608,1309,782],{"className":1310},[658,696],[608,1312,1263],{"className":1313},[1262],[608,1315,1317],{"className":1316},[677],[608,1318,1320],{"className":1319,"style":1270},[681],[608,1321],{},[608,1323],{"className":1324,"style":1325},[966],"margin-right:0.1667em;",[608,1327,1181],{"className":1328},[1329],"minner",[608,1331],{"className":1332,"style":1325},[966],[608,1334,1336,1339],{"className":1335},[658],[608,1337,1157],{"className":1338},[658,894],[608,1340,1342],{"className":1341},[669],[608,1343,1345,1365],{"className":1344},[673,1230],[608,1346,1348,1362],{"className":1347},[677],[608,1349,1351],{"className":1350,"style":1237},[681],[608,1352,1353,1356],{"style":1240},[608,1354],{"className":1355,"style":689},[688],[608,1357,1359],{"className":1358},[693,694,695,696],[608,1360,659],{"className":1361},[658,696],[608,1363,1263],{"className":1364},[1262],[608,1366,1368],{"className":1367},[677],[608,1369,1372],{"className":1370,"style":1371},[681],"height:0.15em;",[608,1373],{},[608,1375,1377,1380],{"className":1376},[658],[608,1378,1157],{"className":1379},[658,894],[608,1381,1383],{"className":1382},[669],[608,1384,1386,1406],{"className":1385},[673,1230],[608,1387,1389,1403],{"className":1388},[677],[608,1390,1392],{"className":1391,"style":1237},[681],[608,1393,1394,1397],{"style":1240},[608,1395],{"className":1396,"style":689},[688],[608,1398,1400],{"className":1399},[693,694,695,696],[608,1401,635],{"className":1402},[658,696],[608,1404,1263],{"className":1405},[1262],[608,1407,1409],{"className":1408},[677],[608,1410,1412],{"className":1411,"style":1371},[681],[608,1413],{},[608,1415,1418,1421],{"className":1416},[1417],"mclose",[608,1419,1198],{"className":1420},[1417],[608,1422,1424],{"className":1423},[669],[608,1425,1427,1448],{"className":1426},[673,1230],[608,1428,1430,1445],{"className":1429},[677],[608,1431,1434],{"className":1432,"style":1433},[681],"height:0.3361em;",[608,1435,1436,1439],{"style":1240},[608,1437],{"className":1438,"style":689},[688],[608,1440,1442],{"className":1441},[693,694,695,696],[608,1443,878],{"className":1444},[658,894,696],[608,1446,1263],{"className":1447},[1262],[608,1449,1451],{"className":1450},[677],[608,1452,1454],{"className":1453,"style":1371},[681],[608,1455],{}," dans la base ",[608,1458,1460,1473],{"className":1459},[611],[608,1461,1463],{"className":1462},[615],[617,1464,1465],{"xmlns":619},[621,1466,1467,1471],{},[624,1468,1469],{},[876,1470,878],{},[637,1472,878],{"encoding":639},[608,1474,1476],{"className":1475,"ariaHidden":645},[644],[608,1477,1479,1482],{"className":1478},[649],[608,1480],{"className":1481,"style":890},[653],[608,1483,878],{"className":1484},[658,894],"\n(les ",[608,1487,1489,1508],{"className":1488},[611],[608,1490,1492],{"className":1491},[615],[617,1493,1494],{"xmlns":619},[621,1495,1496,1505],{},[624,1497,1498],{},[1153,1499,1500,1502],{},[876,1501,1157],{},[876,1503,1504],{},"i",[637,1506,1507],{"encoding":639},"c_i",[608,1509,1511],{"className":1510,"ariaHidden":645},[644],[608,1512,1514,1518],{"className":1513},[649],[608,1515],{"className":1516,"style":1517},[653],"height:0.5806em;vertical-align:-0.15em;",[608,1519,1521,1524],{"className":1520},[658],[608,1522,1157],{"className":1523},[658,894],[608,1525,1527],{"className":1526},[669],[608,1528,1530,1551],{"className":1529},[673,1230],[608,1531,1533,1548],{"className":1532},[677],[608,1534,1537],{"className":1535,"style":1536},[681],"height:0.3117em;",[608,1538,1539,1542],{"style":1240},[608,1540],{"className":1541,"style":689},[688],[608,1543,1545],{"className":1544},[693,694,695,696],[608,1546,1504],{"className":1547},[658,894,696],[608,1549,1263],{"className":1550},[1262],[608,1552,1554],{"className":1553},[677],[608,1555,1557],{"className":1556,"style":1371},[681],[608,1558],{}," étant les ",[603,1561,1562],{},"chiffres",", à ne pas confondre avec la base ",[608,1565,1567,1580],{"className":1566},[611],[608,1568,1570],{"className":1569},[615],[617,1571,1572],{"xmlns":619},[621,1573,1574,1578],{},[624,1575,1576],{},[876,1577,878],{},[637,1579,878],{"encoding":639},[608,1581,1583],{"className":1582,"ariaHidden":645},[644],[608,1584,1586,1589],{"className":1585},[649],[608,1587],{"className":1588,"style":890},[653],[608,1590,878],{"className":1591},[658,894],") :",[608,1594,1597],{"className":1595},[1596],"katex-display",[608,1598,1600,1656],{"className":1599},[611],[608,1601,1603],{"className":1602},[615],[617,1604,1606],{"xmlns":619,"display":1605},"block",[621,1607,1608,1653],{},[624,1609,1610,1613,1616,1638,1644,1647],{},[876,1611,1612],{},"N",[942,1614,1615],{},"=",[1617,1618,1619,1622,1630],"munderover",{},[942,1620,1621],{},"∑",[624,1623,1624,1626,1628],{},[876,1625,1504],{},[942,1627,1615],{},[630,1629,635],{},[624,1631,1632,1634,1636],{},[876,1633,1162],{},[942,1635,1060],{},[630,1637,659],{},[1153,1639,1640,1642],{},[876,1641,1157],{},[876,1643,1504],{},[942,1645,1646],{},"⋅",[627,1648,1649,1651],{},[876,1650,878],{},[876,1652,1504],{},[637,1654,1655],{"encoding":639},"N = \\sum_{i=0}^{n-1} c_i \\cdot b^i",[608,1657,1659,1679,1821],{"className":1658,"ariaHidden":645},[644],[608,1660,1662,1666,1670,1673,1676],{"className":1661},[649],[608,1663],{"className":1664,"style":1665},[653],"height:0.6833em;",[608,1667,1612],{"className":1668,"style":1669},[658,894],"margin-right:0.109em;",[608,1671],{"className":1672,"style":967},[966],[608,1674,1615],{"className":1675},[971],[608,1677],{"className":1678,"style":967},[966],[608,1680,1682,1686,1769,1772,1812,1815,1818],{"className":1681},[649],[608,1683],{"className":1684,"style":1685},[653],"height:3.0788em;vertical-align:-1.2777em;",[608,1687,1691],{"className":1688},[1689,1690],"mop","op-limits",[608,1692,1694,1760],{"className":1693},[673,1230],[608,1695,1697,1757],{"className":1696},[677],[608,1698,1701,1723,1736],{"className":1699,"style":1700},[681],"height:1.8011em;",[608,1702,1704,1708],{"style":1703},"top:-1.8723em;margin-left:0em;",[608,1705],{"className":1706,"style":1707},[688],"height:3.05em;",[608,1709,1711],{"className":1710},[693,694,695,696],[608,1712,1714,1717,1720],{"className":1713},[658,696],[608,1715,1504],{"className":1716},[658,894,696],[608,1718,1615],{"className":1719},[971,696],[608,1721,635],{"className":1722},[658,696],[608,1724,1726,1729],{"style":1725},"top:-3.05em;",[608,1727],{"className":1728,"style":1707},[688],[608,1730,1731],{},[608,1732,1621],{"className":1733},[1689,1734,1735],"op-symbol","large-op",[608,1737,1739,1742],{"style":1738},"top:-4.3em;margin-left:0em;",[608,1740],{"className":1741,"style":1707},[688],[608,1743,1745],{"className":1744},[693,694,695,696],[608,1746,1748,1751,1754],{"className":1747},[658,696],[608,1749,1162],{"className":1750},[658,894,696],[608,1752,1060],{"className":1753},[1086,696],[608,1755,659],{"className":1756},[658,696],[608,1758,1263],{"className":1759},[1262],[608,1761,1763],{"className":1762},[677],[608,1764,1767],{"className":1765,"style":1766},[681],"height:1.2777em;",[608,1768],{},[608,1770],{"className":1771,"style":1325},[966],[608,1773,1775,1778],{"className":1774},[658],[608,1776,1157],{"className":1777},[658,894],[608,1779,1781],{"className":1780},[669],[608,1782,1784,1804],{"className":1783},[673,1230],[608,1785,1787,1801],{"className":1786},[677],[608,1788,1790],{"className":1789,"style":1536},[681],[608,1791,1792,1795],{"style":1240},[608,1793],{"className":1794,"style":689},[688],[608,1796,1798],{"className":1797},[693,694,695,696],[608,1799,1504],{"className":1800},[658,894,696],[608,1802,1263],{"className":1803},[1262],[608,1805,1807],{"className":1806},[677],[608,1808,1810],{"className":1809,"style":1371},[681],[608,1811],{},[608,1813],{"className":1814,"style":1082},[966],[608,1816,1646],{"className":1817},[1086],[608,1819],{"className":1820,"style":1082},[966],[608,1822,1824,1828],{"className":1823},[649],[608,1825],{"className":1826,"style":1827},[653],"height:0.8747em;",[608,1829,1831,1834],{"className":1830},[658],[608,1832,878],{"className":1833},[658,894],[608,1835,1837],{"className":1836},[669],[608,1838,1840],{"className":1839},[673],[608,1841,1843],{"className":1842},[677],[608,1844,1846],{"className":1845,"style":1827},[681],[608,1847,1849,1852],{"style":1848},"top:-3.113em;margin-right:0.05em;",[608,1850],{"className":1851,"style":689},[688],[608,1853,1855],{"className":1854},[693,694,695,696],[608,1856,1504],{"className":1857},[658,894,696],[1859,1860,1861,1880],"table",{},[1862,1863,1864],"thead",{},[1865,1866,1867,1871,1874,1877],"tr",{},[1868,1869,1870],"th",{},"Base",[1868,1872,1873],{},"Nom",[1868,1875,1876],{},"Chiffres utilisés",[1868,1878,1879],{},"Préfixe Python",[1881,1882,1883,1904,1924,1941],"tbody",{},[1865,1884,1885,1888,1891,1899],{},[1886,1887,782],"td",{},[1886,1889,1890],{},"binaire",[1886,1892,1893,1896,1897],{},[1894,1895,635],"code",{},", ",[1894,1898,659],{},[1886,1900,1901],{},[1894,1902,1903],{},"0b",[1865,1905,1906,1909,1912,1919],{},[1886,1907,1908],{},"8",[1886,1910,1911],{},"octal",[1886,1913,1914,1043,1916],{},[1894,1915,635],{},[1894,1917,1918],{},"7",[1886,1920,1921],{},[1894,1922,1923],{},"0o",[1865,1925,1926,1928,1931,1938],{},[1886,1927,632],{},[1886,1929,1930],{},"décimal",[1886,1932,1933,1043,1935],{},[1894,1934,635],{},[1894,1936,1937],{},"9",[1886,1939,1940],{},"aucun",[1865,1942,1943,1946,1949,1962],{},[1886,1944,1945],{},"16",[1886,1947,1948],{},"hexadécimal",[1886,1950,1951,1043,1953,1955,1956,1043,1959],{},[1894,1952,635],{},[1894,1954,1937],{},", puis ",[1894,1957,1958],{},"A",[1894,1960,1961],{},"F",[1886,1963,1964],{},[1894,1965,1966],{},"0x",[599,1968,1969,1970,1043,1972,1974,1975,1043,2003,2032,2033,2036,2037,2066,2067,2036,2069,2097],{},"En hexadécimal, les lettres ",[1894,1971,1958],{},[1894,1973,1961],{}," représentent les valeurs ",[608,1976,1978,1991],{"className":1977},[611],[608,1979,1981],{"className":1980},[615],[617,1982,1983],{"xmlns":619},[621,1984,1985,1989],{},[624,1986,1987],{},[630,1988,632],{},[637,1990,632],{"encoding":639},[608,1992,1994],{"className":1993,"ariaHidden":645},[644],[608,1995,1997,2000],{"className":1996},[649],[608,1998],{"className":1999,"style":981},[653],[608,2001,632],{"className":2002},[658],[608,2004,2006,2020],{"className":2005},[611],[608,2007,2009],{"className":2008},[615],[617,2010,2011],{"xmlns":619},[621,2012,2013,2018],{},[624,2014,2015],{},[630,2016,2017],{},"15",[637,2019,2017],{"encoding":639},[608,2021,2023],{"className":2022,"ariaHidden":645},[644],[608,2024,2026,2029],{"className":2025},[649],[608,2027],{"className":2028,"style":981},[653],[608,2030,2017],{"className":2031},[658],".\nAinsi, ",[1894,2034,2035],{},"B"," vaut ",[608,2038,2040,2054],{"className":2039},[611],[608,2041,2043],{"className":2042},[615],[617,2044,2045],{"xmlns":619},[621,2046,2047,2052],{},[624,2048,2049],{},[630,2050,2051],{},"11",[637,2053,2051],{"encoding":639},[608,2055,2057],{"className":2056,"ariaHidden":645},[644],[608,2058,2060,2063],{"className":2059},[649],[608,2061],{"className":2062,"style":981},[653],[608,2064,2051],{"className":2065},[658]," et ",[1894,2068,1961],{},[608,2070,2072,2085],{"className":2071},[611],[608,2073,2075],{"className":2074},[615],[617,2076,2077],{"xmlns":619},[621,2078,2079,2083],{},[624,2080,2081],{},[630,2082,2017],{},[637,2084,2017],{"encoding":639},[608,2086,2088],{"className":2087,"ariaHidden":645},[644],[608,2089,2091,2094],{"className":2090},[649],[608,2092],{"className":2093,"style":981},[653],[608,2095,2017],{"className":2096},[658],".",[2099,2100,2101],"note",{},[599,2102,2103,2104,2185,2186,2275,2276,1896,2278,2066,2280,2282],{},"Pour lever toute ambiguïté, on note souvent la base en indice :\n",[608,2105,2107,2130],{"className":2106},[611],[608,2108,2110],{"className":2109},[615],[617,2111,2112],{"xmlns":619},[621,2113,2114,2127],{},[624,2115,2116,2118,2121],{},[942,2117,1151],{"stretchy":1150},[630,2119,2120],{},"1011",[1153,2122,2123,2125],{},[942,2124,1198],{"stretchy":1150},[630,2126,782],{},[637,2128,2129],{"encoding":639},"(1011)_2",[608,2131,2133],{"className":2132,"ariaHidden":645},[644],[608,2134,2136,2139,2142,2145],{"className":2135},[649],[608,2137],{"className":2138,"style":1213},[653],[608,2140,1151],{"className":2141},[1217],[608,2143,2120],{"className":2144},[658],[608,2146,2148,2151],{"className":2147},[1417],[608,2149,1198],{"className":2150},[1417],[608,2152,2154],{"className":2153},[669],[608,2155,2157,2177],{"className":2156},[673,1230],[608,2158,2160,2174],{"className":2159},[677],[608,2161,2163],{"className":2162,"style":1237},[681],[608,2164,2165,2168],{"style":1240},[608,2166],{"className":2167,"style":689},[688],[608,2169,2171],{"className":2170},[693,694,695,696],[608,2172,782],{"className":2173},[658,696],[608,2175,1263],{"className":2176},[1262],[608,2178,2180],{"className":2179},[677],[608,2181,2183],{"className":2182,"style":1371},[681],[608,2184],{}," pour le binaire, ",[608,2187,2189,2213],{"className":2188},[611],[608,2190,2192],{"className":2191},[615],[617,2193,2194],{"xmlns":619},[621,2195,2196,2210],{},[624,2197,2198,2200,2202,2204],{},[942,2199,1151],{"stretchy":1150},[630,2201,782],{},[876,2203,2035],{},[1153,2205,2206,2208],{},[942,2207,1198],{"stretchy":1150},[630,2209,1945],{},[637,2211,2212],{"encoding":639},"(2B)_{16}",[608,2214,2216],{"className":2215,"ariaHidden":645},[644],[608,2217,2219,2222,2225,2228,2232],{"className":2218},[649],[608,2220],{"className":2221,"style":1213},[653],[608,2223,1151],{"className":2224},[1217],[608,2226,782],{"className":2227},[658],[608,2229,2035],{"className":2230,"style":2231},[658,894],"margin-right:0.0502em;",[608,2233,2235,2238],{"className":2234},[1417],[608,2236,1198],{"className":2237},[1417],[608,2239,2241],{"className":2240},[669],[608,2242,2244,2267],{"className":2243},[673,1230],[608,2245,2247,2264],{"className":2246},[677],[608,2248,2250],{"className":2249,"style":1237},[681],[608,2251,2252,2255],{"style":1240},[608,2253],{"className":2254,"style":689},[688],[608,2256,2258],{"className":2257},[693,694,695,696],[608,2259,2261],{"className":2260},[658,696],[608,2262,1945],{"className":2263},[658,696],[608,2265,1263],{"className":2266},[1262],[608,2268,2270],{"className":2269},[677],[608,2271,2273],{"className":2272,"style":1371},[681],[608,2274],{}," pour l'hexadécimal. En Python, les\npréfixes ",[1894,2277,1903],{},[1894,2279,1923],{},[1894,2281,1966],{}," jouent le même rôle.",[594,2284,2286],{"id":2285},"du-binaire-au-décimal","Du binaire au décimal",[599,2288,2289,2290,2293],{},"Pour convertir une écriture binaire en décimal, on ",[603,2291,2292],{},"somme les puissances de\n2"," correspondant aux bits valant 1.",[599,2295,2296,2297,2097],{},"Exemple : convertir ",[608,2298,2300,2321],{"className":2299},[611],[608,2301,2303],{"className":2302},[615],[617,2304,2305],{"xmlns":619},[621,2306,2307,2319],{},[624,2308,2309,2311,2313],{},[942,2310,1151],{"stretchy":1150},[630,2312,2120],{},[1153,2314,2315,2317],{},[942,2316,1198],{"stretchy":1150},[630,2318,782],{},[637,2320,2129],{"encoding":639},[608,2322,2324],{"className":2323,"ariaHidden":645},[644],[608,2325,2327,2330,2333,2336],{"className":2326},[649],[608,2328],{"className":2329,"style":1213},[653],[608,2331,1151],{"className":2332},[1217],[608,2334,2120],{"className":2335},[658],[608,2337,2339,2342],{"className":2338},[1417],[608,2340,1198],{"className":2341},[1417],[608,2343,2345],{"className":2344},[669],[608,2346,2348,2368],{"className":2347},[673,1230],[608,2349,2351,2365],{"className":2350},[677],[608,2352,2354],{"className":2353,"style":1237},[681],[608,2355,2356,2359],{"style":1240},[608,2357],{"className":2358,"style":689},[688],[608,2360,2362],{"className":2361},[693,694,695,696],[608,2363,782],{"className":2364},[658,696],[608,2366,1263],{"className":2367},[1262],[608,2369,2371],{"className":2370},[677],[608,2372,2374],{"className":2373,"style":1371},[681],[608,2375],{},[608,2377,2379],{"className":2378},[1596],[608,2380,2382,2474],{"className":2381},[611],[608,2383,2385],{"className":2384},[615],[617,2386,2387],{"xmlns":619,"display":1605},[621,2388,2389,2471],{},[624,2390,2391,2393,2395,2401,2403,2405,2407,2414,2417,2419,2421,2427,2429,2431,2433,2439,2441,2443,2445,2451,2453,2455,2457,2459,2461,2463,2465,2467,2469],{},[942,2392,1151],{"stretchy":1150},[630,2394,2120],{},[1153,2396,2397,2399],{},[942,2398,1198],{"stretchy":1150},[630,2400,782],{},[942,2402,1615],{},[630,2404,659],{},[942,2406,1646],{},[627,2408,2409,2411],{},[630,2410,782],{},[630,2412,2413],{},"3",[942,2415,2416],{},"+",[630,2418,635],{},[942,2420,1646],{},[627,2422,2423,2425],{},[630,2424,782],{},[630,2426,782],{},[942,2428,2416],{},[630,2430,659],{},[942,2432,1646],{},[627,2434,2435,2437],{},[630,2436,782],{},[630,2438,659],{},[942,2440,2416],{},[630,2442,659],{},[942,2444,1646],{},[627,2446,2447,2449],{},[630,2448,782],{},[630,2450,635],{},[942,2452,1615],{},[630,2454,1908],{},[942,2456,2416],{},[630,2458,635],{},[942,2460,2416],{},[630,2462,782],{},[942,2464,2416],{},[630,2466,659],{},[942,2468,1615],{},[630,2470,2051],{},[637,2472,2473],{"encoding":639},"(1011)_2 = 1 \\cdot 2^3 + 0 \\cdot 2^2 + 1 \\cdot 2^1 + 1 \\cdot 2^0 = 8 + 0 + 2 + 1 = 11",[608,2475,2477,2538,2556,2602,2620,2664,2682,2726,2744,2788,2807,2825,2843,2861],{"className":2476,"ariaHidden":645},[644],[608,2478,2480,2483,2486,2489,2529,2532,2535],{"className":2479},[649],[608,2481],{"className":2482,"style":1213},[653],[608,2484,1151],{"className":2485},[1217],[608,2487,2120],{"className":2488},[658],[608,2490,2492,2495],{"className":2491},[1417],[608,2493,1198],{"className":2494},[1417],[608,2496,2498],{"className":2497},[669],[608,2499,2501,2521],{"className":2500},[673,1230],[608,2502,2504,2518],{"className":2503},[677],[608,2505,2507],{"className":2506,"style":1237},[681],[608,2508,2509,2512],{"style":1240},[608,2510],{"className":2511,"style":689},[688],[608,2513,2515],{"className":2514},[693,694,695,696],[608,2516,782],{"className":2517},[658,696],[608,2519,1263],{"className":2520},[1262],[608,2522,2524],{"className":2523},[677],[608,2525,2527],{"className":2526,"style":1371},[681],[608,2528],{},[608,2530],{"className":2531,"style":967},[966],[608,2533,1615],{"className":2534},[971],[608,2536],{"className":2537,"style":967},[966],[608,2539,2541,2544,2547,2550,2553],{"className":2540},[649],[608,2542],{"className":2543,"style":981},[653],[608,2545,659],{"className":2546},[658],[608,2548],{"className":2549,"style":1082},[966],[608,2551,1646],{"className":2552},[1086],[608,2554],{"className":2555,"style":1082},[966],[608,2557,2559,2563,2593,2596,2599],{"className":2558},[649],[608,2560],{"className":2561,"style":2562},[653],"height:0.9474em;vertical-align:-0.0833em;",[608,2564,2566,2569],{"className":2565},[658],[608,2567,782],{"className":2568},[658],[608,2570,2572],{"className":2571},[669],[608,2573,2575],{"className":2574},[673],[608,2576,2578],{"className":2577},[677],[608,2579,2582],{"className":2580,"style":2581},[681],"height:0.8641em;",[608,2583,2584,2587],{"style":1848},[608,2585],{"className":2586,"style":689},[688],[608,2588,2590],{"className":2589},[693,694,695,696],[608,2591,2413],{"className":2592},[658,696],[608,2594],{"className":2595,"style":1082},[966],[608,2597,2416],{"className":2598},[1086],[608,2600],{"className":2601,"style":1082},[966],[608,2603,2605,2608,2611,2614,2617],{"className":2604},[649],[608,2606],{"className":2607,"style":981},[653],[608,2609,635],{"className":2610},[658],[608,2612],{"className":2613,"style":1082},[966],[608,2615,1646],{"className":2616},[1086],[608,2618],{"className":2619,"style":1082},[966],[608,2621,2623,2626,2655,2658,2661],{"className":2622},[649],[608,2624],{"className":2625,"style":2562},[653],[608,2627,2629,2632],{"className":2628},[658],[608,2630,782],{"className":2631},[658],[608,2633,2635],{"className":2634},[669],[608,2636,2638],{"className":2637},[673],[608,2639,2641],{"className":2640},[677],[608,2642,2644],{"className":2643,"style":2581},[681],[608,2645,2646,2649],{"style":1848},[608,2647],{"className":2648,"style":689},[688],[608,2650,2652],{"className":2651},[693,694,695,696],[608,2653,782],{"className":2654},[658,696],[608,2656],{"className":2657,"style":1082},[966],[608,2659,2416],{"className":2660},[1086],[608,2662],{"className":2663,"style":1082},[966],[608,2665,2667,2670,2673,2676,2679],{"className":2666},[649],[608,2668],{"className":2669,"style":981},[653],[608,2671,659],{"className":2672},[658],[608,2674],{"className":2675,"style":1082},[966],[608,2677,1646],{"className":2678},[1086],[608,2680],{"className":2681,"style":1082},[966],[608,2683,2685,2688,2717,2720,2723],{"className":2684},[649],[608,2686],{"className":2687,"style":2562},[653],[608,2689,2691,2694],{"className":2690},[658],[608,2692,782],{"className":2693},[658],[608,2695,2697],{"className":2696},[669],[608,2698,2700],{"className":2699},[673],[608,2701,2703],{"className":2702},[677],[608,2704,2706],{"className":2705,"style":2581},[681],[608,2707,2708,2711],{"style":1848},[608,2709],{"className":2710,"style":689},[688],[608,2712,2714],{"className":2713},[693,694,695,696],[608,2715,659],{"className":2716},[658,696],[608,2718],{"className":2719,"style":1082},[966],[608,2721,2416],{"className":2722},[1086],[608,2724],{"className":2725,"style":1082},[966],[608,2727,2729,2732,2735,2738,2741],{"className":2728},[649],[608,2730],{"className":2731,"style":981},[653],[608,2733,659],{"className":2734},[658],[608,2736],{"className":2737,"style":1082},[966],[608,2739,1646],{"className":2740},[1086],[608,2742],{"className":2743,"style":1082},[966],[608,2745,2747,2750,2779,2782,2785],{"className":2746},[649],[608,2748],{"className":2749,"style":2581},[653],[608,2751,2753,2756],{"className":2752},[658],[608,2754,782],{"className":2755},[658],[608,2757,2759],{"className":2758},[669],[608,2760,2762],{"className":2761},[673],[608,2763,2765],{"className":2764},[677],[608,2766,2768],{"className":2767,"style":2581},[681],[608,2769,2770,2773],{"style":1848},[608,2771],{"className":2772,"style":689},[688],[608,2774,2776],{"className":2775},[693,694,695,696],[608,2777,635],{"className":2778},[658,696],[608,2780],{"className":2781,"style":967},[966],[608,2783,1615],{"className":2784},[971],[608,2786],{"className":2787,"style":967},[966],[608,2789,2791,2795,2798,2801,2804],{"className":2790},[649],[608,2792],{"className":2793,"style":2794},[653],"height:0.7278em;vertical-align:-0.0833em;",[608,2796,1908],{"className":2797},[658],[608,2799],{"className":2800,"style":1082},[966],[608,2802,2416],{"className":2803},[1086],[608,2805],{"className":2806,"style":1082},[966],[608,2808,2810,2813,2816,2819,2822],{"className":2809},[649],[608,2811],{"className":2812,"style":2794},[653],[608,2814,635],{"className":2815},[658],[608,2817],{"className":2818,"style":1082},[966],[608,2820,2416],{"className":2821},[1086],[608,2823],{"className":2824,"style":1082},[966],[608,2826,2828,2831,2834,2837,2840],{"className":2827},[649],[608,2829],{"className":2830,"style":2794},[653],[608,2832,782],{"className":2833},[658],[608,2835],{"className":2836,"style":1082},[966],[608,2838,2416],{"className":2839},[1086],[608,2841],{"className":2842,"style":1082},[966],[608,2844,2846,2849,2852,2855,2858],{"className":2845},[649],[608,2847],{"className":2848,"style":981},[653],[608,2850,659],{"className":2851},[658],[608,2853],{"className":2854,"style":967},[966],[608,2856,1615],{"className":2857},[971],[608,2859],{"className":2860,"style":967},[966],[608,2862,2864,2867],{"className":2863},[649],[608,2865],{"className":2866,"style":981},[653],[608,2868,2051],{"className":2869},[658],[599,2871,2872,2873,2876],{},"Python sait faire cette conversion automatiquement grâce à ",[1894,2874,2875],{},"int()"," avec sa\nbase passée en second argument :",[2878,2879,2884],"pre",{"className":2880,"code":2881,"language":2882,"meta":2883,"style":2883},"language-python shiki shiki-themes github-light github-light github-dark","print(int(\"1011\", 2))  # 11\nprint(int(\"FF\", 16))   # 255\nprint(int(\"17\", 8))    # 15\n","python","",[1894,2885,2886,2917,2941],{"__ignoreMap":2883},[608,2887,2890,2894,2897,2900,2902,2906,2908,2910,2913],{"class":2888,"line":2889},"line",1,[608,2891,2893],{"class":2892},"sBjJW","print",[608,2895,1151],{"class":2896},"sxrX7",[608,2898,2899],{"class":2892},"int",[608,2901,1151],{"class":2896},[608,2903,2905],{"class":2904},"sIIMD","\"1011\"",[608,2907,1896],{"class":2896},[608,2909,782],{"class":2892},[608,2911,2912],{"class":2896},"))  ",[608,2914,2916],{"class":2915},"sCsY4","# 11\n",[608,2918,2920,2922,2924,2926,2928,2931,2933,2935,2938],{"class":2888,"line":2919},2,[608,2921,2893],{"class":2892},[608,2923,1151],{"class":2896},[608,2925,2899],{"class":2892},[608,2927,1151],{"class":2896},[608,2929,2930],{"class":2904},"\"FF\"",[608,2932,1896],{"class":2896},[608,2934,1945],{"class":2892},[608,2936,2937],{"class":2896},"))   ",[608,2939,2940],{"class":2915},"# 255\n",[608,2942,2944,2946,2948,2950,2952,2955,2957,2959,2962],{"class":2888,"line":2943},3,[608,2945,2893],{"class":2892},[608,2947,1151],{"class":2896},[608,2949,2899],{"class":2892},[608,2951,1151],{"class":2896},[608,2953,2954],{"class":2904},"\"17\"",[608,2956,1896],{"class":2896},[608,2958,1908],{"class":2892},[608,2960,2961],{"class":2896},"))    ",[608,2963,2964],{"class":2915},"# 15\n",[2966,2967,2969,3057],"self-eval-qcm",{"correct":659,"slug":2968},"bases-entiers-positifs-c1",[2970,2971,2972],"template",{"v-slot:question":2883},[599,2973,2974,2975,3056],{},"Que vaut ",[608,2976,2978,3001],{"className":2977},[611],[608,2979,2981],{"className":2980},[615],[617,2982,2983],{"xmlns":619},[621,2984,2985,2998],{},[624,2986,2987,2989,2992],{},[942,2988,1151],{"stretchy":1150},[630,2990,2991],{},"1101",[1153,2993,2994,2996],{},[942,2995,1198],{"stretchy":1150},[630,2997,782],{},[637,2999,3000],{"encoding":639},"(1101)_2",[608,3002,3004],{"className":3003,"ariaHidden":645},[644],[608,3005,3007,3010,3013,3016],{"className":3006},[649],[608,3008],{"className":3009,"style":1213},[653],[608,3011,1151],{"className":3012},[1217],[608,3014,2991],{"className":3015},[658],[608,3017,3019,3022],{"className":3018},[1417],[608,3020,1198],{"className":3021},[1417],[608,3023,3025],{"className":3024},[669],[608,3026,3028,3048],{"className":3027},[673,1230],[608,3029,3031,3045],{"className":3030},[677],[608,3032,3034],{"className":3033,"style":1237},[681],[608,3035,3036,3039],{"style":1240},[608,3037],{"className":3038,"style":689},[688],[608,3040,3042],{"className":3041},[693,694,695,696],[608,3043,782],{"className":3044},[658,696],[608,3046,1263],{"className":3047},[1262],[608,3049,3051],{"className":3050},[677],[608,3052,3054],{"className":3053,"style":1371},[681],[608,3055],{}," en décimal ?",[2970,3058,3059],{"v-slot:choices":2883},[3060,3061,3062,3065,3068,3071],"ul",{},[3063,3064,2051],"li",{},[3063,3066,3067],{},"13",[3063,3069,3070],{},"14",[3063,3072,2017],{},[594,3074,3076],{"id":3075},"du-décimal-au-binaire-les-divisions-successives","Du décimal au binaire — les divisions successives",[599,3078,3079,3080,3083,3084,3087],{},"Pour passer du décimal au binaire, on divise le nombre par 2 et on note le\n",[603,3081,3082],{},"reste",". On répète l'opération sur le quotient jusqu'à obtenir 0. La lecture\ndes restes ",[603,3085,3086],{},"de bas en haut"," donne l'écriture binaire.",[599,3089,2296,3090,3118],{},[608,3091,3093,3106],{"className":3092},[611],[608,3094,3096],{"className":3095},[615],[617,3097,3098],{"xmlns":619},[621,3099,3100,3104],{},[624,3101,3102],{},[630,3103,3067],{},[637,3105,3067],{"encoding":639},[608,3107,3109],{"className":3108,"ariaHidden":645},[644],[608,3110,3112,3115],{"className":3111},[649],[608,3113],{"className":3114,"style":981},[653],[608,3116,3067],{"className":3117},[658]," en binaire.",[1859,3120,3121,3134],{},[1862,3122,3123],{},[1865,3124,3125,3128,3131],{},[1868,3126,3127],{},"Division",[1868,3129,3130],{},"Quotient",[1868,3132,3133],{},"Reste",[1881,3135,3136,3256,3372,3487],{},[1865,3137,3138,3192,3223],{},[1886,3139,3140],{},[608,3141,3143,3162],{"className":3142},[611],[608,3144,3146],{"className":3145},[615],[617,3147,3148],{"xmlns":619},[621,3149,3150,3159],{},[624,3151,3152,3154,3157],{},[630,3153,3067],{},[942,3155,3156],{},"÷",[630,3158,782],{},[637,3160,3161],{"encoding":639},"13 \\div 2",[608,3163,3165,3183],{"className":3164,"ariaHidden":645},[644],[608,3166,3168,3171,3174,3177,3180],{"className":3167},[649],[608,3169],{"className":3170,"style":2794},[653],[608,3172,3067],{"className":3173},[658],[608,3175],{"className":3176,"style":1082},[966],[608,3178,3156],{"className":3179},[1086],[608,3181],{"className":3182,"style":1082},[966],[608,3184,3186,3189],{"className":3185},[649],[608,3187],{"className":3188,"style":981},[653],[608,3190,782],{"className":3191},[658],[1886,3193,3194],{},[608,3195,3197,3211],{"className":3196},[611],[608,3198,3200],{"className":3199},[615],[617,3201,3202],{"xmlns":619},[621,3203,3204,3209],{},[624,3205,3206],{},[630,3207,3208],{},"6",[637,3210,3208],{"encoding":639},[608,3212,3214],{"className":3213,"ariaHidden":645},[644],[608,3215,3217,3220],{"className":3216},[649],[608,3218],{"className":3219,"style":981},[653],[608,3221,3208],{"className":3222},[658],[1886,3224,3225],{},[608,3226,3228,3243],{"className":3227},[611],[608,3229,3231],{"className":3230},[615],[617,3232,3233],{"xmlns":619},[621,3234,3235,3240],{},[624,3236,3237],{},[630,3238,659],{"mathvariant":3239},"bold",[637,3241,3242],{"encoding":639},"\\mathbf{1}",[608,3244,3246],{"className":3245,"ariaHidden":645},[644],[608,3247,3249,3252],{"className":3248},[649],[608,3250],{"className":3251,"style":981},[653],[608,3253,659],{"className":3254},[658,3255],"mathbf",[1865,3257,3258,3311,3341],{},[1886,3259,3260],{},[608,3261,3263,3281],{"className":3262},[611],[608,3264,3266],{"className":3265},[615],[617,3267,3268],{"xmlns":619},[621,3269,3270,3278],{},[624,3271,3272,3274,3276],{},[630,3273,3208],{},[942,3275,3156],{},[630,3277,782],{},[637,3279,3280],{"encoding":639},"6 \\div 2",[608,3282,3284,3302],{"className":3283,"ariaHidden":645},[644],[608,3285,3287,3290,3293,3296,3299],{"className":3286},[649],[608,3288],{"className":3289,"style":2794},[653],[608,3291,3208],{"className":3292},[658],[608,3294],{"className":3295,"style":1082},[966],[608,3297,3156],{"className":3298},[1086],[608,3300],{"className":3301,"style":1082},[966],[608,3303,3305,3308],{"className":3304},[649],[608,3306],{"className":3307,"style":981},[653],[608,3309,782],{"className":3310},[658],[1886,3312,3313],{},[608,3314,3316,3329],{"className":3315},[611],[608,3317,3319],{"className":3318},[615],[617,3320,3321],{"xmlns":619},[621,3322,3323,3327],{},[624,3324,3325],{},[630,3326,2413],{},[637,3328,2413],{"encoding":639},[608,3330,3332],{"className":3331,"ariaHidden":645},[644],[608,3333,3335,3338],{"className":3334},[649],[608,3336],{"className":3337,"style":981},[653],[608,3339,2413],{"className":3340},[658],[1886,3342,3343],{},[608,3344,3346,3360],{"className":3345},[611],[608,3347,3349],{"className":3348},[615],[617,3350,3351],{"xmlns":619},[621,3352,3353,3357],{},[624,3354,3355],{},[630,3356,635],{"mathvariant":3239},[637,3358,3359],{"encoding":639},"\\mathbf{0}",[608,3361,3363],{"className":3362,"ariaHidden":645},[644],[608,3364,3366,3369],{"className":3365},[649],[608,3367],{"className":3368,"style":981},[653],[608,3370,635],{"className":3371},[658,3255],[1865,3373,3374,3427,3457],{},[1886,3375,3376],{},[608,3377,3379,3397],{"className":3378},[611],[608,3380,3382],{"className":3381},[615],[617,3383,3384],{"xmlns":619},[621,3385,3386,3394],{},[624,3387,3388,3390,3392],{},[630,3389,2413],{},[942,3391,3156],{},[630,3393,782],{},[637,3395,3396],{"encoding":639},"3 \\div 2",[608,3398,3400,3418],{"className":3399,"ariaHidden":645},[644],[608,3401,3403,3406,3409,3412,3415],{"className":3402},[649],[608,3404],{"className":3405,"style":2794},[653],[608,3407,2413],{"className":3408},[658],[608,3410],{"className":3411,"style":1082},[966],[608,3413,3156],{"className":3414},[1086],[608,3416],{"className":3417,"style":1082},[966],[608,3419,3421,3424],{"className":3420},[649],[608,3422],{"className":3423,"style":981},[653],[608,3425,782],{"className":3426},[658],[1886,3428,3429],{},[608,3430,3432,3445],{"className":3431},[611],[608,3433,3435],{"className":3434},[615],[617,3436,3437],{"xmlns":619},[621,3438,3439,3443],{},[624,3440,3441],{},[630,3442,659],{},[637,3444,659],{"encoding":639},[608,3446,3448],{"className":3447,"ariaHidden":645},[644],[608,3449,3451,3454],{"className":3450},[649],[608,3452],{"className":3453,"style":981},[653],[608,3455,659],{"className":3456},[658],[1886,3458,3459],{},[608,3460,3462,3475],{"className":3461},[611],[608,3463,3465],{"className":3464},[615],[617,3466,3467],{"xmlns":619},[621,3468,3469,3473],{},[624,3470,3471],{},[630,3472,659],{"mathvariant":3239},[637,3474,3242],{"encoding":639},[608,3476,3478],{"className":3477,"ariaHidden":645},[644],[608,3479,3481,3484],{"className":3480},[649],[608,3482],{"className":3483,"style":981},[653],[608,3485,659],{"className":3486},[658,3255],[1865,3488,3489,3542,3572],{},[1886,3490,3491],{},[608,3492,3494,3512],{"className":3493},[611],[608,3495,3497],{"className":3496},[615],[617,3498,3499],{"xmlns":619},[621,3500,3501,3509],{},[624,3502,3503,3505,3507],{},[630,3504,659],{},[942,3506,3156],{},[630,3508,782],{},[637,3510,3511],{"encoding":639},"1 \\div 2",[608,3513,3515,3533],{"className":3514,"ariaHidden":645},[644],[608,3516,3518,3521,3524,3527,3530],{"className":3517},[649],[608,3519],{"className":3520,"style":2794},[653],[608,3522,659],{"className":3523},[658],[608,3525],{"className":3526,"style":1082},[966],[608,3528,3156],{"className":3529},[1086],[608,3531],{"className":3532,"style":1082},[966],[608,3534,3536,3539],{"className":3535},[649],[608,3537],{"className":3538,"style":981},[653],[608,3540,782],{"className":3541},[658],[1886,3543,3544],{},[608,3545,3547,3560],{"className":3546},[611],[608,3548,3550],{"className":3549},[615],[617,3551,3552],{"xmlns":619},[621,3553,3554,3558],{},[624,3555,3556],{},[630,3557,635],{},[637,3559,635],{"encoding":639},[608,3561,3563],{"className":3562,"ariaHidden":645},[644],[608,3564,3566,3569],{"className":3565},[649],[608,3567],{"className":3568,"style":981},[653],[608,3570,635],{"className":3571},[658],[1886,3573,3574],{},[608,3575,3577,3590],{"className":3576},[611],[608,3578,3580],{"className":3579},[615],[617,3581,3582],{"xmlns":619},[621,3583,3584,3588],{},[624,3585,3586],{},[630,3587,659],{"mathvariant":3239},[637,3589,3242],{"encoding":639},[608,3591,3593],{"className":3592,"ariaHidden":645},[644],[608,3594,3596,3599],{"className":3595},[649],[608,3597],{"className":3598,"style":981},[653],[608,3600,659],{"className":3601},[658,3255],[599,3603,3604,3605,2097],{},"En lisant les restes de bas en haut : ",[608,3606,3608,3634],{"className":3607},[611],[608,3609,3611],{"className":3610},[615],[617,3612,3613],{"xmlns":619},[621,3614,3615,3631],{},[624,3616,3617,3619,3621,3623,3625],{},[630,3618,3067],{},[942,3620,1615],{},[942,3622,1151],{"stretchy":1150},[630,3624,2991],{},[1153,3626,3627,3629],{},[942,3628,1198],{"stretchy":1150},[630,3630,782],{},[637,3632,3633],{"encoding":639},"13 = (1101)_2",[608,3635,3637,3655],{"className":3636,"ariaHidden":645},[644],[608,3638,3640,3643,3646,3649,3652],{"className":3639},[649],[608,3641],{"className":3642,"style":981},[653],[608,3644,3067],{"className":3645},[658],[608,3647],{"className":3648,"style":967},[966],[608,3650,1615],{"className":3651},[971],[608,3653],{"className":3654,"style":967},[966],[608,3656,3658,3661,3664,3667],{"className":3657},[649],[608,3659],{"className":3660,"style":1213},[653],[608,3662,1151],{"className":3663},[1217],[608,3665,2991],{"className":3666},[658],[608,3668,3670,3673],{"className":3669},[1417],[608,3671,1198],{"className":3672},[1417],[608,3674,3676],{"className":3675},[669],[608,3677,3679,3699],{"className":3678},[673,1230],[608,3680,3682,3696],{"className":3681},[677],[608,3683,3685],{"className":3684,"style":1237},[681],[608,3686,3687,3690],{"style":1240},[608,3688],{"className":3689,"style":689},[688],[608,3691,3693],{"className":3692},[693,694,695,696],[608,3694,782],{"className":3695},[658,696],[608,3697,1263],{"className":3698},[1262],[608,3700,3702],{"className":3701},[677],[608,3703,3705],{"className":3704,"style":1371},[681],[608,3706],{},[3708,3709,3710],"python-snippet",{},[2878,3711,3713],{"className":2880,"code":3712,"language":2882,"meta":2883,"style":2883},"n = 13\nprint(f\"{n} en binaire : {bin(n)}\")\nprint(f\"{n} en hexadécimal : {hex(n)}\")\nprint(f\"{n} en octal : {oct(n)}\")\n",[1894,3714,3715,3726,3762,3792],{"__ignoreMap":2883},[608,3716,3717,3720,3723],{"class":2888,"line":2889},[608,3718,3719],{"class":2896},"n ",[608,3721,1615],{"class":3722},"s8jYJ",[608,3724,3725],{"class":2892}," 13\n",[608,3727,3728,3730,3732,3735,3738,3741,3743,3746,3749,3752,3755,3757,3759],{"class":2888,"line":2919},[608,3729,2893],{"class":2892},[608,3731,1151],{"class":2896},[608,3733,3734],{"class":3722},"f",[608,3736,3737],{"class":2904},"\"",[608,3739,3740],{"class":2892},"{",[608,3742,1162],{"class":2896},[608,3744,3745],{"class":2892},"}",[608,3747,3748],{"class":2904}," en binaire : ",[608,3750,3751],{"class":2892},"{bin",[608,3753,3754],{"class":2896},"(n)",[608,3756,3745],{"class":2892},[608,3758,3737],{"class":2904},[608,3760,3761],{"class":2896},")\n",[608,3763,3764,3766,3768,3770,3772,3774,3776,3778,3781,3784,3786,3788,3790],{"class":2888,"line":2943},[608,3765,2893],{"class":2892},[608,3767,1151],{"class":2896},[608,3769,3734],{"class":3722},[608,3771,3737],{"class":2904},[608,3773,3740],{"class":2892},[608,3775,1162],{"class":2896},[608,3777,3745],{"class":2892},[608,3779,3780],{"class":2904}," en hexadécimal : ",[608,3782,3783],{"class":2892},"{hex",[608,3785,3754],{"class":2896},[608,3787,3745],{"class":2892},[608,3789,3737],{"class":2904},[608,3791,3761],{"class":2896},[608,3793,3795,3797,3799,3801,3803,3805,3807,3809,3812,3815,3817,3819,3821],{"class":2888,"line":3794},4,[608,3796,2893],{"class":2892},[608,3798,1151],{"class":2896},[608,3800,3734],{"class":3722},[608,3802,3737],{"class":2904},[608,3804,3740],{"class":2892},[608,3806,1162],{"class":2896},[608,3808,3745],{"class":2892},[608,3810,3811],{"class":2904}," en octal : ",[608,3813,3814],{"class":2892},"{oct",[608,3816,3754],{"class":2896},[608,3818,3745],{"class":2892},[608,3820,3737],{"class":2904},[608,3822,3761],{"class":2896},[599,3824,3825],{},"Sortie attendue :",[2878,3827,3832],{"className":3828,"code":3830,"language":3831},[3829],"language-text","13 en binaire : 0b1101\n13 en hexadécimal : 0xd\n13 en octal : 0o15\n","text",[1894,3833,3830],{"__ignoreMap":2883},[594,3835,3837],{"id":3836},"le-pont-avec-lhexadécimal","Le pont avec l'hexadécimal",[599,3839,3840,3841,3844],{},"L'hexadécimal a un atout précieux : ",[603,3842,3843],{},"chaque chiffre hexadécimal correspond\nexactement à 4 bits",". La conversion binaire ↔ hexadécimal se fait donc par\nblocs, sans aucun calcul.",[1859,3846,3847,3861],{},[1862,3848,3849],{},[1865,3850,3851,3854,3857,3859],{},[1868,3852,3853],{},"Hex",[1868,3855,3856],{},"Binaire",[1868,3858,3853],{},[1868,3860,3856],{},[1881,3862,3863,3883,3903,3923,3942,3964,3985,4006],{},[1865,3864,3865,3869,3874,3878],{},[1886,3866,3867],{},[1894,3868,635],{},[1886,3870,3871],{},[1894,3872,3873],{},"0000",[1886,3875,3876],{},[1894,3877,1908],{},[1886,3879,3880],{},[1894,3881,3882],{},"1000",[1865,3884,3885,3889,3894,3898],{},[1886,3886,3887],{},[1894,3888,659],{},[1886,3890,3891],{},[1894,3892,3893],{},"0001",[1886,3895,3896],{},[1894,3897,1937],{},[1886,3899,3900],{},[1894,3901,3902],{},"1001",[1865,3904,3905,3909,3914,3918],{},[1886,3906,3907],{},[1894,3908,782],{},[1886,3910,3911],{},[1894,3912,3913],{},"0010",[1886,3915,3916],{},[1894,3917,1958],{},[1886,3919,3920],{},[1894,3921,3922],{},"1010",[1865,3924,3925,3929,3934,3938],{},[1886,3926,3927],{},[1894,3928,2413],{},[1886,3930,3931],{},[1894,3932,3933],{},"0011",[1886,3935,3936],{},[1894,3937,2035],{},[1886,3939,3940],{},[1894,3941,2120],{},[1865,3943,3944,3949,3954,3959],{},[1886,3945,3946],{},[1894,3947,3948],{},"4",[1886,3950,3951],{},[1894,3952,3953],{},"0100",[1886,3955,3956],{},[1894,3957,3958],{},"C",[1886,3960,3961],{},[1894,3962,3963],{},"1100",[1865,3965,3966,3971,3976,3981],{},[1886,3967,3968],{},[1894,3969,3970],{},"5",[1886,3972,3973],{},[1894,3974,3975],{},"0101",[1886,3977,3978],{},[1894,3979,3980],{},"D",[1886,3982,3983],{},[1894,3984,2991],{},[1865,3986,3987,3991,3996,4001],{},[1886,3988,3989],{},[1894,3990,3208],{},[1886,3992,3993],{},[1894,3994,3995],{},"0110",[1886,3997,3998],{},[1894,3999,4000],{},"E",[1886,4002,4003],{},[1894,4004,4005],{},"1110",[1865,4007,4008,4012,4017,4021],{},[1886,4009,4010],{},[1894,4011,1918],{},[1886,4013,4014],{},[1894,4015,4016],{},"0111",[1886,4018,4019],{},[1894,4020,1961],{},[1886,4022,4023],{},[1894,4024,4025],{},"1111",[599,4027,4028,4029,2097],{},"Exemple : ",[608,4030,4032,4079],{"className":4031},[611],[608,4033,4035],{"className":4034},[615],[617,4036,4037],{"xmlns":619},[621,4038,4039,4076],{},[624,4040,4041,4043,4045,4047,4053,4055,4057,4059,4063,4065,4071,4073],{},[942,4042,1151],{"stretchy":1150},[630,4044,782],{},[876,4046,1961],{},[1153,4048,4049,4051],{},[942,4050,1198],{"stretchy":1150},[630,4052,1945],{},[942,4054,1615],{},[942,4056,1151],{"stretchy":1150},[630,4058,3913],{},[4060,4061,4062],"mtext",{}," ",[630,4064,4025],{},[1153,4066,4067,4069],{},[942,4068,1198],{"stretchy":1150},[630,4070,782],{},[942,4072,1615],{},[630,4074,4075],{},"47",[637,4077,4078],{"encoding":639},"(2F)_{16} = (0010\\ 1111)_2 = 47",[608,4080,4082,4150,4217],{"className":4081,"ariaHidden":645},[644],[608,4083,4085,4088,4091,4094,4098,4141,4144,4147],{"className":4084},[649],[608,4086],{"className":4087,"style":1213},[653],[608,4089,1151],{"className":4090},[1217],[608,4092,782],{"className":4093},[658],[608,4095,1961],{"className":4096,"style":4097},[658,894],"margin-right:0.1389em;",[608,4099,4101,4104],{"className":4100},[1417],[608,4102,1198],{"className":4103},[1417],[608,4105,4107],{"className":4106},[669],[608,4108,4110,4133],{"className":4109},[673,1230],[608,4111,4113,4130],{"className":4112},[677],[608,4114,4116],{"className":4115,"style":1237},[681],[608,4117,4118,4121],{"style":1240},[608,4119],{"className":4120,"style":689},[688],[608,4122,4124],{"className":4123},[693,694,695,696],[608,4125,4127],{"className":4126},[658,696],[608,4128,1945],{"className":4129},[658,696],[608,4131,1263],{"className":4132},[1262],[608,4134,4136],{"className":4135},[677],[608,4137,4139],{"className":4138,"style":1371},[681],[608,4140],{},[608,4142],{"className":4143,"style":967},[966],[608,4145,1615],{"className":4146},[971],[608,4148],{"className":4149,"style":967},[966],[608,4151,4153,4156,4159,4162,4165,4168,4208,4211,4214],{"className":4152},[649],[608,4154],{"className":4155,"style":1213},[653],[608,4157,1151],{"className":4158},[1217],[608,4160,3913],{"className":4161},[658],[608,4163,4062],{"className":4164},[966],[608,4166,4025],{"className":4167},[658],[608,4169,4171,4174],{"className":4170},[1417],[608,4172,1198],{"className":4173},[1417],[608,4175,4177],{"className":4176},[669],[608,4178,4180,4200],{"className":4179},[673,1230],[608,4181,4183,4197],{"className":4182},[677],[608,4184,4186],{"className":4185,"style":1237},[681],[608,4187,4188,4191],{"style":1240},[608,4189],{"className":4190,"style":689},[688],[608,4192,4194],{"className":4193},[693,694,695,696],[608,4195,782],{"className":4196},[658,696],[608,4198,1263],{"className":4199},[1262],[608,4201,4203],{"className":4202},[677],[608,4204,4206],{"className":4205,"style":1371},[681],[608,4207],{},[608,4209],{"className":4210,"style":967},[966],[608,4212,1615],{"className":4213},[971],[608,4215],{"className":4216,"style":967},[966],[608,4218,4220,4223],{"className":4219},[649],[608,4221],{"className":4222,"style":981},[653],[608,4224,4075],{"className":4225},[658],[599,4227,4228,4229,4232],{},"C'est pour cela qu'on rencontre l'hexadécimal partout en informatique :\nadresses mémoire, couleurs CSS (",[1894,4230,4231],{},"#FF8800","), valeurs d'un octet… Lire un octet\nen hexadécimal, c'est lire 2 chiffres au lieu de 8 bits.",[2966,4234,4236,4330],{"correct":782,"slug":4235},"bases-entiers-positifs-c2",[2970,4237,4238],{"v-slot:question":2883},[599,4239,4240,4241,4329],{},"Combien de bits sont nécessaires pour écrire ",[608,4242,4244,4268],{"className":4243},[611],[608,4245,4247],{"className":4246},[615],[617,4248,4249],{"xmlns":619},[621,4250,4251,4265],{},[624,4252,4253,4255,4257,4259],{},[942,4254,1151],{"stretchy":1150},[876,4256,1958],{},[630,4258,2413],{},[1153,4260,4261,4263],{},[942,4262,1198],{"stretchy":1150},[630,4264,1945],{},[637,4266,4267],{"encoding":639},"(A3)_{16}",[608,4269,4271],{"className":4270,"ariaHidden":645},[644],[608,4272,4274,4277,4280,4283,4286],{"className":4273},[649],[608,4275],{"className":4276,"style":1213},[653],[608,4278,1151],{"className":4279},[1217],[608,4281,1958],{"className":4282},[658,894],[608,4284,2413],{"className":4285},[658],[608,4287,4289,4292],{"className":4288},[1417],[608,4290,1198],{"className":4291},[1417],[608,4293,4295],{"className":4294},[669],[608,4296,4298,4321],{"className":4297},[673,1230],[608,4299,4301,4318],{"className":4300},[677],[608,4302,4304],{"className":4303,"style":1237},[681],[608,4305,4306,4309],{"style":1240},[608,4307],{"className":4308,"style":689},[688],[608,4310,4312],{"className":4311},[693,694,695,696],[608,4313,4315],{"className":4314},[658,696],[608,4316,1945],{"className":4317},[658,696],[608,4319,1263],{"className":4320},[1262],[608,4322,4324],{"className":4323},[677],[608,4325,4327],{"className":4326,"style":1371},[681],[608,4328],{}," ?",[2970,4331,4332],{"v-slot:choices":2883},[3060,4333,4334,4336,4338,4340],{},[3063,4335,3948],{},[3063,4337,3208],{},[3063,4339,1908],{},[3063,4341,1945],{},[594,4343,4345],{"id":4344},"combien-de-bits-pour-un-nombre","Combien de bits pour un nombre ?",[599,4347,4348,4349,4378,4379,4439,4440,1043,4468,2097],{},"Avec ",[608,4350,4352,4365],{"className":4351},[611],[608,4353,4355],{"className":4354},[615],[617,4356,4357],{"xmlns":619},[621,4358,4359,4363],{},[624,4360,4361],{},[876,4362,1162],{},[637,4364,1162],{"encoding":639},[608,4366,4368],{"className":4367,"ariaHidden":645},[644],[608,4369,4371,4375],{"className":4370},[649],[608,4372],{"className":4373,"style":4374},[653],"height:0.4306em;",[608,4376,1162],{"className":4377},[658,894]," bits, on peut représenter ",[608,4380,4382,4400],{"className":4381},[611],[608,4383,4385],{"className":4384},[615],[617,4386,4387],{"xmlns":619},[621,4388,4389,4397],{},[624,4390,4391],{},[627,4392,4393,4395],{},[630,4394,782],{},[876,4396,1162],{},[637,4398,4399],{"encoding":639},"2^n",[608,4401,4403],{"className":4402,"ariaHidden":645},[644],[608,4404,4406,4410],{"className":4405},[649],[608,4407],{"className":4408,"style":4409},[653],"height:0.6644em;",[608,4411,4413,4416],{"className":4412},[658],[608,4414,782],{"className":4415},[658],[608,4417,4419],{"className":4418},[669],[608,4420,4422],{"className":4421},[673],[608,4423,4425],{"className":4424},[677],[608,4426,4428],{"className":4427,"style":4409},[681],[608,4429,4430,4433],{"style":684},[608,4431],{"className":4432,"style":689},[688],[608,4434,4436],{"className":4435},[693,694,695,696],[608,4437,1162],{"className":4438},[658,894,696]," valeurs différentes, soit les entiers\nde ",[608,4441,4443,4456],{"className":4442},[611],[608,4444,4446],{"className":4445},[615],[617,4447,4448],{"xmlns":619},[621,4449,4450,4454],{},[624,4451,4452],{},[630,4453,635],{},[637,4455,635],{"encoding":639},[608,4457,4459],{"className":4458,"ariaHidden":645},[644],[608,4460,4462,4465],{"className":4461},[649],[608,4463],{"className":4464,"style":981},[653],[608,4466,635],{"className":4467},[658],[608,4469,4471,4493],{"className":4470},[611],[608,4472,4474],{"className":4473},[615],[617,4475,4476],{"xmlns":619},[621,4477,4478,4490],{},[624,4479,4480,4486,4488],{},[627,4481,4482,4484],{},[630,4483,782],{},[876,4485,1162],{},[942,4487,1060],{},[630,4489,659],{},[637,4491,4492],{"encoding":639},"2^n - 1",[608,4494,4496,4541],{"className":4495,"ariaHidden":645},[644],[608,4497,4499,4503,4532,4535,4538],{"className":4498},[649],[608,4500],{"className":4501,"style":4502},[653],"height:0.7477em;vertical-align:-0.0833em;",[608,4504,4506,4509],{"className":4505},[658],[608,4507,782],{"className":4508},[658],[608,4510,4512],{"className":4511},[669],[608,4513,4515],{"className":4514},[673],[608,4516,4518],{"className":4517},[677],[608,4519,4521],{"className":4520,"style":4409},[681],[608,4522,4523,4526],{"style":684},[608,4524],{"className":4525,"style":689},[688],[608,4527,4529],{"className":4528},[693,694,695,696],[608,4530,1162],{"className":4531},[658,894,696],[608,4533],{"className":4534,"style":1082},[966],[608,4536,1060],{"className":4537},[1086],[608,4539],{"className":4540,"style":1082},[966],[608,4542,4544,4547],{"className":4543},[649],[608,4545],{"className":4546,"style":981},[653],[608,4548,659],{"className":4549},[658],[1859,4551,4552,4565],{},[1862,4553,4554],{},[1865,4555,4556,4559,4562],{},[1868,4557,4558],{},"Bits",[1868,4560,4561],{},"Valeurs représentables",[1868,4563,4564],{},"Maximum",[1881,4566,4567,4684,4804,4949],{},[1865,4568,4569,4571,4654],{},[1886,4570,3948],{},[1886,4572,4573],{},[608,4574,4576,4598],{"className":4575},[611],[608,4577,4579],{"className":4578},[615],[617,4580,4581],{"xmlns":619},[621,4582,4583,4595],{},[624,4584,4585,4591,4593],{},[627,4586,4587,4589],{},[630,4588,782],{},[630,4590,3948],{},[942,4592,1615],{},[630,4594,1945],{},[637,4596,4597],{"encoding":639},"2^4 = 16",[608,4599,4601,4645],{"className":4600,"ariaHidden":645},[644],[608,4602,4604,4607,4636,4639,4642],{"className":4603},[649],[608,4605],{"className":4606,"style":654},[653],[608,4608,4610,4613],{"className":4609},[658],[608,4611,782],{"className":4612},[658],[608,4614,4616],{"className":4615},[669],[608,4617,4619],{"className":4618},[673],[608,4620,4622],{"className":4621},[677],[608,4623,4625],{"className":4624,"style":654},[681],[608,4626,4627,4630],{"style":684},[608,4628],{"className":4629,"style":689},[688],[608,4631,4633],{"className":4632},[693,694,695,696],[608,4634,3948],{"className":4635},[658,696],[608,4637],{"className":4638,"style":967},[966],[608,4640,1615],{"className":4641},[971],[608,4643],{"className":4644,"style":967},[966],[608,4646,4648,4651],{"className":4647},[649],[608,4649],{"className":4650,"style":981},[653],[608,4652,1945],{"className":4653},[658],[1886,4655,4656],{},[608,4657,4659,4672],{"className":4658},[611],[608,4660,4662],{"className":4661},[615],[617,4663,4664],{"xmlns":619},[621,4665,4666,4670],{},[624,4667,4668],{},[630,4669,2017],{},[637,4671,2017],{"encoding":639},[608,4673,4675],{"className":4674,"ariaHidden":645},[644],[608,4676,4678,4681],{"className":4677},[649],[608,4679],{"className":4680,"style":981},[653],[608,4682,2017],{"className":4683},[658],[1865,4685,4686,4689,4773],{},[1886,4687,4688],{},"8 (octet)",[1886,4690,4691],{},[608,4692,4694,4717],{"className":4693},[611],[608,4695,4697],{"className":4696},[615],[617,4698,4699],{"xmlns":619},[621,4700,4701,4714],{},[624,4702,4703,4709,4711],{},[627,4704,4705,4707],{},[630,4706,782],{},[630,4708,1908],{},[942,4710,1615],{},[630,4712,4713],{},"256",[637,4715,4716],{"encoding":639},"2^8 = 256",[608,4718,4720,4764],{"className":4719,"ariaHidden":645},[644],[608,4721,4723,4726,4755,4758,4761],{"className":4722},[649],[608,4724],{"className":4725,"style":654},[653],[608,4727,4729,4732],{"className":4728},[658],[608,4730,782],{"className":4731},[658],[608,4733,4735],{"className":4734},[669],[608,4736,4738],{"className":4737},[673],[608,4739,4741],{"className":4740},[677],[608,4742,4744],{"className":4743,"style":654},[681],[608,4745,4746,4749],{"style":684},[608,4747],{"className":4748,"style":689},[688],[608,4750,4752],{"className":4751},[693,694,695,696],[608,4753,1908],{"className":4754},[658,696],[608,4756],{"className":4757,"style":967},[966],[608,4759,1615],{"className":4760},[971],[608,4762],{"className":4763,"style":967},[966],[608,4765,4767,4770],{"className":4766},[649],[608,4768],{"className":4769,"style":981},[653],[608,4771,4713],{"className":4772},[658],[1886,4774,4775],{},[608,4776,4778,4792],{"className":4777},[611],[608,4779,4781],{"className":4780},[615],[617,4782,4783],{"xmlns":619},[621,4784,4785,4790],{},[624,4786,4787],{},[630,4788,4789],{},"255",[637,4791,4789],{"encoding":639},[608,4793,4795],{"className":4794,"ariaHidden":645},[644],[608,4796,4798,4801],{"className":4797},[649],[608,4799],{"className":4800,"style":981},[653],[608,4802,4789],{"className":4803},[658],[1865,4805,4806,4808,4907],{},[1886,4807,1945],{},[1886,4809,4810],{},[608,4811,4813,4842],{"className":4812},[611],[608,4814,4816],{"className":4815},[615],[617,4817,4818],{"xmlns":619},[621,4819,4820,4839],{},[624,4821,4822,4828,4830,4833,4836],{},[627,4823,4824,4826],{},[630,4825,782],{},[630,4827,1945],{},[942,4829,1615],{},[630,4831,4832],{},"65",[4060,4834,4835],{}," ",[630,4837,4838],{},"536",[637,4840,4841],{"encoding":639},"2^{16} = 65\\,536",[608,4843,4845,4892],{"className":4844,"ariaHidden":645},[644],[608,4846,4848,4851,4883,4886,4889],{"className":4847},[649],[608,4849],{"className":4850,"style":654},[653],[608,4852,4854,4857],{"className":4853},[658],[608,4855,782],{"className":4856},[658],[608,4858,4860],{"className":4859},[669],[608,4861,4863],{"className":4862},[673],[608,4864,4866],{"className":4865},[677],[608,4867,4869],{"className":4868,"style":654},[681],[608,4870,4871,4874],{"style":684},[608,4872],{"className":4873,"style":689},[688],[608,4875,4877],{"className":4876},[693,694,695,696],[608,4878,4880],{"className":4879},[658,696],[608,4881,1945],{"className":4882},[658,696],[608,4884],{"className":4885,"style":967},[966],[608,4887,1615],{"className":4888},[971],[608,4890],{"className":4891,"style":967},[966],[608,4893,4895,4898,4901,4904],{"className":4894},[649],[608,4896],{"className":4897,"style":981},[653],[608,4899,4832],{"className":4900},[658],[608,4902],{"className":4903,"style":1325},[966],[608,4905,4838],{"className":4906},[658],[1886,4908,4909],{},[608,4910,4912,4931],{"className":4911},[611],[608,4913,4915],{"className":4914},[615],[617,4916,4917],{"xmlns":619},[621,4918,4919,4928],{},[624,4920,4921,4923,4925],{},[630,4922,4832],{},[4060,4924,4835],{},[630,4926,4927],{},"535",[637,4929,4930],{"encoding":639},"65\\,535",[608,4932,4934],{"className":4933,"ariaHidden":645},[644],[608,4935,4937,4940,4943,4946],{"className":4936},[649],[608,4938],{"className":4939,"style":981},[653],[608,4941,4832],{"className":4942},[658],[608,4944],{"className":4945,"style":1325},[966],[608,4947,4927],{"className":4948},[658],[1865,4950,4951,4954,5109],{},[1886,4952,4953],{},"32",[1886,4955,4956],{},[608,4957,4959,4991],{"className":4958},[611],[608,4960,4962],{"className":4961},[615],[617,4963,4964],{"xmlns":619},[621,4965,4966,4988],{},[624,4967,4968,4974,4977,4980,4982],{},[627,4969,4970,4972],{},[630,4971,782],{},[630,4973,4953],{},[942,4975,4976],{},"≈",[630,4978,4979],{},"4,3",[942,4981,1646],{},[627,4983,4984,4986],{},[630,4985,632],{},[630,4987,1937],{},[637,4989,4990],{"encoding":639},"2^{32} \\approx 4{,}3 \\cdot 10^9",[608,4992,4994,5041,5071],{"className":4993,"ariaHidden":645},[644],[608,4995,4997,5000,5032,5035,5038],{"className":4996},[649],[608,4998],{"className":4999,"style":654},[653],[608,5001,5003,5006],{"className":5002},[658],[608,5004,782],{"className":5005},[658],[608,5007,5009],{"className":5008},[669],[608,5010,5012],{"className":5011},[673],[608,5013,5015],{"className":5014},[677],[608,5016,5018],{"className":5017,"style":654},[681],[608,5019,5020,5023],{"style":684},[608,5021],{"className":5022,"style":689},[688],[608,5024,5026],{"className":5025},[693,694,695,696],[608,5027,5029],{"className":5028},[658,696],[608,5030,4953],{"className":5031},[658,696],[608,5033],{"className":5034,"style":967},[966],[608,5036,4976],{"className":5037},[971],[608,5039],{"className":5040,"style":967},[966],[608,5042,5044,5048,5051,5059,5062,5065,5068],{"className":5043},[649],[608,5045],{"className":5046,"style":5047},[653],"height:0.8389em;vertical-align:-0.1944em;",[608,5049,3948],{"className":5050},[658],[608,5052,5054],{"className":5053},[658],[608,5055,5058],{"className":5056},[5057],"mpunct",",",[608,5060,2413],{"className":5061},[658],[608,5063],{"className":5064,"style":1082},[966],[608,5066,1646],{"className":5067},[1086],[608,5069],{"className":5070,"style":1082},[966],[608,5072,5074,5077,5080],{"className":5073},[649],[608,5075],{"className":5076,"style":654},[653],[608,5078,659],{"className":5079},[658],[608,5081,5083,5086],{"className":5082},[658],[608,5084,635],{"className":5085},[658],[608,5087,5089],{"className":5088},[669],[608,5090,5092],{"className":5091},[673],[608,5093,5095],{"className":5094},[677],[608,5096,5098],{"className":5097,"style":654},[681],[608,5099,5100,5103],{"style":684},[608,5101],{"className":5102,"style":689},[688],[608,5104,5106],{"className":5105},[693,694,695,696],[608,5107,1937],{"className":5108},[658,696],[1886,5110,5111],{},"environ 4 milliards",[599,5113,5114,5115,5143,5144,5275,5276,5279],{},"Inversement, pour écrire un entier ",[608,5116,5118,5131],{"className":5117},[611],[608,5119,5121],{"className":5120},[615],[617,5122,5123],{"xmlns":619},[621,5124,5125,5129],{},[624,5126,5127],{},[876,5128,1612],{},[637,5130,1612],{"encoding":639},[608,5132,5134],{"className":5133,"ariaHidden":645},[644],[608,5135,5137,5140],{"className":5136},[649],[608,5138],{"className":5139,"style":1665},[653],[608,5141,1612],{"className":5142,"style":1669},[658,894]," positif, il faut ",[608,5145,5147,5187],{"className":5146},[611],[608,5148,5150],{"className":5149},[615],[617,5151,5152],{"xmlns":619},[621,5153,5154,5184],{},[624,5155,5156,5159,5171,5173,5175,5177,5180,5182],{},[942,5157,5158],{"stretchy":1150},"⌊",[1153,5160,5161,5169],{},[624,5162,5163,5166],{},[876,5164,5165],{},"log",[942,5167,5168],{},"⁡",[630,5170,782],{},[942,5172,1151],{"stretchy":1150},[876,5174,1612],{},[942,5176,1198],{"stretchy":1150},[942,5178,5179],{"stretchy":1150},"⌋",[942,5181,2416],{},[630,5183,659],{},[637,5185,5186],{"encoding":639},"\\lfloor \\log_2(N) \\rfloor + 1",[608,5188,5190,5266],{"className":5189,"ariaHidden":645},[644],[608,5191,5193,5196,5199,5247,5250,5253,5257,5260,5263],{"className":5192},[649],[608,5194],{"className":5195,"style":1213},[653],[608,5197,5158],{"className":5198},[1217],[608,5200,5202,5210],{"className":5201},[1689],[608,5203,5205,5206],{"className":5204},[1689],"lo",[608,5207,5209],{"style":5208},"margin-right:0.0139em;","g",[608,5211,5213],{"className":5212},[669],[608,5214,5216,5238],{"className":5215},[673,1230],[608,5217,5219,5235],{"className":5218},[677],[608,5220,5223],{"className":5221,"style":5222},[681],"height:0.207em;",[608,5224,5226,5229],{"style":5225},"top:-2.4559em;margin-right:0.05em;",[608,5227],{"className":5228,"style":689},[688],[608,5230,5232],{"className":5231},[693,694,695,696],[608,5233,782],{"className":5234},[658,696],[608,5236,1263],{"className":5237},[1262],[608,5239,5241],{"className":5240},[677],[608,5242,5245],{"className":5243,"style":5244},[681],"height:0.2441em;",[608,5246],{},[608,5248,1151],{"className":5249},[1217],[608,5251,1612],{"className":5252,"style":1669},[658,894],[608,5254,5256],{"className":5255},[1417],")⌋",[608,5258],{"className":5259,"style":1082},[966],[608,5261,2416],{"className":5262},[1086],[608,5264],{"className":5265,"style":1082},[966],[608,5267,5269,5272],{"className":5268},[649],[608,5270],{"className":5271,"style":981},[653],[608,5273,659],{"className":5274},[658],"\nbits. En pratique, vous pouvez aussi compter en utilisant ",[1894,5277,5278],{},"bin(N)"," :",[2878,5281,5283],{"className":2880,"code":5282,"language":2882,"meta":2883,"style":2883},"n = 1000\nprint(bin(n))            # 0b1111101000\nprint(len(bin(n)) - 2)   # 10 — on retire le préfixe \"0b\"\n",[1894,5284,5285,5294,5309],{"__ignoreMap":2883},[608,5286,5287,5289,5291],{"class":2888,"line":2889},[608,5288,3719],{"class":2896},[608,5290,1615],{"class":3722},[608,5292,5293],{"class":2892}," 1000\n",[608,5295,5296,5298,5300,5303,5306],{"class":2888,"line":2919},[608,5297,2893],{"class":2892},[608,5299,1151],{"class":2896},[608,5301,5302],{"class":2892},"bin",[608,5304,5305],{"class":2896},"(n))            ",[608,5307,5308],{"class":2915},"# 0b1111101000\n",[608,5310,5311,5313,5315,5318,5320,5322,5325,5328,5331,5334],{"class":2888,"line":2943},[608,5312,2893],{"class":2892},[608,5314,1151],{"class":2896},[608,5316,5317],{"class":2892},"len",[608,5319,1151],{"class":2896},[608,5321,5302],{"class":2892},[608,5323,5324],{"class":2896},"(n)) ",[608,5326,5327],{"class":3722},"-",[608,5329,5330],{"class":2892}," 2",[608,5332,5333],{"class":2896},")   ",[608,5335,5336],{"class":2915},"# 10 — on retire le préfixe \"0b\"\n",[594,5338,5340],{"id":5339},"pièges-courants","Pièges courants",[5342,5343,5344],"warning",{},[3060,5345,5346,5466,5535],{},[3063,5347,5348,5349,5351,5352,5355,5356,5436,5437,5465],{},"Ne pas confondre la ",[603,5350,649],{}," et le ",[603,5353,5354],{},"nombre de chiffres",". ",[608,5357,5359,5381],{"className":5358},[611],[608,5360,5362],{"className":5361},[615],[617,5363,5364],{"xmlns":619},[621,5365,5366,5378],{},[624,5367,5368,5370,5372],{},[942,5369,1151],{"stretchy":1150},[630,5371,632],{},[1153,5373,5374,5376],{},[942,5375,1198],{"stretchy":1150},[630,5377,782],{},[637,5379,5380],{"encoding":639},"(10)_2",[608,5382,5384],{"className":5383,"ariaHidden":645},[644],[608,5385,5387,5390,5393,5396],{"className":5386},[649],[608,5388],{"className":5389,"style":1213},[653],[608,5391,1151],{"className":5392},[1217],[608,5394,632],{"className":5395},[658],[608,5397,5399,5402],{"className":5398},[1417],[608,5400,1198],{"className":5401},[1417],[608,5403,5405],{"className":5404},[669],[608,5406,5408,5428],{"className":5407},[673,1230],[608,5409,5411,5425],{"className":5410},[677],[608,5412,5414],{"className":5413,"style":1237},[681],[608,5415,5416,5419],{"style":1240},[608,5417],{"className":5418,"style":689},[688],[608,5420,5422],{"className":5421},[693,694,695,696],[608,5423,782],{"className":5424},[658,696],[608,5426,1263],{"className":5427},[1262],[608,5429,5431],{"className":5430},[677],[608,5432,5434],{"className":5433,"style":1371},[681],[608,5435],{}," vaut\n",[608,5438,5440,5453],{"className":5439},[611],[608,5441,5443],{"className":5442},[615],[617,5444,5445],{"xmlns":619},[621,5446,5447,5451],{},[624,5448,5449],{},[630,5450,782],{},[637,5452,782],{"encoding":639},[608,5454,5456],{"className":5455,"ariaHidden":645},[644],[608,5457,5459,5462],{"className":5458},[649],[608,5460],{"className":5461,"style":981},[653],[608,5463,782],{"className":5464},[658],", pas dix.",[3063,5467,5468,5469,5472,5473,5501,5502,5505,5506,5534],{},"En Python, ",[1894,5470,5471],{},"0b10"," est un entier dont la valeur est ",[608,5474,5476,5489],{"className":5475},[611],[608,5477,5479],{"className":5478},[615],[617,5480,5481],{"xmlns":619},[621,5482,5483,5487],{},[624,5484,5485],{},[630,5486,782],{},[637,5488,782],{"encoding":639},[608,5490,5492],{"className":5491,"ariaHidden":645},[644],[608,5493,5495,5498],{"className":5494},[649],[608,5496],{"className":5497,"style":981},[653],[608,5499,782],{"className":5500},[658],". Quand vous\nl'affichez avec ",[1894,5503,5504],{},"print(0b10)",", Python affiche ",[608,5507,5509,5522],{"className":5508},[611],[608,5510,5512],{"className":5511},[615],[617,5513,5514],{"xmlns":619},[621,5515,5516,5520],{},[624,5517,5518],{},[630,5519,782],{},[637,5521,782],{"encoding":639},[608,5523,5525],{"className":5524,"ariaHidden":645},[644],[608,5526,5528,5531],{"className":5527},[649],[608,5529],{"className":5530,"style":981},[653],[608,5532,782],{"className":5533},[658]," en décimal par défaut.",[3063,5536,5537,5538,5541,5542,5545],{},"L'écriture ",[1894,5539,5540],{},"010"," dans un code Python ancien (Python 2) était un octal —\nsource de bugs. En Python 3, il faut écrire ",[1894,5543,5544],{},"0o10"," explicitement.",[594,5547,5549],{"id":5548},"pour-aller-plus-loin","Pour aller plus loin",[599,5551,5552,5553,5556,5557,5619],{},"Python gère ",[603,5554,5555],{},"les entiers de taille arbitraire"," : vous pouvez calculer\n",[608,5558,5560,5578],{"className":5559},[611],[608,5561,5563],{"className":5562},[615],[617,5564,5565],{"xmlns":619},[621,5566,5567,5575],{},[624,5568,5569],{},[627,5570,5571,5573],{},[630,5572,782],{},[630,5574,3882],{},[637,5576,5577],{"encoding":639},"2^{1000}",[608,5579,5581],{"className":5580,"ariaHidden":645},[644],[608,5582,5584,5587],{"className":5583},[649],[608,5585],{"className":5586,"style":654},[653],[608,5588,5590,5593],{"className":5589},[658],[608,5591,782],{"className":5592},[658],[608,5594,5596],{"className":5595},[669],[608,5597,5599],{"className":5598},[673],[608,5600,5602],{"className":5601},[677],[608,5603,5605],{"className":5604,"style":654},[681],[608,5606,5607,5610],{"style":684},[608,5608],{"className":5609,"style":689},[688],[608,5611,5613],{"className":5612},[693,694,695,696],[608,5614,5616],{"className":5615},[658,696],[608,5617,3882],{"className":5618},[658,696]," sans erreur, contrairement à beaucoup d'autres langages où les\nentiers sont bornés à 32 ou 64 bits. Vous reverrez cette notion dans le cours\nsuivant sur les entiers relatifs.",[5621,5622,5623],"style",{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}",{"title":2883,"searchDepth":2919,"depth":2919,"links":5625},[5626,5627,5629,5630,5631,5632,5633,5634],{"id":596,"depth":2919,"text":597},{"id":861,"depth":2919,"text":5628},"Le principe d'une base bbb",{"id":2285,"depth":2919,"text":2286},{"id":3075,"depth":2919,"text":3076},{"id":3836,"depth":2919,"text":3837},{"id":4344,"depth":2919,"text":4345},{"id":5339,"depth":2919,"text":5340},{"id":5548,"depth":2919,"text":5549},"Comment écrire un nombre en binaire, en hexadécimal ou en octal, et comment convertir d'une base à l'autre.",null,"md",{},true,[5641,5646],{"slug":2968,"kind":5642,"question":5643,"choices":5644,"correct":659,"multi":5645,"explanation":2883},"qcm","Que vaut (1101)2(1101)_2(1101)2​ en décimal ?",[2051,3067,3070,2017],false,{"slug":4235,"kind":5642,"question":5647,"choices":5648,"correct":782,"multi":5645,"explanation":2883},"Combien de bits sont nécessaires pour écrire (A3)16(A3)_{16}(A3)16​ ?",[3948,3208,1908,1945],{"title":263,"description":5635},"KHe64FYpinZWaTTv2GNqgCh3u5JXUV_9SiELR5sZQsM",{"id":5652,"title":267,"bo":5653,"body":5655,"description":8814,"eval":5636,"extension":5637,"labo":5636,"meta":8815,"navigation":5639,"path":268,"quizzes":8816,"readingTime":2889,"seo":8822,"stem":269,"tp":5636,"__hash__":8823},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.md",[5654],"PB03",{"type":591,"value":5656,"toc":8802},[5657,5659,5706,5713,5717,5730,5733,5811,5818,5835,5844,5848,5956,5992,6065,6186,6203,6265,6568,6763,6766,7094,7210,7311,7443,7445,7451,7455,7486,7682,8175,8182,8202,8206,8209,8551,8556,8560,8566,8646,8649,8707,8709,8790,8792,8799],[594,5658,597],{"id":596},[599,5660,5661,5662,5665,5666,5700,5701,5355,5704],{},"Vous savez désormais représenter un entier positif en binaire. Mais comment\nfait-on pour coder un nombre ",[603,5663,5664],{},"négatif"," comme ",[608,5667,5669,5685],{"className":5668},[611],[608,5670,5672],{"className":5671},[615],[617,5673,5674],{"xmlns":619},[621,5675,5676,5682],{},[624,5677,5678,5680],{},[942,5679,1060],{},[630,5681,3970],{},[637,5683,5684],{"encoding":639},"-5",[608,5686,5688],{"className":5687,"ariaHidden":645},[644],[608,5689,5691,5694,5697],{"className":5690},[649],[608,5692],{"className":5693,"style":2794},[653],[608,5695,1060],{"className":5696},[658],[608,5698,3970],{"className":5699},[658]," ? Il n'y a pas de signe\nmoins disponible — seuls les bits 0 et 1 existent. Plusieurs solutions ont été\nimaginées au fil de l'histoire de l'informatique, mais une seule s'est imposée\nuniversellement : ",[603,5702,5703],{},"le complément à deux",[857,5705],{"code":5654},[599,5707,5708,5709,5712],{},"Ce codage a une propriété quasi magique : il permet d'",[603,5710,5711],{},"additionner"," un\npositif et un négatif comme s'il s'agissait de deux entiers ordinaires, sans\ndistinction de signe. Les circuits arithmétiques s'en trouvent considérablement\nsimplifiés.",[594,5714,5716],{"id":5715},"le-bit-de-signe-première-idée","Le bit de signe — première idée",[599,5718,5719,5720,5723,5724,5726,5727,5729],{},"L'idée la plus naïve consiste à réserver le ",[603,5721,5722],{},"bit de poids fort"," comme bit de\nsigne : ",[1894,5725,635],{}," pour le positif, ",[1894,5728,659],{}," pour le négatif. Les autres bits codent la\nvaleur absolue.",[599,5731,5732],{},"Sur 4 bits :",[3060,5734,5735,5774],{},[3063,5736,5737,5739,5740,2097],{},[1894,5738,3975],{}," représenterait ",[608,5741,5743,5759],{"className":5742},[611],[608,5744,5746],{"className":5745},[615],[617,5747,5748],{"xmlns":619},[621,5749,5750,5756],{},[624,5751,5752,5754],{},[942,5753,2416],{},[630,5755,3970],{},[637,5757,5758],{"encoding":639},"+5",[608,5760,5762],{"className":5761,"ariaHidden":645},[644],[608,5763,5765,5768,5771],{"className":5764},[649],[608,5766],{"className":5767,"style":2794},[653],[608,5769,2416],{"className":5770},[658],[608,5772,3970],{"className":5773},[658],[3063,5775,5776,5739,5778,2097],{},[1894,5777,2991],{},[608,5779,5781,5796],{"className":5780},[611],[608,5782,5784],{"className":5783},[615],[617,5785,5786],{"xmlns":619},[621,5787,5788,5794],{},[624,5789,5790,5792],{},[942,5791,1060],{},[630,5793,3970],{},[637,5795,5684],{"encoding":639},[608,5797,5799],{"className":5798,"ariaHidden":645},[644],[608,5800,5802,5805,5808],{"className":5801},[649],[608,5803],{"className":5804,"style":2794},[653],[608,5806,1060],{"className":5807},[658],[608,5809,3970],{"className":5810},[658],[599,5812,5813,5814,5817],{},"Cette approche, appelée ",[603,5815,5816],{},"signe et valeur absolue",", semble naturelle… mais\nelle souffre de défauts rédhibitoires :",[3060,5819,5820,5832],{},[3063,5821,5822,5823,5826,5827,2066,5829,5831],{},"Il existe ",[603,5824,5825],{},"deux zéros"," : ",[1894,5828,3873],{},[1894,5830,3882],{},", ce qui complique les\ncomparaisons.",[3063,5833,5834],{},"L'addition d'un positif et d'un négatif ne fonctionne plus de façon\nélémentaire : il faut un circuit dédié pour gérer le signe.",[5342,5836,5837],{},[599,5838,5839,5840,5843],{},"Ce n'est ",[603,5841,5842],{},"pas"," ce codage qui est utilisé en pratique. Il est présenté ici\nuniquement pour comprendre pourquoi le complément à deux a été inventé.",[594,5845,5847],{"id":5846},"le-complément-à-deux","Le complément à deux",[599,5849,5850,5851,5879,5880,5883,5884,5913,5914,5942,5943,5946,5947,5949,5950,5952,5953,2097],{},"Sur ",[608,5852,5854,5867],{"className":5853},[611],[608,5855,5857],{"className":5856},[615],[617,5858,5859],{"xmlns":619},[621,5860,5861,5865],{},[624,5862,5863],{},[876,5864,1162],{},[637,5866,1162],{"encoding":639},[608,5868,5870],{"className":5869,"ariaHidden":645},[644],[608,5871,5873,5876],{"className":5872},[649],[608,5874],{"className":5875,"style":4374},[653],[608,5877,1162],{"className":5878},[658,894]," bits, le ",[603,5881,5882],{},"complément à deux"," d'un entier ",[608,5885,5887,5901],{"className":5886},[611],[608,5888,5890],{"className":5889},[615],[617,5891,5892],{"xmlns":619},[621,5893,5894,5899],{},[624,5895,5896],{},[876,5897,5898],{},"x",[637,5900,5898],{"encoding":639},[608,5902,5904],{"className":5903,"ariaHidden":645},[644],[608,5905,5907,5910],{"className":5906},[649],[608,5908],{"className":5909,"style":4374},[653],[608,5911,5898],{"className":5912},[658,894]," se définit ainsi : on\nécrit ",[608,5915,5917,5930],{"className":5916},[611],[608,5918,5920],{"className":5919},[615],[617,5921,5922],{"xmlns":619},[621,5923,5924,5928],{},[624,5925,5926],{},[876,5927,5898],{},[637,5929,5898],{"encoding":639},[608,5931,5933],{"className":5932,"ariaHidden":645},[644],[608,5934,5936,5939],{"className":5935},[649],[608,5937],{"className":5938,"style":4374},[653],[608,5940,5898],{"className":5941},[658,894]," comme un entier non signé, on ",[603,5944,5945],{},"inverse tous les bits"," (les ",[1894,5948,635],{},"\ndeviennent ",[1894,5951,659],{}," et inversement), puis on ",[603,5954,5955],{},"ajoute 1",[599,5957,5958,5959,2097],{},"Exemple sur 8 bits : représenter ",[608,5960,5962,5977],{"className":5961},[611],[608,5963,5965],{"className":5964},[615],[617,5966,5967],{"xmlns":619},[621,5968,5969,5975],{},[624,5970,5971,5973],{},[942,5972,1060],{},[630,5974,3970],{},[637,5976,5684],{"encoding":639},[608,5978,5980],{"className":5979,"ariaHidden":645},[644],[608,5981,5983,5986,5989],{"className":5982},[649],[608,5984],{"className":5985,"style":2794},[653],[608,5987,1060],{"className":5988},[658],[608,5990,3970],{"className":5991},[658],[1859,5993,5994,6004],{},[1862,5995,5996],{},[1865,5997,5998,6001],{},[1868,5999,6000],{},"Étape",[1868,6002,6003],{},"Valeur",[1881,6005,6006,6045,6055],{},[1865,6007,6008,6040],{},[1886,6009,6010,6011,6039],{},"1. Écrire ",[608,6012,6014,6027],{"className":6013},[611],[608,6015,6017],{"className":6016},[615],[617,6018,6019],{"xmlns":619},[621,6020,6021,6025],{},[624,6022,6023],{},[630,6024,3970],{},[637,6026,3970],{"encoding":639},[608,6028,6030],{"className":6029,"ariaHidden":645},[644],[608,6031,6033,6036],{"className":6032},[649],[608,6034],{"className":6035,"style":981},[653],[608,6037,3970],{"className":6038},[658]," sur 8 bits",[1886,6041,6042],{},[1894,6043,6044],{},"0000 0101",[1865,6046,6047,6050],{},[1886,6048,6049],{},"2. Inverser tous les bits",[1886,6051,6052],{},[1894,6053,6054],{},"1111 1010",[1865,6056,6057,6060],{},[1886,6058,6059],{},"3. Ajouter 1",[1886,6061,6062],{},[1894,6063,6064],{},"1111 1011",[599,6066,6067,6068,6185],{},"Donc ",[608,6069,6071,6103],{"className":6070},[611],[608,6072,6074],{"className":6073},[615],[617,6075,6076],{"xmlns":619},[621,6077,6078,6100],{},[624,6079,6080,6082,6084,6086,6088,6090,6092,6094],{},[942,6081,1060],{},[630,6083,3970],{},[942,6085,1615],{},[942,6087,1151],{"stretchy":1150},[630,6089,4025],{},[4060,6091,4835],{},[630,6093,2120],{},[1153,6095,6096,6098],{},[942,6097,1198],{"stretchy":1150},[630,6099,782],{},[637,6101,6102],{"encoding":639},"-5 = (1111\\,1011)_2",[608,6104,6106,6127],{"className":6105,"ariaHidden":645},[644],[608,6107,6109,6112,6115,6118,6121,6124],{"className":6108},[649],[608,6110],{"className":6111,"style":2794},[653],[608,6113,1060],{"className":6114},[658],[608,6116,3970],{"className":6117},[658],[608,6119],{"className":6120,"style":967},[966],[608,6122,1615],{"className":6123},[971],[608,6125],{"className":6126,"style":967},[966],[608,6128,6130,6133,6136,6139,6142,6145],{"className":6129},[649],[608,6131],{"className":6132,"style":1213},[653],[608,6134,1151],{"className":6135},[1217],[608,6137,4025],{"className":6138},[658],[608,6140],{"className":6141,"style":1325},[966],[608,6143,2120],{"className":6144},[658],[608,6146,6148,6151],{"className":6147},[1417],[608,6149,1198],{"className":6150},[1417],[608,6152,6154],{"className":6153},[669],[608,6155,6157,6177],{"className":6156},[673,1230],[608,6158,6160,6174],{"className":6159},[677],[608,6161,6163],{"className":6162,"style":1237},[681],[608,6164,6165,6168],{"style":1240},[608,6166],{"className":6167,"style":689},[688],[608,6169,6171],{"className":6170},[693,694,695,696],[608,6172,782],{"className":6173},[658,696],[608,6175,1263],{"className":6176},[1262],[608,6178,6180],{"className":6179},[677],[608,6181,6183],{"className":6182,"style":1371},[681],[608,6184],{}," en complément à deux sur 8 bits.",[2099,6187,6188],{},[599,6189,6190,6191,6193,6194,6196,6197,6199,6200,6202],{},"Le ",[603,6192,5722],{}," indique toujours le signe : ",[1894,6195,635],{}," pour les positifs,\n",[1894,6198,659],{}," pour les négatifs. Mais contrairement au codage signe\u002Fvaleur absolue, les\nautres bits ne sont ",[603,6201,5842],{}," la valeur absolue — ils résultent de la double\nopération « inversion + 1 ».",[594,6204,6206,6207],{"id":6205},"pourquoi-ça-marche-laddition-modulo-2n2n2n","Pourquoi ça marche — l'addition modulo ",[608,6208,6210,6227],{"className":6209},[611],[608,6211,6213],{"className":6212},[615],[617,6214,6215],{"xmlns":619},[621,6216,6217,6225],{},[624,6218,6219],{},[627,6220,6221,6223],{},[630,6222,782],{},[876,6224,1162],{},[637,6226,4399],{"encoding":639},[608,6228,6230],{"className":6229,"ariaHidden":645},[644],[608,6231,6233,6236],{"className":6232},[649],[608,6234],{"className":6235,"style":4409},[653],[608,6237,6239,6242],{"className":6238},[658],[608,6240,782],{"className":6241},[658],[608,6243,6245],{"className":6244},[669],[608,6246,6248],{"className":6247},[673],[608,6249,6251],{"className":6250},[677],[608,6252,6254],{"className":6253,"style":4409},[681],[608,6255,6256,6259],{"style":684},[608,6257],{"className":6258,"style":689},[688],[608,6260,6262],{"className":6261},[693,694,695,696],[608,6263,1162],{"className":6264},[658,894,696],[599,6266,5850,6267,6295,6296,6357,6358,6393,6394,6475,6476,6534,6535,2097],{},[608,6268,6270,6283],{"className":6269},[611],[608,6271,6273],{"className":6272},[615],[617,6274,6275],{"xmlns":619},[621,6276,6277,6281],{},[624,6278,6279],{},[876,6280,1162],{},[637,6282,1162],{"encoding":639},[608,6284,6286],{"className":6285,"ariaHidden":645},[644],[608,6287,6289,6292],{"className":6288},[649],[608,6290],{"className":6291,"style":4374},[653],[608,6293,1162],{"className":6294},[658,894]," bits, l'arithmétique se fait ",[603,6297,6298,6299],{},"modulo ",[608,6300,6302,6319],{"className":6301},[611],[608,6303,6305],{"className":6304},[615],[617,6306,6307],{"xmlns":619},[621,6308,6309,6317],{},[624,6310,6311],{},[627,6312,6313,6315],{},[630,6314,782],{},[876,6316,1162],{},[637,6318,4399],{"encoding":639},[608,6320,6322],{"className":6321,"ariaHidden":645},[644],[608,6323,6325,6328],{"className":6324},[649],[608,6326],{"className":6327,"style":4409},[653],[608,6329,6331,6334],{"className":6330},[658],[608,6332,782],{"className":6333},[658],[608,6335,6337],{"className":6336},[669],[608,6338,6340],{"className":6339},[673],[608,6341,6343],{"className":6342},[677],[608,6344,6346],{"className":6345,"style":4409},[681],[608,6347,6348,6351],{"style":684},[608,6349],{"className":6350,"style":689},[688],[608,6352,6354],{"className":6353},[693,694,695,696],[608,6355,1162],{"className":6356},[658,894,696],". Coder un entier négatif\n",[608,6359,6361,6377],{"className":6360},[611],[608,6362,6364],{"className":6363},[615],[617,6365,6366],{"xmlns":619},[621,6367,6368,6374],{},[624,6369,6370,6372],{},[942,6371,1060],{},[876,6373,5898],{},[637,6375,6376],{"encoding":639},"-x",[608,6378,6380],{"className":6379,"ariaHidden":645},[644],[608,6381,6383,6387,6390],{"className":6382},[649],[608,6384],{"className":6385,"style":6386},[653],"height:0.6667em;vertical-align:-0.0833em;",[608,6388,1060],{"className":6389},[658],[608,6391,5898],{"className":6392},[658,894]," revient à utiliser la valeur positive ",[608,6395,6397,6419],{"className":6396},[611],[608,6398,6400],{"className":6399},[615],[617,6401,6402],{"xmlns":619},[621,6403,6404,6416],{},[624,6405,6406,6412,6414],{},[627,6407,6408,6410],{},[630,6409,782],{},[876,6411,1162],{},[942,6413,1060],{},[876,6415,5898],{},[637,6417,6418],{"encoding":639},"2^n - x",[608,6420,6422,6466],{"className":6421,"ariaHidden":645},[644],[608,6423,6425,6428,6457,6460,6463],{"className":6424},[649],[608,6426],{"className":6427,"style":4502},[653],[608,6429,6431,6434],{"className":6430},[658],[608,6432,782],{"className":6433},[658],[608,6435,6437],{"className":6436},[669],[608,6438,6440],{"className":6439},[673],[608,6441,6443],{"className":6442},[677],[608,6444,6446],{"className":6445,"style":4409},[681],[608,6447,6448,6451],{"style":684},[608,6449],{"className":6450,"style":689},[688],[608,6452,6454],{"className":6453},[693,694,695,696],[608,6455,1162],{"className":6456},[658,894,696],[608,6458],{"className":6459,"style":1082},[966],[608,6461,1060],{"className":6462},[1086],[608,6464],{"className":6465,"style":1082},[966],[608,6467,6469,6472],{"className":6468},[649],[608,6470],{"className":6471,"style":4374},[653],[608,6473,5898],{"className":6474},[658,894]," qui, modulo ",[608,6477,6479,6496],{"className":6478},[611],[608,6480,6482],{"className":6481},[615],[617,6483,6484],{"xmlns":619},[621,6485,6486,6494],{},[624,6487,6488],{},[627,6489,6490,6492],{},[630,6491,782],{},[876,6493,1162],{},[637,6495,4399],{"encoding":639},[608,6497,6499],{"className":6498,"ariaHidden":645},[644],[608,6500,6502,6505],{"className":6501},[649],[608,6503],{"className":6504,"style":4409},[653],[608,6506,6508,6511],{"className":6507},[658],[608,6509,782],{"className":6510},[658],[608,6512,6514],{"className":6513},[669],[608,6515,6517],{"className":6516},[673],[608,6518,6520],{"className":6519},[677],[608,6521,6523],{"className":6522,"style":4409},[681],[608,6524,6525,6528],{"style":684},[608,6526],{"className":6527,"style":689},[688],[608,6529,6531],{"className":6530},[693,694,695,696],[608,6532,1162],{"className":6533},[658,894,696],", se\ncomporte exactement comme ",[608,6536,6538,6553],{"className":6537},[611],[608,6539,6541],{"className":6540},[615],[617,6542,6543],{"xmlns":619},[621,6544,6545,6551],{},[624,6546,6547,6549],{},[942,6548,1060],{},[876,6550,5898],{},[637,6552,6376],{"encoding":639},[608,6554,6556],{"className":6555,"ariaHidden":645},[644],[608,6557,6559,6562,6565],{"className":6558},[649],[608,6560],{"className":6561,"style":6386},[653],[608,6563,1060],{"className":6564},[658],[608,6566,5898],{"className":6567},[658,894],[599,6569,6570,6571,6604,6605,6762],{},"Vérification : sur 8 bits, ",[608,6572,6574,6589],{"className":6573},[611],[608,6575,6577],{"className":6576},[615],[617,6578,6579],{"xmlns":619},[621,6580,6581,6587],{},[624,6582,6583,6585],{},[942,6584,1060],{},[630,6586,3970],{},[637,6588,5684],{"encoding":639},[608,6590,6592],{"className":6591,"ariaHidden":645},[644],[608,6593,6595,6598,6601],{"className":6594},[649],[608,6596],{"className":6597,"style":2794},[653],[608,6599,1060],{"className":6600},[658],[608,6602,3970],{"className":6603},[658]," correspond à ",[608,6606,6608,6647],{"className":6607},[611],[608,6609,6611],{"className":6610},[615],[617,6612,6613],{"xmlns":619},[621,6614,6615,6644],{},[624,6616,6617,6619,6621,6623,6625,6628,6630,6632,6634,6636,6638],{},[630,6618,4713],{},[942,6620,1060],{},[630,6622,3970],{},[942,6624,1615],{},[630,6626,6627],{},"251",[942,6629,1615],{},[942,6631,1151],{"stretchy":1150},[630,6633,4025],{},[4060,6635,4835],{},[630,6637,2120],{},[1153,6639,6640,6642],{},[942,6641,1198],{"stretchy":1150},[630,6643,782],{},[637,6645,6646],{"encoding":639},"256 - 5 = 251 = (1111\\,1011)_2",[608,6648,6650,6668,6686,6704],{"className":6649,"ariaHidden":645},[644],[608,6651,6653,6656,6659,6662,6665],{"className":6652},[649],[608,6654],{"className":6655,"style":2794},[653],[608,6657,4713],{"className":6658},[658],[608,6660],{"className":6661,"style":1082},[966],[608,6663,1060],{"className":6664},[1086],[608,6666],{"className":6667,"style":1082},[966],[608,6669,6671,6674,6677,6680,6683],{"className":6670},[649],[608,6672],{"className":6673,"style":981},[653],[608,6675,3970],{"className":6676},[658],[608,6678],{"className":6679,"style":967},[966],[608,6681,1615],{"className":6682},[971],[608,6684],{"className":6685,"style":967},[966],[608,6687,6689,6692,6695,6698,6701],{"className":6688},[649],[608,6690],{"className":6691,"style":981},[653],[608,6693,6627],{"className":6694},[658],[608,6696],{"className":6697,"style":967},[966],[608,6699,1615],{"className":6700},[971],[608,6702],{"className":6703,"style":967},[966],[608,6705,6707,6710,6713,6716,6719,6722],{"className":6706},[649],[608,6708],{"className":6709,"style":1213},[653],[608,6711,1151],{"className":6712},[1217],[608,6714,4025],{"className":6715},[658],[608,6717],{"className":6718,"style":1325},[966],[608,6720,2120],{"className":6721},[658],[608,6723,6725,6728],{"className":6724},[1417],[608,6726,1198],{"className":6727},[1417],[608,6729,6731],{"className":6730},[669],[608,6732,6734,6754],{"className":6733},[673,1230],[608,6735,6737,6751],{"className":6736},[677],[608,6738,6740],{"className":6739,"style":1237},[681],[608,6741,6742,6745],{"style":1240},[608,6743],{"className":6744,"style":689},[688],[608,6746,6748],{"className":6747},[693,694,695,696],[608,6749,782],{"className":6750},[658,696],[608,6752,1263],{"className":6753},[1262],[608,6755,6757],{"className":6756},[677],[608,6758,6760],{"className":6759,"style":1371},[681],[608,6761],{},".\nC'est bien le résultat trouvé plus haut.",[599,6764,6765],{},"Et l'addition fonctionne :",[608,6767,6769],{"className":6768},[1596],[608,6770,6772,6848],{"className":6771},[611],[608,6773,6775],{"className":6774},[615],[617,6776,6777],{"xmlns":619,"display":1605},[621,6778,6779,6845],{},[624,6780,6781,6783,6785,6787,6789,6791,6793,6795,6797,6799,6801,6803,6809,6811,6813,6815,6817,6819,6825,6827,6829,6831,6833,6835,6837,6839],{},[630,6782,3970],{},[942,6784,2416],{},[942,6786,1151],{"stretchy":1150},[942,6788,1060],{},[630,6790,3970],{},[942,6792,1198],{"stretchy":1150},[942,6794,1615],{},[942,6796,1151],{"stretchy":1150},[630,6798,3873],{},[4060,6800,4835],{},[630,6802,3975],{},[1153,6804,6805,6807],{},[942,6806,1198],{"stretchy":1150},[630,6808,782],{},[942,6810,2416],{},[942,6812,1151],{"stretchy":1150},[630,6814,4025],{},[4060,6816,4835],{},[630,6818,2120],{},[1153,6820,6821,6823],{},[942,6822,1198],{"stretchy":1150},[630,6824,782],{},[942,6826,1615],{},[942,6828,1151],{"stretchy":1150},[630,6830,659],{},[4060,6832,4835],{},[630,6834,3873],{},[4060,6836,4835],{},[630,6838,3873],{},[1153,6840,6841,6843],{},[942,6842,1198],{"stretchy":1150},[630,6844,782],{},[637,6846,6847],{"encoding":639},"5 + (-5) = (0000\\,0101)_2 + (1111\\,1011)_2 = (1\\,0000\\,0000)_2",[608,6849,6851,6869,6896,6963,7030],{"className":6850,"ariaHidden":645},[644],[608,6852,6854,6857,6860,6863,6866],{"className":6853},[649],[608,6855],{"className":6856,"style":2794},[653],[608,6858,3970],{"className":6859},[658],[608,6861],{"className":6862,"style":1082},[966],[608,6864,2416],{"className":6865},[1086],[608,6867],{"className":6868,"style":1082},[966],[608,6870,6872,6875,6878,6881,6884,6887,6890,6893],{"className":6871},[649],[608,6873],{"className":6874,"style":1213},[653],[608,6876,1151],{"className":6877},[1217],[608,6879,1060],{"className":6880},[658],[608,6882,3970],{"className":6883},[658],[608,6885,1198],{"className":6886},[1417],[608,6888],{"className":6889,"style":967},[966],[608,6891,1615],{"className":6892},[971],[608,6894],{"className":6895,"style":967},[966],[608,6897,6899,6902,6905,6908,6911,6914,6954,6957,6960],{"className":6898},[649],[608,6900],{"className":6901,"style":1213},[653],[608,6903,1151],{"className":6904},[1217],[608,6906,3873],{"className":6907},[658],[608,6909],{"className":6910,"style":1325},[966],[608,6912,3975],{"className":6913},[658],[608,6915,6917,6920],{"className":6916},[1417],[608,6918,1198],{"className":6919},[1417],[608,6921,6923],{"className":6922},[669],[608,6924,6926,6946],{"className":6925},[673,1230],[608,6927,6929,6943],{"className":6928},[677],[608,6930,6932],{"className":6931,"style":1237},[681],[608,6933,6934,6937],{"style":1240},[608,6935],{"className":6936,"style":689},[688],[608,6938,6940],{"className":6939},[693,694,695,696],[608,6941,782],{"className":6942},[658,696],[608,6944,1263],{"className":6945},[1262],[608,6947,6949],{"className":6948},[677],[608,6950,6952],{"className":6951,"style":1371},[681],[608,6953],{},[608,6955],{"className":6956,"style":1082},[966],[608,6958,2416],{"className":6959},[1086],[608,6961],{"className":6962,"style":1082},[966],[608,6964,6966,6969,6972,6975,6978,6981,7021,7024,7027],{"className":6965},[649],[608,6967],{"className":6968,"style":1213},[653],[608,6970,1151],{"className":6971},[1217],[608,6973,4025],{"className":6974},[658],[608,6976],{"className":6977,"style":1325},[966],[608,6979,2120],{"className":6980},[658],[608,6982,6984,6987],{"className":6983},[1417],[608,6985,1198],{"className":6986},[1417],[608,6988,6990],{"className":6989},[669],[608,6991,6993,7013],{"className":6992},[673,1230],[608,6994,6996,7010],{"className":6995},[677],[608,6997,6999],{"className":6998,"style":1237},[681],[608,7000,7001,7004],{"style":1240},[608,7002],{"className":7003,"style":689},[688],[608,7005,7007],{"className":7006},[693,694,695,696],[608,7008,782],{"className":7009},[658,696],[608,7011,1263],{"className":7012},[1262],[608,7014,7016],{"className":7015},[677],[608,7017,7019],{"className":7018,"style":1371},[681],[608,7020],{},[608,7022],{"className":7023,"style":967},[966],[608,7025,1615],{"className":7026},[971],[608,7028],{"className":7029,"style":967},[966],[608,7031,7033,7036,7039,7042,7045,7048,7051,7054],{"className":7032},[649],[608,7034],{"className":7035,"style":1213},[653],[608,7037,1151],{"className":7038},[1217],[608,7040,659],{"className":7041},[658],[608,7043],{"className":7044,"style":1325},[966],[608,7046,3873],{"className":7047},[658],[608,7049],{"className":7050,"style":1325},[966],[608,7052,3873],{"className":7053},[658],[608,7055,7057,7060],{"className":7056},[1417],[608,7058,1198],{"className":7059},[1417],[608,7061,7063],{"className":7062},[669],[608,7064,7066,7086],{"className":7065},[673,1230],[608,7067,7069,7083],{"className":7068},[677],[608,7070,7072],{"className":7071,"style":1237},[681],[608,7073,7074,7077],{"style":1240},[608,7075],{"className":7076,"style":689},[688],[608,7078,7080],{"className":7079},[693,694,695,696],[608,7081,782],{"className":7082},[658,696],[608,7084,1263],{"className":7085},[1262],[608,7087,7089],{"className":7088},[677],[608,7090,7092],{"className":7091,"style":1371},[681],[608,7093],{},[599,7095,7096,7097,7209],{},"Le 9ᵉ bit déborde de l'octet et est ignoré : il reste ",[608,7098,7100,7130],{"className":7099},[611],[608,7101,7103],{"className":7102},[615],[617,7104,7105],{"xmlns":619},[621,7106,7107,7127],{},[624,7108,7109,7111,7113,7115,7117,7123,7125],{},[942,7110,1151],{"stretchy":1150},[630,7112,3873],{},[4060,7114,4835],{},[630,7116,3873],{},[1153,7118,7119,7121],{},[942,7120,1198],{"stretchy":1150},[630,7122,782],{},[942,7124,1615],{},[630,7126,635],{},[637,7128,7129],{"encoding":639},"(0000\\,0000)_2 = 0",[608,7131,7133,7200],{"className":7132,"ariaHidden":645},[644],[608,7134,7136,7139,7142,7145,7148,7151,7191,7194,7197],{"className":7135},[649],[608,7137],{"className":7138,"style":1213},[653],[608,7140,1151],{"className":7141},[1217],[608,7143,3873],{"className":7144},[658],[608,7146],{"className":7147,"style":1325},[966],[608,7149,3873],{"className":7150},[658],[608,7152,7154,7157],{"className":7153},[1417],[608,7155,1198],{"className":7156},[1417],[608,7158,7160],{"className":7159},[669],[608,7161,7163,7183],{"className":7162},[673,1230],[608,7164,7166,7180],{"className":7165},[677],[608,7167,7169],{"className":7168,"style":1237},[681],[608,7170,7171,7174],{"style":1240},[608,7172],{"className":7173,"style":689},[688],[608,7175,7177],{"className":7176},[693,694,695,696],[608,7178,782],{"className":7179},[658,696],[608,7181,1263],{"className":7182},[1262],[608,7184,7186],{"className":7185},[677],[608,7187,7189],{"className":7188,"style":1371},[681],[608,7190],{},[608,7192],{"className":7193,"style":967},[966],[608,7195,1615],{"className":7196},[971],[608,7198],{"className":7199,"style":967},[966],[608,7201,7203,7206],{"className":7202},[649],[608,7204],{"className":7205,"style":981},[653],[608,7207,635],{"className":7208},[658],".\nL'opération donne bien zéro.",[7211,7212,7213],"tip",{},[599,7214,7215,7218,7219,7222,7223,7226,7227,7229,7230,7288,7289,7292,7293,7296,7297,2066,7299,7301,7302,7305,7306,7308,7309,2097],{},[603,7216,7217],{},"Deux opérateurs Python utilisés ci-dessous"," :\n",[1894,7220,7221],{},"a \u003C\u003C n"," décale les bits de ",[1894,7224,7225],{},"a"," de ",[1894,7228,1162],{}," rangs vers la gauche (équivaut à\nmultiplier par ",[608,7231,7233,7250],{"className":7232},[611],[608,7234,7236],{"className":7235},[615],[617,7237,7238],{"xmlns":619},[621,7239,7240,7248],{},[624,7241,7242],{},[627,7243,7244,7246],{},[630,7245,782],{},[876,7247,1162],{},[637,7249,4399],{"encoding":639},[608,7251,7253],{"className":7252,"ariaHidden":645},[644],[608,7254,7256,7259],{"className":7255},[649],[608,7257],{"className":7258,"style":4409},[653],[608,7260,7262,7265],{"className":7261},[658],[608,7263,782],{"className":7264},[658],[608,7266,7268],{"className":7267},[669],[608,7269,7271],{"className":7270},[673],[608,7272,7274],{"className":7273},[677],[608,7275,7277],{"className":7276,"style":4409},[681],[608,7278,7279,7282],{"style":684},[608,7280],{"className":7281,"style":689},[688],[608,7283,7285],{"className":7284},[693,694,695,696],[608,7286,1162],{"className":7287},[658,894,696],") ; ",[1894,7290,7291],{},"a & b"," réalise un ",[603,7294,7295],{},"ET bit-à-bit"," (chaque bit du\nrésultat vaut 1 seulement si les deux bits correspondants de ",[1894,7298,7225],{},[1894,7300,878],{},"\nvalent 1). Combinés, ",[1894,7303,7304],{},"x & ((1 \u003C\u003C n) - 1)"," ne garde que les ",[1894,7307,1162],{}," bits de poids\nfaible de ",[1894,7310,5898],{},[3708,7312,7313],{},[2878,7314,7316],{"className":2880,"code":7315,"language":2882,"meta":2883,"style":2883},"# Python gère les entiers signés en complément à deux conceptuellement\n# mais sans largeur fixe. On simule un complément à deux sur 8 bits.\nn_bits = 8\nx = -5\n# Conversion en complément à deux sur 8 bits :\ncode = x & ((1 \u003C\u003C n_bits) - 1)  # masque les 8 bits de poids faible\nprint(f\"{x} sur {n_bits} bits : {bin(code)} ({code} non signé)\")\n",[1894,7317,7318,7323,7328,7338,7351,7357,7393],{"__ignoreMap":2883},[608,7319,7320],{"class":2888,"line":2889},[608,7321,7322],{"class":2915},"# Python gère les entiers signés en complément à deux conceptuellement\n",[608,7324,7325],{"class":2888,"line":2919},[608,7326,7327],{"class":2915},"# mais sans largeur fixe. On simule un complément à deux sur 8 bits.\n",[608,7329,7330,7333,7335],{"class":2888,"line":2943},[608,7331,7332],{"class":2896},"n_bits ",[608,7334,1615],{"class":3722},[608,7336,7337],{"class":2892}," 8\n",[608,7339,7340,7343,7345,7348],{"class":2888,"line":3794},[608,7341,7342],{"class":2896},"x ",[608,7344,1615],{"class":3722},[608,7346,7347],{"class":3722}," -",[608,7349,7350],{"class":2892},"5\n",[608,7352,7354],{"class":2888,"line":7353},5,[608,7355,7356],{"class":2915},"# Conversion en complément à deux sur 8 bits :\n",[608,7358,7360,7363,7365,7368,7371,7374,7376,7379,7382,7384,7387,7390],{"class":2888,"line":7359},6,[608,7361,7362],{"class":2896},"code ",[608,7364,1615],{"class":3722},[608,7366,7367],{"class":2896}," x ",[608,7369,7370],{"class":3722},"&",[608,7372,7373],{"class":2896}," ((",[608,7375,659],{"class":2892},[608,7377,7378],{"class":3722}," \u003C\u003C",[608,7380,7381],{"class":2896}," n_bits) ",[608,7383,5327],{"class":3722},[608,7385,7386],{"class":2892}," 1",[608,7388,7389],{"class":2896},")  ",[608,7391,7392],{"class":2915},"# masque les 8 bits de poids faible\n",[608,7394,7396,7398,7400,7402,7404,7406,7408,7410,7413,7415,7418,7420,7423,7425,7428,7430,7432,7434,7436,7438,7441],{"class":2888,"line":7395},7,[608,7397,2893],{"class":2892},[608,7399,1151],{"class":2896},[608,7401,3734],{"class":3722},[608,7403,3737],{"class":2904},[608,7405,3740],{"class":2892},[608,7407,5898],{"class":2896},[608,7409,3745],{"class":2892},[608,7411,7412],{"class":2904}," sur ",[608,7414,3740],{"class":2892},[608,7416,7417],{"class":2896},"n_bits",[608,7419,3745],{"class":2892},[608,7421,7422],{"class":2904}," bits : ",[608,7424,3751],{"class":2892},[608,7426,7427],{"class":2896},"(code)",[608,7429,3745],{"class":2892},[608,7431,842],{"class":2904},[608,7433,3740],{"class":2892},[608,7435,1894],{"class":2896},[608,7437,3745],{"class":2892},[608,7439,7440],{"class":2904}," non signé)\"",[608,7442,3761],{"class":2896},[599,7444,3825],{},[2878,7446,7449],{"className":7447,"code":7448,"language":3831},[3829],"-5 sur 8 bits : 0b11111011 (251 non signé)\n",[1894,7450,7448],{"__ignoreMap":2883},[594,7452,7454],{"id":7453},"étendue-représentable","Étendue représentable",[599,7456,5850,7457,7485],{},[608,7458,7460,7473],{"className":7459},[611],[608,7461,7463],{"className":7462},[615],[617,7464,7465],{"xmlns":619},[621,7466,7467,7471],{},[624,7468,7469],{},[876,7470,1162],{},[637,7472,1162],{"encoding":639},[608,7474,7476],{"className":7475,"ariaHidden":645},[644],[608,7477,7479,7482],{"className":7478},[649],[608,7480],{"className":7481,"style":4374},[653],[608,7483,1162],{"className":7484},[658,894]," bits en complément à deux, on couvre l'intervalle :",[608,7487,7489],{"className":7488},[1596],[608,7490,7492,7541],{"className":7491},[611],[608,7493,7495],{"className":7494},[615],[617,7496,7497],{"xmlns":619,"display":1605},[621,7498,7499,7538],{},[624,7500,7501,7503,7515,7518,7520,7522,7534,7536],{},[942,7502,1060],{},[627,7504,7505,7507],{},[630,7506,782],{},[624,7508,7509,7511,7513],{},[876,7510,1162],{},[942,7512,1060],{},[630,7514,659],{},[942,7516,7517],{},"≤",[876,7519,5898],{},[942,7521,7517],{},[627,7523,7524,7526],{},[630,7525,782],{},[624,7527,7528,7530,7532],{},[876,7529,1162],{},[942,7531,1060],{},[630,7533,659],{},[942,7535,1060],{},[630,7537,659],{},[637,7539,7540],{"encoding":639},"-2^{n-1} \\leq x \\leq 2^{n-1} - 1",[608,7542,7544,7601,7620,7673],{"className":7543,"ariaHidden":645},[644],[608,7545,7547,7551,7554,7592,7595,7598],{"className":7546},[649],[608,7548],{"className":7549,"style":7550},[653],"height:1.0001em;vertical-align:-0.136em;",[608,7552,1060],{"className":7553},[658],[608,7555,7557,7560],{"className":7556},[658],[608,7558,782],{"className":7559},[658],[608,7561,7563],{"className":7562},[669],[608,7564,7566],{"className":7565},[673],[608,7567,7569],{"className":7568},[677],[608,7570,7572],{"className":7571,"style":2581},[681],[608,7573,7574,7577],{"style":1848},[608,7575],{"className":7576,"style":689},[688],[608,7578,7580],{"className":7579},[693,694,695,696],[608,7581,7583,7586,7589],{"className":7582},[658,696],[608,7584,1162],{"className":7585},[658,894,696],[608,7587,1060],{"className":7588},[1086,696],[608,7590,659],{"className":7591},[658,696],[608,7593],{"className":7594,"style":967},[966],[608,7596,7517],{"className":7597},[971],[608,7599],{"className":7600,"style":967},[966],[608,7602,7604,7608,7611,7614,7617],{"className":7603},[649],[608,7605],{"className":7606,"style":7607},[653],"height:0.7719em;vertical-align:-0.136em;",[608,7609,5898],{"className":7610},[658,894],[608,7612],{"className":7613,"style":967},[966],[608,7615,7517],{"className":7616},[971],[608,7618],{"className":7619,"style":967},[966],[608,7621,7623,7626,7664,7667,7670],{"className":7622},[649],[608,7624],{"className":7625,"style":2562},[653],[608,7627,7629,7632],{"className":7628},[658],[608,7630,782],{"className":7631},[658],[608,7633,7635],{"className":7634},[669],[608,7636,7638],{"className":7637},[673],[608,7639,7641],{"className":7640},[677],[608,7642,7644],{"className":7643,"style":2581},[681],[608,7645,7646,7649],{"style":1848},[608,7647],{"className":7648,"style":689},[688],[608,7650,7652],{"className":7651},[693,694,695,696],[608,7653,7655,7658,7661],{"className":7654},[658,696],[608,7656,1162],{"className":7657},[658,894,696],[608,7659,1060],{"className":7660},[1086,696],[608,7662,659],{"className":7663},[658,696],[608,7665],{"className":7666,"style":1082},[966],[608,7668,1060],{"className":7669},[1086],[608,7671],{"className":7672,"style":1082},[966],[608,7674,7676,7679],{"className":7675},[649],[608,7677],{"className":7678,"style":981},[653],[608,7680,659],{"className":7681},[658],[1859,7683,7684,7695],{},[1862,7685,7686],{},[1865,7687,7688,7690,7693],{},[1868,7689,4558],{},[1868,7691,7692],{},"Minimum",[1868,7694,4564],{},[1881,7696,7697,7775,7873,8013],{},[1865,7698,7699,7701,7738],{},[1886,7700,1908],{},[1886,7702,7703],{},[608,7704,7706,7723],{"className":7705},[611],[608,7707,7709],{"className":7708},[615],[617,7710,7711],{"xmlns":619},[621,7712,7713,7720],{},[624,7714,7715,7717],{},[942,7716,1060],{},[630,7718,7719],{},"128",[637,7721,7722],{"encoding":639},"-128",[608,7724,7726],{"className":7725,"ariaHidden":645},[644],[608,7727,7729,7732,7735],{"className":7728},[649],[608,7730],{"className":7731,"style":2794},[653],[608,7733,1060],{"className":7734},[658],[608,7736,7719],{"className":7737},[658],[1886,7739,7740],{},[608,7741,7743,7760],{"className":7742},[611],[608,7744,7746],{"className":7745},[615],[617,7747,7748],{"xmlns":619},[621,7749,7750,7757],{},[624,7751,7752,7754],{},[942,7753,2416],{},[630,7755,7756],{},"127",[637,7758,7759],{"encoding":639},"+127",[608,7761,7763],{"className":7762,"ariaHidden":645},[644],[608,7764,7766,7769,7772],{"className":7765},[649],[608,7767],{"className":7768,"style":2794},[653],[608,7770,2416],{"className":7771},[658],[608,7773,7756],{"className":7774},[658],[1865,7776,7777,7779,7826],{},[1886,7778,1945],{},[1886,7780,7781],{},[608,7782,7784,7805],{"className":7783},[611],[608,7785,7787],{"className":7786},[615],[617,7788,7789],{"xmlns":619},[621,7790,7791,7802],{},[624,7792,7793,7795,7797,7799],{},[942,7794,1060],{},[630,7796,4953],{},[4060,7798,4835],{},[630,7800,7801],{},"768",[637,7803,7804],{"encoding":639},"-32\\,768",[608,7806,7808],{"className":7807,"ariaHidden":645},[644],[608,7809,7811,7814,7817,7820,7823],{"className":7810},[649],[608,7812],{"className":7813,"style":2794},[653],[608,7815,1060],{"className":7816},[658],[608,7818,4953],{"className":7819},[658],[608,7821],{"className":7822,"style":1325},[966],[608,7824,7801],{"className":7825},[658],[1886,7827,7828],{},[608,7829,7831,7852],{"className":7830},[611],[608,7832,7834],{"className":7833},[615],[617,7835,7836],{"xmlns":619},[621,7837,7838,7849],{},[624,7839,7840,7842,7844,7846],{},[942,7841,2416],{},[630,7843,4953],{},[4060,7845,4835],{},[630,7847,7848],{},"767",[637,7850,7851],{"encoding":639},"+32\\,767",[608,7853,7855],{"className":7854,"ariaHidden":645},[644],[608,7856,7858,7861,7864,7867,7870],{"className":7857},[649],[608,7859],{"className":7860,"style":2794},[653],[608,7862,2416],{"className":7863},[658],[608,7865,4953],{"className":7866},[658],[608,7868],{"className":7869,"style":1325},[966],[608,7871,7848],{"className":7872},[658],[1865,7874,7875,7877,7946],{},[1886,7876,4953],{},[1886,7878,7879],{},[608,7880,7882,7913],{"className":7881},[611],[608,7883,7885],{"className":7884},[615],[617,7886,7887],{"xmlns":619},[621,7888,7889,7910],{},[624,7890,7891,7893,7895,7897,7900,7902,7905,7907],{},[942,7892,1060],{},[630,7894,782],{},[4060,7896,4835],{},[630,7898,7899],{},"147",[4060,7901,4835],{},[630,7903,7904],{},"483",[4060,7906,4835],{},[630,7908,7909],{},"648",[637,7911,7912],{"encoding":639},"-2\\,147\\,483\\,648",[608,7914,7916],{"className":7915,"ariaHidden":645},[644],[608,7917,7919,7922,7925,7928,7931,7934,7937,7940,7943],{"className":7918},[649],[608,7920],{"className":7921,"style":2794},[653],[608,7923,1060],{"className":7924},[658],[608,7926,782],{"className":7927},[658],[608,7929],{"className":7930,"style":1325},[966],[608,7932,7899],{"className":7933},[658],[608,7935],{"className":7936,"style":1325},[966],[608,7938,7904],{"className":7939},[658],[608,7941],{"className":7942,"style":1325},[966],[608,7944,7909],{"className":7945},[658],[1886,7947,7948],{},[608,7949,7951,7980],{"className":7950},[611],[608,7952,7954],{"className":7953},[615],[617,7955,7956],{"xmlns":619},[621,7957,7958,7977],{},[624,7959,7960,7962,7964,7966,7968,7970,7972,7974],{},[942,7961,2416],{},[630,7963,782],{},[4060,7965,4835],{},[630,7967,7899],{},[4060,7969,4835],{},[630,7971,7904],{},[4060,7973,4835],{},[630,7975,7976],{},"647",[637,7978,7979],{"encoding":639},"+2\\,147\\,483\\,647",[608,7981,7983],{"className":7982,"ariaHidden":645},[644],[608,7984,7986,7989,7992,7995,7998,8001,8004,8007,8010],{"className":7985},[649],[608,7987],{"className":7988,"style":2794},[653],[608,7990,2416],{"className":7991},[658],[608,7993,782],{"className":7994},[658],[608,7996],{"className":7997,"style":1325},[966],[608,7999,7899],{"className":8000},[658],[608,8002],{"className":8003,"style":1325},[966],[608,8005,7904],{"className":8006},[658],[608,8008],{"className":8009,"style":1325},[966],[608,8011,7976],{"className":8012},[658],[1865,8014,8015,8018,8089],{},[1886,8016,8017],{},"64",[1886,8019,8020],{},[608,8021,8023,8044],{"className":8022},[611],[608,8024,8026],{"className":8025},[615],[617,8027,8028],{"xmlns":619},[621,8029,8030,8041],{},[624,8031,8032,8034],{},[942,8033,1060],{},[627,8035,8036,8038],{},[630,8037,782],{},[630,8039,8040],{},"63",[637,8042,8043],{"encoding":639},"-2^{63}",[608,8045,8047],{"className":8046,"ariaHidden":645},[644],[608,8048,8050,8054,8057],{"className":8049},[649],[608,8051],{"className":8052,"style":8053},[653],"height:0.8974em;vertical-align:-0.0833em;",[608,8055,1060],{"className":8056},[658],[608,8058,8060,8063],{"className":8059},[658],[608,8061,782],{"className":8062},[658],[608,8064,8066],{"className":8065},[669],[608,8067,8069],{"className":8068},[673],[608,8070,8072],{"className":8071},[677],[608,8073,8075],{"className":8074,"style":654},[681],[608,8076,8077,8080],{"style":684},[608,8078],{"className":8079,"style":689},[688],[608,8081,8083],{"className":8082},[693,694,695,696],[608,8084,8086],{"className":8085},[658,696],[608,8087,8040],{"className":8088},[658,696],[1886,8090,8091],{},[608,8092,8094,8116],{"className":8093},[611],[608,8095,8097],{"className":8096},[615],[617,8098,8099],{"xmlns":619},[621,8100,8101,8113],{},[624,8102,8103,8109,8111],{},[627,8104,8105,8107],{},[630,8106,782],{},[630,8108,8040],{},[942,8110,1060],{},[630,8112,659],{},[637,8114,8115],{"encoding":639},"2^{63} - 1",[608,8117,8119,8166],{"className":8118,"ariaHidden":645},[644],[608,8120,8122,8125,8157,8160,8163],{"className":8121},[649],[608,8123],{"className":8124,"style":8053},[653],[608,8126,8128,8131],{"className":8127},[658],[608,8129,782],{"className":8130},[658],[608,8132,8134],{"className":8133},[669],[608,8135,8137],{"className":8136},[673],[608,8138,8140],{"className":8139},[677],[608,8141,8143],{"className":8142,"style":654},[681],[608,8144,8145,8148],{"style":684},[608,8146],{"className":8147,"style":689},[688],[608,8149,8151],{"className":8150},[693,694,695,696],[608,8152,8154],{"className":8153},[658,696],[608,8155,8040],{"className":8156},[658,696],[608,8158],{"className":8159,"style":1082},[966],[608,8161,1060],{"className":8162},[1086],[608,8164],{"className":8165,"style":1082},[966],[608,8167,8169,8172],{"className":8168},[649],[608,8170],{"className":8171,"style":981},[653],[608,8173,659],{"className":8174},[658],[599,8176,8177,8178,8181],{},"On remarque que l'intervalle n'est ",[603,8179,8180],{},"pas symétrique"," : il y a un entier\nnégatif de plus que d'entiers positifs. C'est la conséquence du fait qu'il\nn'existe qu'un seul zéro (qui est compté comme positif au sens large).",[2966,8183,8185,8190],{"correct":2413,"slug":8184},"entiers-relatifs-c1",[2970,8186,8187],{"v-slot:question":2883},[599,8188,8189],{},"Sur 8 bits en complément à deux, combien d'entiers distincts peut-on représenter ?",[2970,8191,8192],{"v-slot:choices":2883},[3060,8193,8194,8196,8198,8200],{},[3063,8195,7756],{},[3063,8197,7719],{},[3063,8199,4789],{},[3063,8201,4713],{},[594,8203,8205],{"id":8204},"bits-nécessaires-pour-une-somme-ou-un-produit","Bits nécessaires pour une somme ou un produit",[599,8207,8208],{},"Combien de bits faut-il prévoir si l'on additionne ou si l'on multiplie deux\nentiers ?",[3060,8210,8211,8298],{},[3063,8212,8213,8216,8217,8245,8246,8297],{},[603,8214,8215],{},"Somme"," de deux entiers de ",[608,8218,8220,8233],{"className":8219},[611],[608,8221,8223],{"className":8222},[615],[617,8224,8225],{"xmlns":619},[621,8226,8227,8231],{},[624,8228,8229],{},[876,8230,1162],{},[637,8232,1162],{"encoding":639},[608,8234,8236],{"className":8235,"ariaHidden":645},[644],[608,8237,8239,8242],{"className":8238},[649],[608,8240],{"className":8241,"style":4374},[653],[608,8243,1162],{"className":8244},[658,894]," bits : il peut y avoir une retenue qui\najoute 1 bit. Le résultat tient sur ",[608,8247,8249,8267],{"className":8248},[611],[608,8250,8252],{"className":8251},[615],[617,8253,8254],{"xmlns":619},[621,8255,8256,8264],{},[624,8257,8258,8260,8262],{},[876,8259,1162],{},[942,8261,2416],{},[630,8263,659],{},[637,8265,8266],{"encoding":639},"n+1",[608,8268,8270,8288],{"className":8269,"ariaHidden":645},[644],[608,8271,8273,8276,8279,8282,8285],{"className":8272},[649],[608,8274],{"className":8275,"style":6386},[653],[608,8277,1162],{"className":8278},[658,894],[608,8280],{"className":8281,"style":1082},[966],[608,8283,2416],{"className":8284},[1086],[608,8286],{"className":8287,"style":1082},[966],[608,8289,8291,8294],{"className":8290},[649],[608,8292],{"className":8293,"style":981},[653],[608,8295,659],{"className":8296},[658]," bits.",[3063,8299,8300,8216,8303,8331,8332,8516,8517,8297],{},[603,8301,8302],{},"Produit",[608,8304,8306,8319],{"className":8305},[611],[608,8307,8309],{"className":8308},[615],[617,8310,8311],{"xmlns":619},[621,8312,8313,8317],{},[624,8314,8315],{},[876,8316,1162],{},[637,8318,1162],{"encoding":639},[608,8320,8322],{"className":8321,"ariaHidden":645},[644],[608,8323,8325,8328],{"className":8324},[649],[608,8326],{"className":8327,"style":4374},[653],[608,8329,1162],{"className":8330},[658,894]," bits : le résultat peut atteindre\n",[608,8333,8335,8377],{"className":8334},[611],[608,8336,8338],{"className":8337},[615],[617,8339,8340],{"xmlns":619},[621,8341,8342,8374],{},[624,8343,8344,8346,8352,8354,8356,8362,8364],{},[942,8345,1151],{"stretchy":1150},[627,8347,8348,8350],{},[630,8349,782],{},[876,8351,1162],{},[942,8353,1060],{},[630,8355,659],{},[627,8357,8358,8360],{},[942,8359,1198],{"stretchy":1150},[630,8361,782],{},[942,8363,4976],{},[627,8365,8366,8368],{},[630,8367,782],{},[624,8369,8370,8372],{},[630,8371,782],{},[876,8373,1162],{},[637,8375,8376],{"encoding":639},"(2^n - 1)^2 \\approx 2^{2n}",[608,8378,8380,8427,8475],{"className":8379,"ariaHidden":645},[644],[608,8381,8383,8386,8389,8418,8421,8424],{"className":8382},[649],[608,8384],{"className":8385,"style":1213},[653],[608,8387,1151],{"className":8388},[1217],[608,8390,8392,8395],{"className":8391},[658],[608,8393,782],{"className":8394},[658],[608,8396,8398],{"className":8397},[669],[608,8399,8401],{"className":8400},[673],[608,8402,8404],{"className":8403},[677],[608,8405,8407],{"className":8406,"style":4409},[681],[608,8408,8409,8412],{"style":684},[608,8410],{"className":8411,"style":689},[688],[608,8413,8415],{"className":8414},[693,694,695,696],[608,8416,1162],{"className":8417},[658,894,696],[608,8419],{"className":8420,"style":1082},[966],[608,8422,1060],{"className":8423},[1086],[608,8425],{"className":8426,"style":1082},[966],[608,8428,8430,8434,8437,8466,8469,8472],{"className":8429},[649],[608,8431],{"className":8432,"style":8433},[653],"height:1.0641em;vertical-align:-0.25em;",[608,8435,659],{"className":8436},[658],[608,8438,8440,8443],{"className":8439},[1417],[608,8441,1198],{"className":8442},[1417],[608,8444,8446],{"className":8445},[669],[608,8447,8449],{"className":8448},[673],[608,8450,8452],{"className":8451},[677],[608,8453,8455],{"className":8454,"style":654},[681],[608,8456,8457,8460],{"style":684},[608,8458],{"className":8459,"style":689},[688],[608,8461,8463],{"className":8462},[693,694,695,696],[608,8464,782],{"className":8465},[658,696],[608,8467],{"className":8468,"style":967},[966],[608,8470,4976],{"className":8471},[971],[608,8473],{"className":8474,"style":967},[966],[608,8476,8478,8481],{"className":8477},[649],[608,8479],{"className":8480,"style":654},[653],[608,8482,8484,8487],{"className":8483},[658],[608,8485,782],{"className":8486},[658],[608,8488,8490],{"className":8489},[669],[608,8491,8493],{"className":8492},[673],[608,8494,8496],{"className":8495},[677],[608,8497,8499],{"className":8498,"style":654},[681],[608,8500,8501,8504],{"style":684},[608,8502],{"className":8503,"style":689},[688],[608,8505,8507],{"className":8506},[693,694,695,696],[608,8508,8510,8513],{"className":8509},[658,696],[608,8511,782],{"className":8512},[658,696],[608,8514,1162],{"className":8515},[658,894,696],", donc il faut prévoir ",[608,8518,8520,8536],{"className":8519},[611],[608,8521,8523],{"className":8522},[615],[617,8524,8525],{"xmlns":619},[621,8526,8527,8533],{},[624,8528,8529,8531],{},[630,8530,782],{},[876,8532,1162],{},[637,8534,8535],{"encoding":639},"2n",[608,8537,8539],{"className":8538,"ariaHidden":645},[644],[608,8540,8542,8545,8548],{"className":8541},[649],[608,8543],{"className":8544,"style":981},[653],[608,8546,782],{"className":8547},[658],[608,8549,1162],{"className":8550},[658,894],[7211,8552,8553],{},[599,8554,8555],{},"En pratique, sur un processeur 64 bits, multiplier deux entiers de 32 bits\ntient encore largement dans 64 bits, donc sans débordement.",[594,8557,8559],{"id":8558},"le-cas-python-entiers-de-taille-arbitraire","Le cas Python — entiers de taille arbitraire",[599,8561,8562,8563,8565],{},"Python (à partir de la version 3) ne limite ",[603,8564,5842],{}," la taille des entiers : ils\npeuvent croître autant que la mémoire le permet.",[2878,8567,8569],{"className":2880,"code":8568,"language":2882,"meta":2883,"style":2883},"print(2 ** 100)\n# 1267650600228229401496703205376\n\nprint(-1 \u003C\u003C 8)\n# -256\n\nprint(bin(-5))\n# -0b101  — Python affiche le signe, pas le complément à deux\n",[1894,8570,8571,8587,8592,8597,8614,8619,8623,8640],{"__ignoreMap":2883},[608,8572,8573,8575,8577,8579,8582,8585],{"class":2888,"line":2889},[608,8574,2893],{"class":2892},[608,8576,1151],{"class":2896},[608,8578,782],{"class":2892},[608,8580,8581],{"class":3722}," **",[608,8583,8584],{"class":2892}," 100",[608,8586,3761],{"class":2896},[608,8588,8589],{"class":2888,"line":2919},[608,8590,8591],{"class":2915},"# 1267650600228229401496703205376\n",[608,8593,8594],{"class":2888,"line":2943},[608,8595,8596],{"emptyLinePlaceholder":5639},"\n",[608,8598,8599,8601,8603,8605,8607,8609,8612],{"class":2888,"line":3794},[608,8600,2893],{"class":2892},[608,8602,1151],{"class":2896},[608,8604,5327],{"class":3722},[608,8606,659],{"class":2892},[608,8608,7378],{"class":3722},[608,8610,8611],{"class":2892}," 8",[608,8613,3761],{"class":2896},[608,8615,8616],{"class":2888,"line":7353},[608,8617,8618],{"class":2915},"# -256\n",[608,8620,8621],{"class":2888,"line":7359},[608,8622,8596],{"emptyLinePlaceholder":5639},[608,8624,8625,8627,8629,8631,8633,8635,8637],{"class":2888,"line":7395},[608,8626,2893],{"class":2892},[608,8628,1151],{"class":2896},[608,8630,5302],{"class":2892},[608,8632,1151],{"class":2896},[608,8634,5327],{"class":3722},[608,8636,3970],{"class":2892},[608,8638,8639],{"class":2896},"))\n",[608,8641,8643],{"class":2888,"line":8642},8,[608,8644,8645],{"class":2915},"# -0b101  — Python affiche le signe, pas le complément à deux\n",[599,8647,8648],{},"Cette commodité a un coût : un entier Python n'a pas une taille fixe en\nmémoire, donc les opérations sont un peu plus lentes que sur un entier 32 ou\n64 bits d'un langage bas niveau comme C.",[2966,8650,8652,8691],{"correct":2413,"slug":8651},"entiers-relatifs-c2",[2970,8653,8654],{"v-slot:question":2883},[599,8655,2974,8656,8690],{},[608,8657,8659,8675],{"className":8658},[611],[608,8660,8662],{"className":8661},[615],[617,8663,8664],{"xmlns":619},[621,8665,8666,8672],{},[624,8667,8668,8670],{},[942,8669,1060],{},[630,8671,659],{},[637,8673,8674],{"encoding":639},"-1",[608,8676,8678],{"className":8677,"ariaHidden":645},[644],[608,8679,8681,8684,8687],{"className":8680},[649],[608,8682],{"className":8683,"style":2794},[653],[608,8685,1060],{"className":8686},[658],[608,8688,659],{"className":8689},[658]," en complément à deux sur 8 bits ?",[2970,8692,8693],{"v-slot:choices":2883},[3060,8694,8695,8698,8701,8704],{},[3063,8696,8697],{},"0000 0001",[3063,8699,8700],{},"1000 0001",[3063,8702,8703],{},"1111 1110",[3063,8705,8706],{},"1111 1111",[594,8708,5340],{"id":5339},[5342,8710,8711],{},[3060,8712,8713,8722,8729],{},[3063,8714,6190,8715,8718,8719,8721],{},[603,8716,8717],{},"bit de poids fort vaut 1"," pour tout négatif, mais ce n'est ",[603,8720,5842],{},"\nl'opposé en valeur absolue : il faut vraiment refaire « inverser + 1 ».",[3063,8723,8724,8725,8728],{},"Sur un type entier de taille fixe (C, Java), additionner deux grands\npositifs peut donner un négatif à cause du ",[603,8726,8727],{},"débordement"," (overflow). Python\nvous protège de cela.",[3063,8730,8731,8732,2097],{},"Pour passer de la représentation interne (positive) au nombre signé, on\nutilise toujours la convention : si le bit de poids fort est 1, on soustrait\n",[608,8733,8735,8752],{"className":8734},[611],[608,8736,8738],{"className":8737},[615],[617,8739,8740],{"xmlns":619},[621,8741,8742,8750],{},[624,8743,8744],{},[627,8745,8746,8748],{},[630,8747,782],{},[876,8749,1162],{},[637,8751,4399],{"encoding":639},[608,8753,8755],{"className":8754,"ariaHidden":645},[644],[608,8756,8758,8761],{"className":8757},[649],[608,8759],{"className":8760,"style":4409},[653],[608,8762,8764,8767],{"className":8763},[658],[608,8765,782],{"className":8766},[658],[608,8768,8770],{"className":8769},[669],[608,8771,8773],{"className":8772},[673],[608,8774,8776],{"className":8775},[677],[608,8777,8779],{"className":8778,"style":4409},[681],[608,8780,8781,8784],{"style":684},[608,8782],{"className":8783,"style":689},[688],[608,8785,8787],{"className":8786},[693,694,695,696],[608,8788,1162],{"className":8789},[658,894,696],[594,8791,5549],{"id":5548},[599,8793,8794,8795,8798],{},"Le complément à deux explique aussi pourquoi le décalage à droite d'un nombre\nnégatif est subtil : il faut ",[603,8796,8797],{},"propager le bit de signe"," pour préserver la\nsémantique de division entière. Vous croiserez cette opération en bas niveau\n(C, assembleur) ou en manipulation de bits.",[5621,8800,8801],{},"html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":8803},[8804,8805,8806,8807,8809,8810,8811,8812,8813],{"id":596,"depth":2919,"text":597},{"id":5715,"depth":2919,"text":5716},{"id":5846,"depth":2919,"text":5847},{"id":6205,"depth":2919,"text":8808},"Pourquoi ça marche — l'addition modulo 2n2^n2n",{"id":7453,"depth":2919,"text":7454},{"id":8204,"depth":2919,"text":8205},{"id":8558,"depth":2919,"text":8559},{"id":5339,"depth":2919,"text":5340},{"id":5548,"depth":2919,"text":5549},"Comment représenter un entier négatif en binaire et pourquoi le complément à deux s'est imposé.",{},[8817,8819],{"slug":8184,"kind":5642,"question":8189,"choices":8818,"correct":2413,"multi":5645,"explanation":2883},[7756,7719,4789,4713],{"slug":8651,"kind":5642,"question":8820,"choices":8821,"correct":2413,"multi":5645,"explanation":2883},"Que vaut −1-1−1 en complément à deux sur 8 bits ?",[8697,8700,8703,8706],{"title":267,"description":8814},"hrB49T9jmSbyZSG7hdTQsZO1Znz_hHVLK_SUDFpArlc",{"id":8825,"title":271,"bo":8826,"body":8828,"description":11691,"eval":5636,"extension":5637,"labo":5636,"meta":11692,"navigation":5639,"path":272,"quizzes":11693,"readingTime":2889,"seo":11699,"stem":273,"tp":5636,"__hash__":11700},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.md",[8827],"PB02",{"type":591,"value":8829,"toc":11678},[8830,8832,8999,9013,9017,9020,9211,9214,9373,9380,9558,9648,9652,9659,9789,9800,9925,9934,9978,10187,10379,10460,10612,10616,10665,10749,10817,10820,10826,10857,10957,10961,11003,11076,11083,11116,11129,11142,11145,11301,11371,11400,11420,11422,11496,11498,11501,11621,11669,11675],[594,8831,597],{"id":596},[599,8833,8834,8835,8874,8875,8984,8985,8988,8989,8992,8993,8996,8997],{},"Les entiers, c'est bien — mais comment représente-t-on ",[608,8836,8838,8853],{"className":8837},[611],[608,8839,8841],{"className":8840},[615],[617,8842,8843],{"xmlns":619},[621,8844,8845,8850],{},[624,8846,8847],{},[630,8848,8849],{},"3,14",[637,8851,8852],{"encoding":639},"3{,}14",[608,8854,8856],{"className":8855,"ariaHidden":645},[644],[608,8857,8859,8862,8865,8871],{"className":8858},[649],[608,8860],{"className":8861,"style":5047},[653],[608,8863,2413],{"className":8864},[658],[608,8866,8868],{"className":8867},[658],[608,8869,5058],{"className":8870},[5057],[608,8872,3070],{"className":8873},[658]," ou\n",[608,8876,8878,8897],{"className":8877},[611],[608,8879,8881],{"className":8880},[615],[617,8882,8883],{"xmlns":619},[621,8884,8885,8894],{},[624,8886,8887],{},[8888,8889,8890,8892],"mfrac",{},[630,8891,659],{},[630,8893,2413],{},[637,8895,8896],{"encoding":639},"\\frac{1}{3}",[608,8898,8900],{"className":8899,"ariaHidden":645},[644],[608,8901,8903,8907],{"className":8902},[649],[608,8904],{"className":8905,"style":8906},[653],"height:1.1901em;vertical-align:-0.345em;",[608,8908,8910,8914,8981],{"className":8909},[658],[608,8911],{"className":8912},[1217,8913],"nulldelimiter",[608,8915,8917],{"className":8916},[8888],[608,8918,8920,8972],{"className":8919},[673,1230],[608,8921,8923,8969],{"className":8922},[677],[608,8924,8927,8943,8954],{"className":8925,"style":8926},[681],"height:0.8451em;",[608,8928,8930,8934],{"style":8929},"top:-2.655em;",[608,8931],{"className":8932,"style":8933},[688],"height:3em;",[608,8935,8937],{"className":8936},[693,694,695,696],[608,8938,8940],{"className":8939},[658,696],[608,8941,2413],{"className":8942},[658,696],[608,8944,8946,8949],{"style":8945},"top:-3.23em;",[608,8947],{"className":8948,"style":8933},[688],[608,8950],{"className":8951,"style":8953},[8952],"frac-line","border-bottom-width:0.04em;",[608,8955,8957,8960],{"style":8956},"top:-3.394em;",[608,8958],{"className":8959,"style":8933},[688],[608,8961,8963],{"className":8962},[693,694,695,696],[608,8964,8966],{"className":8965},[658,696],[608,8967,659],{"className":8968},[658,696],[608,8970,1263],{"className":8971},[1262],[608,8973,8975],{"className":8974},[677],[608,8976,8979],{"className":8977,"style":8978},[681],"height:0.345em;",[608,8980],{},[608,8982],{"className":8983},[1417,8913]," ? Les nombres réels posent un problème fondamental : il en existe\nune ",[603,8986,8987],{},"infinité"," entre deux entiers consécutifs, alors qu'une machine ne\ndispose que d'un nombre fini de bits. Il faudra donc ",[603,8990,8991],{},"approximer",". C'est ce\nque fait la représentation en ",[603,8994,8995],{},"virgule flottante",", normalisée par le standard\nIEEE 754 et utilisée par tous les processeurs modernes. ",[857,8998],{"code":8827},[599,9000,9001,9002,9005,9006,9008,9009,9012],{},"Cette approximation a un effet déroutant : en Python, ",[1894,9003,9004],{},"0.1 + 0.2"," ne vaut\n",[603,9007,5842],{}," exactement ",[1894,9010,9011],{},"0.3",". Comprendre pourquoi vous évitera bien des bugs.",[594,9014,9016],{"id":9015},"lanalogie-avec-la-notation-scientifique","L'analogie avec la notation scientifique",[599,9018,9019],{},"En mathématiques, on note souvent un grand ou un petit nombre en notation\nscientifique :",[608,9021,9023],{"className":9022},[1596],[608,9024,9026,9069],{"className":9025},[611],[608,9027,9029],{"className":9028},[615],[617,9030,9031],{"xmlns":619,"display":1605},[621,9032,9033,9066],{},[624,9034,9035,9038,9040,9047,9050,9053,9055],{},[630,9036,9037],{},"6,022",[942,9039,1646],{},[627,9041,9042,9044],{},[630,9043,632],{},[630,9045,9046],{},"23",[966,9048],{"width":9049},"2em",[630,9051,9052],{},"1,6",[942,9054,1646],{},[627,9056,9057,9059],{},[630,9058,632],{},[624,9060,9061,9063],{},[942,9062,1060],{},[630,9064,9065],{},"19",[637,9067,9068],{"encoding":639},"6{,}022 \\cdot 10^{23} \\qquad 1{,}6 \\cdot 10^{-19}",[608,9070,9072,9100,9167],{"className":9071,"ariaHidden":645},[644],[608,9073,9075,9078,9081,9087,9091,9094,9097],{"className":9074},[649],[608,9076],{"className":9077,"style":5047},[653],[608,9079,3208],{"className":9080},[658],[608,9082,9084],{"className":9083},[658],[608,9085,5058],{"className":9086},[5057],[608,9088,9090],{"className":9089},[658],"022",[608,9092],{"className":9093,"style":1082},[966],[608,9095,1646],{"className":9096},[1086],[608,9098],{"className":9099,"style":1082},[966],[608,9101,9103,9107,9110,9142,9146,9149,9155,9158,9161,9164],{"className":9102},[649],[608,9104],{"className":9105,"style":9106},[653],"height:1.0585em;vertical-align:-0.1944em;",[608,9108,659],{"className":9109},[658],[608,9111,9113,9116],{"className":9112},[658],[608,9114,635],{"className":9115},[658],[608,9117,9119],{"className":9118},[669],[608,9120,9122],{"className":9121},[673],[608,9123,9125],{"className":9124},[677],[608,9126,9128],{"className":9127,"style":2581},[681],[608,9129,9130,9133],{"style":1848},[608,9131],{"className":9132,"style":689},[688],[608,9134,9136],{"className":9135},[693,694,695,696],[608,9137,9139],{"className":9138},[658,696],[608,9140,9046],{"className":9141},[658,696],[608,9143],{"className":9144,"style":9145},[966],"margin-right:2em;",[608,9147,659],{"className":9148},[658],[608,9150,9152],{"className":9151},[658],[608,9153,5058],{"className":9154},[5057],[608,9156,3208],{"className":9157},[658],[608,9159],{"className":9160,"style":1082},[966],[608,9162,1646],{"className":9163},[1086],[608,9165],{"className":9166,"style":1082},[966],[608,9168,9170,9173,9176],{"className":9169},[649],[608,9171],{"className":9172,"style":2581},[653],[608,9174,659],{"className":9175},[658],[608,9177,9179,9182],{"className":9178},[658],[608,9180,635],{"className":9181},[658],[608,9183,9185],{"className":9184},[669],[608,9186,9188],{"className":9187},[673],[608,9189,9191],{"className":9190},[677],[608,9192,9194],{"className":9193,"style":2581},[681],[608,9195,9196,9199],{"style":1848},[608,9197],{"className":9198,"style":689},[688],[608,9200,9202],{"className":9201},[693,694,695,696],[608,9203,9205,9208],{"className":9204},[658,696],[608,9206,1060],{"className":9207},[658,696],[608,9209,9065],{"className":9210},[658,696],[599,9212,9213],{},"Trois éléments structurent cette écriture :",[3060,9215,9216,9300,9305],{},[3063,9217,9218,9219,9222,9223,9261,9262,5058],{},"une ",[603,9220,9221],{},"mantisse"," : ici ",[608,9224,9226,9240],{"className":9225},[611],[608,9227,9229],{"className":9228},[615],[617,9230,9231],{"xmlns":619},[621,9232,9233,9237],{},[624,9234,9235],{},[630,9236,9037],{},[637,9238,9239],{"encoding":639},"6{,}022",[608,9241,9243],{"className":9242,"ariaHidden":645},[644],[608,9244,9246,9249,9252,9258],{"className":9245},[649],[608,9247],{"className":9248,"style":5047},[653],[608,9250,3208],{"className":9251},[658],[608,9253,9255],{"className":9254},[658],[608,9256,5058],{"className":9257},[5057],[608,9259,9090],{"className":9260},[658]," ou ",[608,9263,9265,9279],{"className":9264},[611],[608,9266,9268],{"className":9267},[615],[617,9269,9270],{"xmlns":619},[621,9271,9272,9276],{},[624,9273,9274],{},[630,9275,9052],{},[637,9277,9278],{"encoding":639},"1{,}6",[608,9280,9282],{"className":9281,"ariaHidden":645},[644],[608,9283,9285,9288,9291,9297],{"className":9284},[649],[608,9286],{"className":9287,"style":5047},[653],[608,9289,659],{"className":9290},[658],[608,9292,9294],{"className":9293},[658],[608,9295,5058],{"className":9296},[5057],[608,9298,3208],{"className":9299},[658],[3063,9301,9218,9302,9304],{},[603,9303,649],{}," : ici 10,",[3063,9306,9307,9308,9222,9311,9261,9339,2097],{},"un ",[603,9309,9310],{},"exposant",[608,9312,9314,9327],{"className":9313},[611],[608,9315,9317],{"className":9316},[615],[617,9318,9319],{"xmlns":619},[621,9320,9321,9325],{},[624,9322,9323],{},[630,9324,9046],{},[637,9326,9046],{"encoding":639},[608,9328,9330],{"className":9329,"ariaHidden":645},[644],[608,9331,9333,9336],{"className":9332},[649],[608,9334],{"className":9335,"style":981},[653],[608,9337,9046],{"className":9338},[658],[608,9340,9342,9358],{"className":9341},[611],[608,9343,9345],{"className":9344},[615],[617,9346,9347],{"xmlns":619},[621,9348,9349,9355],{},[624,9350,9351,9353],{},[942,9352,1060],{},[630,9354,9065],{},[637,9356,9357],{"encoding":639},"-19",[608,9359,9361],{"className":9360,"ariaHidden":645},[644],[608,9362,9364,9367,9370],{"className":9363},[649],[608,9365],{"className":9366,"style":2794},[653],[608,9368,1060],{"className":9369},[658],[608,9371,9065],{"className":9372},[658],[599,9374,9375,9376,9379],{},"La virgule flottante reprend exactement cette idée, ",[603,9377,9378],{},"en base 2",". Un flottant\ns'écrit alors comme :",[608,9381,9383],{"className":9382},[1596],[608,9384,9386,9429],{"className":9385},[611],[608,9387,9389],{"className":9388},[615],[617,9390,9391],{"xmlns":619,"display":1605},[621,9392,9393,9426],{},[624,9394,9395,9397,9399,9401,9403,9405,9412,9414,9417,9419],{},[876,9396,5898],{},[942,9398,1615],{},[942,9400,1151],{"stretchy":1150},[942,9402,1060],{},[630,9404,659],{},[627,9406,9407,9409],{},[942,9408,1198],{"stretchy":1150},[876,9410,9411],{},"s",[942,9413,1646],{},[876,9415,9416],{},"m",[942,9418,1646],{},[627,9420,9421,9423],{},[630,9422,782],{},[876,9424,9425],{},"e",[637,9427,9428],{"encoding":639},"x = (-1)^s \\cdot m \\cdot 2^e",[608,9430,9432,9450,9504,9523],{"className":9431,"ariaHidden":645},[644],[608,9433,9435,9438,9441,9444,9447],{"className":9434},[649],[608,9436],{"className":9437,"style":4374},[653],[608,9439,5898],{"className":9440},[658,894],[608,9442],{"className":9443,"style":967},[966],[608,9445,1615],{"className":9446},[971],[608,9448],{"className":9449,"style":967},[966],[608,9451,9453,9456,9459,9462,9465,9495,9498,9501],{"className":9452},[649],[608,9454],{"className":9455,"style":1213},[653],[608,9457,1151],{"className":9458},[1217],[608,9460,1060],{"className":9461},[658],[608,9463,659],{"className":9464},[658],[608,9466,9468,9471],{"className":9467},[1417],[608,9469,1198],{"className":9470},[1417],[608,9472,9474],{"className":9473},[669],[608,9475,9477],{"className":9476},[673],[608,9478,9480],{"className":9479},[677],[608,9481,9484],{"className":9482,"style":9483},[681],"height:0.7144em;",[608,9485,9486,9489],{"style":1848},[608,9487],{"className":9488,"style":689},[688],[608,9490,9492],{"className":9491},[693,694,695,696],[608,9493,9411],{"className":9494},[658,894,696],[608,9496],{"className":9497,"style":1082},[966],[608,9499,1646],{"className":9500},[1086],[608,9502],{"className":9503,"style":1082},[966],[608,9505,9507,9511,9514,9517,9520],{"className":9506},[649],[608,9508],{"className":9509,"style":9510},[653],"height:0.4445em;",[608,9512,9416],{"className":9513},[658,894],[608,9515],{"className":9516,"style":1082},[966],[608,9518,1646],{"className":9519},[1086],[608,9521],{"className":9522,"style":1082},[966],[608,9524,9526,9529],{"className":9525},[649],[608,9527],{"className":9528,"style":9483},[653],[608,9530,9532,9535],{"className":9531},[658],[608,9533,782],{"className":9534},[658],[608,9536,9538],{"className":9537},[669],[608,9539,9541],{"className":9540},[673],[608,9542,9544],{"className":9543},[677],[608,9545,9547],{"className":9546,"style":9483},[681],[608,9548,9549,9552],{"style":1848},[608,9550],{"className":9551,"style":689},[688],[608,9553,9555],{"className":9554},[693,694,695,696],[608,9556,9425],{"className":9557},[658,894,696],[599,9559,9560,9561,9589,9590,9618,9619,9647],{},"où ",[608,9562,9564,9577],{"className":9563},[611],[608,9565,9567],{"className":9566},[615],[617,9568,9569],{"xmlns":619},[621,9570,9571,9575],{},[624,9572,9573],{},[876,9574,9411],{},[637,9576,9411],{"encoding":639},[608,9578,9580],{"className":9579,"ariaHidden":645},[644],[608,9581,9583,9586],{"className":9582},[649],[608,9584],{"className":9585,"style":4374},[653],[608,9587,9411],{"className":9588},[658,894]," est le bit de signe, ",[608,9591,9593,9606],{"className":9592},[611],[608,9594,9596],{"className":9595},[615],[617,9597,9598],{"xmlns":619},[621,9599,9600,9604],{},[624,9601,9602],{},[876,9603,9416],{},[637,9605,9416],{"encoding":639},[608,9607,9609],{"className":9608,"ariaHidden":645},[644],[608,9610,9612,9615],{"className":9611},[649],[608,9613],{"className":9614,"style":4374},[653],[608,9616,9416],{"className":9617},[658,894]," la mantisse (en binaire, entre 1 et 2 environ),\net ",[608,9620,9622,9635],{"className":9621},[611],[608,9623,9625],{"className":9624},[615],[617,9626,9627],{"xmlns":619},[621,9628,9629,9633],{},[624,9630,9631],{},[876,9632,9425],{},[637,9634,9425],{"encoding":639},[608,9636,9638],{"className":9637,"ariaHidden":645},[644],[608,9639,9641,9644],{"className":9640},[649],[608,9642],{"className":9643,"style":4374},[653],[608,9645,9425],{"className":9646},[658,894]," l'exposant (un entier signé).",[594,9649,9651],{"id":9650},"la-norme-ieee-754-vue-simplifiée","La norme IEEE 754, vue simplifiée",[599,9653,9654,9655,9658],{},"La norme IEEE 754 fixe la répartition des bits. En ",[603,9656,9657],{},"simple précision"," (32\nbits) :",[1859,9660,9661,9673],{},[1862,9662,9663],{},[1865,9664,9665,9668,9670],{},[1868,9666,9667],{},"Champ",[1868,9669,4558],{},[1868,9671,9672],{},"Rôle",[1881,9674,9675,9713,9751],{},[1865,9676,9677,9708,9710],{},[1886,9678,9679,9680],{},"Signe ",[608,9681,9683,9696],{"className":9682},[611],[608,9684,9686],{"className":9685},[615],[617,9687,9688],{"xmlns":619},[621,9689,9690,9694],{},[624,9691,9692],{},[876,9693,9411],{},[637,9695,9411],{"encoding":639},[608,9697,9699],{"className":9698,"ariaHidden":645},[644],[608,9700,9702,9705],{"className":9701},[649],[608,9703],{"className":9704,"style":4374},[653],[608,9706,9411],{"className":9707},[658,894],[1886,9709,659],{},[1886,9711,9712],{},"0 = positif, 1 = négatif",[1865,9714,9715,9746,9748],{},[1886,9716,9717,9718],{},"Exposant ",[608,9719,9721,9734],{"className":9720},[611],[608,9722,9724],{"className":9723},[615],[617,9725,9726],{"xmlns":619},[621,9727,9728,9732],{},[624,9729,9730],{},[876,9731,9425],{},[637,9733,9425],{"encoding":639},[608,9735,9737],{"className":9736,"ariaHidden":645},[644],[608,9738,9740,9743],{"className":9739},[649],[608,9741],{"className":9742,"style":4374},[653],[608,9744,9425],{"className":9745},[658,894],[1886,9747,1908],{},[1886,9749,9750],{},"Codé avec un biais de 127",[1865,9752,9753,9784,9786],{},[1886,9754,9755,9756],{},"Mantisse ",[608,9757,9759,9772],{"className":9758},[611],[608,9760,9762],{"className":9761},[615],[617,9763,9764],{"xmlns":619},[621,9765,9766,9770],{},[624,9767,9768],{},[876,9769,9416],{},[637,9771,9416],{"encoding":639},[608,9773,9775],{"className":9774,"ariaHidden":645},[644],[608,9776,9778,9781],{"className":9777},[649],[608,9779],{"className":9780,"style":4374},[653],[608,9782,9416],{"className":9783},[658,894],[1886,9785,9046],{},[1886,9787,9788],{},"Partie fractionnaire après le « 1, » implicite",[599,9790,9791,9792,9795,9796,9799],{},"En ",[603,9793,9794],{},"double précision"," (64 bits, le ",[1894,9797,9798],{},"float"," Python par défaut) :",[1859,9801,9802,9812],{},[1862,9803,9804],{},[1865,9805,9806,9808,9810],{},[1868,9807,9667],{},[1868,9809,4558],{},[1868,9811,9672],{},[1881,9813,9814,9850,9887],{},[1865,9815,9816,9846,9848],{},[1886,9817,9679,9818],{},[608,9819,9821,9834],{"className":9820},[611],[608,9822,9824],{"className":9823},[615],[617,9825,9826],{"xmlns":619},[621,9827,9828,9832],{},[624,9829,9830],{},[876,9831,9411],{},[637,9833,9411],{"encoding":639},[608,9835,9837],{"className":9836,"ariaHidden":645},[644],[608,9838,9840,9843],{"className":9839},[649],[608,9841],{"className":9842,"style":4374},[653],[608,9844,9411],{"className":9845},[658,894],[1886,9847,659],{},[1886,9849,9712],{},[1865,9851,9852,9882,9884],{},[1886,9853,9717,9854],{},[608,9855,9857,9870],{"className":9856},[611],[608,9858,9860],{"className":9859},[615],[617,9861,9862],{"xmlns":619},[621,9863,9864,9868],{},[624,9865,9866],{},[876,9867,9425],{},[637,9869,9425],{"encoding":639},[608,9871,9873],{"className":9872,"ariaHidden":645},[644],[608,9874,9876,9879],{"className":9875},[649],[608,9877],{"className":9878,"style":4374},[653],[608,9880,9425],{"className":9881},[658,894],[1886,9883,2051],{},[1886,9885,9886],{},"Biais de 1023",[1865,9888,9889,9919,9922],{},[1886,9890,9755,9891],{},[608,9892,9894,9907],{"className":9893},[611],[608,9895,9897],{"className":9896},[615],[617,9898,9899],{"xmlns":619},[621,9900,9901,9905],{},[624,9902,9903],{},[876,9904,9416],{},[637,9906,9416],{"encoding":639},[608,9908,9910],{"className":9909,"ariaHidden":645},[644],[608,9911,9913,9916],{"className":9912},[649],[608,9914],{"className":9915,"style":4374},[653],[608,9917,9416],{"className":9918},[658,894],[1886,9920,9921],{},"52",[1886,9923,9924],{},"Partie fractionnaire",[2099,9926,9927],{},[599,9928,9929,9930,9933],{},"Le programme officiel ne vous demande ",[603,9931,9932],{},"aucune connaissance précise"," des bits\nde la norme IEEE 754. Ce qui compte, c'est de retenir le principe : signe,\nmantisse, exposant — et les conséquences sur la précision.",[594,9935,9937,9938,9977],{"id":9936},"pourquoi-010101-pose-problème","Pourquoi ",[608,9939,9941,9956],{"className":9940},[611],[608,9942,9944],{"className":9943},[615],[617,9945,9946],{"xmlns":619},[621,9947,9948,9953],{},[624,9949,9950],{},[630,9951,9952],{},"0,1",[637,9954,9955],{"encoding":639},"0{,}1",[608,9957,9959],{"className":9958,"ariaHidden":645},[644],[608,9960,9962,9965,9968,9974],{"className":9961},[649],[608,9963],{"className":9964,"style":5047},[653],[608,9966,635],{"className":9967},[658],[608,9969,9971],{"className":9970},[658],[608,9972,5058],{"className":9973},[5057],[608,9975,659],{"className":9976},[658]," pose problème",[599,9979,9980,9981,10079,10080,10082,10083,10085,10086,10183,10184,5279],{},"En base 10, ",[608,9982,9984,10002],{"className":9983},[611],[608,9985,9987],{"className":9986},[615],[617,9988,9989],{"xmlns":619},[621,9990,9991,9999],{},[624,9992,9993],{},[8888,9994,9995,9997],{},[630,9996,659],{},[630,9998,632],{},[637,10000,10001],{"encoding":639},"\\frac{1}{10}",[608,10003,10005],{"className":10004,"ariaHidden":645},[644],[608,10006,10008,10011],{"className":10007},[649],[608,10009],{"className":10010,"style":8906},[653],[608,10012,10014,10017,10076],{"className":10013},[658],[608,10015],{"className":10016},[1217,8913],[608,10018,10020],{"className":10019},[8888],[608,10021,10023,10068],{"className":10022},[673,1230],[608,10024,10026,10065],{"className":10025},[677],[608,10027,10029,10043,10051],{"className":10028,"style":8926},[681],[608,10030,10031,10034],{"style":8929},[608,10032],{"className":10033,"style":8933},[688],[608,10035,10037],{"className":10036},[693,694,695,696],[608,10038,10040],{"className":10039},[658,696],[608,10041,632],{"className":10042},[658,696],[608,10044,10045,10048],{"style":8945},[608,10046],{"className":10047,"style":8933},[688],[608,10049],{"className":10050,"style":8953},[8952],[608,10052,10053,10056],{"style":8956},[608,10054],{"className":10055,"style":8933},[688],[608,10057,10059],{"className":10058},[693,694,695,696],[608,10060,10062],{"className":10061},[658,696],[608,10063,659],{"className":10064},[658,696],[608,10066,1263],{"className":10067},[1262],[608,10069,10071],{"className":10070},[677],[608,10072,10074],{"className":10073,"style":8978},[681],[608,10075],{},[608,10077],{"className":10078},[1417,8913]," s'écrit simplement ",[1894,10081,9952],{},". Mais en ",[603,10084,837],{},",\n",[608,10087,10089,10106],{"className":10088},[611],[608,10090,10092],{"className":10091},[615],[617,10093,10094],{"xmlns":619},[621,10095,10096,10104],{},[624,10097,10098],{},[8888,10099,10100,10102],{},[630,10101,659],{},[630,10103,632],{},[637,10105,10001],{"encoding":639},[608,10107,10109],{"className":10108,"ariaHidden":645},[644],[608,10110,10112,10115],{"className":10111},[649],[608,10113],{"className":10114,"style":8906},[653],[608,10116,10118,10121,10180],{"className":10117},[658],[608,10119],{"className":10120},[1217,8913],[608,10122,10124],{"className":10123},[8888],[608,10125,10127,10172],{"className":10126},[673,1230],[608,10128,10130,10169],{"className":10129},[677],[608,10131,10133,10147,10155],{"className":10132,"style":8926},[681],[608,10134,10135,10138],{"style":8929},[608,10136],{"className":10137,"style":8933},[688],[608,10139,10141],{"className":10140},[693,694,695,696],[608,10142,10144],{"className":10143},[658,696],[608,10145,632],{"className":10146},[658,696],[608,10148,10149,10152],{"style":8945},[608,10150],{"className":10151,"style":8933},[688],[608,10153],{"className":10154,"style":8953},[8952],[608,10156,10157,10160],{"style":8956},[608,10158],{"className":10159,"style":8933},[688],[608,10161,10163],{"className":10162},[693,694,695,696],[608,10164,10166],{"className":10165},[658,696],[608,10167,659],{"className":10168},[658,696],[608,10170,1263],{"className":10171},[1262],[608,10173,10175],{"className":10174},[677],[608,10176,10178],{"className":10177,"style":8978},[681],[608,10179],{},[608,10181],{"className":10182},[1417,8913]," a un développement ",[603,10185,10186],{},"infini périodique",[608,10188,10190],{"className":10189},[1596],[608,10191,10193,10226],{"className":10192},[611],[608,10194,10196],{"className":10195},[615],[617,10197,10198],{"xmlns":619,"display":1605},[621,10199,10200,10223],{},[624,10201,10202,10208,10210,10212,10215,10217],{},[8888,10203,10204,10206],{},[630,10205,659],{},[630,10207,632],{},[942,10209,1615],{},[942,10211,1151],{"stretchy":1150},[630,10213,10214],{},"0,0001100110011",[942,10216,1181],{},[1153,10218,10219,10221],{},[942,10220,1198],{"stretchy":1150},[630,10222,782],{},[637,10224,10225],{"encoding":639},"\\frac{1}{10} = (0{,}0001100110011\\ldots)_2",[608,10227,10229,10311],{"className":10228,"ariaHidden":645},[644],[608,10230,10232,10236,10302,10305,10308],{"className":10231},[649],[608,10233],{"className":10234,"style":10235},[653],"height:2.0074em;vertical-align:-0.686em;",[608,10237,10239,10242,10299],{"className":10238},[658],[608,10240],{"className":10241},[1217,8913],[608,10243,10245],{"className":10244},[8888],[608,10246,10248,10290],{"className":10247},[673,1230],[608,10249,10251,10287],{"className":10250},[677],[608,10252,10255,10267,10275],{"className":10253,"style":10254},[681],"height:1.3214em;",[608,10256,10258,10261],{"style":10257},"top:-2.314em;",[608,10259],{"className":10260,"style":8933},[688],[608,10262,10264],{"className":10263},[658],[608,10265,632],{"className":10266},[658],[608,10268,10269,10272],{"style":8945},[608,10270],{"className":10271,"style":8933},[688],[608,10273],{"className":10274,"style":8953},[8952],[608,10276,10278,10281],{"style":10277},"top:-3.677em;",[608,10279],{"className":10280,"style":8933},[688],[608,10282,10284],{"className":10283},[658],[608,10285,659],{"className":10286},[658],[608,10288,1263],{"className":10289},[1262],[608,10291,10293],{"className":10292},[677],[608,10294,10297],{"className":10295,"style":10296},[681],"height:0.686em;",[608,10298],{},[608,10300],{"className":10301},[1417,8913],[608,10303],{"className":10304,"style":967},[966],[608,10306,1615],{"className":10307},[971],[608,10309],{"className":10310,"style":967},[966],[608,10312,10314,10317,10320,10323,10329,10333,10336,10339],{"className":10313},[649],[608,10315],{"className":10316,"style":1213},[653],[608,10318,1151],{"className":10319},[1217],[608,10321,635],{"className":10322},[658],[608,10324,10326],{"className":10325},[658],[608,10327,5058],{"className":10328},[5057],[608,10330,10332],{"className":10331},[658],"0001100110011",[608,10334],{"className":10335,"style":1325},[966],[608,10337,1181],{"className":10338},[1329],[608,10340,10342,10345],{"className":10341},[1417],[608,10343,1198],{"className":10344},[1417],[608,10346,10348],{"className":10347},[669],[608,10349,10351,10371],{"className":10350},[673,1230],[608,10352,10354,10368],{"className":10353},[677],[608,10355,10357],{"className":10356,"style":1237},[681],[608,10358,10359,10362],{"style":1240},[608,10360],{"className":10361,"style":689},[688],[608,10363,10365],{"className":10364},[693,694,695,696],[608,10366,782],{"className":10367},[658,696],[608,10369,1263],{"className":10370},[1262],[608,10372,10374],{"className":10373},[677],[608,10375,10377],{"className":10376,"style":1371},[681],[608,10378],{},[599,10380,10381,10382,7226,10385,10422,10423,2097],{},"Comme la mantisse a une taille bornée (52 bits en double précision), la machine\ntronque ce développement infini. Le résultat est une ",[603,10383,10384],{},"approximation très\nproche",[608,10386,10388,10401],{"className":10387},[611],[608,10389,10391],{"className":10390},[615],[617,10392,10393],{"xmlns":619},[621,10394,10395,10399],{},[624,10396,10397],{},[630,10398,9952],{},[637,10400,9955],{"encoding":639},[608,10402,10404],{"className":10403,"ariaHidden":645},[644],[608,10405,10407,10410,10413,10419],{"className":10406},[649],[608,10408],{"className":10409,"style":5047},[653],[608,10411,635],{"className":10412},[658],[608,10414,10416],{"className":10415},[658],[608,10417,5058],{"className":10418},[5057],[608,10420,659],{"className":10421},[658],", mais pas exactement ",[608,10424,10426,10439],{"className":10425},[611],[608,10427,10429],{"className":10428},[615],[617,10430,10431],{"xmlns":619},[621,10432,10433,10437],{},[624,10434,10435],{},[630,10436,9952],{},[637,10438,9955],{"encoding":639},[608,10440,10442],{"className":10441,"ariaHidden":645},[644],[608,10443,10445,10448,10451,10457],{"className":10444},[649],[608,10446],{"className":10447,"style":5047},[653],[608,10449,635],{"className":10450},[658],[608,10452,10454],{"className":10453},[658],[608,10455,5058],{"className":10456},[5057],[608,10458,659],{"className":10459},[658],[7211,10461,10462],{},[599,10463,10464,10465,10562,10563,10611],{},"La même chose se produit en base 10 avec ",[608,10466,10468,10485],{"className":10467},[611],[608,10469,10471],{"className":10470},[615],[617,10472,10473],{"xmlns":619},[621,10474,10475,10483],{},[624,10476,10477],{},[8888,10478,10479,10481],{},[630,10480,659],{},[630,10482,2413],{},[637,10484,8896],{"encoding":639},[608,10486,10488],{"className":10487,"ariaHidden":645},[644],[608,10489,10491,10494],{"className":10490},[649],[608,10492],{"className":10493,"style":8906},[653],[608,10495,10497,10500,10559],{"className":10496},[658],[608,10498],{"className":10499},[1217,8913],[608,10501,10503],{"className":10502},[8888],[608,10504,10506,10551],{"className":10505},[673,1230],[608,10507,10509,10548],{"className":10508},[677],[608,10510,10512,10526,10534],{"className":10511,"style":8926},[681],[608,10513,10514,10517],{"style":8929},[608,10515],{"className":10516,"style":8933},[688],[608,10518,10520],{"className":10519},[693,694,695,696],[608,10521,10523],{"className":10522},[658,696],[608,10524,2413],{"className":10525},[658,696],[608,10527,10528,10531],{"style":8945},[608,10529],{"className":10530,"style":8933},[688],[608,10532],{"className":10533,"style":8953},[8952],[608,10535,10536,10539],{"style":8956},[608,10537],{"className":10538,"style":8933},[688],[608,10540,10542],{"className":10541},[693,694,695,696],[608,10543,10545],{"className":10544},[658,696],[608,10546,659],{"className":10547},[658,696],[608,10549,1263],{"className":10550},[1262],[608,10552,10554],{"className":10553},[677],[608,10555,10557],{"className":10556,"style":8978},[681],[608,10558],{},[608,10560],{"className":10561},[1417,8913]," : son écriture\ndécimale ",[608,10564,10566,10583],{"className":10565},[611],[608,10567,10569],{"className":10568},[615],[617,10570,10571],{"xmlns":619},[621,10572,10573,10580],{},[624,10574,10575,10578],{},[630,10576,10577],{},"0,3333",[942,10579,1181],{},[637,10581,10582],{"encoding":639},"0{,}3333\\ldots",[608,10584,10586],{"className":10585,"ariaHidden":645},[644],[608,10587,10589,10592,10595,10601,10605,10608],{"className":10588},[649],[608,10590],{"className":10591,"style":5047},[653],[608,10593,635],{"className":10594},[658],[608,10596,10598],{"className":10597},[658],[608,10599,5058],{"className":10600},[5057],[608,10602,10604],{"className":10603},[658],"3333",[608,10606],{"className":10607,"style":1325},[966],[608,10609,1181],{"className":10610},[1329]," ne s'arrête jamais. Si on devait l'écrire avec un\nnombre fini de chiffres, on obtiendrait une approximation, jamais la valeur\nexacte. La base 2 a le même genre de souci, mais pour d'autres fractions —\nnotamment celles dont le dénominateur n'est pas une puissance de 2.",[594,10613,10615],{"id":10614},"la-preuve-par-python","La preuve par Python",[2878,10617,10619],{"className":2880,"code":10618,"language":2882,"meta":2883,"style":2883},"print(0.1 + 0.2)       # 0.30000000000000004\nprint(0.1 + 0.2 == 0.3)  # False\n",[1894,10620,10621,10642],{"__ignoreMap":2883},[608,10622,10623,10625,10627,10630,10633,10636,10639],{"class":2888,"line":2889},[608,10624,2893],{"class":2892},[608,10626,1151],{"class":2896},[608,10628,10629],{"class":2892},"0.1",[608,10631,10632],{"class":3722}," +",[608,10634,10635],{"class":2892}," 0.2",[608,10637,10638],{"class":2896},")       ",[608,10640,10641],{"class":2915},"# 0.30000000000000004\n",[608,10643,10644,10646,10648,10650,10652,10654,10657,10660,10662],{"class":2888,"line":2919},[608,10645,2893],{"class":2892},[608,10647,1151],{"class":2896},[608,10649,10629],{"class":2892},[608,10651,10632],{"class":3722},[608,10653,10635],{"class":2892},[608,10655,10656],{"class":3722}," ==",[608,10658,10659],{"class":2892}," 0.3",[608,10661,7389],{"class":2896},[608,10663,10664],{"class":2915},"# False\n",[599,10666,10667,10668,10741,10742,10745,10746,10748],{},"L'écart est minuscule — de l'ordre de ",[608,10669,10671,10694],{"className":10670},[611],[608,10672,10674],{"className":10673},[615],[617,10675,10676],{"xmlns":619},[621,10677,10678,10691],{},[624,10679,10680],{},[627,10681,10682,10684],{},[630,10683,632],{},[624,10685,10686,10688],{},[942,10687,1060],{},[630,10689,10690],{},"17",[637,10692,10693],{"encoding":639},"10^{-17}",[608,10695,10697],{"className":10696,"ariaHidden":645},[644],[608,10698,10700,10703,10706],{"className":10699},[649],[608,10701],{"className":10702,"style":654},[653],[608,10704,659],{"className":10705},[658],[608,10707,10709,10712],{"className":10708},[658],[608,10710,635],{"className":10711},[658],[608,10713,10715],{"className":10714},[669],[608,10716,10718],{"className":10717},[673],[608,10719,10721],{"className":10720},[677],[608,10722,10724],{"className":10723,"style":654},[681],[608,10725,10726,10729],{"style":684},[608,10727],{"className":10728,"style":689},[688],[608,10730,10732],{"className":10731},[693,694,695,696],[608,10733,10735,10738],{"className":10734},[658,696],[608,10736,1060],{"className":10737},[658,696],[608,10739,10690],{"className":10740},[658,696]," — mais il est ",[603,10743,10744],{},"non nul",", et\nil suffit à faire échouer un test d'égalité strict. Ce comportement n'est\n",[603,10747,5842],{}," un bug de Python : c'est la conséquence directe de la représentation\nbinaire des flottants.",[3708,10750,10751],{},[2878,10752,10754],{"className":2880,"code":10753,"language":2882,"meta":2883,"style":2883},"# Affichage très précis pour voir la \"vraie\" valeur stockée\nfrom decimal import Decimal\nprint(Decimal(0.1))\nprint(Decimal(0.2))\nprint(Decimal(0.1) + Decimal(0.2))\n",[1894,10755,10756,10761,10775,10786,10797],{"__ignoreMap":2883},[608,10757,10758],{"class":2888,"line":2889},[608,10759,10760],{"class":2915},"# Affichage très précis pour voir la \"vraie\" valeur stockée\n",[608,10762,10763,10766,10769,10772],{"class":2888,"line":2919},[608,10764,10765],{"class":3722},"from",[608,10767,10768],{"class":2896}," decimal ",[608,10770,10771],{"class":3722},"import",[608,10773,10774],{"class":2896}," Decimal\n",[608,10776,10777,10779,10782,10784],{"class":2888,"line":2943},[608,10778,2893],{"class":2892},[608,10780,10781],{"class":2896},"(Decimal(",[608,10783,10629],{"class":2892},[608,10785,8639],{"class":2896},[608,10787,10788,10790,10792,10795],{"class":2888,"line":3794},[608,10789,2893],{"class":2892},[608,10791,10781],{"class":2896},[608,10793,10794],{"class":2892},"0.2",[608,10796,8639],{"class":2896},[608,10798,10799,10801,10803,10805,10808,10810,10813,10815],{"class":2888,"line":7353},[608,10800,2893],{"class":2892},[608,10802,10781],{"class":2896},[608,10804,10629],{"class":2892},[608,10806,10807],{"class":2896},") ",[608,10809,2416],{"class":3722},[608,10811,10812],{"class":2896}," Decimal(",[608,10814,10794],{"class":2892},[608,10816,8639],{"class":2896},[599,10818,10819],{},"Sortie attendue (extrait) :",[2878,10821,10824],{"className":10822,"code":10823,"language":3831},[3829],"0.1000000000000000055511151231257827021181583404541015625\n0.200000000000000011102230246251565404236316680908203125\n0.3000000000000000166533453694\n",[1894,10825,10823],{"__ignoreMap":2883},[599,10827,10828,10829,10832,10833,2066,10835,10837,10838,10841,10842,10845,10846,10849,10850,10853,10854,10856],{},"Les deux premières lignes affichent la ",[603,10830,10831],{},"valeur binaire exacte"," des floats\n",[1894,10834,10629],{},[1894,10836,10794],{}," (le constructeur ",[1894,10839,10840],{},"Decimal(float)"," n'arrondit pas). La 3ᵉ ligne\ncorrespond à l'",[603,10843,10844],{},"addition"," : Python repasse alors par le contexte ",[1894,10847,10848],{},"Decimal","\nde précision par défaut, qui arrondit à ",[603,10851,10852],{},"28 chiffres significatifs",". La\nvaleur reste différente de ",[1894,10855,9011],{}," exact — ce qui confirme l'erreur de\nreprésentation binaire.",[2966,10858,10860,10941],{"correct":659,"slug":10859},"flottants-c1",[2970,10861,10862],{"v-slot:question":2883},[599,10863,10864,10865,4329],{},"Que renvoie en Python l'expression ",[608,10866,10868,10892],{"className":10867},[611],[608,10869,10871],{"className":10870},[615],[617,10872,10873],{"xmlns":619},[621,10874,10875,10889],{},[624,10876,10877,10879,10881,10883,10885,10887],{},[630,10878,10629],{},[942,10880,2416],{},[630,10882,10794],{},[942,10884,1615],{},[942,10886,1615],{},[630,10888,9011],{},[637,10890,10891],{"encoding":639},"0.1 + 0.2 == 0.3",[608,10893,10895,10913,10932],{"className":10894,"ariaHidden":645},[644],[608,10896,10898,10901,10904,10907,10910],{"className":10897},[649],[608,10899],{"className":10900,"style":2794},[653],[608,10902,10629],{"className":10903},[658],[608,10905],{"className":10906,"style":1082},[966],[608,10908,2416],{"className":10909},[1086],[608,10911],{"className":10912,"style":1082},[966],[608,10914,10916,10919,10922,10925,10929],{"className":10915},[649],[608,10917],{"className":10918,"style":981},[653],[608,10920,10794],{"className":10921},[658],[608,10923],{"className":10924,"style":967},[966],[608,10926,10928],{"className":10927},[971],"==",[608,10930],{"className":10931,"style":967},[966],[608,10933,10935,10938],{"className":10934},[649],[608,10936],{"className":10937,"style":981},[653],[608,10939,9011],{"className":10940},[658],[2970,10942,10943],{"v-slot:choices":2883},[3060,10944,10945,10948,10951,10954],{},[3063,10946,10947],{},"True",[3063,10949,10950],{},"False",[3063,10952,10953],{},"une erreur",[3063,10955,10956],{},"cela dépend",[594,10958,10960],{"id":10959},"comment-comparer-deux-flottants","Comment comparer deux flottants",[599,10962,10963,10964,10967,10968,10971,10972,11002],{},"La règle d'or est simple : ",[603,10965,10966],{},"on ne teste jamais l'égalité stricte de deux\nflottants",". À la place, on vérifie qu'ils sont ",[603,10969,10970],{},"suffisamment proches",", à une\ntolérance ",[608,10973,10975,10990],{"className":10974},[611],[608,10976,10978],{"className":10977},[615],[617,10979,10980],{"xmlns":619},[621,10981,10982,10987],{},[624,10983,10984],{},[876,10985,10986],{},"ε",[637,10988,10989],{"encoding":639},"\\varepsilon",[608,10991,10993],{"className":10992,"ariaHidden":645},[644],[608,10994,10996,10999],{"className":10995},[649],[608,10997],{"className":10998,"style":4374},[653],[608,11000,10986],{"className":11001},[658,894]," près.",[2878,11004,11006],{"className":2880,"code":11005,"language":2882,"meta":2883,"style":2883},"def proche(a, b, eps=1e-9):\n    return abs(a - b) \u003C eps\n\nprint(proche(0.1 + 0.2, 0.3))  # True\n",[1894,11007,11008,11028,11050,11054],{"__ignoreMap":2883},[608,11009,11010,11013,11017,11020,11022,11025],{"class":2888,"line":2889},[608,11011,11012],{"class":3722},"def",[608,11014,11016],{"class":11015},"snPdu"," proche",[608,11018,11019],{"class":2896},"(a, b, eps",[608,11021,1615],{"class":3722},[608,11023,11024],{"class":2892},"1e-9",[608,11026,11027],{"class":2896},"):\n",[608,11029,11030,11033,11036,11039,11041,11044,11047],{"class":2888,"line":2919},[608,11031,11032],{"class":3722},"    return",[608,11034,11035],{"class":2892}," abs",[608,11037,11038],{"class":2896},"(a ",[608,11040,5327],{"class":3722},[608,11042,11043],{"class":2896}," b) ",[608,11045,11046],{"class":3722},"\u003C",[608,11048,11049],{"class":2896}," eps\n",[608,11051,11052],{"class":2888,"line":2943},[608,11053,8596],{"emptyLinePlaceholder":5639},[608,11055,11056,11058,11061,11063,11065,11067,11069,11071,11073],{"class":2888,"line":3794},[608,11057,2893],{"class":2892},[608,11059,11060],{"class":2896},"(proche(",[608,11062,10629],{"class":2892},[608,11064,10632],{"class":3722},[608,11066,10635],{"class":2892},[608,11068,1896],{"class":2896},[608,11070,9011],{"class":2892},[608,11072,2912],{"class":2896},[608,11074,11075],{"class":2915},"# True\n",[599,11077,11078,11079,11082],{},"Python fournit aussi ",[1894,11080,11081],{},"math.isclose"," pour cette comparaison, plus robuste :",[2878,11084,11086],{"className":2880,"code":11085,"language":2882,"meta":2883,"style":2883},"import math\nprint(math.isclose(0.1 + 0.2, 0.3))  # True\n",[1894,11087,11088,11095],{"__ignoreMap":2883},[608,11089,11090,11092],{"class":2888,"line":2889},[608,11091,10771],{"class":3722},[608,11093,11094],{"class":2896}," math\n",[608,11096,11097,11099,11102,11104,11106,11108,11110,11112,11114],{"class":2888,"line":2919},[608,11098,2893],{"class":2892},[608,11100,11101],{"class":2896},"(math.isclose(",[608,11103,10629],{"class":2892},[608,11105,10632],{"class":3722},[608,11107,10635],{"class":2892},[608,11109,1896],{"class":2896},[608,11111,9011],{"class":2892},[608,11113,2912],{"class":2896},[608,11115,11075],{"class":2915},[5342,11117,11118],{},[599,11119,11120,11121,11124,11125,11128],{},"Cette précaution est ",[603,11122,11123],{},"systématique",". Que vous compariez le résultat d'un\ncalcul de physique, une distance, une fraction… utilisez toujours ",[1894,11126,11127],{},"isclose"," ou\nune tolérance explicite quand des flottants sont en jeu.",[594,11130,11132,11133,1896,11136,1896,11139],{"id":11131},"valeurs-spéciales-inf-inf-nan","Valeurs spéciales — ",[1894,11134,11135],{},"inf",[1894,11137,11138],{},"-inf",[1894,11140,11141],{},"nan",[599,11143,11144],{},"IEEE 754 prévoit quelques valeurs particulières :",[3060,11146,11147,11152,11157],{},[3063,11148,11149,11151],{},[1894,11150,11135],{}," (infini positif) : résultat d'un dépassement ou d'une division par\nzéro positive.",[3063,11153,11154,11156],{},[1894,11155,11138],{}," (infini négatif).",[3063,11158,11159,842,11161,11164,11165,8874,11200,2097],{},[1894,11160,11141],{},[844,11162,11163],{},"Not a Number",") : résultat d'une opération invalide, comme ",[608,11166,11168,11188],{"className":11167},[611],[608,11169,11171],{"className":11170},[615],[617,11172,11173],{"xmlns":619},[621,11174,11175,11185],{},[624,11176,11177,11179,11183],{},[630,11178,635],{},[876,11180,11182],{"mathvariant":11181},"normal","\u002F",[630,11184,635],{},[637,11186,11187],{"encoding":639},"0\u002F0",[608,11189,11191],{"className":11190,"ariaHidden":645},[644],[608,11192,11194,11197],{"className":11193},[649],[608,11195],{"className":11196,"style":1213},[653],[608,11198,11187],{"className":11199},[658],[608,11201,11203,11224],{"className":11202},[611],[608,11204,11206],{"className":11205},[615],[617,11207,11208],{"xmlns":619},[621,11209,11210,11221],{},[624,11211,11212],{},[11213,11214,11215],"msqrt",{},[624,11216,11217,11219],{},[942,11218,1060],{},[630,11220,659],{},[637,11222,11223],{"encoding":639},"\\sqrt{-1}",[608,11225,11227],{"className":11226,"ariaHidden":645},[644],[608,11228,11230,11234],{"className":11229},[649],[608,11231],{"className":11232,"style":11233},[653],"height:1.04em;vertical-align:-0.1744em;",[608,11235,11238],{"className":11236},[658,11237],"sqrt",[608,11239,11241,11292],{"className":11240},[673,1230],[608,11242,11244,11289],{"className":11243},[677],[608,11245,11248,11266],{"className":11246,"style":11247},[681],"height:0.8656em;",[608,11249,11253,11256],{"className":11250,"style":11252},[11251],"svg-align","top:-3em;",[608,11254],{"className":11255,"style":8933},[688],[608,11257,11260,11263],{"className":11258,"style":11259},[658],"padding-left:0.833em;",[608,11261,1060],{"className":11262},[658],[608,11264,659],{"className":11265},[658],[608,11267,11269,11272],{"style":11268},"top:-2.8256em;",[608,11270],{"className":11271,"style":8933},[688],[608,11273,11277],{"className":11274,"style":11276},[11275],"hide-tail","min-width:0.853em;height:1.08em;",[11278,11279,11285],"svg",{"xmlns":11280,"width":11281,"height":11282,"viewBox":11283,"preserveAspectRatio":11284},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[11286,11287],"path",{"d":11288},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[608,11290,1263],{"className":11291},[1262],[608,11293,11295],{"className":11294},[677],[608,11296,11299],{"className":11297,"style":11298},[681],"height:0.1744em;",[608,11300],{},[2878,11302,11304],{"className":2880,"code":11303,"language":2882,"meta":2883,"style":2883},"import math\nprint(math.inf)              # inf — constante du module math\nprint(1 \u002F 0.0)               # ZeroDivisionError en Python pur (et non inf)\nprint(float('nan') == float('nan'))  # False — un nan n'est jamais égal à lui-même !\n",[1894,11305,11306,11312,11322,11342],{"__ignoreMap":2883},[608,11307,11308,11310],{"class":2888,"line":2889},[608,11309,10771],{"class":3722},[608,11311,11094],{"class":2896},[608,11313,11314,11316,11319],{"class":2888,"line":2919},[608,11315,2893],{"class":2892},[608,11317,11318],{"class":2896},"(math.inf)              ",[608,11320,11321],{"class":2915},"# inf — constante du module math\n",[608,11323,11324,11326,11328,11330,11333,11336,11339],{"class":2888,"line":2943},[608,11325,2893],{"class":2892},[608,11327,1151],{"class":2896},[608,11329,659],{"class":2892},[608,11331,11332],{"class":3722}," \u002F",[608,11334,11335],{"class":2892}," 0.0",[608,11337,11338],{"class":2896},")               ",[608,11340,11341],{"class":2915},"# ZeroDivisionError en Python pur (et non inf)\n",[608,11343,11344,11346,11348,11350,11352,11355,11357,11359,11362,11364,11366,11368],{"class":2888,"line":3794},[608,11345,2893],{"class":2892},[608,11347,1151],{"class":2896},[608,11349,9798],{"class":2892},[608,11351,1151],{"class":2896},[608,11353,11354],{"class":2904},"'nan'",[608,11356,10807],{"class":2896},[608,11358,10928],{"class":3722},[608,11360,11361],{"class":2892}," float",[608,11363,1151],{"class":2896},[608,11365,11354],{"class":2904},[608,11367,2912],{"class":2896},[608,11369,11370],{"class":2915},"# False — un nan n'est jamais égal à lui-même !\n",[2099,11372,11373],{},[599,11374,11375,11376,11379,11380,11386,11387,11389,11390,11393,11394,11396,11397,11399],{},"En Python pur, ",[1894,11377,11378],{},"1 \u002F 0.0"," ",[603,11381,11382,11383],{},"lève une ",[1894,11384,11385],{},"ZeroDivisionError"," — l'infini ne sort\npas naturellement de la division. Pour obtenir ",[1894,11388,11135],{},", on passe par la\nconstante ",[1894,11391,11392],{},"math.inf"," (ou par des bibliothèques comme NumPy, où ",[1894,11395,11378],{},"\nrenvoie effectivement ",[1894,11398,11135],{},").",[2966,11401,11403,11408],{"correct":782,"slug":11402},"flottants-c2",[2970,11404,11405],{"v-slot:question":2883},[599,11406,11407],{},"Combien de bits utilise un float en double précision ?",[2970,11409,11410],{"v-slot:choices":2883},[3060,11411,11412,11414,11416,11418],{},[3063,11413,1945],{},[3063,11415,4953],{},[3063,11417,8017],{},[3063,11419,7719],{},[594,11421,5340],{"id":5339},[5342,11423,11424],{},[3060,11425,11426,11438,11441],{},[3063,11427,11428,11431,11432,11435,11436,2097],{},[603,11429,11430],{},"Ne jamais"," tester ",[1894,11433,11434],{},"a == b"," pour des flottants. Utiliser ",[1894,11437,11081],{},[3063,11439,11440],{},"Additionner beaucoup de petits flottants peut perdre en précision (les\npetits termes deviennent négligeables face au cumul).",[3063,11442,11443,11444,11447,11448,842,11450,11453,11454,11457,11458,11495],{},"Convertir une ",[1894,11445,11446],{},"str"," en ",[1894,11449,9798],{},[1894,11451,11452],{},"float(\"0.1\")",") crée la ",[603,11455,11456],{},"même\napproximation"," : il n'y a pas de chemin pour obtenir un ",[608,11459,11461,11474],{"className":11460},[611],[608,11462,11464],{"className":11463},[615],[617,11465,11466],{"xmlns":619},[621,11467,11468,11472],{},[624,11469,11470],{},[630,11471,9952],{},[637,11473,9955],{"encoding":639},[608,11475,11477],{"className":11476,"ariaHidden":645},[644],[608,11478,11480,11483,11486,11492],{"className":11479},[649],[608,11481],{"className":11482,"style":5047},[653],[608,11484,635],{"className":11485},[658],[608,11487,11489],{"className":11488},[658],[608,11490,5058],{"className":11491},[5057],[608,11493,659],{"className":11494},[658]," exact en\nflottant binaire.",[594,11497,5549],{"id":5548},[599,11499,11500],{},"Pour des calculs financiers ou scientifiques exigeants, Python fournit deux\nalternatives :",[3060,11502,11503,11510],{},[3063,11504,11505,11506,11509],{},"Le module ",[1894,11507,11508],{},"decimal"," (calcul en base 10, à précision arbitraire) — bien adapté\naux montants monétaires.",[3063,11511,11505,11512,11515,11516,11620],{},[1894,11513,11514],{},"fractions"," (rationnels exacts ",[608,11517,11519,11538],{"className":11518},[611],[608,11520,11522],{"className":11521},[615],[617,11523,11524],{"xmlns":619},[621,11525,11526,11535],{},[624,11527,11528],{},[8888,11529,11530,11532],{},[876,11531,599],{},[876,11533,11534],{},"q",[637,11536,11537],{"encoding":639},"\\frac{p}{q}",[608,11539,11541],{"className":11540,"ariaHidden":645},[644],[608,11542,11544,11548],{"className":11543},[649],[608,11545],{"className":11546,"style":11547},[653],"height:1.2286em;vertical-align:-0.4811em;",[608,11549,11551,11554,11617],{"className":11550},[658],[608,11552],{"className":11553},[1217,8913],[608,11555,11557],{"className":11556},[8888],[608,11558,11560,11608],{"className":11559},[673,1230],[608,11561,11563,11605],{"className":11562},[677],[608,11564,11567,11582,11590],{"className":11565,"style":11566},[681],"height:0.7475em;",[608,11568,11569,11572],{"style":8929},[608,11570],{"className":11571,"style":8933},[688],[608,11573,11575],{"className":11574},[693,694,695,696],[608,11576,11578],{"className":11577},[658,696],[608,11579,11534],{"className":11580,"style":11581},[658,894,696],"margin-right:0.0359em;",[608,11583,11584,11587],{"style":8945},[608,11585],{"className":11586,"style":8933},[688],[608,11588],{"className":11589,"style":8953},[8952],[608,11591,11593,11596],{"style":11592},"top:-3.4461em;",[608,11594],{"className":11595,"style":8933},[688],[608,11597,11599],{"className":11598},[693,694,695,696],[608,11600,11602],{"className":11601},[658,696],[608,11603,599],{"className":11604},[658,894,696],[608,11606,1263],{"className":11607},[1262],[608,11609,11611],{"className":11610},[677],[608,11612,11615],{"className":11613,"style":11614},[681],"height:0.4811em;",[608,11616],{},[608,11618],{"className":11619},[1417,8913],") — utile pour les\ncalculs symboliques où les approximations sont interdites.",[2878,11622,11624],{"className":2880,"code":11623,"language":2882,"meta":2883,"style":2883},"from fractions import Fraction\nprint(Fraction(1, 10) + Fraction(2, 10))  # 3\u002F10\n",[1894,11625,11626,11638],{"__ignoreMap":2883},[608,11627,11628,11630,11633,11635],{"class":2888,"line":2889},[608,11629,10765],{"class":3722},[608,11631,11632],{"class":2896}," fractions ",[608,11634,10771],{"class":3722},[608,11636,11637],{"class":2896}," Fraction\n",[608,11639,11640,11642,11645,11647,11649,11651,11653,11655,11658,11660,11662,11664,11666],{"class":2888,"line":2919},[608,11641,2893],{"class":2892},[608,11643,11644],{"class":2896},"(Fraction(",[608,11646,659],{"class":2892},[608,11648,1896],{"class":2896},[608,11650,632],{"class":2892},[608,11652,10807],{"class":2896},[608,11654,2416],{"class":3722},[608,11656,11657],{"class":2896}," Fraction(",[608,11659,782],{"class":2892},[608,11661,1896],{"class":2896},[608,11663,632],{"class":2892},[608,11665,2912],{"class":2896},[608,11667,11668],{"class":2915},"# 3\u002F10\n",[599,11670,11671,11672,2097],{},"Ces alternatives sont plus lentes, mais elles fournissent le résultat\n",[603,11673,11674],{},"exact",[5621,11676,11677],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":2883,"searchDepth":2919,"depth":2919,"links":11679},[11680,11681,11682,11683,11685,11686,11687,11689,11690],{"id":596,"depth":2919,"text":597},{"id":9015,"depth":2919,"text":9016},{"id":9650,"depth":2919,"text":9651},{"id":9936,"depth":2919,"text":11684},"Pourquoi 0,10{,}10,1 pose problème",{"id":10614,"depth":2919,"text":10615},{"id":10959,"depth":2919,"text":10960},{"id":11131,"depth":2919,"text":11688},"Valeurs spéciales — inf, -inf, nan",{"id":5339,"depth":2919,"text":5340},{"id":5548,"depth":2919,"text":5549},"Comment la machine approxime les nombres réels et pourquoi 0.1 + 0.2 ne vaut pas exactement 0.3.",{},[11694,11697],{"slug":10859,"kind":5642,"question":11695,"choices":11696,"correct":659,"multi":5645,"explanation":2883},"Que renvoie en Python l'expression 0.1+0.2==0.30.1 + 0.2 == 0.30.1+0.2==0.3 ?",[10947,10950,10953,10956],{"slug":11402,"kind":5642,"question":11407,"choices":11698,"correct":782,"multi":5645,"explanation":2883},[1945,4953,8017,7719],{"title":271,"description":11691},"xlRY1HXwAAnvG_w8rV44fLO50aNDFDiEibupS_Wxhio",{"id":11702,"title":275,"bo":11703,"body":11705,"description":12947,"eval":5636,"extension":5637,"labo":5636,"meta":12948,"navigation":5639,"path":276,"quizzes":12949,"readingTime":2889,"seo":12956,"stem":277,"tp":5636,"__hash__":12957},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages.md",[11704],"PB04",{"type":591,"value":11706,"toc":12936},[11707,11709,11721,11724,11728,11824,11902,11905,11977,11988,11998,12058,12078,12102,12106,12132,12189,12196,12203,12207,12217,12232,12321,12332,12336,12342,12345,12363,12442,12584,12586,12592,12623,12636,12660,12664,12674,12696,12705,12723,12727,12730,12834,12841,12891,12893,12928,12930,12933],[594,11708,597],{"id":596},[599,11710,11711,11712,11715,11716,5355,11719],{},"Vous savez désormais comment la machine code des nombres. Mais comment\nreprésente-t-elle un mot, une phrase, un emoji ? Aucun « A » ne vit dans la\nmémoire d'un ordinateur : seuls vivent des bits. Il faut donc une ",[603,11713,11714],{},"convention","\nqui associe à chaque caractère un numéro, lui-même représentable en binaire.\nCette convention s'appelle un ",[603,11717,11718],{},"encodage de caractères",[857,11720],{"code":11704},[599,11722,11723],{},"Plusieurs encodages ont coexisté dans l'histoire — chacun avec ses avantages et\nses limites. Comprendre lequel est utilisé est essentiel pour ouvrir\ncorrectement un fichier texte, transmettre des données, ou simplement éviter\nqu'un accent ne se transforme en hiéroglyphe.",[594,11725,11727],{"id":11726},"ascii-lancêtre","ASCII — l'ancêtre",[599,11729,11730,11731,842,11734,11737,11738,11741,11742,11823],{},"L'",[603,11732,11733],{},"ASCII",[844,11735,11736],{},"American Standard Code for Information Interchange",") date de 1963.\nIl code chaque caractère sur ",[603,11739,11740],{},"7 bits",", soit ",[608,11743,11745,11767],{"className":11744},[611],[608,11746,11748],{"className":11747},[615],[617,11749,11750],{"xmlns":619},[621,11751,11752,11764],{},[624,11753,11754,11760,11762],{},[627,11755,11756,11758],{},[630,11757,782],{},[630,11759,1918],{},[942,11761,1615],{},[630,11763,7719],{},[637,11765,11766],{"encoding":639},"2^7 = 128",[608,11768,11770,11814],{"className":11769,"ariaHidden":645},[644],[608,11771,11773,11776,11805,11808,11811],{"className":11772},[649],[608,11774],{"className":11775,"style":654},[653],[608,11777,11779,11782],{"className":11778},[658],[608,11780,782],{"className":11781},[658],[608,11783,11785],{"className":11784},[669],[608,11786,11788],{"className":11787},[673],[608,11789,11791],{"className":11790},[677],[608,11792,11794],{"className":11793,"style":654},[681],[608,11795,11796,11799],{"style":684},[608,11797],{"className":11798,"style":689},[688],[608,11800,11802],{"className":11801},[693,694,695,696],[608,11803,1918],{"className":11804},[658,696],[608,11806],{"className":11807,"style":967},[966],[608,11809,1615],{"className":11810},[971],[608,11812],{"className":11813,"style":967},[966],[608,11815,11817,11820],{"className":11816},[649],[608,11818],{"className":11819,"style":981},[653],[608,11821,7719],{"className":11822},[658]," valeurs possibles, de\n0 à 127.",[1859,11825,11826,11836],{},[1862,11827,11828],{},[1865,11829,11830,11833],{},[1868,11831,11832],{},"Plage",[1868,11834,11835],{},"Contenu",[1881,11837,11838,11846,11863,11876,11889],{},[1865,11839,11840,11843],{},[1886,11841,11842],{},"0–31",[1886,11844,11845],{},"Caractères de contrôle (retour chariot, tabulation…)",[1865,11847,11848,11851],{},[1886,11849,11850],{},"32–47",[1886,11852,11853,11854,1896,11857,1896,11859,11862],{},"Espace et ponctuation (",[1894,11855,11856],{},"!",[1894,11858,3737],{},[1894,11860,11861],{},"#",", etc.)",[1865,11864,11865,11868],{},[1886,11866,11867],{},"48–57",[1886,11869,11870,11871,11873,11874],{},"Chiffres ",[1894,11872,635],{},"–",[1894,11875,1937],{},[1865,11877,11878,11881],{},[1886,11879,11880],{},"65–90",[1886,11882,11883,11884,11873,11886],{},"Lettres majuscules ",[1894,11885,1958],{},[1894,11887,11888],{},"Z",[1865,11890,11891,11894],{},[1886,11892,11893],{},"97–122",[1886,11895,11896,11897,11873,11899],{},"Lettres minuscules ",[1894,11898,7225],{},[1894,11900,11901],{},"z",[599,11903,11904],{},"Quelques codes à connaître :",[1859,11906,11907,11920],{},[1862,11908,11909],{},[1865,11910,11911,11914,11917],{},[1868,11912,11913],{},"Caractère",[1868,11915,11916],{},"Code décimal",[1868,11918,11919],{},"Code binaire",[1881,11921,11922,11935,11949,11963],{},[1865,11923,11924,11928,11930],{},[1886,11925,11926],{},[1894,11927,1958],{},[1886,11929,4832],{},[1886,11931,11932],{},[1894,11933,11934],{},"1000001",[1865,11936,11937,11941,11944],{},[1886,11938,11939],{},[1894,11940,7225],{},[1886,11942,11943],{},"97",[1886,11945,11946],{},[1894,11947,11948],{},"1100001",[1865,11950,11951,11955,11958],{},[1886,11952,11953],{},[1894,11954,635],{},[1886,11956,11957],{},"48",[1886,11959,11960],{},[1894,11961,11962],{},"0110000",[1865,11964,11965,11970,11972],{},[1886,11966,11967],{},[1894,11968,11969],{},"(espace)",[1886,11971,4953],{},[1886,11973,11974],{},[1894,11975,11976],{},"0100000",[2099,11978,11979],{},[599,11980,11981,11982,11984,11985,11987],{},"Remarquez qu'il y a exactement 32 d'écart entre une majuscule et la minuscule\ncorrespondante : ",[1894,11983,1958],{}," vaut 65 et ",[1894,11986,7225],{}," vaut 97. C'est pour cela que basculer un\ncaractère en minuscule revient, en interne, à mettre un bit à 1.",[599,11989,11990,11991,2066,11994,11997],{},"En Python, les fonctions ",[1894,11992,11993],{},"ord()",[1894,11995,11996],{},"chr()"," permettent de naviguer entre un\ncaractère et son code :",[2878,11999,12001],{"className":2880,"code":12000,"language":2882,"meta":2883,"style":2883},"print(ord('A'))   # 65\nprint(chr(97))    # 'a'\nprint(ord('0'))   # 48\n",[1894,12002,12003,12022,12040],{"__ignoreMap":2883},[608,12004,12005,12007,12009,12012,12014,12017,12019],{"class":2888,"line":2889},[608,12006,2893],{"class":2892},[608,12008,1151],{"class":2896},[608,12010,12011],{"class":2892},"ord",[608,12013,1151],{"class":2896},[608,12015,12016],{"class":2904},"'A'",[608,12018,2937],{"class":2896},[608,12020,12021],{"class":2915},"# 65\n",[608,12023,12024,12026,12028,12031,12033,12035,12037],{"class":2888,"line":2919},[608,12025,2893],{"class":2892},[608,12027,1151],{"class":2896},[608,12029,12030],{"class":2892},"chr",[608,12032,1151],{"class":2896},[608,12034,11943],{"class":2892},[608,12036,2961],{"class":2896},[608,12038,12039],{"class":2915},"# 'a'\n",[608,12041,12042,12044,12046,12048,12050,12053,12055],{"class":2888,"line":2943},[608,12043,2893],{"class":2892},[608,12045,1151],{"class":2896},[608,12047,12011],{"class":2892},[608,12049,1151],{"class":2896},[608,12051,12052],{"class":2904},"'0'",[608,12054,2937],{"class":2896},[608,12056,12057],{"class":2915},"# 48\n",[599,12059,12060,12061,12064,12065,1896,12068,1896,12071,12074,12075,12077],{},"Limites évidentes de l'ASCII : ",[603,12062,12063],{},"pas d'accent",", pas de ",[1894,12066,12067],{},"é",[1894,12069,12070],{},"à",[1894,12072,12073],{},"ç",". Le ",[1894,12076,1157],{},"\nfrançais cédille n'existe pas. L'ASCII est resté valable pour l'anglais, mais\nintenable pour le reste du monde.",[2966,12079,12081,12088],{"correct":659,"slug":12080},"textes-encodages-c1",[2970,12082,12083],{"v-slot:question":2883},[599,12084,12085,12086,4329],{},"Quel est le code ASCII de la lettre ",[1894,12087,11888],{},[2970,12089,12090],{"v-slot:choices":2883},[3060,12091,12092,12094,12097,12099],{},[3063,12093,8017],{},[3063,12095,12096],{},"90",[3063,12098,11943],{},[3063,12100,12101],{},"122",[594,12103,12105],{"id":12104},"iso-8859-1-lextension-européenne","ISO-8859-1 — l'extension européenne",[599,12107,11730,12108,12111,12112,12115,12116,12119,12120,1896,12122,1896,12124,1896,12126,10085,12129,1181],{},[603,12109,12110],{},"ISO-8859-1"," (parfois appelée ",[603,12113,12114],{},"Latin-1",") étend l'ASCII à ",[603,12117,12118],{},"8 bits"," et\ndéfinit les 128 caractères supplémentaires (codes 128 à 255). On y trouve les\ncaractères accentués des langues d'Europe occidentale : ",[1894,12121,12067],{},[1894,12123,12070],{},[1894,12125,12073],{},[1894,12127,12128],{},"ñ",[1894,12130,12131],{},"ö",[1859,12133,12134,12145],{},[1862,12135,12136],{},[1865,12137,12138,12140,12142],{},[1868,12139,11913],{},[1868,12141,11916],{},[1868,12143,12144],{},"Code hexa",[1881,12146,12147,12161,12175],{},[1865,12148,12149,12153,12156],{},[1886,12150,12151],{},[1894,12152,12067],{},[1886,12154,12155],{},"233",[1886,12157,12158],{},[1894,12159,12160],{},"0xE9",[1865,12162,12163,12167,12170],{},[1886,12164,12165],{},[1894,12166,12070],{},[1886,12168,12169],{},"224",[1886,12171,12172],{},[1894,12173,12174],{},"0xE0",[1865,12176,12177,12181,12184],{},[1886,12178,12179],{},[1894,12180,12073],{},[1886,12182,12183],{},"231",[1886,12185,12186],{},[1894,12187,12188],{},"0xE7",[599,12190,12191,12192,12195],{},"ISO-8859-1 reste ",[603,12193,12194],{},"rétrocompatible"," avec l'ASCII : les 128 premiers codes sont\nidentiques. Un fichier ASCII pur reste valide en ISO-8859-1.",[599,12197,12198,12199,12202],{},"Mais ISO-8859-1 ne couvre ",[603,12200,12201],{},"que"," l'Europe de l'Ouest. Pour le grec, le russe,\nl'arabe, le chinois… il a fallu d'autres variantes (ISO-8859-2, ISO-8859-5,\netc.). Cette multiplication des encodages a vite tourné au cauchemar : un même\noctet ne signifiait plus la même chose selon le contexte.",[594,12204,12206],{"id":12205},"unicode-un-caractère-un-numéro-universel","Unicode — un caractère, un numéro universel",[599,12208,12209,12212,12213,12216],{},[603,12210,12211],{},"Unicode"," est né dans les années 1990 d'une ambition simple : attribuer un\nnuméro unique à ",[603,12214,12215],{},"tous"," les caractères de toutes les langues du monde, plus\nles symboles, les emojis, les caractères mathématiques, etc.",[599,12218,12219,12220,12223,12224,12227,12228,12231],{},"Chaque caractère reçoit un ",[603,12221,12222],{},"code point"," (point de code), noté typiquement\n",[1894,12225,12226],{},"U+XXXX"," où ",[1894,12229,12230],{},"XXXX"," est sa valeur en hexadécimal :",[1859,12233,12234,12246],{},[1862,12235,12236],{},[1865,12237,12238,12240,12243],{},[1868,12239,11913],{},[1868,12241,12242],{},"Code point",[1868,12244,12245],{},"Décimal",[1881,12247,12248,12261,12274,12290,12306],{},[1865,12249,12250,12254,12259],{},[1886,12251,12252],{},[1894,12253,1958],{},[1886,12255,12256],{},[1894,12257,12258],{},"U+0041",[1886,12260,4832],{},[1865,12262,12263,12267,12272],{},[1886,12264,12265],{},[1894,12266,12067],{},[1886,12268,12269],{},[1894,12270,12271],{},"U+00E9",[1886,12273,12155],{},[1865,12275,12276,12282,12287],{},[1886,12277,12278,12281],{},[1894,12279,12280],{},"λ"," (lambda grec)",[1886,12283,12284],{},[1894,12285,12286],{},"U+03BB",[1886,12288,12289],{},"955",[1865,12291,12292,12298,12303],{},[1886,12293,12294,12297],{},[1894,12295,12296],{},"日"," (idéogramme japonais « jour »)",[1886,12299,12300],{},[1894,12301,12302],{},"U+65E5",[1886,12304,12305],{},"26 085",[1865,12307,12308,12313,12318],{},[1886,12309,12310],{},[1894,12311,12312],{},"😀",[1886,12314,12315],{},[1894,12316,12317],{},"U+1F600",[1886,12319,12320],{},"128 512",[599,12322,12323,12324,12327,12328,12331],{},"Unicode définit aujourd'hui plus de 150 000 caractères. Mais Unicode n'est\nqu'un ",[603,12325,12326],{},"catalogue"," : il dit quel numéro a chaque caractère, pas comment\nécrire ce numéro en mémoire. Pour cela, il faut un ",[603,12329,12330],{},"encodage",", parmi\nplusieurs disponibles.",[594,12333,12335],{"id":12334},"utf-8-lencodage-qui-a-tout-emporté","UTF-8 — l'encodage qui a tout emporté",[599,12337,12338,12341],{},[603,12339,12340],{},"UTF-8"," est de loin l'encodage Unicode le plus utilisé aujourd'hui — sur le\nWeb, dans les fichiers texte, dans la quasi-totalité des langages modernes.",[599,12343,12344],{},"Ses propriétés :",[3060,12346,12347,12354,12360],{},[3063,12348,12349,12350,12353],{},"Il code chaque caractère sur ",[603,12351,12352],{},"1 à 4 octets"," selon sa valeur Unicode.",[3063,12355,12356,12357,12359],{},"Il est ",[603,12358,12194],{}," avec l'ASCII : un fichier ASCII pur est aussi un\nfichier UTF-8 valide.",[3063,12361,12362],{},"Les caractères courants en anglais tiennent sur 1 octet, les caractères\naccentués courants sur 2 octets, les idéogrammes sur 3, certains emojis sur 4.",[1859,12364,12365,12376],{},[1862,12366,12367],{},[1865,12368,12369,12371,12373],{},[1868,12370,11913],{},[1868,12372,12242],{},[1868,12374,12375],{},"Octets en UTF-8",[1881,12377,12378,12394,12410,12426],{},[1865,12379,12380,12384,12388],{},[1886,12381,12382],{},[1894,12383,1958],{},[1886,12385,12386],{},[1894,12387,12258],{},[1886,12389,12390,12393],{},[1894,12391,12392],{},"41"," (1 octet)",[1865,12395,12396,12400,12404],{},[1886,12397,12398],{},[1894,12399,12067],{},[1886,12401,12402],{},[1894,12403,12271],{},[1886,12405,12406,12409],{},[1894,12407,12408],{},"C3 A9"," (2 octets)",[1865,12411,12412,12416,12420],{},[1886,12413,12414],{},[1894,12415,12296],{},[1886,12417,12418],{},[1894,12419,12302],{},[1886,12421,12422,12425],{},[1894,12423,12424],{},"E6 97 A5"," (3 octets)",[1865,12427,12428,12432,12436],{},[1886,12429,12430],{},[1894,12431,12312],{},[1886,12433,12434],{},[1894,12435,12317],{},[1886,12437,12438,12441],{},[1894,12439,12440],{},"F0 9F 98 80"," (4 octets)",[3708,12443,12444],{},[2878,12445,12447],{"className":2880,"code":12446,"language":2882,"meta":2883,"style":2883},"# Voir l'encodage UTF-8 d'une chaîne en hexadécimal\nmot = \"café\"\noctets = mot.encode(\"utf-8\")\nprint(f\"Chaîne   : {mot}\")\nprint(f\"Octets   : {octets}\")\nprint(f\"En hexa  : {octets.hex(' ')}\")\nprint(f\"Longueur : {len(octets)} octets pour {len(mot)} caractères\")\n",[1894,12448,12449,12454,12464,12479,12501,12523,12550],{"__ignoreMap":2883},[608,12450,12451],{"class":2888,"line":2889},[608,12452,12453],{"class":2915},"# Voir l'encodage UTF-8 d'une chaîne en hexadécimal\n",[608,12455,12456,12459,12461],{"class":2888,"line":2919},[608,12457,12458],{"class":2896},"mot ",[608,12460,1615],{"class":3722},[608,12462,12463],{"class":2904}," \"café\"\n",[608,12465,12466,12469,12471,12474,12477],{"class":2888,"line":2943},[608,12467,12468],{"class":2896},"octets ",[608,12470,1615],{"class":3722},[608,12472,12473],{"class":2896}," mot.encode(",[608,12475,12476],{"class":2904},"\"utf-8\"",[608,12478,3761],{"class":2896},[608,12480,12481,12483,12485,12487,12490,12492,12495,12497,12499],{"class":2888,"line":3794},[608,12482,2893],{"class":2892},[608,12484,1151],{"class":2896},[608,12486,3734],{"class":3722},[608,12488,12489],{"class":2904},"\"Chaîne   : ",[608,12491,3740],{"class":2892},[608,12493,12494],{"class":2896},"mot",[608,12496,3745],{"class":2892},[608,12498,3737],{"class":2904},[608,12500,3761],{"class":2896},[608,12502,12503,12505,12507,12509,12512,12514,12517,12519,12521],{"class":2888,"line":7353},[608,12504,2893],{"class":2892},[608,12506,1151],{"class":2896},[608,12508,3734],{"class":3722},[608,12510,12511],{"class":2904},"\"Octets   : ",[608,12513,3740],{"class":2892},[608,12515,12516],{"class":2896},"octets",[608,12518,3745],{"class":2892},[608,12520,3737],{"class":2904},[608,12522,3761],{"class":2896},[608,12524,12525,12527,12529,12531,12534,12536,12539,12542,12544,12546,12548],{"class":2888,"line":7359},[608,12526,2893],{"class":2892},[608,12528,1151],{"class":2896},[608,12530,3734],{"class":3722},[608,12532,12533],{"class":2904},"\"En hexa  : ",[608,12535,3740],{"class":2892},[608,12537,12538],{"class":2896},"octets.hex(",[608,12540,12541],{"class":2904},"' '",[608,12543,1198],{"class":2896},[608,12545,3745],{"class":2892},[608,12547,3737],{"class":2904},[608,12549,3761],{"class":2896},[608,12551,12552,12554,12556,12558,12561,12564,12567,12569,12572,12574,12577,12579,12582],{"class":2888,"line":7395},[608,12553,2893],{"class":2892},[608,12555,1151],{"class":2896},[608,12557,3734],{"class":3722},[608,12559,12560],{"class":2904},"\"Longueur : ",[608,12562,12563],{"class":2892},"{len",[608,12565,12566],{"class":2896},"(octets)",[608,12568,3745],{"class":2892},[608,12570,12571],{"class":2904}," octets pour ",[608,12573,12563],{"class":2892},[608,12575,12576],{"class":2896},"(mot)",[608,12578,3745],{"class":2892},[608,12580,12581],{"class":2904}," caractères\"",[608,12583,3761],{"class":2896},[599,12585,3825],{},[2878,12587,12590],{"className":12588,"code":12589,"language":3831},[3829],"Chaîne   : café\nOctets   : b'caf\\xc3\\xa9'\nEn hexa  : 63 61 66 c3 a9\nLongueur : 5 octets pour 4 caractères\n",[1894,12591,12589],{"__ignoreMap":2883},[2099,12593,12594],{},[599,12595,12596,12597,12600,12601,12604,12605,12608,12609,12612,12613,12616,12617,1896,12619,1896,12621,11399],{},"Le préfixe ",[1894,12598,12599],{},"b'...'"," signale à Python que l'objet est une ",[603,12602,12603],{},"suite d'octets","\n(",[1894,12606,12607],{},"bytes",") — pas une chaîne de caractères. Les octets non imprimables comme\n",[1894,12610,12611],{},"0xC3"," sont affichés sous la forme ",[1894,12614,12615],{},"\\xc3"," ; les octets correspondant à des\nlettres ASCII sont affichés tels quels (",[1894,12618,1157],{},[1894,12620,7225],{},[1894,12622,3734],{},[599,12624,6190,12625,12627,12628,12631,12632,12635],{},[1894,12626,12067],{}," occupe ",[603,12629,12630],{},"2 octets"," en UTF-8 alors qu'il n'occupe ",[603,12633,12634],{},"qu'un seul caractère\nlogique"," dans la chaîne Python.",[2966,12637,12639,12648],{"correct":782,"slug":12638},"textes-encodages-c2",[2970,12640,12641],{"v-slot:question":2883},[599,12642,12643,12644,12647],{},"Combien d'octets occupe la chaîne ",[1894,12645,12646],{},"\\\"é\\\""," en UTF-8 ?",[2970,12649,12650],{"v-slot:choices":2883},[3060,12651,12652,12654,12656,12658],{},[3063,12653,659],{},[3063,12655,782],{},[3063,12657,2413],{},[3063,12659,3948],{},[594,12661,12663],{"id":12662},"le-bug-classique-des-encodages-mélangés","Le bug classique des encodages mélangés",[599,12665,12666,12667,12670,12671,4329],{},"Imaginez un fichier ",[1894,12668,12669],{},"notes.txt"," enregistré en UTF-8 par votre éditeur de texte.\nVous l'ouvrez ensuite avec un logiciel qui croit lire de l'ISO-8859-1. Que se\npasse-t-il pour le mot ",[1894,12672,12673],{},"café",[3060,12675,12676,12685],{},[3063,12677,12678,12679,12681,12682,2097],{},"En UTF-8, ",[1894,12680,12067],{}," est stocké comme les octets ",[1894,12683,12684],{},"0xC3 0xA9",[3063,12686,12687,12688,12691,12692,12695],{},"Lus en ISO-8859-1, ces deux octets sont interprétés comme deux caractères\nséparés : ",[1894,12689,12690],{},"Ã"," (code 195) et ",[1894,12693,12694],{},"©"," (code 169).",[599,12697,12698,12699,12701,12702,12704],{},"Résultat à l'écran : ",[1894,12700,12673],{}," devient ",[1894,12703,12673],{},". C'est le signe quasi universel\nd'un fichier UTF-8 mal interprété.",[5342,12706,12707],{},[599,12708,12709,12710,1896,12713,9261,12716,12719,12720,2097],{},"Quand vous voyez des séquences comme ",[1894,12711,12712],{},"é",[1894,12714,12715],{},"à ",[1894,12717,12718],{},"ç"," à la place d'accents,\nc'est presque toujours un fichier UTF-8 lu comme ISO-8859-1. Le contenu n'est\npas corrompu — il est juste ",[603,12721,12722],{},"mal interprété",[594,12724,12726],{"id":12725},"en-python-préciser-lencodage-à-louverture","En Python — préciser l'encodage à l'ouverture",[599,12728,12729],{},"Python permet de préciser l'encodage à l'ouverture d'un fichier :",[2878,12731,12733],{"className":2880,"code":12732,"language":2882,"meta":2883,"style":2883},"# Lecture explicite en UTF-8\nwith open(\"notes.txt\", encoding=\"utf-8\") as f:\n    contenu = f.read()\n\n# Conversion en bytes puis dans un autre encodage\ntexte = \"café\"\noctets_utf8 = texte.encode(\"utf-8\")     # b'caf\\xc3\\xa9'\noctets_latin = texte.encode(\"latin-1\")  # b'caf\\xe9'\n",[1894,12734,12735,12740,12771,12781,12785,12790,12799,12817],{"__ignoreMap":2883},[608,12736,12737],{"class":2888,"line":2889},[608,12738,12739],{"class":2915},"# Lecture explicite en UTF-8\n",[608,12741,12742,12745,12748,12750,12753,12755,12759,12761,12763,12765,12768],{"class":2888,"line":2919},[608,12743,12744],{"class":3722},"with",[608,12746,12747],{"class":2892}," open",[608,12749,1151],{"class":2896},[608,12751,12752],{"class":2904},"\"notes.txt\"",[608,12754,1896],{"class":2896},[608,12756,12758],{"class":12757},"sP4rz","encoding",[608,12760,1615],{"class":3722},[608,12762,12476],{"class":2904},[608,12764,10807],{"class":2896},[608,12766,12767],{"class":3722},"as",[608,12769,12770],{"class":2896}," f:\n",[608,12772,12773,12776,12778],{"class":2888,"line":2943},[608,12774,12775],{"class":2896},"    contenu ",[608,12777,1615],{"class":3722},[608,12779,12780],{"class":2896}," f.read()\n",[608,12782,12783],{"class":2888,"line":3794},[608,12784,8596],{"emptyLinePlaceholder":5639},[608,12786,12787],{"class":2888,"line":7353},[608,12788,12789],{"class":2915},"# Conversion en bytes puis dans un autre encodage\n",[608,12791,12792,12795,12797],{"class":2888,"line":7359},[608,12793,12794],{"class":2896},"texte ",[608,12796,1615],{"class":3722},[608,12798,12463],{"class":2904},[608,12800,12801,12804,12806,12809,12811,12814],{"class":2888,"line":7395},[608,12802,12803],{"class":2896},"octets_utf8 ",[608,12805,1615],{"class":3722},[608,12807,12808],{"class":2896}," texte.encode(",[608,12810,12476],{"class":2904},[608,12812,12813],{"class":2896},")     ",[608,12815,12816],{"class":2915},"# b'caf\\xc3\\xa9'\n",[608,12818,12819,12822,12824,12826,12829,12831],{"class":2888,"line":8642},[608,12820,12821],{"class":2896},"octets_latin ",[608,12823,1615],{"class":3722},[608,12825,12808],{"class":2896},[608,12827,12828],{"class":2904},"\"latin-1\"",[608,12830,7389],{"class":2896},[608,12832,12833],{"class":2915},"# b'caf\\xe9'\n",[599,12835,12836,12837,12840],{},"Inversement, ",[1894,12838,12839],{},"decode()"," reconstruit la chaîne à partir des octets, en précisant\nl'encodage utilisé.",[2878,12842,12844],{"className":2880,"code":12843,"language":2882,"meta":2883,"style":2883},"octets = b'caf\\xc3\\xa9'\nprint(octets.decode(\"utf-8\"))    # 'café' — correct\nprint(octets.decode(\"latin-1\"))  # 'café' — buggué\n",[1894,12845,12846,12864,12878],{"__ignoreMap":2883},[608,12847,12848,12850,12852,12855,12858,12861],{"class":2888,"line":2889},[608,12849,12468],{"class":2896},[608,12851,1615],{"class":3722},[608,12853,12854],{"class":3722}," b",[608,12856,12857],{"class":2904},"'caf",[608,12859,12860],{"class":2892},"\\xc3\\xa9",[608,12862,12863],{"class":2904},"'\n",[608,12865,12866,12868,12871,12873,12875],{"class":2888,"line":2919},[608,12867,2893],{"class":2892},[608,12869,12870],{"class":2896},"(octets.decode(",[608,12872,12476],{"class":2904},[608,12874,2961],{"class":2896},[608,12876,12877],{"class":2915},"# 'café' — correct\n",[608,12879,12880,12882,12884,12886,12888],{"class":2888,"line":2943},[608,12881,2893],{"class":2892},[608,12883,12870],{"class":2896},[608,12885,12828],{"class":2904},[608,12887,2912],{"class":2896},[608,12889,12890],{"class":2915},"# 'café' — buggué\n",[594,12892,5340],{"id":5339},[5342,12894,12895],{},[3060,12896,12897,12908,12925],{},[3063,12898,12899,12900,12903,12904,12907],{},"Toujours ",[603,12901,12902],{},"préciser l'encodage"," quand vous ouvrez un fichier texte\n(",[1894,12905,12906],{},"open(\"fichier.txt\", encoding=\"utf-8\")","). Les valeurs par défaut varient\nd'un OS à l'autre.",[3063,12909,12910,12911,12913,12914,12917,12918,842,12921,12924],{},"Le nombre d'",[603,12912,12516],{}," d'une chaîne (",[1894,12915,12916],{},"len(s.encode())",") n'est pas forcément\nle nombre de ",[603,12919,12920],{},"caractères",[1894,12922,12923],{},"len(s)",") — vrai uniquement en ASCII.",[3063,12926,12927],{},"Un caractère emoji peut occuper jusqu'à 4 octets en UTF-8, et certains\ncombinent plusieurs code points (famille, drapeaux…).",[594,12929,5549],{"id":5548},[599,12931,12932],{},"UTF-8 n'est pas le seul encodage Unicode. UTF-16 est utilisé en interne par\nJava, JavaScript et Windows ; UTF-32 (4 octets par caractère, fixe) est rare\nmais simple. Le Web s'est concentré sur UTF-8 pour sa compacité et sa\nrétrocompatibilité ASCII — c'est aujourd'hui le standard de facto.",[5621,12934,12935],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":2883,"searchDepth":2919,"depth":2919,"links":12937},[12938,12939,12940,12941,12942,12943,12944,12945,12946],{"id":596,"depth":2919,"text":597},{"id":11726,"depth":2919,"text":11727},{"id":12104,"depth":2919,"text":12105},{"id":12205,"depth":2919,"text":12206},{"id":12334,"depth":2919,"text":12335},{"id":12662,"depth":2919,"text":12663},{"id":12725,"depth":2919,"text":12726},{"id":5339,"depth":2919,"text":5340},{"id":5548,"depth":2919,"text":5549},"Comment la machine représente des caractères, et pourquoi un mauvais encodage transforme « café » en « café ».",{},[12950,12953],{"slug":12080,"kind":5642,"question":12951,"choices":12952,"correct":659,"multi":5645,"explanation":2883},"Quel est le code ASCII de la lettre Z ?",[8017,12096,11943,12101],{"slug":12638,"kind":5642,"question":12954,"choices":12955,"correct":782,"multi":5645,"explanation":2883},"Combien d'octets occupe la chaîne \\\"é\\\" en UTF-8 ?",[659,782,2413,3948],{"title":275,"description":12947},"rohEeSq4sjpiIUwb-VoySg5X5AvihJ1DA5e1fyhmuOw",{"id":12959,"title":279,"bo":12960,"body":12962,"description":14796,"eval":5636,"extension":5637,"labo":5636,"meta":14797,"navigation":5639,"path":280,"quizzes":14798,"readingTime":2889,"seo":14807,"stem":281,"tp":5636,"__hash__":14808},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens.md",[12961],"PB05",{"type":591,"value":12963,"toc":14777},[12964,12966,12983,12996,13004,13008,13011,13174,13257,13259,13265,13269,13276,13283,13440,13446,13605,13611,13697,13790,13797,13842,14000,14009,14043,14104,14124,14128,14142,14160,14167,14191,14236,14254,14280,14284,14301,14327,14334,14392,14406,14415,14419,14426,14446,14662,14672,14674,14754,14756,14774],[594,12965,597],{"id":596},[599,12967,12968,12969,12972,12973,12976,12977,9261,12980,2097],{},"Tout programme prend, à un moment ou un autre, des ",[603,12970,12971],{},"décisions"," : « si ce\nnombre est positif, faire ceci, sinon cela ». Ces décisions s'appuient sur des\n",[603,12974,12975],{},"conditions",", qui ne peuvent avoir que deux résultats : ",[603,12978,12979],{},"vrai",[603,12981,12982],{},"faux",[599,12984,12985,12986,12989,12990,12993,12994],{},"Cette logique binaire, formalisée au XIXᵉ siècle par le mathématicien anglais\nGeorge Boole, porte aujourd'hui son nom : on parle d'",[603,12987,12988],{},"algèbre de Boole"," et\nde ",[603,12991,12992],{},"valeurs booléennes",". Elle est partout en informatique — des tests\nd'égalité jusqu'aux portes logiques qui constituent physiquement le\nprocesseur. ",[857,12995],{"code":12961},[599,12997,12998,12999,2066,13001,13003],{},"En Python, ces deux valeurs s'écrivent ",[1894,13000,10947],{},[1894,13002,10950],{}," (avec une majuscule\ninitiale, c'est important).",[594,13005,13007],{"id":13006},"les-trois-opérateurs-de-base","Les trois opérateurs de base",[599,13009,13010],{},"L'algèbre de Boole repose sur trois opérateurs principaux. En Python, ils\ns'écrivent en toutes lettres.",[1859,13012,13013,13029],{},[1862,13014,13015],{},[1865,13016,13017,13020,13023,13026],{},[1868,13018,13019],{},"Opérateur logique",[1868,13021,13022],{},"Python",[1868,13024,13025],{},"Notation mathématique",[1868,13027,13028],{},"Effet",[1881,13030,13031,13081,13129],{},[1865,13032,13033,13036,13041,13074],{},[1886,13034,13035],{},"ET",[1886,13037,13038],{},[1894,13039,13040],{},"and",[1886,13042,13043],{},[608,13044,13046,13061],{"className":13045},[611],[608,13047,13049],{"className":13048},[615],[617,13050,13051],{"xmlns":619},[621,13052,13053,13058],{},[624,13054,13055],{},[942,13056,13057],{},"∧",[637,13059,13060],{"encoding":639},"\\land",[608,13062,13064],{"className":13063,"ariaHidden":645},[644],[608,13065,13067,13071],{"className":13066},[649],[608,13068],{"className":13069,"style":13070},[653],"height:0.5556em;",[608,13072,13057],{"className":13073},[658],[1886,13075,13076,13077,13080],{},"vrai si ",[603,13078,13079],{},"les deux"," opérandes sont vrais",[1865,13082,13083,13086,13091,13123],{},[1886,13084,13085],{},"OU",[1886,13087,13088],{},[1894,13089,13090],{},"or",[1886,13092,13093],{},[608,13094,13096,13111],{"className":13095},[611],[608,13097,13099],{"className":13098},[615],[617,13100,13101],{"xmlns":619},[621,13102,13103,13108],{},[624,13104,13105],{},[942,13106,13107],{},"∨",[637,13109,13110],{"encoding":639},"\\lor",[608,13112,13114],{"className":13113,"ariaHidden":645},[644],[608,13115,13117,13120],{"className":13116},[649],[608,13118],{"className":13119,"style":13070},[653],[608,13121,13107],{"className":13122},[658],[1886,13124,13076,13125,13128],{},[603,13126,13127],{},"au moins un"," des opérandes est vrai",[1865,13130,13131,13134,13139,13171],{},[1886,13132,13133],{},"NON",[1886,13135,13136],{},[1894,13137,13138],{},"not",[1886,13140,13141],{},[608,13142,13144,13159],{"className":13143},[611],[608,13145,13147],{"className":13146},[615],[617,13148,13149],{"xmlns":619},[621,13150,13151,13156],{},[624,13152,13153],{},[876,13154,13155],{"mathvariant":11181},"¬",[637,13157,13158],{"encoding":639},"\\lnot",[608,13160,13162],{"className":13161,"ariaHidden":645},[644],[608,13163,13165,13168],{"className":13164},[649],[608,13166],{"className":13167,"style":4374},[653],[608,13169,13155],{"className":13170},[658],[1886,13172,13173],{},"inverse la valeur",[3708,13175,13176],{},[2878,13177,13179],{"className":2880,"code":13178,"language":2882,"meta":2883,"style":2883},"print(True and False)   # False\nprint(True or False)    # True\nprint(not True)         # False\nprint(not (5 > 3))      # False, car 5 > 3 est True\n",[1894,13180,13181,13199,13217,13233],{"__ignoreMap":2883},[608,13182,13183,13185,13187,13189,13192,13195,13197],{"class":2888,"line":2889},[608,13184,2893],{"class":2892},[608,13186,1151],{"class":2896},[608,13188,10947],{"class":2892},[608,13190,13191],{"class":3722}," and",[608,13193,13194],{"class":2892}," False",[608,13196,5333],{"class":2896},[608,13198,10664],{"class":2915},[608,13200,13201,13203,13205,13207,13210,13212,13215],{"class":2888,"line":2919},[608,13202,2893],{"class":2892},[608,13204,1151],{"class":2896},[608,13206,10947],{"class":2892},[608,13208,13209],{"class":3722}," or",[608,13211,13194],{"class":2892},[608,13213,13214],{"class":2896},")    ",[608,13216,11075],{"class":2915},[608,13218,13219,13221,13223,13225,13228,13231],{"class":2888,"line":2943},[608,13220,2893],{"class":2892},[608,13222,1151],{"class":2896},[608,13224,13138],{"class":3722},[608,13226,13227],{"class":2892}," True",[608,13229,13230],{"class":2896},")         ",[608,13232,10664],{"class":2915},[608,13234,13235,13237,13239,13241,13243,13245,13248,13251,13254],{"class":2888,"line":3794},[608,13236,2893],{"class":2892},[608,13238,1151],{"class":2896},[608,13240,13138],{"class":3722},[608,13242,842],{"class":2896},[608,13244,3970],{"class":2892},[608,13246,13247],{"class":3722}," >",[608,13249,13250],{"class":2892}," 3",[608,13252,13253],{"class":2896},"))      ",[608,13255,13256],{"class":2915},"# False, car 5 > 3 est True\n",[599,13258,3825],{},[2878,13260,13263],{"className":13261,"code":13262,"language":3831},[3829],"False\nTrue\nFalse\nFalse\n",[1894,13264,13262],{"__ignoreMap":2883},[594,13266,13268],{"id":13267},"tables-de-vérité","Tables de vérité",[599,13270,13271,13272,13275],{},"Une ",[603,13273,13274],{},"table de vérité"," énumère toutes les combinaisons possibles des\nopérandes et donne la valeur du résultat. C'est l'outil de référence pour\nanalyser une expression booléenne.",[13277,13278,13280,13282],"h3",{"id":13279},"and-et-logique",[1894,13281,13040],{}," (ET logique)",[1859,13284,13285,13402],{},[1862,13286,13287],{},[1865,13288,13289,13319,13349],{},[1868,13290,13291],{},[608,13292,13294,13307],{"className":13293},[611],[608,13295,13297],{"className":13296},[615],[617,13298,13299],{"xmlns":619},[621,13300,13301,13305],{},[624,13302,13303],{},[876,13304,1958],{},[637,13306,1958],{"encoding":639},[608,13308,13310],{"className":13309,"ariaHidden":645},[644],[608,13311,13313,13316],{"className":13312},[649],[608,13314],{"className":13315,"style":1665},[653],[608,13317,1958],{"className":13318},[658,894],[1868,13320,13321],{},[608,13322,13324,13337],{"className":13323},[611],[608,13325,13327],{"className":13326},[615],[617,13328,13329],{"xmlns":619},[621,13330,13331,13335],{},[624,13332,13333],{},[876,13334,2035],{},[637,13336,2035],{"encoding":639},[608,13338,13340],{"className":13339,"ariaHidden":645},[644],[608,13341,13343,13346],{"className":13342},[649],[608,13344],{"className":13345,"style":1665},[653],[608,13347,2035],{"className":13348,"style":2231},[658,894],[1868,13350,13351],{},[608,13352,13354,13372],{"className":13353},[611],[608,13355,13357],{"className":13356},[615],[617,13358,13359],{"xmlns":619},[621,13360,13361,13369],{},[624,13362,13363,13365,13367],{},[876,13364,1958],{},[942,13366,13057],{},[876,13368,2035],{},[637,13370,13371],{"encoding":639},"A \\land B",[608,13373,13375,13393],{"className":13374,"ariaHidden":645},[644],[608,13376,13378,13381,13384,13387,13390],{"className":13377},[649],[608,13379],{"className":13380,"style":1665},[653],[608,13382,1958],{"className":13383},[658,894],[608,13385],{"className":13386,"style":1082},[966],[608,13388,13057],{"className":13389},[1086],[608,13391],{"className":13392,"style":1082},[966],[608,13394,13396,13399],{"className":13395},[649],[608,13397],{"className":13398,"style":1665},[653],[608,13400,2035],{"className":13401,"style":2231},[658,894],[1881,13403,13404,13413,13422,13430],{},[1865,13405,13406,13409,13411],{},[1886,13407,13408],{},"Faux",[1886,13410,13408],{},[1886,13412,13408],{},[1865,13414,13415,13417,13420],{},[1886,13416,13408],{},[1886,13418,13419],{},"Vrai",[1886,13421,13408],{},[1865,13423,13424,13426,13428],{},[1886,13425,13419],{},[1886,13427,13408],{},[1886,13429,13408],{},[1865,13431,13432,13434,13436],{},[1886,13433,13419],{},[1886,13435,13419],{},[1886,13437,13438],{},[603,13439,13419],{},[13277,13441,13443,13445],{"id":13442},"or-ou-logique",[1894,13444,13090],{}," (OU logique)",[1859,13447,13448,13565],{},[1862,13449,13450],{},[1865,13451,13452,13482,13512],{},[1868,13453,13454],{},[608,13455,13457,13470],{"className":13456},[611],[608,13458,13460],{"className":13459},[615],[617,13461,13462],{"xmlns":619},[621,13463,13464,13468],{},[624,13465,13466],{},[876,13467,1958],{},[637,13469,1958],{"encoding":639},[608,13471,13473],{"className":13472,"ariaHidden":645},[644],[608,13474,13476,13479],{"className":13475},[649],[608,13477],{"className":13478,"style":1665},[653],[608,13480,1958],{"className":13481},[658,894],[1868,13483,13484],{},[608,13485,13487,13500],{"className":13486},[611],[608,13488,13490],{"className":13489},[615],[617,13491,13492],{"xmlns":619},[621,13493,13494,13498],{},[624,13495,13496],{},[876,13497,2035],{},[637,13499,2035],{"encoding":639},[608,13501,13503],{"className":13502,"ariaHidden":645},[644],[608,13504,13506,13509],{"className":13505},[649],[608,13507],{"className":13508,"style":1665},[653],[608,13510,2035],{"className":13511,"style":2231},[658,894],[1868,13513,13514],{},[608,13515,13517,13535],{"className":13516},[611],[608,13518,13520],{"className":13519},[615],[617,13521,13522],{"xmlns":619},[621,13523,13524,13532],{},[624,13525,13526,13528,13530],{},[876,13527,1958],{},[942,13529,13107],{},[876,13531,2035],{},[637,13533,13534],{"encoding":639},"A \\lor B",[608,13536,13538,13556],{"className":13537,"ariaHidden":645},[644],[608,13539,13541,13544,13547,13550,13553],{"className":13540},[649],[608,13542],{"className":13543,"style":1665},[653],[608,13545,1958],{"className":13546},[658,894],[608,13548],{"className":13549,"style":1082},[966],[608,13551,13107],{"className":13552},[1086],[608,13554],{"className":13555,"style":1082},[966],[608,13557,13559,13562],{"className":13558},[649],[608,13560],{"className":13561,"style":1665},[653],[608,13563,2035],{"className":13564,"style":2231},[658,894],[1881,13566,13567,13575,13585,13595],{},[1865,13568,13569,13571,13573],{},[1886,13570,13408],{},[1886,13572,13408],{},[1886,13574,13408],{},[1865,13576,13577,13579,13581],{},[1886,13578,13408],{},[1886,13580,13419],{},[1886,13582,13583],{},[603,13584,13419],{},[1865,13586,13587,13589,13591],{},[1886,13588,13419],{},[1886,13590,13408],{},[1886,13592,13593],{},[603,13594,13419],{},[1865,13596,13597,13599,13601],{},[1886,13598,13419],{},[1886,13600,13419],{},[1886,13602,13603],{},[603,13604,13419],{},[13277,13606,13608,13610],{"id":13607},"not-non-logique",[1894,13609,13138],{}," (NON logique)",[1859,13612,13613,13683],{},[1862,13614,13615],{},[1865,13616,13617,13647],{},[1868,13618,13619],{},[608,13620,13622,13635],{"className":13621},[611],[608,13623,13625],{"className":13624},[615],[617,13626,13627],{"xmlns":619},[621,13628,13629,13633],{},[624,13630,13631],{},[876,13632,1958],{},[637,13634,1958],{"encoding":639},[608,13636,13638],{"className":13637,"ariaHidden":645},[644],[608,13639,13641,13644],{"className":13640},[649],[608,13642],{"className":13643,"style":1665},[653],[608,13645,1958],{"className":13646},[658,894],[1868,13648,13649],{},[608,13650,13652,13668],{"className":13651},[611],[608,13653,13655],{"className":13654},[615],[617,13656,13657],{"xmlns":619},[621,13658,13659,13665],{},[624,13660,13661,13663],{},[876,13662,13155],{"mathvariant":11181},[876,13664,1958],{},[637,13666,13667],{"encoding":639},"\\lnot A",[608,13669,13671],{"className":13670,"ariaHidden":645},[644],[608,13672,13674,13677,13680],{"className":13673},[649],[608,13675],{"className":13676,"style":1665},[653],[608,13678,13155],{"className":13679},[658],[608,13681,1958],{"className":13682},[658,894],[1881,13684,13685,13691],{},[1865,13686,13687,13689],{},[1886,13688,13408],{},[1886,13690,13419],{},[1865,13692,13693,13695],{},[1886,13694,13419],{},[1886,13696,13408],{},[2099,13698,13699],{},[599,13700,13701,13702,13730,13731,13789],{},"Pour ",[608,13703,13705,13718],{"className":13704},[611],[608,13706,13708],{"className":13707},[615],[617,13709,13710],{"xmlns":619},[621,13711,13712,13716],{},[624,13713,13714],{},[876,13715,1162],{},[637,13717,1162],{"encoding":639},[608,13719,13721],{"className":13720,"ariaHidden":645},[644],[608,13722,13724,13727],{"className":13723},[649],[608,13725],{"className":13726,"style":4374},[653],[608,13728,1162],{"className":13729},[658,894]," variables booléennes, la table de vérité contient ",[608,13732,13734,13751],{"className":13733},[611],[608,13735,13737],{"className":13736},[615],[617,13738,13739],{"xmlns":619},[621,13740,13741,13749],{},[624,13742,13743],{},[627,13744,13745,13747],{},[630,13746,782],{},[876,13748,1162],{},[637,13750,4399],{"encoding":639},[608,13752,13754],{"className":13753,"ariaHidden":645},[644],[608,13755,13757,13760],{"className":13756},[649],[608,13758],{"className":13759,"style":4409},[653],[608,13761,13763,13766],{"className":13762},[658],[608,13764,782],{"className":13765},[658],[608,13767,13769],{"className":13768},[669],[608,13770,13772],{"className":13771},[673],[608,13773,13775],{"className":13774},[677],[608,13776,13778],{"className":13777,"style":4409},[681],[608,13779,13780,13783],{"style":684},[608,13781],{"className":13782,"style":689},[688],[608,13784,13786],{"className":13785},[693,694,695,696],[608,13787,1162],{"className":13788},[658,894,696]," lignes —\ntoutes les combinaisons possibles. Pour 3 variables, 8 lignes ; pour 4\nvariables, 16. La taille double à chaque ajout.",[594,13791,13793,13794],{"id":13792},"le-ou-exclusif-xor","Le ou exclusif — ",[1894,13795,13796],{},"xor",[599,13798,13799,13800,13803,13804,13806,13807,13837,13838,13841],{},"Il existe un quatrième opérateur très utile : le ",[603,13801,13802],{},"ou exclusif",", noté ",[1894,13805,13796],{},"\nou ",[608,13808,13810,13825],{"className":13809},[611],[608,13811,13813],{"className":13812},[615],[617,13814,13815],{"xmlns":619},[621,13816,13817,13822],{},[624,13818,13819],{},[942,13820,13821],{},"⊕",[637,13823,13824],{"encoding":639},"\\oplus",[608,13826,13828],{"className":13827,"ariaHidden":645},[644],[608,13829,13831,13834],{"className":13830},[649],[608,13832],{"className":13833,"style":6386},[653],[608,13835,13821],{"className":13836},[658],". Il vaut ",[603,13839,13840],{},"vrai si exactement un"," des opérandes est vrai (et donc\nfaux si les deux sont identiques).",[1859,13843,13844,13962],{},[1862,13845,13846],{},[1865,13847,13848,13878,13908],{},[1868,13849,13850],{},[608,13851,13853,13866],{"className":13852},[611],[608,13854,13856],{"className":13855},[615],[617,13857,13858],{"xmlns":619},[621,13859,13860,13864],{},[624,13861,13862],{},[876,13863,1958],{},[637,13865,1958],{"encoding":639},[608,13867,13869],{"className":13868,"ariaHidden":645},[644],[608,13870,13872,13875],{"className":13871},[649],[608,13873],{"className":13874,"style":1665},[653],[608,13876,1958],{"className":13877},[658,894],[1868,13879,13880],{},[608,13881,13883,13896],{"className":13882},[611],[608,13884,13886],{"className":13885},[615],[617,13887,13888],{"xmlns":619},[621,13889,13890,13894],{},[624,13891,13892],{},[876,13893,2035],{},[637,13895,2035],{"encoding":639},[608,13897,13899],{"className":13898,"ariaHidden":645},[644],[608,13900,13902,13905],{"className":13901},[649],[608,13903],{"className":13904,"style":1665},[653],[608,13906,2035],{"className":13907,"style":2231},[658,894],[1868,13909,13910],{},[608,13911,13913,13931],{"className":13912},[611],[608,13914,13916],{"className":13915},[615],[617,13917,13918],{"xmlns":619},[621,13919,13920,13928],{},[624,13921,13922,13924,13926],{},[876,13923,1958],{},[942,13925,13821],{},[876,13927,2035],{},[637,13929,13930],{"encoding":639},"A \\oplus B",[608,13932,13934,13953],{"className":13933,"ariaHidden":645},[644],[608,13935,13937,13941,13944,13947,13950],{"className":13936},[649],[608,13938],{"className":13939,"style":13940},[653],"height:0.7667em;vertical-align:-0.0833em;",[608,13942,1958],{"className":13943},[658,894],[608,13945],{"className":13946,"style":1082},[966],[608,13948,13821],{"className":13949},[1086],[608,13951],{"className":13952,"style":1082},[966],[608,13954,13956,13959],{"className":13955},[649],[608,13957],{"className":13958,"style":1665},[653],[608,13960,2035],{"className":13961,"style":2231},[658,894],[1881,13963,13964,13972,13982,13992],{},[1865,13965,13966,13968,13970],{},[1886,13967,13408],{},[1886,13969,13408],{},[1886,13971,13408],{},[1865,13973,13974,13976,13978],{},[1886,13975,13408],{},[1886,13977,13419],{},[1886,13979,13980],{},[603,13981,13419],{},[1865,13983,13984,13986,13988],{},[1886,13985,13419],{},[1886,13987,13408],{},[1886,13989,13990],{},[603,13991,13419],{},[1865,13993,13994,13996,13998],{},[1886,13995,13419],{},[1886,13997,13419],{},[1886,13999,13408],{},[599,14001,14002,14003,14005,14006,5279],{},"Python ne fournit pas d'opérateur ",[1894,14004,13796],{}," en toutes lettres pour les booléens,\nmais on peut l'écrire avec ",[1894,14007,14008],{},"!=",[2878,14010,14012],{"className":2880,"code":14011,"language":2882,"meta":2883,"style":2883},"A, B = True, False\nprint(A != B)   # True — équivaut à A xor B\n",[1894,14013,14014,14028],{"__ignoreMap":2883},[608,14015,14016,14019,14021,14023,14025],{"class":2888,"line":2889},[608,14017,14018],{"class":2896},"A, B ",[608,14020,1615],{"class":3722},[608,14022,13227],{"class":2892},[608,14024,1896],{"class":2896},[608,14026,14027],{"class":2892},"False\n",[608,14029,14030,14032,14035,14037,14040],{"class":2888,"line":2919},[608,14031,2893],{"class":2892},[608,14033,14034],{"class":2896},"(A ",[608,14036,14008],{"class":3722},[608,14038,14039],{"class":2896}," B)   ",[608,14041,14042],{"class":2915},"# True — équivaut à A xor B\n",[599,14044,6190,14045,14047,14048,14051,14052,14103],{},[1894,14046,13796],{}," est très utile pour l'",[603,14049,14050],{},"addition binaire"," : sur 1 bit, ",[608,14053,14055,14073],{"className":14054},[611],[608,14056,14058],{"className":14057},[615],[617,14059,14060],{"xmlns":619},[621,14061,14062,14070],{},[624,14063,14064,14066,14068],{},[630,14065,659],{},[942,14067,13821],{},[630,14069,659],{},[637,14071,14072],{"encoding":639},"1 \\oplus 1",[608,14074,14076,14094],{"className":14075,"ariaHidden":645},[644],[608,14077,14079,14082,14085,14088,14091],{"className":14078},[649],[608,14080],{"className":14081,"style":2794},[653],[608,14083,659],{"className":14084},[658],[608,14086],{"className":14087,"style":1082},[966],[608,14089,13821],{"className":14090},[1086],[608,14092],{"className":14093,"style":1082},[966],[608,14095,14097,14100],{"className":14096},[649],[608,14098],{"className":14099,"style":981},[653],[608,14101,659],{"className":14102},[658],"\nvaut 0 avec une retenue. C'est exactement ce dont a besoin un additionneur\nmatériel.",[2966,14105,14107,14112],{"correct":782,"slug":14106},"booleens-c1",[2970,14108,14109],{"v-slot:question":2883},[599,14110,14111],{},"Quel opérateur retourne vrai uniquement si A et B sont différents ?",[2970,14113,14114],{"v-slot:choices":2883},[3060,14115,14116,14118,14120,14122],{},[3063,14117,13040],{},[3063,14119,13090],{},[3063,14121,13796],{},[3063,14123,13138],{},[594,14125,14127],{"id":14126},"expressions-booléennes","Expressions booléennes",[599,14129,14130,14131,1896,14133,10085,14135,14137,14138,14141],{},"On peut composer des expressions plus complexes en combinant ",[1894,14132,13040],{},[1894,14134,13090],{},[1894,14136,13138],{},". Comme en arithmétique, ",[603,14139,14140],{},"l'ordre des opérations"," compte :",[14143,14144,14145,14150,14155],"ol",{},[3063,14146,14147,14149],{},[1894,14148,13138],{}," est appliqué en premier (priorité la plus haute).",[3063,14151,14152,14153,2097],{},"Puis ",[1894,14154,13040],{},[3063,14156,14152,14157,14159],{},[1894,14158,13090],{}," (priorité la plus basse).",[599,14161,14162,14163,14166],{},"Exemple : pour évaluer ",[1894,14164,14165],{},"True or False and not False",", on lit dans cet ordre :",[14143,14168,14169,14177,14184],{},[3063,14170,14171,14174,14175],{},[1894,14172,14173],{},"not False"," → ",[1894,14176,10947],{},[3063,14178,14179,14174,14182],{},[1894,14180,14181],{},"False and True",[1894,14183,10950],{},[3063,14185,14186,14174,14189],{},[1894,14187,14188],{},"True or False",[1894,14190,10947],{},[7211,14192,14193,14200],{},[599,14194,14195,14196,14199],{},"En cas de doute, ",[603,14197,14198],{},"mettez des parenthèses",". Elles ne coûtent rien et\nclarifient la lecture du code :",[2878,14201,14203],{"className":2880,"code":14202,"language":2882,"meta":2883,"style":2883},"if (age >= 18) and (permis or accompagnateur):\n    conduire()\n",[1894,14204,14205,14231],{"__ignoreMap":2883},[608,14206,14207,14210,14213,14216,14219,14221,14223,14226,14228],{"class":2888,"line":2889},[608,14208,14209],{"class":3722},"if",[608,14211,14212],{"class":2896}," (age ",[608,14214,14215],{"class":3722},">=",[608,14217,14218],{"class":2892}," 18",[608,14220,10807],{"class":2896},[608,14222,13040],{"class":3722},[608,14224,14225],{"class":2896}," (permis ",[608,14227,13090],{"class":3722},[608,14229,14230],{"class":2896}," accompagnateur):\n",[608,14232,14233],{"class":2888,"line":2919},[608,14234,14235],{"class":2896},"    conduire()\n",[2966,14237,14239,14246],{"correct":659,"slug":14238},"booleens-c2",[2970,14240,14241],{"v-slot:question":2883},[599,14242,2974,14243,4329],{},[1894,14244,14245],{},"not (True or False)",[2970,14247,14248],{"v-slot:choices":2883},[3060,14249,14250,14252],{},[3063,14251,10947],{},[3063,14253,10950],{},[2966,14255,14257,14272],{"correct":635,"slug":14256},"booleens-c3",[2970,14258,14259],{"v-slot:question":2883},[599,14260,14261,14262,14265,14266,14268,14269,14271],{},"Sans parenthèses, que vaut ",[1894,14263,14264],{},"True or False and False"," (rappel : ",[1894,14267,13040],{}," est prioritaire sur ",[1894,14270,13090],{},") ?",[2970,14273,14274],{"v-slot:choices":2883},[3060,14275,14276,14278],{},[3063,14277,10947],{},[3063,14279,10950],{},[594,14281,14283],{"id":14282},"le-caractère-séquentiel-des-opérateurs","Le caractère séquentiel des opérateurs",[599,14285,14286,14287,14290,14291,14294,14295,9261,14298,2097],{},"Une particularité importante : Python évalue les opérateurs booléens ",[603,14288,14289],{},"de\ngauche à droite",", et il ",[603,14292,14293],{},"s'arrête dès qu'il connaît le résultat",". On parle\nd'",[603,14296,14297],{},"évaluation paresseuse",[603,14299,14300],{},"séquentielle",[3060,14302,14303,14317],{},[3063,14304,13701,14305,14308,14309,14311,14312,11379,14315,2097],{},[1894,14306,14307],{},"A and B"," : si ",[1894,14310,1958],{}," est faux, le résultat est forcément faux. Python ne\nva même ",[603,14313,14314],{},"pas évaluer",[1894,14316,2035],{},[3063,14318,13701,14319,14308,14322,14324,14325,2097],{},[1894,14320,14321],{},"A or B",[1894,14323,1958],{}," est vrai, le résultat est forcément vrai. Python ne\nva même pas évaluer ",[1894,14326,2035],{},[599,14328,14329,14330,14333],{},"Cette propriété est exploitée comme une ",[603,14331,14332],{},"protection"," courante :",[2878,14335,14337],{"className":2880,"code":14336,"language":2882,"meta":2883,"style":2883},"liste = []\n# Sans court-circuit, on aurait une erreur en accédant à liste[0]\nif liste and liste[0] > 5:\n    print(\"Premier élément grand\")\n",[1894,14338,14339,14349,14354,14380],{"__ignoreMap":2883},[608,14340,14341,14344,14346],{"class":2888,"line":2889},[608,14342,14343],{"class":2896},"liste ",[608,14345,1615],{"class":3722},[608,14347,14348],{"class":2896}," []\n",[608,14350,14351],{"class":2888,"line":2919},[608,14352,14353],{"class":2915},"# Sans court-circuit, on aurait une erreur en accédant à liste[0]\n",[608,14355,14356,14358,14361,14363,14366,14368,14371,14374,14377],{"class":2888,"line":2943},[608,14357,14209],{"class":3722},[608,14359,14360],{"class":2896}," liste ",[608,14362,13040],{"class":3722},[608,14364,14365],{"class":2896}," liste[",[608,14367,635],{"class":2892},[608,14369,14370],{"class":2896},"] ",[608,14372,14373],{"class":3722},">",[608,14375,14376],{"class":2892}," 5",[608,14378,14379],{"class":2896},":\n",[608,14381,14382,14385,14387,14390],{"class":2888,"line":3794},[608,14383,14384],{"class":2892},"    print",[608,14386,1151],{"class":2896},[608,14388,14389],{"class":2904},"\"Premier élément grand\"",[608,14391,3761],{"class":2896},[599,14393,14394,14395,14398,14399,14401,14402,14405],{},"Si ",[1894,14396,14397],{},"liste"," est vide, ",[1894,14400,14397],{}," est considéré comme faux, et Python ne tente même\npas d'évaluer ",[1894,14403,14404],{},"liste[0]"," qui aurait planté.",[5342,14407,14408],{},[599,14409,14410,14411,14414],{},"Cette évaluation séquentielle peut surprendre quand un opérande contient un\n",[603,14412,14413],{},"effet de bord"," (appel de fonction, modification d'une variable). L'effet ne\nse produira pas si l'opérande n'est pas évalué. Toujours en avoir conscience.",[594,14416,14418],{"id":14417},"une-application-directe-laddition-binaire","Une application directe — l'addition binaire",[599,14420,14421,14422,2066,14424,5279],{},"L'addition de deux bits se décompose élégamment avec ",[1894,14423,13796],{},[1894,14425,13040],{},[3060,14427,14428,14437],{},[3063,14429,6190,14430,14433,14434,2097],{},[603,14431,14432],{},"bit de résultat"," est donné par ",[1894,14435,14436],{},"A xor B",[3063,14438,14439,14440,14443,14444,2097],{},"La ",[603,14441,14442],{},"retenue"," est donnée par ",[1894,14445,14307],{},[1859,14447,14448,14618],{},[1862,14449,14450],{},[1865,14451,14452,14482,14512,14565],{},[1868,14453,14454],{},[608,14455,14457,14470],{"className":14456},[611],[608,14458,14460],{"className":14459},[615],[617,14461,14462],{"xmlns":619},[621,14463,14464,14468],{},[624,14465,14466],{},[876,14467,1958],{},[637,14469,1958],{"encoding":639},[608,14471,14473],{"className":14472,"ariaHidden":645},[644],[608,14474,14476,14479],{"className":14475},[649],[608,14477],{"className":14478,"style":1665},[653],[608,14480,1958],{"className":14481},[658,894],[1868,14483,14484],{},[608,14485,14487,14500],{"className":14486},[611],[608,14488,14490],{"className":14489},[615],[617,14491,14492],{"xmlns":619},[621,14493,14494,14498],{},[624,14495,14496],{},[876,14497,2035],{},[637,14499,2035],{"encoding":639},[608,14501,14503],{"className":14502,"ariaHidden":645},[644],[608,14504,14506,14509],{"className":14505},[649],[608,14507],{"className":14508,"style":1665},[653],[608,14510,2035],{"className":14511,"style":2231},[658,894],[1868,14513,14514,14515],{},"Somme = ",[608,14516,14518,14535],{"className":14517},[611],[608,14519,14521],{"className":14520},[615],[617,14522,14523],{"xmlns":619},[621,14524,14525,14533],{},[624,14526,14527,14529,14531],{},[876,14528,1958],{},[942,14530,13821],{},[876,14532,2035],{},[637,14534,13930],{"encoding":639},[608,14536,14538,14556],{"className":14537,"ariaHidden":645},[644],[608,14539,14541,14544,14547,14550,14553],{"className":14540},[649],[608,14542],{"className":14543,"style":13940},[653],[608,14545,1958],{"className":14546},[658,894],[608,14548],{"className":14549,"style":1082},[966],[608,14551,13821],{"className":14552},[1086],[608,14554],{"className":14555,"style":1082},[966],[608,14557,14559,14562],{"className":14558},[649],[608,14560],{"className":14561,"style":1665},[653],[608,14563,2035],{"className":14564,"style":2231},[658,894],[1868,14566,14567,14568],{},"Retenue = ",[608,14569,14571,14588],{"className":14570},[611],[608,14572,14574],{"className":14573},[615],[617,14575,14576],{"xmlns":619},[621,14577,14578,14586],{},[624,14579,14580,14582,14584],{},[876,14581,1958],{},[942,14583,13057],{},[876,14585,2035],{},[637,14587,13371],{"encoding":639},[608,14589,14591,14609],{"className":14590,"ariaHidden":645},[644],[608,14592,14594,14597,14600,14603,14606],{"className":14593},[649],[608,14595],{"className":14596,"style":1665},[653],[608,14598,1958],{"className":14599},[658,894],[608,14601],{"className":14602,"style":1082},[966],[608,14604,13057],{"className":14605},[1086],[608,14607],{"className":14608,"style":1082},[966],[608,14610,14612,14615],{"className":14611},[649],[608,14613],{"className":14614,"style":1665},[653],[608,14616,2035],{"className":14617,"style":2231},[658,894],[1881,14619,14620,14630,14640,14650],{},[1865,14621,14622,14624,14626,14628],{},[1886,14623,635],{},[1886,14625,635],{},[1886,14627,635],{},[1886,14629,635],{},[1865,14631,14632,14634,14636,14638],{},[1886,14633,635],{},[1886,14635,659],{},[1886,14637,659],{},[1886,14639,635],{},[1865,14641,14642,14644,14646,14648],{},[1886,14643,659],{},[1886,14645,635],{},[1886,14647,659],{},[1886,14649,635],{},[1865,14651,14652,14654,14656,14658],{},[1886,14653,659],{},[1886,14655,659],{},[1886,14657,635],{},[1886,14659,14660],{},[603,14661,659],{},[599,14663,14664,14665,14668,14669,2097],{},"C'est ce circuit minimal, appelé ",[603,14666,14667],{},"demi-additionneur",", qu'on assemble pour\nconstruire l'additionneur complet d'un processeur. La logique booléenne n'est\ndonc pas qu'abstraite : elle est ",[603,14670,14671],{},"gravée dans le silicium",[594,14673,5340],{"id":5339},[5342,14675,14676],{},[3060,14677,14678,14694,14717,14731],{},[3063,14679,5468,14680,2066,14682,14684,14685,5355,14688,14690,14691,14693],{},[1894,14681,10947],{},[1894,14683,10950],{}," prennent une ",[603,14686,14687],{},"majuscule initiale",[1894,14689,645],{},"\n(avec t minuscule) n'est ",[603,14692,5842],{}," une valeur valide.",[3063,14695,14696,14697,14699,14700,14702,14703,14706,14707,14709,14710,14713,14714,14716],{},"Ne pas confondre ",[1894,14698,1615],{}," (affectation) et ",[1894,14701,10928],{}," (test d'égalité) : ",[1894,14704,14705],{},"x = 5"," met 5\ndans ",[1894,14708,5898],{}," ; ",[1894,14711,14712],{},"x == 5"," teste si ",[1894,14715,5898],{}," vaut 5 et retourne un booléen.",[3063,14718,14719,14720,14723,14724,14727,14728,2097],{},"Sans parenthèses, ",[1894,14721,14722],{},"not a == b"," se lit ",[1894,14725,14726],{},"not(a == b)",", pas ",[1894,14729,14730],{},"(not a) == b",[3063,14732,14733,14734,1896,14736,1896,14739,14742,14743,14746,14747,1896,14749,1896,14751,14753],{},"Les opérateurs ",[1894,14735,7370],{},[1894,14737,14738],{},"|",[1894,14740,14741],{},"^"," existent aussi mais agissent ",[603,14744,14745],{},"bit à bit"," sur\ndes entiers — c'est différent de ",[1894,14748,13040],{},[1894,14750,13090],{},[1894,14752,13796],{}," sur des booléens.",[594,14755,5549],{"id":5548},[599,14757,14758,14759,1896,14761,2066,14763,14765,14766,14769,14770,14773],{},"Toute fonction logique, aussi complexe soit-elle, peut être réécrite\nuniquement avec ",[1894,14760,13138],{},[1894,14762,13040],{},[1894,14764,13090],{}," (on parle d'",[603,14767,14768],{},"ensemble fonctionnellement\ncomplet","). En électronique, l'opérateur ",[1894,14771,14772],{},"nand"," (NON-ET) suffit à lui seul à\ntout reconstruire — d'où son utilisation massive dans la conception des puces.",[5621,14775,14776],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":2883,"searchDepth":2919,"depth":2919,"links":14778},[14779,14780,14781,14789,14791,14792,14793,14794,14795],{"id":596,"depth":2919,"text":597},{"id":13006,"depth":2919,"text":13007},{"id":13267,"depth":2919,"text":13268,"children":14782},[14783,14785,14787],{"id":13279,"depth":2943,"text":14784},"and (ET logique)",{"id":13442,"depth":2943,"text":14786},"or (OU logique)",{"id":13607,"depth":2943,"text":14788},"not (NON logique)",{"id":13792,"depth":2919,"text":14790},"Le ou exclusif — xor",{"id":14126,"depth":2919,"text":14127},{"id":14282,"depth":2919,"text":14283},{"id":14417,"depth":2919,"text":14418},{"id":5339,"depth":2919,"text":5340},{"id":5548,"depth":2919,"text":5549},"Le calcul à deux états — vrai ou faux — qui structure toute condition et tout test en programmation.",{},[14799,14801,14804],{"slug":14106,"kind":5642,"question":14111,"choices":14800,"correct":782,"multi":5645,"explanation":2883},[13040,13090,13796,13138],{"slug":14238,"kind":5642,"question":14802,"choices":14803,"correct":659,"multi":5645,"explanation":2883},"Que vaut not (True or False) ?",[10947,10950],{"slug":14256,"kind":5642,"question":14805,"choices":14806,"correct":635,"multi":5645,"explanation":2883},"Sans parenthèses, que vaut True or False and False (rappel : and est prioritaire sur or) ?",[10947,10950],{"title":279,"description":14796},"5uP-GwBvt02044m2-qUAT-95DiG_Ji3GQhhd7Xj3DPw",{"id":14810,"title":257,"bo":5636,"body":14811,"description":14915,"eval":5636,"extension":5637,"labo":5636,"meta":14916,"navigation":5639,"path":258,"quizzes":14918,"readingTime":2889,"seo":14919,"stem":259,"tp":5636,"__hash__":14920},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F0.index.md",{"type":591,"value":14812,"toc":14911},[14813,14817,14824,14859,14863,14905],[594,14814,14816],{"id":14815},"au-programme","Au programme",[599,14818,14819,14820,14823],{},"Un ordinateur ne stocke et ne manipule, au plus bas niveau, ",[603,14821,14822],{},"que des suites de\nbits"," — des 0 et des 1. Tout ce que vous voyez à l'écran, qu'il s'agisse d'un\nnombre, d'un mot, d'une couleur ou d'une condition logique, n'existe en mémoire\nque sous cette forme. Ce chapitre vous donne les clés de cette traduction.",[599,14825,14826,14827,14830,14831,14834,14835,14838,14839,14841,14842,14844,14845,14848,14849,5826,14852,1896,14854,1896,14856,14858],{},"Vous découvrirez d'abord ",[603,14828,14829],{},"les bases de numération"," (binaire, hexadécimal,\noctal) et comment passer de l'une à l'autre. Vous verrez ensuite comment la\nmachine code les ",[603,14832,14833],{},"entiers relatifs"," grâce au complément à deux, puis comment\nelle approxime les ",[603,14836,14837],{},"nombres réels"," avec les flottants — et pourquoi\n",[1894,14840,9004],{}," ne vaut pas exactement ",[1894,14843,9011],{},". Vous apprendrez à reconnaître les\n",[603,14846,14847],{},"encodages de texte"," (ASCII, ISO-8859-1, Unicode\u002FUTF-8) et à comprendre les\nfameux caractères « bizarres » qui apparaissent parfois dans un fichier mal\nouvert. Enfin, vous formaliserez le ",[603,14850,14851],{},"calcul booléen",[1894,14853,13040],{},[1894,14855,13090],{},[1894,14857,13138],{},",\nles tables de vérité et les expressions logiques qui structurent toute\nprogrammation.",[594,14860,14862],{"id":14861},"plan-du-chapitre","Plan du chapitre",[14143,14864,14865,14873,14881,14889,14897],{},[3063,14866,14867,14870,14871],{},[603,14868,14869],{},"Bases et entiers positifs"," — binaire, hexadécimal, octal, conversions. ",[857,14872],{"code":589},[3063,14874,14875,14878,14879],{},[603,14876,14877],{},"Entiers relatifs"," — bit de signe et complément à deux. ",[857,14880],{"code":5654},[3063,14882,14883,14886,14887],{},[603,14884,14885],{},"Nombres flottants"," — virgule flottante, IEEE 754 et imprécisions. ",[857,14888],{"code":8827},[3063,14890,14891,14894,14895],{},[603,14892,14893],{},"Textes et encodages"," — ASCII, ISO-8859-1, Unicode, UTF-8. ",[857,14896],{"code":11704},[3063,14898,14899,14902,14903],{},[603,14900,14901],{},"Valeurs booléennes"," — opérateurs logiques et tables de vérité. ",[857,14904],{"code":12961},[599,14906,14907,14908,14910],{},"À la fin du chapitre, vous saurez convertir un nombre entre bases, expliquer\npourquoi un ",[1894,14909,9798],{}," peut « mentir », identifier l'encodage d'un fichier texte\net dresser la table de vérité d'une expression logique.",{"title":2883,"searchDepth":2919,"depth":2919,"links":14912},[14913,14914],{"id":14815,"depth":2919,"text":14816},{"id":14861,"depth":2919,"text":14862},"Comment une machine, qui ne sait manipuler que des 0 et des 1, représente-t-elle des nombres, du texte et des valeurs logiques ?",{"icon":14917},"i-lucide-binary",[],{"title":257,"description":14915},"Mnv0xqqyUP3JZs-FwTEvZFUY6cC9CLexMKS8k9MLMWM",[14922,16058,16934,17561,17837,18294],{"id":14923,"title":14924,"bo":14925,"body":14926,"description":16051,"eval":5636,"extension":5637,"labo":5636,"meta":16052,"navigation":5645,"path":16053,"quizzes":16054,"readingTime":2889,"seo":16055,"stem":16056,"tp":5636,"__hash__":16057},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs.memo.md","Bases de numération et entiers positifs — l'essentiel",[589],{"type":591,"value":14927,"toc":16044},[14928,14932,14995,14999,15484,15488,15742,15744,15956,15960,16041],[594,14929,14931],{"id":14930},"en-une-phrase","En une phrase",[599,14933,897,14934,14962,14963,14991,14992,2097],{},[608,14935,14937,14950],{"className":14936},[611],[608,14938,14940],{"className":14939},[615],[617,14941,14942],{"xmlns":619},[621,14943,14944,14948],{},[624,14945,14946],{},[876,14947,878],{},[637,14949,878],{"encoding":639},[608,14951,14953],{"className":14952,"ariaHidden":645},[644],[608,14954,14956,14959],{"className":14955},[649],[608,14957],{"className":14958,"style":890},[653],[608,14960,878],{"className":14961},[658,894],", chaque chiffre occupe une position qui correspond à une\npuissance de ",[608,14964,14966,14979],{"className":14965},[611],[608,14967,14969],{"className":14968},[615],[617,14970,14971],{"xmlns":619},[621,14972,14973,14977],{},[624,14974,14975],{},[876,14976,878],{},[637,14978,878],{"encoding":639},[608,14980,14982],{"className":14981,"ariaHidden":645},[644],[608,14983,14985,14988],{"className":14984},[649],[608,14986],{"className":14987,"style":890},[653],[608,14989,878],{"className":14990},[658,894]," — ",[603,14993,14994],{},"un nombre vaut la somme de ses chiffres pondérés par les\npuissances de la base",[594,14996,14998],{"id":14997},"à-mémoriser","À mémoriser",[3060,15000,15001,15011,15021,15035,15283,15481],{},[3063,15002,15003,15004,1896,15006,15008,15009,2097],{},"Base 2 : chiffres ",[1894,15005,635],{},[1894,15007,659],{},". Préfixe Python ",[1894,15010,1903],{},[3063,15012,15013,15014,1043,15016,15018,15019,2097],{},"Base 8 : chiffres ",[1894,15015,635],{},[1894,15017,1918],{},". Préfixe ",[1894,15020,1923],{},[3063,15022,15023,15024,1043,15026,15028,15029,11873,15031,15018,15033,2097],{},"Base 16 : chiffres ",[1894,15025,635],{},[1894,15027,1937],{}," puis ",[1894,15030,1958],{},[1894,15032,1961],{},[1894,15034,1966],{},[3063,15036,15037,15282],{},[608,15038,15040,15091],{"className":15039},[611],[608,15041,15043],{"className":15042},[615],[617,15044,15045],{"xmlns":619},[621,15046,15047,15088],{},[624,15048,15049,15051,15053,15074,15080,15082],{},[876,15050,1612],{},[942,15052,1615],{},[15054,15055,15056,15058,15066],"msubsup",{},[942,15057,1621],{},[624,15059,15060,15062,15064],{},[876,15061,1504],{},[942,15063,1615],{},[630,15065,635],{},[624,15067,15068,15070,15072],{},[876,15069,1162],{},[942,15071,1060],{},[630,15073,659],{},[1153,15075,15076,15078],{},[876,15077,878],{},[876,15079,1504],{},[942,15081,1646],{},[627,15083,15084,15086],{},[876,15085,878],{},[876,15087,1504],{},[637,15089,15090],{"encoding":639},"N = \\sum_{i=0}^{n-1} b_i \\cdot b^i",[608,15092,15094,15112,15246],{"className":15093,"ariaHidden":645},[644],[608,15095,15097,15100,15103,15106,15109],{"className":15096},[649],[608,15098],{"className":15099,"style":1665},[653],[608,15101,1612],{"className":15102,"style":1669},[658,894],[608,15104],{"className":15105,"style":967},[966],[608,15107,1615],{"className":15108},[971],[608,15110],{"className":15111,"style":967},[966],[608,15113,15115,15119,15194,15197,15237,15240,15243],{"className":15114},[649],[608,15116],{"className":15117,"style":15118},[653],"height:1.2537em;vertical-align:-0.2997em;",[608,15120,15122,15127],{"className":15121},[1689],[608,15123,1621],{"className":15124,"style":15126},[1689,1734,15125],"small-op","position:relative;top:0em;",[608,15128,15130],{"className":15129},[669],[608,15131,15133,15185],{"className":15132},[673,1230],[608,15134,15136,15182],{"className":15135},[677],[608,15137,15140,15161],{"className":15138,"style":15139},[681],"height:0.954em;",[608,15141,15143,15146],{"style":15142},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[608,15144],{"className":15145,"style":689},[688],[608,15147,15149],{"className":15148},[693,694,695,696],[608,15150,15152,15155,15158],{"className":15151},[658,696],[608,15153,1504],{"className":15154},[658,894,696],[608,15156,1615],{"className":15157},[971,696],[608,15159,635],{"className":15160},[658,696],[608,15162,15164,15167],{"style":15163},"top:-3.2029em;margin-right:0.05em;",[608,15165],{"className":15166,"style":689},[688],[608,15168,15170],{"className":15169},[693,694,695,696],[608,15171,15173,15176,15179],{"className":15172},[658,696],[608,15174,1162],{"className":15175},[658,894,696],[608,15177,1060],{"className":15178},[1086,696],[608,15180,659],{"className":15181},[658,696],[608,15183,1263],{"className":15184},[1262],[608,15186,15188],{"className":15187},[677],[608,15189,15192],{"className":15190,"style":15191},[681],"height:0.2997em;",[608,15193],{},[608,15195],{"className":15196,"style":1325},[966],[608,15198,15200,15203],{"className":15199},[658],[608,15201,878],{"className":15202},[658,894],[608,15204,15206],{"className":15205},[669],[608,15207,15209,15229],{"className":15208},[673,1230],[608,15210,15212,15226],{"className":15211},[677],[608,15213,15215],{"className":15214,"style":1536},[681],[608,15216,15217,15220],{"style":1240},[608,15218],{"className":15219,"style":689},[688],[608,15221,15223],{"className":15222},[693,694,695,696],[608,15224,1504],{"className":15225},[658,894,696],[608,15227,1263],{"className":15228},[1262],[608,15230,15232],{"className":15231},[677],[608,15233,15235],{"className":15234,"style":1371},[681],[608,15236],{},[608,15238],{"className":15239,"style":1082},[966],[608,15241,1646],{"className":15242},[1086],[608,15244],{"className":15245,"style":1082},[966],[608,15247,15249,15253],{"className":15248},[649],[608,15250],{"className":15251,"style":15252},[653],"height:0.8247em;",[608,15254,15256,15259],{"className":15255},[658],[608,15257,878],{"className":15258},[658,894],[608,15260,15262],{"className":15261},[669],[608,15263,15265],{"className":15264},[673],[608,15266,15268],{"className":15267},[677],[608,15269,15271],{"className":15270,"style":15252},[681],[608,15272,15273,15276],{"style":684},[608,15274],{"className":15275,"style":689},[688],[608,15277,15279],{"className":15278},[693,694,695,696],[608,15280,1504],{"className":15281},[658,894,696]," — formule de conversion vers le décimal.",[3063,15284,4348,15285,15313,15314,15372,15373,1043,15401,2097],{},[608,15286,15288,15301],{"className":15287},[611],[608,15289,15291],{"className":15290},[615],[617,15292,15293],{"xmlns":619},[621,15294,15295,15299],{},[624,15296,15297],{},[876,15298,1162],{},[637,15300,1162],{"encoding":639},[608,15302,15304],{"className":15303,"ariaHidden":645},[644],[608,15305,15307,15310],{"className":15306},[649],[608,15308],{"className":15309,"style":4374},[653],[608,15311,1162],{"className":15312},[658,894]," bits, on code ",[608,15315,15317,15334],{"className":15316},[611],[608,15318,15320],{"className":15319},[615],[617,15321,15322],{"xmlns":619},[621,15323,15324,15332],{},[624,15325,15326],{},[627,15327,15328,15330],{},[630,15329,782],{},[876,15331,1162],{},[637,15333,4399],{"encoding":639},[608,15335,15337],{"className":15336,"ariaHidden":645},[644],[608,15338,15340,15343],{"className":15339},[649],[608,15341],{"className":15342,"style":4409},[653],[608,15344,15346,15349],{"className":15345},[658],[608,15347,782],{"className":15348},[658],[608,15350,15352],{"className":15351},[669],[608,15353,15355],{"className":15354},[673],[608,15356,15358],{"className":15357},[677],[608,15359,15361],{"className":15360,"style":4409},[681],[608,15362,15363,15366],{"style":684},[608,15364],{"className":15365,"style":689},[688],[608,15367,15369],{"className":15368},[693,694,695,696],[608,15370,1162],{"className":15371},[658,894,696]," valeurs, de ",[608,15374,15376,15389],{"className":15375},[611],[608,15377,15379],{"className":15378},[615],[617,15380,15381],{"xmlns":619},[621,15382,15383,15387],{},[624,15384,15385],{},[630,15386,635],{},[637,15388,635],{"encoding":639},[608,15390,15392],{"className":15391,"ariaHidden":645},[644],[608,15393,15395,15398],{"className":15394},[649],[608,15396],{"className":15397,"style":981},[653],[608,15399,635],{"className":15400},[658],[608,15402,15404,15425],{"className":15403},[611],[608,15405,15407],{"className":15406},[615],[617,15408,15409],{"xmlns":619},[621,15410,15411,15423],{},[624,15412,15413,15419,15421],{},[627,15414,15415,15417],{},[630,15416,782],{},[876,15418,1162],{},[942,15420,1060],{},[630,15422,659],{},[637,15424,4492],{"encoding":639},[608,15426,15428,15472],{"className":15427,"ariaHidden":645},[644],[608,15429,15431,15434,15463,15466,15469],{"className":15430},[649],[608,15432],{"className":15433,"style":4502},[653],[608,15435,15437,15440],{"className":15436},[658],[608,15438,782],{"className":15439},[658],[608,15441,15443],{"className":15442},[669],[608,15444,15446],{"className":15445},[673],[608,15447,15449],{"className":15448},[677],[608,15450,15452],{"className":15451,"style":4409},[681],[608,15453,15454,15457],{"style":684},[608,15455],{"className":15456,"style":689},[688],[608,15458,15460],{"className":15459},[693,694,695,696],[608,15461,1162],{"className":15462},[658,894,696],[608,15464],{"className":15465,"style":1082},[966],[608,15467,1060],{"className":15468},[1086],[608,15470],{"className":15471,"style":1082},[966],[608,15473,15475,15478],{"className":15474},[649],[608,15476],{"className":15477,"style":981},[653],[608,15479,659],{"className":15480},[658],[3063,15482,15483],{},"1 chiffre hexadécimal = 4 bits exactement.",[594,15485,15487],{"id":15486},"à-savoir-reconnaître","À savoir reconnaître",[1859,15489,15490,15500],{},[1862,15491,15492],{},[1865,15493,15494,15497],{},[1868,15495,15496],{},"Situation",[1868,15498,15499],{},"Action à mener",[1881,15501,15502,15570,15578,15586],{},[1865,15503,15504,15507],{},[1886,15505,15506],{},"Convertir binaire → décimal",[1886,15508,15509,15510,15569],{},"Somme des ",[608,15511,15513,15531],{"className":15512},[611],[608,15514,15516],{"className":15515},[615],[617,15517,15518],{"xmlns":619},[621,15519,15520,15528],{},[624,15521,15522],{},[627,15523,15524,15526],{},[630,15525,782],{},[876,15527,1504],{},[637,15529,15530],{"encoding":639},"2^i",[608,15532,15534],{"className":15533,"ariaHidden":645},[644],[608,15535,15537,15540],{"className":15536},[649],[608,15538],{"className":15539,"style":15252},[653],[608,15541,15543,15546],{"className":15542},[658],[608,15544,782],{"className":15545},[658],[608,15547,15549],{"className":15548},[669],[608,15550,15552],{"className":15551},[673],[608,15553,15555],{"className":15554},[677],[608,15556,15558],{"className":15557,"style":15252},[681],[608,15559,15560,15563],{"style":684},[608,15561],{"className":15562,"style":689},[688],[608,15564,15566],{"className":15565},[693,694,695,696],[608,15567,1504],{"className":15568},[658,894,696]," pour les bits à 1.",[1865,15571,15572,15575],{},[1886,15573,15574],{},"Convertir décimal → binaire",[1886,15576,15577],{},"Divisions successives par 2, lecture des restes de bas en haut.",[1865,15579,15580,15583],{},[1886,15581,15582],{},"Convertir binaire ↔ hexa",[1886,15584,15585],{},"Regrouper par paquets de 4 bits, ou utiliser le tableau de correspondance.",[1865,15587,15588,15619],{},[1886,15589,15590,15591],{},"Compter le nombre de bits d'un entier ",[608,15592,15594,15607],{"className":15593},[611],[608,15595,15597],{"className":15596},[615],[617,15598,15599],{"xmlns":619},[621,15600,15601,15605],{},[624,15602,15603],{},[876,15604,1612],{},[637,15606,1612],{"encoding":639},[608,15608,15610],{"className":15609,"ariaHidden":645},[644],[608,15611,15613,15616],{"className":15612},[649],[608,15614],{"className":15615,"style":1665},[653],[608,15617,1612],{"className":15618,"style":1669},[658,894],[1886,15620,15621,15737,15738,15741],{},[608,15622,15624,15656],{"className":15623},[611],[608,15625,15627],{"className":15626},[615],[617,15628,15629],{"xmlns":619},[621,15630,15631,15653],{},[624,15632,15633,15635,15645,15647,15649,15651],{},[942,15634,5158],{"stretchy":1150},[1153,15636,15637,15643],{},[624,15638,15639,15641],{},[876,15640,5165],{},[942,15642,5168],{},[630,15644,782],{},[876,15646,1612],{},[942,15648,5179],{"stretchy":1150},[942,15650,2416],{},[630,15652,659],{},[637,15654,15655],{"encoding":639},"\\lfloor \\log_2 N \\rfloor + 1",[608,15657,15659,15728],{"className":15658,"ariaHidden":645},[644],[608,15660,15662,15665,15668,15710,15713,15716,15719,15722,15725],{"className":15661},[649],[608,15663],{"className":15664,"style":1213},[653],[608,15666,5158],{"className":15667},[1217],[608,15669,15671,15676],{"className":15670},[1689],[608,15672,5205,15674],{"className":15673},[1689],[608,15675,5209],{"style":5208},[608,15677,15679],{"className":15678},[669],[608,15680,15682,15702],{"className":15681},[673,1230],[608,15683,15685,15699],{"className":15684},[677],[608,15686,15688],{"className":15687,"style":5222},[681],[608,15689,15690,15693],{"style":5225},[608,15691],{"className":15692,"style":689},[688],[608,15694,15696],{"className":15695},[693,694,695,696],[608,15697,782],{"className":15698},[658,696],[608,15700,1263],{"className":15701},[1262],[608,15703,15705],{"className":15704},[677],[608,15706,15708],{"className":15707,"style":5244},[681],[608,15709],{},[608,15711],{"className":15712,"style":1325},[966],[608,15714,1612],{"className":15715,"style":1669},[658,894],[608,15717,5179],{"className":15718},[1417],[608,15720],{"className":15721,"style":1082},[966],[608,15723,2416],{"className":15724},[1086],[608,15726],{"className":15727,"style":1082},[966],[608,15729,15731,15734],{"className":15730},[649],[608,15732],{"className":15733,"style":981},[653],[608,15735,659],{"className":15736},[658],", ou ",[1894,15739,15740],{},"len(bin(N)) - 2"," en Python.",[594,15743,5340],{"id":5339},[3060,15745,15746,15837,15953],{},[3063,15747,15748,2036,15750,14727,15778,14991,15806,15808,15809,2097],{},[1894,15749,5471],{},[608,15751,15753,15766],{"className":15752},[611],[608,15754,15756],{"className":15755},[615],[617,15757,15758],{"xmlns":619},[621,15759,15760,15764],{},[624,15761,15762],{},[630,15763,782],{},[637,15765,782],{"encoding":639},[608,15767,15769],{"className":15768,"ariaHidden":645},[644],[608,15770,15772,15775],{"className":15771},[649],[608,15773],{"className":15774,"style":981},[653],[608,15776,782],{"className":15777},[658],[608,15779,15781,15794],{"className":15780},[611],[608,15782,15784],{"className":15783},[615],[617,15785,15786],{"xmlns":619},[621,15787,15788,15792],{},[624,15789,15790],{},[630,15791,632],{},[637,15793,632],{"encoding":639},[608,15795,15797],{"className":15796,"ariaHidden":645},[644],[608,15798,15800,15803],{"className":15799},[649],[608,15801],{"className":15802,"style":981},[653],[608,15804,632],{"className":15805},[658],[1894,15807,5504],{}," affiche ",[608,15810,15812,15825],{"className":15811},[611],[608,15813,15815],{"className":15814},[615],[617,15816,15817],{"xmlns":619},[621,15818,15819,15823],{},[624,15820,15821],{},[630,15822,782],{},[637,15824,782],{"encoding":639},[608,15826,15828],{"className":15827,"ariaHidden":645},[644],[608,15829,15831,15834],{"className":15830},[649],[608,15832],{"className":15833,"style":981},[653],[608,15835,782],{"className":15836},[658],[3063,15838,14696,15839,2066,15841,5826,15844,15924,15925,2097],{},[603,15840,649],{},[603,15842,15843],{},"chiffre",[608,15845,15847,15869],{"className":15846},[611],[608,15848,15850],{"className":15849},[615],[617,15851,15852],{"xmlns":619},[621,15853,15854,15866],{},[624,15855,15856,15858,15860],{},[942,15857,1151],{"stretchy":1150},[630,15859,632],{},[1153,15861,15862,15864],{},[942,15863,1198],{"stretchy":1150},[876,15865,878],{},[637,15867,15868],{"encoding":639},"(10)_b",[608,15870,15872],{"className":15871,"ariaHidden":645},[644],[608,15873,15875,15878,15881,15884],{"className":15874},[649],[608,15876],{"className":15877,"style":1213},[653],[608,15879,1151],{"className":15880},[1217],[608,15882,632],{"className":15883},[658],[608,15885,15887,15890],{"className":15886},[1417],[608,15888,1198],{"className":15889},[1417],[608,15891,15893],{"className":15892},[669],[608,15894,15896,15916],{"className":15895},[673,1230],[608,15897,15899,15913],{"className":15898},[677],[608,15900,15902],{"className":15901,"style":1433},[681],[608,15903,15904,15907],{"style":1240},[608,15905],{"className":15906,"style":689},[688],[608,15908,15910],{"className":15909},[693,694,695,696],[608,15911,878],{"className":15912},[658,894,696],[608,15914,1263],{"className":15915},[1262],[608,15917,15919],{"className":15918},[677],[608,15920,15922],{"className":15921,"style":1371},[681],[608,15923],{}," vaut toujours ",[608,15926,15928,15941],{"className":15927},[611],[608,15929,15931],{"className":15930},[615],[617,15932,15933],{"xmlns":619},[621,15934,15935,15939],{},[624,15936,15937],{},[876,15938,878],{},[637,15940,878],{"encoding":639},[608,15942,15944],{"className":15943,"ariaHidden":645},[644],[608,15945,15947,15950],{"className":15946},[649],[608,15948],{"className":15949,"style":890},[653],[608,15951,878],{"className":15952},[658,894],[3063,15954,15955],{},"En binaire, on lit de la droite (poids faible) vers la gauche (poids fort).",[594,15957,15959],{"id":15958},"syntaxe-python","Syntaxe Python",[2878,15961,15963],{"className":2880,"code":15962,"language":2882,"meta":2883,"style":2883},"int(\"1011\", 2)   # 11\nint(\"2F\", 16)    # 47\nbin(13)          # '0b1101'\nhex(255)         # '0xff'\noct(15)          # '0o17'\n",[1894,15964,15965,15981,15999,16013,16027],{"__ignoreMap":2883},[608,15966,15967,15969,15971,15973,15975,15977,15979],{"class":2888,"line":2889},[608,15968,2899],{"class":2892},[608,15970,1151],{"class":2896},[608,15972,2905],{"class":2904},[608,15974,1896],{"class":2896},[608,15976,782],{"class":2892},[608,15978,5333],{"class":2896},[608,15980,2916],{"class":2915},[608,15982,15983,15985,15987,15990,15992,15994,15996],{"class":2888,"line":2919},[608,15984,2899],{"class":2892},[608,15986,1151],{"class":2896},[608,15988,15989],{"class":2904},"\"2F\"",[608,15991,1896],{"class":2896},[608,15993,1945],{"class":2892},[608,15995,13214],{"class":2896},[608,15997,15998],{"class":2915},"# 47\n",[608,16000,16001,16003,16005,16007,16010],{"class":2888,"line":2943},[608,16002,5302],{"class":2892},[608,16004,1151],{"class":2896},[608,16006,3067],{"class":2892},[608,16008,16009],{"class":2896},")          ",[608,16011,16012],{"class":2915},"# '0b1101'\n",[608,16014,16015,16018,16020,16022,16024],{"class":2888,"line":3794},[608,16016,16017],{"class":2892},"hex",[608,16019,1151],{"class":2896},[608,16021,4789],{"class":2892},[608,16023,13230],{"class":2896},[608,16025,16026],{"class":2915},"# '0xff'\n",[608,16028,16029,16032,16034,16036,16038],{"class":2888,"line":7353},[608,16030,16031],{"class":2892},"oct",[608,16033,1151],{"class":2896},[608,16035,2017],{"class":2892},[608,16037,16009],{"class":2896},[608,16039,16040],{"class":2915},"# '0o17'\n",[5621,16042,16043],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":16045},[16046,16047,16048,16049,16050],{"id":14930,"depth":2919,"text":14931},{"id":14997,"depth":2919,"text":14998},{"id":15486,"depth":2919,"text":15487},{"id":5339,"depth":2919,"text":5340},{"id":15958,"depth":2919,"text":15959},"Mémo de révision sur les bases binaire, hexadécimale, octale et leurs conversions.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbases-entiers-positifs.memo",[],{"title":14924,"description":16051},"1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs.memo","cEzQeAoR_6g9f16k8fHg3RCQc91qV69ioPQoBdqd_V8",{"id":16059,"title":16060,"bo":16061,"body":16062,"description":16927,"eval":5636,"extension":5637,"labo":5636,"meta":16928,"navigation":5645,"path":16929,"quizzes":16930,"readingTime":2889,"seo":16931,"stem":16932,"tp":5636,"__hash__":16933},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.memo.md","Entiers relatifs et complément à deux — l'essentiel",[5654],{"type":591,"value":16063,"toc":16920},[16064,16066,16214,16216,16505,16507,16815,16817,16871,16873,16917],[594,16065,14931],{"id":14930},[599,16067,6190,16068,16070,16071,7412,16104,16132,16133,16213],{},[603,16069,5882],{}," code un entier négatif ",[608,16072,16074,16089],{"className":16073},[611],[608,16075,16077],{"className":16076},[615],[617,16078,16079],{"xmlns":619},[621,16080,16081,16087],{},[624,16082,16083,16085],{},[942,16084,1060],{},[876,16086,5898],{},[637,16088,6376],{"encoding":639},[608,16090,16092],{"className":16091,"ariaHidden":645},[644],[608,16093,16095,16098,16101],{"className":16094},[649],[608,16096],{"className":16097,"style":6386},[653],[608,16099,1060],{"className":16100},[658],[608,16102,5898],{"className":16103},[658,894],[608,16105,16107,16120],{"className":16106},[611],[608,16108,16110],{"className":16109},[615],[617,16111,16112],{"xmlns":619},[621,16113,16114,16118],{},[624,16115,16116],{},[876,16117,1162],{},[637,16119,1162],{"encoding":639},[608,16121,16123],{"className":16122,"ariaHidden":645},[644],[608,16124,16126,16129],{"className":16125},[649],[608,16127],{"className":16128,"style":4374},[653],[608,16130,1162],{"className":16131},[658,894]," bits comme\n",[608,16134,16136,16157],{"className":16135},[611],[608,16137,16139],{"className":16138},[615],[617,16140,16141],{"xmlns":619},[621,16142,16143,16155],{},[624,16144,16145,16151,16153],{},[627,16146,16147,16149],{},[630,16148,782],{},[876,16150,1162],{},[942,16152,1060],{},[876,16154,5898],{},[637,16156,6418],{"encoding":639},[608,16158,16160,16204],{"className":16159,"ariaHidden":645},[644],[608,16161,16163,16166,16195,16198,16201],{"className":16162},[649],[608,16164],{"className":16165,"style":4502},[653],[608,16167,16169,16172],{"className":16168},[658],[608,16170,782],{"className":16171},[658],[608,16173,16175],{"className":16174},[669],[608,16176,16178],{"className":16177},[673],[608,16179,16181],{"className":16180},[677],[608,16182,16184],{"className":16183,"style":4409},[681],[608,16185,16186,16189],{"style":684},[608,16187],{"className":16188,"style":689},[688],[608,16190,16192],{"className":16191},[693,694,695,696],[608,16193,1162],{"className":16194},[658,894,696],[608,16196],{"className":16197,"style":1082},[966],[608,16199,1060],{"className":16200},[1086],[608,16202],{"className":16203,"style":1082},[966],[608,16205,16207,16210],{"className":16206},[649],[608,16208],{"className":16209,"style":4374},[653],[608,16211,5898],{"className":16212},[658,894]," — ce qui permet de faire des additions sans circuit dédié au signe.",[594,16215,14998],{"id":14997},[3060,16217,16218,16425,16493,16502],{},[3063,16219,5850,16220,16248,16249,2097],{},[608,16221,16223,16236],{"className":16222},[611],[608,16224,16226],{"className":16225},[615],[617,16227,16228],{"xmlns":619},[621,16229,16230,16234],{},[624,16231,16232],{},[876,16233,1162],{},[637,16235,1162],{"encoding":639},[608,16237,16239],{"className":16238,"ariaHidden":645},[644],[608,16240,16242,16245],{"className":16241},[649],[608,16243],{"className":16244,"style":4374},[653],[608,16246,1162],{"className":16247},[658,894]," bits en complément à deux : intervalle ",[608,16250,16252,16304],{"className":16251},[611],[608,16253,16255],{"className":16254},[615],[617,16256,16257],{"xmlns":619},[621,16258,16259,16301],{},[624,16260,16261,16264,16266,16278,16280,16282,16294,16296,16298],{},[942,16262,16263],{"stretchy":1150},"[",[942,16265,1060],{},[627,16267,16268,16270],{},[630,16269,782],{},[624,16271,16272,16274,16276],{},[876,16273,1162],{},[942,16275,1060],{},[630,16277,659],{},[942,16279,5058],{"separator":645},[4060,16281,4062],{},[627,16283,16284,16286],{},[630,16285,782],{},[624,16287,16288,16290,16292],{},[876,16289,1162],{},[942,16291,1060],{},[630,16293,659],{},[942,16295,1060],{},[630,16297,659],{},[942,16299,16300],{"stretchy":1150},"]",[637,16302,16303],{"encoding":639},"[-2^{n-1},\\ 2^{n-1} - 1]",[608,16305,16307,16413],{"className":16306,"ariaHidden":645},[644],[608,16308,16310,16313,16316,16319,16357,16360,16363,16366,16404,16407,16410],{"className":16309},[649],[608,16311],{"className":16312,"style":8433},[653],[608,16314,16263],{"className":16315},[1217],[608,16317,1060],{"className":16318},[658],[608,16320,16322,16325],{"className":16321},[658],[608,16323,782],{"className":16324},[658],[608,16326,16328],{"className":16327},[669],[608,16329,16331],{"className":16330},[673],[608,16332,16334],{"className":16333},[677],[608,16335,16337],{"className":16336,"style":654},[681],[608,16338,16339,16342],{"style":684},[608,16340],{"className":16341,"style":689},[688],[608,16343,16345],{"className":16344},[693,694,695,696],[608,16346,16348,16351,16354],{"className":16347},[658,696],[608,16349,1162],{"className":16350},[658,894,696],[608,16352,1060],{"className":16353},[1086,696],[608,16355,659],{"className":16356},[658,696],[608,16358,5058],{"className":16359},[5057],[608,16361,4062],{"className":16362},[966],[608,16364],{"className":16365,"style":1325},[966],[608,16367,16369,16372],{"className":16368},[658],[608,16370,782],{"className":16371},[658],[608,16373,16375],{"className":16374},[669],[608,16376,16378],{"className":16377},[673],[608,16379,16381],{"className":16380},[677],[608,16382,16384],{"className":16383,"style":654},[681],[608,16385,16386,16389],{"style":684},[608,16387],{"className":16388,"style":689},[688],[608,16390,16392],{"className":16391},[693,694,695,696],[608,16393,16395,16398,16401],{"className":16394},[658,696],[608,16396,1162],{"className":16397},[658,894,696],[608,16399,1060],{"className":16400},[1086,696],[608,16402,659],{"className":16403},[658,696],[608,16405],{"className":16406,"style":1082},[966],[608,16408,1060],{"className":16409},[1086],[608,16411],{"className":16412,"style":1082},[966],[608,16414,16416,16419,16422],{"className":16415},[649],[608,16417],{"className":16418,"style":1213},[653],[608,16420,659],{"className":16421},[658],[608,16423,16300],{"className":16424},[1417],[3063,16426,16427,16463,16464,16492],{},[603,16428,16429,16430],{},"Méthode pour coder ",[608,16431,16433,16448],{"className":16432},[611],[608,16434,16436],{"className":16435},[615],[617,16437,16438],{"xmlns":619},[621,16439,16440,16446],{},[624,16441,16442,16444],{},[942,16443,1060],{},[876,16445,5898],{},[637,16447,6376],{"encoding":639},[608,16449,16451],{"className":16450,"ariaHidden":645},[644],[608,16452,16454,16457,16460],{"className":16453},[649],[608,16455],{"className":16456,"style":6386},[653],[608,16458,1060],{"className":16459},[658],[608,16461,5898],{"className":16462},[658,894]," : écrire ",[608,16465,16467,16480],{"className":16466},[611],[608,16468,16470],{"className":16469},[615],[617,16471,16472],{"xmlns":619},[621,16473,16474,16478],{},[624,16475,16476],{},[876,16477,5898],{},[637,16479,5898],{"encoding":639},[608,16481,16483],{"className":16482,"ariaHidden":645},[644],[608,16484,16486,16489],{"className":16485},[649],[608,16487],{"className":16488,"style":4374},[653],[608,16490,5898],{"className":16491},[658,894]," en binaire, inverser tous les bits,\najouter 1.",[3063,16494,16495,16496,16498,16499,16501],{},"Le bit de poids fort = bit de signe (",[1894,16497,635],{}," positif, ",[1894,16500,659],{}," négatif).",[3063,16503,16504],{},"Tailles courantes : 8, 16, 32, 64 bits.",[594,16506,15487],{"id":15486},[1859,16508,16509,16517],{},[1862,16510,16511],{},[1865,16512,16513,16515],{},[1868,16514,15496],{},[1868,16516,15499],{},[1881,16518,16519,16595,16662,16747],{},[1865,16520,16521,16586],{},[1886,16522,16523,16524,7412,16557,16585],{},"Coder ",[608,16525,16527,16542],{"className":16526},[611],[608,16528,16530],{"className":16529},[615],[617,16531,16532],{"xmlns":619},[621,16533,16534,16540],{},[624,16535,16536,16538],{},[942,16537,1060],{},[876,16539,5898],{},[637,16541,6376],{"encoding":639},[608,16543,16545],{"className":16544,"ariaHidden":645},[644],[608,16546,16548,16551,16554],{"className":16547},[649],[608,16549],{"className":16550,"style":6386},[653],[608,16552,1060],{"className":16553},[658],[608,16555,5898],{"className":16556},[658,894],[608,16558,16560,16573],{"className":16559},[611],[608,16561,16563],{"className":16562},[615],[617,16564,16565],{"xmlns":619},[621,16566,16567,16571],{},[624,16568,16569],{},[876,16570,1162],{},[637,16572,1162],{"encoding":639},[608,16574,16576],{"className":16575,"ariaHidden":645},[644],[608,16577,16579,16582],{"className":16578},[649],[608,16580],{"className":16581,"style":4374},[653],[608,16583,1162],{"className":16584},[658,894]," bits",[1886,16587,16588,16591,16592,2097],{},[1894,16589,16590],{},"not(x)"," bit à bit puis ",[1894,16593,16594],{},"+1",[1865,16596,16597,16600],{},[1886,16598,16599],{},"Lire un négatif",[1886,16601,16602,16603,16661],{},"Bit de poids fort à 1, soustraire ",[608,16604,16606,16623],{"className":16605},[611],[608,16607,16609],{"className":16608},[615],[617,16610,16611],{"xmlns":619},[621,16612,16613,16621],{},[624,16614,16615],{},[627,16616,16617,16619],{},[630,16618,782],{},[876,16620,1162],{},[637,16622,4399],{"encoding":639},[608,16624,16626],{"className":16625,"ariaHidden":645},[644],[608,16627,16629,16632],{"className":16628},[649],[608,16630],{"className":16631,"style":4409},[653],[608,16633,16635,16638],{"className":16634},[658],[608,16636,782],{"className":16637},[658],[608,16639,16641],{"className":16640},[669],[608,16642,16644],{"className":16643},[673],[608,16645,16647],{"className":16646},[677],[608,16648,16650],{"className":16649,"style":4409},[681],[608,16651,16652,16655],{"style":684},[608,16653],{"className":16654,"style":689},[688],[608,16656,16658],{"className":16657},[693,694,695,696],[608,16659,1162],{"className":16660},[658,894,696]," à la valeur lue.",[1865,16663,16664,16695],{},[1886,16665,16666,16667,16585],{},"Estimer la taille d'une somme de ",[608,16668,16670,16683],{"className":16669},[611],[608,16671,16673],{"className":16672},[615],[617,16674,16675],{"xmlns":619},[621,16676,16677,16681],{},[624,16678,16679],{},[876,16680,1162],{},[637,16682,1162],{"encoding":639},[608,16684,16686],{"className":16685,"ariaHidden":645},[644],[608,16687,16689,16692],{"className":16688},[649],[608,16690],{"className":16691,"style":4374},[653],[608,16693,1162],{"className":16694},[658,894],[1886,16696,16697,8297],{},[608,16698,16700,16717],{"className":16699},[611],[608,16701,16703],{"className":16702},[615],[617,16704,16705],{"xmlns":619},[621,16706,16707,16715],{},[624,16708,16709,16711,16713],{},[876,16710,1162],{},[942,16712,2416],{},[630,16714,659],{},[637,16716,8266],{"encoding":639},[608,16718,16720,16738],{"className":16719,"ariaHidden":645},[644],[608,16721,16723,16726,16729,16732,16735],{"className":16722},[649],[608,16724],{"className":16725,"style":6386},[653],[608,16727,1162],{"className":16728},[658,894],[608,16730],{"className":16731,"style":1082},[966],[608,16733,2416],{"className":16734},[1086],[608,16736],{"className":16737,"style":1082},[966],[608,16739,16741,16744],{"className":16740},[649],[608,16742],{"className":16743,"style":981},[653],[608,16745,659],{"className":16746},[658],[1865,16748,16749,16780],{},[1886,16750,16751,16752,16585],{},"Estimer la taille d'un produit de ",[608,16753,16755,16768],{"className":16754},[611],[608,16756,16758],{"className":16757},[615],[617,16759,16760],{"xmlns":619},[621,16761,16762,16766],{},[624,16763,16764],{},[876,16765,1162],{},[637,16767,1162],{"encoding":639},[608,16769,16771],{"className":16770,"ariaHidden":645},[644],[608,16772,16774,16777],{"className":16773},[649],[608,16775],{"className":16776,"style":4374},[653],[608,16778,1162],{"className":16779},[658,894],[1886,16781,16782,8297],{},[608,16783,16785,16800],{"className":16784},[611],[608,16786,16788],{"className":16787},[615],[617,16789,16790],{"xmlns":619},[621,16791,16792,16798],{},[624,16793,16794,16796],{},[630,16795,782],{},[876,16797,1162],{},[637,16799,8535],{"encoding":639},[608,16801,16803],{"className":16802,"ariaHidden":645},[644],[608,16804,16806,16809,16812],{"className":16805},[649],[608,16807],{"className":16808,"style":981},[653],[608,16810,782],{"className":16811},[658],[608,16813,1162],{"className":16814},[658,894],[594,16816,5340],{"id":5339},[3060,16818,16819,16825,16865],{},[3063,16820,16821,16822,16824],{},"L'intervalle n'est ",[603,16823,8180],{}," : un négatif de plus que de positifs.",[3063,16826,16827,16860,16861,16864],{},[608,16828,16830,16845],{"className":16829},[611],[608,16831,16833],{"className":16832},[615],[617,16834,16835],{"xmlns":619},[621,16836,16837,16843],{},[624,16838,16839,16841],{},[942,16840,1060],{},[630,16842,659],{},[637,16844,8674],{"encoding":639},[608,16846,16848],{"className":16847,"ariaHidden":645},[644],[608,16849,16851,16854,16857],{"className":16850},[649],[608,16852],{"className":16853,"style":2794},[653],[608,16855,1060],{"className":16856},[658],[608,16858,659],{"className":16859},[658]," en complément à deux vaut ",[1894,16862,16863],{},"1111 ... 1111"," (tous les bits à 1).",[3063,16866,16867,16868,16870],{},"Python n'a ",[603,16869,5842],{}," de taille fixe pour ses entiers : pas de débordement.",[594,16872,15959],{"id":15958},[2878,16874,16876],{"className":2880,"code":16875,"language":2882,"meta":2883,"style":2883},"x = -5\ncode = x & 0xFF      # complément à deux sur 8 bits → 251\nbin(code)            # '0b11111011'\n",[1894,16877,16878,16888,16907],{"__ignoreMap":2883},[608,16879,16880,16882,16884,16886],{"class":2888,"line":2889},[608,16881,7342],{"class":2896},[608,16883,1615],{"class":3722},[608,16885,7347],{"class":3722},[608,16887,7350],{"class":2892},[608,16889,16890,16892,16894,16896,16898,16901,16904],{"class":2888,"line":2919},[608,16891,7362],{"class":2896},[608,16893,1615],{"class":3722},[608,16895,7367],{"class":2896},[608,16897,7370],{"class":3722},[608,16899,16900],{"class":3722}," 0x",[608,16902,16903],{"class":2892},"FF",[608,16905,16906],{"class":2915},"      # complément à deux sur 8 bits → 251\n",[608,16908,16909,16911,16914],{"class":2888,"line":2943},[608,16910,5302],{"class":2892},[608,16912,16913],{"class":2896},"(code)            ",[608,16915,16916],{"class":2915},"# '0b11111011'\n",[5621,16918,16919],{},"html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":16921},[16922,16923,16924,16925,16926],{"id":14930,"depth":2919,"text":14931},{"id":14997,"depth":2919,"text":14998},{"id":15486,"depth":2919,"text":15487},{"id":5339,"depth":2919,"text":5340},{"id":15958,"depth":2919,"text":15959},"Mémo de révision sur la représentation binaire des entiers signés.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs.memo",[],{"title":16060,"description":16927},"1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.memo","_qSVkfWrJe7L9M8vAcZQh6-D5nu1dmDQ2Y70e_mtBlo",{"id":16935,"title":16936,"bo":16937,"body":16938,"description":17554,"eval":5636,"extension":5637,"labo":5636,"meta":17555,"navigation":5645,"path":17556,"quizzes":17557,"readingTime":2889,"seo":17558,"stem":17559,"tp":5636,"__hash__":17560},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.memo.md","Nombres flottants — l'essentiel",[8827],{"type":591,"value":16939,"toc":17547},[16940,16942,17101,17103,17333,17335,17391,17393,17452,17454,17544],[594,16941,14931],{"id":14930},[599,16943,16944,16945,16948,16949,17097,17098,2097],{},"Un ",[603,16946,16947],{},"flottant"," approxime un nombre réel par ",[608,16950,16952,16988],{"className":16951},[611],[608,16953,16955],{"className":16954},[615],[617,16956,16957],{"xmlns":619},[621,16958,16959,16985],{},[624,16960,16961,16963,16965,16967,16973,16975,16977,16979],{},[942,16962,1151],{"stretchy":1150},[942,16964,1060],{},[630,16966,659],{},[627,16968,16969,16971],{},[942,16970,1198],{"stretchy":1150},[876,16972,9411],{},[942,16974,1646],{},[876,16976,9416],{},[942,16978,1646],{},[627,16980,16981,16983],{},[630,16982,782],{},[876,16984,9425],{},[637,16986,16987],{"encoding":639},"(-1)^s \\cdot m \\cdot 2^e",[608,16989,16991,17044,17062],{"className":16990,"ariaHidden":645},[644],[608,16992,16994,16997,17000,17003,17006,17035,17038,17041],{"className":16993},[649],[608,16995],{"className":16996,"style":1213},[653],[608,16998,1151],{"className":16999},[1217],[608,17001,1060],{"className":17002},[658],[608,17004,659],{"className":17005},[658],[608,17007,17009,17012],{"className":17008},[1417],[608,17010,1198],{"className":17011},[1417],[608,17013,17015],{"className":17014},[669],[608,17016,17018],{"className":17017},[673],[608,17019,17021],{"className":17020},[677],[608,17022,17024],{"className":17023,"style":4409},[681],[608,17025,17026,17029],{"style":684},[608,17027],{"className":17028,"style":689},[688],[608,17030,17032],{"className":17031},[693,694,695,696],[608,17033,9411],{"className":17034},[658,894,696],[608,17036],{"className":17037,"style":1082},[966],[608,17039,1646],{"className":17040},[1086],[608,17042],{"className":17043,"style":1082},[966],[608,17045,17047,17050,17053,17056,17059],{"className":17046},[649],[608,17048],{"className":17049,"style":9510},[653],[608,17051,9416],{"className":17052},[658,894],[608,17054],{"className":17055,"style":1082},[966],[608,17057,1646],{"className":17058},[1086],[608,17060],{"className":17061,"style":1082},[966],[608,17063,17065,17068],{"className":17064},[649],[608,17066],{"className":17067,"style":4409},[653],[608,17069,17071,17074],{"className":17070},[658],[608,17072,782],{"className":17073},[658],[608,17075,17077],{"className":17076},[669],[608,17078,17080],{"className":17079},[673],[608,17081,17083],{"className":17082},[677],[608,17084,17086],{"className":17085,"style":4409},[681],[608,17087,17088,17091],{"style":684},[608,17089],{"className":17090,"style":689},[688],[608,17092,17094],{"className":17093},[693,694,695,696],[608,17095,9425],{"className":17096},[658,894,696]," — d'où\ndes erreurs d'arrondi inévitables, dont la plus célèbre est\n",[1894,17099,17100],{},"0.1 + 0.2 != 0.3",[594,17102,14998],{"id":14997},[3060,17104,17105,17202,17207,17310,17320],{},[3063,17106,17107,17108,1896,17139,1896,17170,17201],{},"Décomposition : ",[603,17109,17110,17111],{},"signe ",[608,17112,17114,17127],{"className":17113},[611],[608,17115,17117],{"className":17116},[615],[617,17118,17119],{"xmlns":619},[621,17120,17121,17125],{},[624,17122,17123],{},[876,17124,9411],{},[637,17126,9411],{"encoding":639},[608,17128,17130],{"className":17129,"ariaHidden":645},[644],[608,17131,17133,17136],{"className":17132},[649],[608,17134],{"className":17135,"style":4374},[653],[608,17137,9411],{"className":17138},[658,894],[603,17140,17141,17142],{},"mantisse ",[608,17143,17145,17158],{"className":17144},[611],[608,17146,17148],{"className":17147},[615],[617,17149,17150],{"xmlns":619},[621,17151,17152,17156],{},[624,17153,17154],{},[876,17155,9416],{},[637,17157,9416],{"encoding":639},[608,17159,17161],{"className":17160,"ariaHidden":645},[644],[608,17162,17164,17167],{"className":17163},[649],[608,17165],{"className":17166,"style":4374},[653],[608,17168,9416],{"className":17169},[658,894],[603,17171,17172,17173],{},"exposant ",[608,17174,17176,17189],{"className":17175},[611],[608,17177,17179],{"className":17178},[615],[617,17180,17181],{"xmlns":619},[621,17182,17183,17187],{},[624,17184,17185],{},[876,17186,9425],{},[637,17188,9425],{"encoding":639},[608,17190,17192],{"className":17191,"ariaHidden":645},[644],[608,17193,17195,17198],{"className":17194},[649],[608,17196],{"className":17197,"style":4374},[653],[608,17199,9425],{"className":17200},[658,894],", base 2.",[3063,17203,17204,17205,11399],{},"Tailles IEEE 754 : simple précision (32 bits), double précision (64 bits — Python ",[1894,17206,9798],{},[3063,17208,17209,17306,17307,17309],{},[608,17210,17212,17229],{"className":17211},[611],[608,17213,17215],{"className":17214},[615],[617,17216,17217],{"xmlns":619},[621,17218,17219,17227],{},[624,17220,17221],{},[8888,17222,17223,17225],{},[630,17224,659],{},[630,17226,632],{},[637,17228,10001],{"encoding":639},[608,17230,17232],{"className":17231,"ariaHidden":645},[644],[608,17233,17235,17238],{"className":17234},[649],[608,17236],{"className":17237,"style":8906},[653],[608,17239,17241,17244,17303],{"className":17240},[658],[608,17242],{"className":17243},[1217,8913],[608,17245,17247],{"className":17246},[8888],[608,17248,17250,17295],{"className":17249},[673,1230],[608,17251,17253,17292],{"className":17252},[677],[608,17254,17256,17270,17278],{"className":17255,"style":8926},[681],[608,17257,17258,17261],{"style":8929},[608,17259],{"className":17260,"style":8933},[688],[608,17262,17264],{"className":17263},[693,694,695,696],[608,17265,17267],{"className":17266},[658,696],[608,17268,632],{"className":17269},[658,696],[608,17271,17272,17275],{"style":8945},[608,17273],{"className":17274,"style":8933},[688],[608,17276],{"className":17277,"style":8953},[8952],[608,17279,17280,17283],{"style":8956},[608,17281],{"className":17282,"style":8933},[688],[608,17284,17286],{"className":17285},[693,694,695,696],[608,17287,17289],{"className":17288},[658,696],[608,17290,659],{"className":17291},[658,696],[608,17293,1263],{"className":17294},[1262],[608,17296,17298],{"className":17297},[677],[608,17299,17301],{"className":17300,"style":8978},[681],[608,17302],{},[608,17304],{"className":17305},[1417,8913]," n'a ",[603,17308,5842],{}," d'écriture finie en binaire.",[3063,17311,17312,11431,17314,17316,17317,17319],{},[603,17313,11430],{},[1894,17315,11434],{}," pour deux flottants : utiliser ",[1894,17318,11081],{}," ou une tolérance.",[3063,17321,17322,17323,1896,17325,1896,17327,17329,17330,11399],{},"Valeurs spéciales : ",[1894,17324,11135],{},[1894,17326,11138],{},[1894,17328,11141],{}," (et ",[1894,17331,17332],{},"nan != nan",[594,17334,15487],{"id":15486},[1859,17336,17337,17345],{},[1862,17338,17339],{},[1865,17340,17341,17343],{},[1868,17342,15496],{},[1868,17344,15499],{},[1881,17346,17347,17360,17371,17381],{},[1865,17348,17349,17352],{},[1886,17350,17351],{},"Comparer deux flottants",[1886,17353,17354,9261,17357,2097],{},[1894,17355,17356],{},"math.isclose(a, b)",[1894,17358,17359],{},"abs(a - b) \u003C eps",[1865,17361,17362,17365],{},[1886,17363,17364],{},"Calcul monétaire exact",[1886,17366,17367,17368,2097],{},"Utiliser ",[1894,17369,17370],{},"decimal.Decimal",[1865,17372,17373,17376],{},[1886,17374,17375],{},"Calcul rationnel exact",[1886,17377,17367,17378,2097],{},[1894,17379,17380],{},"fractions.Fraction",[1865,17382,17383,17386],{},[1886,17384,17385],{},"Vérifier que c'est bien un nombre",[1886,17387,17388,2097],{},[1894,17389,17390],{},"not math.isnan(x) and math.isfinite(x)",[594,17392,5340],{"id":5339},[3060,17394,17395,17402,17405],{},[3063,17396,17397,17399,17400,2097],{},[1894,17398,10891],{}," renvoie ",[1894,17401,10950],{},[3063,17403,17404],{},"Sommer beaucoup de petits termes perd de la précision (les petits sont écrasés par les grands).",[3063,17406,17407,17408,11447,17411,17413,17414,17451],{},"Convertir ",[1894,17409,17410],{},"\"0.1\"",[1894,17412,9798],{}," recrée la même approximation — pas de chemin pour avoir un ",[608,17415,17417,17430],{"className":17416},[611],[608,17418,17420],{"className":17419},[615],[617,17421,17422],{"xmlns":619},[621,17423,17424,17428],{},[624,17425,17426],{},[630,17427,9952],{},[637,17429,9955],{"encoding":639},[608,17431,17433],{"className":17432,"ariaHidden":645},[644],[608,17434,17436,17439,17442,17448],{"className":17435},[649],[608,17437],{"className":17438,"style":5047},[653],[608,17440,635],{"className":17441},[658],[608,17443,17445],{"className":17444},[658],[608,17446,5058],{"className":17447},[5057],[608,17449,659],{"className":17450},[658]," exact.",[594,17453,15959],{"id":15958},[2878,17455,17457],{"className":2880,"code":17456,"language":2882,"meta":2883,"style":2883},"import math\nmath.isclose(0.1 + 0.2, 0.3)   # True\nmath.inf                       # infini positif\nmath.isnan(float('nan'))       # True\n\nfrom decimal import Decimal\nDecimal('0.1') + Decimal('0.2')  # Decimal('0.3') — exact\n",[1894,17458,17459,17465,17484,17492,17508,17512,17522],{"__ignoreMap":2883},[608,17460,17461,17463],{"class":2888,"line":2889},[608,17462,10771],{"class":3722},[608,17464,11094],{"class":2896},[608,17466,17467,17470,17472,17474,17476,17478,17480,17482],{"class":2888,"line":2919},[608,17468,17469],{"class":2896},"math.isclose(",[608,17471,10629],{"class":2892},[608,17473,10632],{"class":3722},[608,17475,10635],{"class":2892},[608,17477,1896],{"class":2896},[608,17479,9011],{"class":2892},[608,17481,5333],{"class":2896},[608,17483,11075],{"class":2915},[608,17485,17486,17489],{"class":2888,"line":2943},[608,17487,17488],{"class":2896},"math.inf                       ",[608,17490,17491],{"class":2915},"# infini positif\n",[608,17493,17494,17497,17499,17501,17503,17506],{"class":2888,"line":3794},[608,17495,17496],{"class":2896},"math.isnan(",[608,17498,9798],{"class":2892},[608,17500,1151],{"class":2896},[608,17502,11354],{"class":2904},[608,17504,17505],{"class":2896},"))       ",[608,17507,11075],{"class":2915},[608,17509,17510],{"class":2888,"line":7353},[608,17511,8596],{"emptyLinePlaceholder":5639},[608,17513,17514,17516,17518,17520],{"class":2888,"line":7359},[608,17515,10765],{"class":3722},[608,17517,10768],{"class":2896},[608,17519,10771],{"class":3722},[608,17521,10774],{"class":2896},[608,17523,17524,17527,17530,17532,17534,17536,17539,17541],{"class":2888,"line":7395},[608,17525,17526],{"class":2896},"Decimal(",[608,17528,17529],{"class":2904},"'0.1'",[608,17531,10807],{"class":2896},[608,17533,2416],{"class":3722},[608,17535,10812],{"class":2896},[608,17537,17538],{"class":2904},"'0.2'",[608,17540,7389],{"class":2896},[608,17542,17543],{"class":2915},"# Decimal('0.3') — exact\n",[5621,17545,17546],{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki 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var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":17548},[17549,17550,17551,17552,17553],{"id":14930,"depth":2919,"text":14931},{"id":14997,"depth":2919,"text":14998},{"id":15486,"depth":2919,"text":15487},{"id":5339,"depth":2919,"text":5340},{"id":15958,"depth":2919,"text":15959},"Mémo de révision sur la représentation IEEE 754 et les approximations.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants.memo",[],{"title":16936,"description":17554},"1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.memo","uJ-K2xS15p4OyKBDiBFPTwe4ZXL00h0tJGWzTxjacTs",{"id":17562,"title":17563,"bo":17564,"body":17565,"description":17830,"eval":5636,"extension":5637,"labo":5636,"meta":17831,"navigation":5645,"path":17832,"quizzes":17833,"readingTime":2889,"seo":17834,"stem":17835,"tp":5636,"__hash__":17836},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages.memo.md","Textes et encodages — l'essentiel",[11704],{"type":591,"value":17566,"toc":17823},[17567,17569,17574,17576,17614,17616,17688,17690,17714,17716,17820],[594,17568,14931],{"id":14930},[599,17570,16944,17571,17573],{},[603,17572,12330],{}," associe à chaque caractère une suite d'octets — différents\nencodages traduisent les mêmes caractères différemment, d'où les bugs quand on\nmélange.",[594,17575,14998],{"id":14997},[3060,17577,17578,17583,17588,17597,17602],{},[3063,17579,17580,17582],{},[603,17581,11733],{}," : 7 bits, 128 caractères. Lettres latines non accentuées,\nchiffres, ponctuation. Pas d'accents.",[3063,17584,17585,17587],{},[603,17586,12110],{}," (Latin-1) : 8 bits, ASCII + caractères d'Europe de l'Ouest.",[3063,17589,17590,17592,17593,11379,17595,2097],{},[603,17591,12211],{}," : catalogue universel — chaque caractère a un ",[844,17594,12222],{},[1894,17596,12226],{},[3063,17598,17599,17601],{},[603,17600,12340],{}," : encodage Unicode dominant. 1 à 4 octets par caractère.\nRétrocompatible ASCII.",[3063,17603,17604,17605,17607,17608,17610,17611,17613],{},"Codes ASCII clés : ",[1894,17606,1958],{}," = 65, ",[1894,17609,7225],{}," = 97, ",[1894,17612,635],{}," = 48, espace = 32.",[594,17615,15487],{"id":15486},[1859,17617,17618,17626],{},[1862,17619,17620],{},[1865,17621,17622,17624],{},[1868,17623,15496],{},[1868,17625,15499],{},[1881,17627,17628,17638,17648,17658,17668,17678],{},[1865,17629,17630,17635],{},[1886,17631,17632,17634],{},[1894,17633,12673],{}," apparaît dans le texte",[1886,17636,17637],{},"Fichier UTF-8 lu comme ISO-8859-1.",[1865,17639,17640,17643],{},[1886,17641,17642],{},"Ouvrir un fichier texte en Python",[1886,17644,17645,2097],{},[1894,17646,17647],{},"open(..., encoding=\"utf-8\")",[1865,17649,17650,17653],{},[1886,17651,17652],{},"Convertir une chaîne en octets",[1886,17654,17655,2097],{},[1894,17656,17657],{},"s.encode(\"utf-8\")",[1865,17659,17660,17663],{},[1886,17661,17662],{},"Convertir des octets en chaîne",[1886,17664,17665,2097],{},[1894,17666,17667],{},"b.decode(\"utf-8\")",[1865,17669,17670,17673],{},[1886,17671,17672],{},"Récupérer le code d'un caractère",[1886,17674,17675,2097],{},[1894,17676,17677],{},"ord(c)",[1865,17679,17680,17683],{},[1886,17681,17682],{},"Récupérer le caractère d'un code",[1886,17684,17685,2097],{},[1894,17686,17687],{},"chr(n)",[594,17689,5340],{"id":5339},[3060,17691,17692,17700,17707],{},[3063,17693,12678,17694,17696,17697,17699],{},[1894,17695,12923],{}," (nb caractères) ≠ ",[1894,17698,12916],{}," (nb octets).",[3063,17701,17702,17703,17706],{},"ISO-8859-1 ne contient pas tous les caractères français (le ",[1894,17704,17705],{},"œ",", par\nexemple, n'y est pas).",[3063,17708,17709,17710,17713],{},"Toujours préciser ",[1894,17711,17712],{},"encoding=\"utf-8\""," à l'ouverture d'un fichier texte.",[594,17715,15959],{"id":15958},[2878,17717,17719],{"className":2880,"code":17718,"language":2882,"meta":2883,"style":2883},"ord('é')                # 233\nchr(233)                # 'é'\n\"café\".encode(\"utf-8\")  # b'caf\\xc3\\xa9'\nb'caf\\xc3\\xa9'.decode(\"utf-8\")  # 'café'\n\nwith open(\"notes.txt\", encoding=\"utf-8\") as f:\n    contenu = f.read()\n",[1894,17720,17721,17736,17749,17763,17784,17788,17812],{"__ignoreMap":2883},[608,17722,17723,17725,17727,17730,17733],{"class":2888,"line":2889},[608,17724,12011],{"class":2892},[608,17726,1151],{"class":2896},[608,17728,17729],{"class":2904},"'é'",[608,17731,17732],{"class":2896},")                ",[608,17734,17735],{"class":2915},"# 233\n",[608,17737,17738,17740,17742,17744,17746],{"class":2888,"line":2919},[608,17739,12030],{"class":2892},[608,17741,1151],{"class":2896},[608,17743,12155],{"class":2892},[608,17745,17732],{"class":2896},[608,17747,17748],{"class":2915},"# 'é'\n",[608,17750,17751,17754,17757,17759,17761],{"class":2888,"line":2943},[608,17752,17753],{"class":2904},"\"café\"",[608,17755,17756],{"class":2896},".encode(",[608,17758,12476],{"class":2904},[608,17760,7389],{"class":2896},[608,17762,12816],{"class":2915},[608,17764,17765,17767,17769,17771,17774,17777,17779,17781],{"class":2888,"line":3794},[608,17766,878],{"class":3722},[608,17768,12857],{"class":2904},[608,17770,12860],{"class":2892},[608,17772,17773],{"class":2904},"'",[608,17775,17776],{"class":2896},".decode(",[608,17778,12476],{"class":2904},[608,17780,7389],{"class":2896},[608,17782,17783],{"class":2915},"# 'café'\n",[608,17785,17786],{"class":2888,"line":7353},[608,17787,8596],{"emptyLinePlaceholder":5639},[608,17789,17790,17792,17794,17796,17798,17800,17802,17804,17806,17808,17810],{"class":2888,"line":7359},[608,17791,12744],{"class":3722},[608,17793,12747],{"class":2892},[608,17795,1151],{"class":2896},[608,17797,12752],{"class":2904},[608,17799,1896],{"class":2896},[608,17801,12758],{"class":12757},[608,17803,1615],{"class":3722},[608,17805,12476],{"class":2904},[608,17807,10807],{"class":2896},[608,17809,12767],{"class":3722},[608,17811,12770],{"class":2896},[608,17813,17814,17816,17818],{"class":2888,"line":7395},[608,17815,12775],{"class":2896},[608,17817,1615],{"class":3722},[608,17819,12780],{"class":2896},[5621,17821,17822],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":17824},[17825,17826,17827,17828,17829],{"id":14930,"depth":2919,"text":14931},{"id":14997,"depth":2919,"text":14998},{"id":15486,"depth":2919,"text":15487},{"id":5339,"depth":2919,"text":5340},{"id":15958,"depth":2919,"text":15959},"Mémo de révision sur ASCII, ISO-8859-1, Unicode et UTF-8.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Ftextes-encodages.memo",[],{"title":17563,"description":17830},"1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages.memo","IOyQuG9fETr1PKFvuODWLQCg847LsYpHuWqcOP6Tnt8",{"id":17838,"title":17839,"bo":17840,"body":17841,"description":18287,"eval":5636,"extension":5637,"labo":5636,"meta":18288,"navigation":5645,"path":18289,"quizzes":18290,"readingTime":2889,"seo":18291,"stem":18292,"tp":5636,"__hash__":18293},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens.memo.md","Valeurs booléennes — l'essentiel",[12961],{"type":591,"value":17842,"toc":18280},[17843,17845,17858,17860,18026,18028,18159,18161,18201,18203,18277],[594,17844,14931],{"id":14930},[599,17846,13271,17847,17850,17851,1896,17853,1896,17855,17857],{},[603,17848,17849],{},"valeur booléenne"," vaut vrai ou faux ; les opérateurs ",[1894,17852,13040],{},[1894,17854,13090],{},[1894,17856,13138],{},"\npermettent de combiner des conditions pour en faire des décisions.",[594,17859,14998],{"id":14997},[3060,17861,17862,17870,17994,18004],{},[3063,17863,17864,17865,2066,17867,17869],{},"En Python : ",[1894,17866,10947],{},[1894,17868,10950],{}," (majuscule initiale obligatoire).",[3063,17871,17872,17873],{},"Tables de vérité élémentaires :\n",[3060,17874,17875,17940,17948,17982],{},[3063,17876,17877,17879,17880,11379,17883,2066,17911,17939],{},[1894,17878,14307],{}," : vrai ",[603,17881,17882],{},"uniquement si",[608,17884,17886,17899],{"className":17885},[611],[608,17887,17889],{"className":17888},[615],[617,17890,17891],{"xmlns":619},[621,17892,17893,17897],{},[624,17894,17895],{},[876,17896,1958],{},[637,17898,1958],{"encoding":639},[608,17900,17902],{"className":17901,"ariaHidden":645},[644],[608,17903,17905,17908],{"className":17904},[649],[608,17906],{"className":17907,"style":1665},[653],[608,17909,1958],{"className":17910},[658,894],[608,17912,17914,17927],{"className":17913},[611],[608,17915,17917],{"className":17916},[615],[617,17918,17919],{"xmlns":619},[621,17920,17921,17925],{},[624,17922,17923],{},[876,17924,2035],{},[637,17926,2035],{"encoding":639},[608,17928,17930],{"className":17929,"ariaHidden":645},[644],[608,17931,17933,17936],{"className":17932},[649],[608,17934],{"className":17935,"style":1665},[653],[608,17937,2035],{"className":17938,"style":2231},[658,894]," sont vrais.",[3063,17941,17942,17879,17944,17947],{},[1894,17943,14321],{},[603,17945,17946],{},"si au moins un"," des deux est vrai.",[3063,17949,17950,17953,17954,2097],{},[1894,17951,17952],{},"not A"," : inverse ",[608,17955,17957,17970],{"className":17956},[611],[608,17958,17960],{"className":17959},[615],[617,17961,17962],{"xmlns":619},[621,17963,17964,17968],{},[624,17965,17966],{},[876,17967,1958],{},[637,17969,1958],{"encoding":639},[608,17971,17973],{"className":17972,"ariaHidden":645},[644],[608,17974,17976,17979],{"className":17975},[649],[608,17977],{"className":17978,"style":1665},[653],[608,17980,1958],{"className":17981},[658,894],[3063,17983,17984,17986,17987,17990,17991,17947],{},[1894,17985,14436],{}," (écrit ",[1894,17988,17989],{},"A != B",") : vrai ",[603,17992,17993],{},"si exactement un",[3063,17995,17996,17997,17999,18000,17999,18002,2097],{},"Priorité : ",[1894,17998,13138],{}," > ",[1894,18001,13040],{},[1894,18003,13090],{},[3063,18005,18006,18007,18009,18010,18012,18013,18015,18016,18018,18019,18012,18021,18015,18023,18025],{},"Évaluation ",[603,18008,14300],{}," (court-circuit) : ",[1894,18011,14307],{}," n'évalue pas ",[1894,18014,2035],{}," si ",[1894,18017,1958],{}," est faux ; ",[1894,18020,14321],{},[1894,18022,2035],{},[1894,18024,1958],{}," est vrai.",[594,18027,15487],{"id":15486},[1859,18029,18030,18038],{},[1862,18031,18032],{},[1865,18033,18034,18036],{},[1868,18035,15496],{},[1868,18037,15499],{},[1881,18039,18040,18135,18146],{},[1865,18041,18042,18074],{},[1886,18043,18044,18045,18073],{},"Dresser une table de vérité à ",[608,18046,18048,18061],{"className":18047},[611],[608,18049,18051],{"className":18050},[615],[617,18052,18053],{"xmlns":619},[621,18054,18055,18059],{},[624,18056,18057],{},[876,18058,1162],{},[637,18060,1162],{"encoding":639},[608,18062,18064],{"className":18063,"ariaHidden":645},[644],[608,18065,18067,18070],{"className":18066},[649],[608,18068],{"className":18069,"style":4374},[653],[608,18071,1162],{"className":18072},[658,894]," variables",[1886,18075,18076,18134],{},[608,18077,18079,18096],{"className":18078},[611],[608,18080,18082],{"className":18081},[615],[617,18083,18084],{"xmlns":619},[621,18085,18086,18094],{},[624,18087,18088],{},[627,18089,18090,18092],{},[630,18091,782],{},[876,18093,1162],{},[637,18095,4399],{"encoding":639},[608,18097,18099],{"className":18098,"ariaHidden":645},[644],[608,18100,18102,18105],{"className":18101},[649],[608,18103],{"className":18104,"style":4409},[653],[608,18106,18108,18111],{"className":18107},[658],[608,18109,782],{"className":18110},[658],[608,18112,18114],{"className":18113},[669],[608,18115,18117],{"className":18116},[673],[608,18118,18120],{"className":18119},[677],[608,18121,18123],{"className":18122,"style":4409},[681],[608,18124,18125,18128],{"style":684},[608,18126],{"className":18127,"style":689},[688],[608,18129,18131],{"className":18130},[693,694,695,696],[608,18132,1162],{"className":18133},[658,894,696]," lignes, énumérer toutes les combinaisons.",[1865,18136,18137,18140],{},[1886,18138,18139],{},"Protéger un accès",[1886,18141,18142,18145],{},[1894,18143,18144],{},"if liste and liste[0] > 5:"," — court-circuit.",[1865,18147,18148,18151],{},[1886,18149,18150],{},"Addition binaire 1 bit",[1886,18152,18153,18154,18156,18157,2097],{},"somme = ",[1894,18155,14436],{},", retenue = ",[1894,18158,14307],{},[594,18160,5340],{"id":5339},[3060,18162,18163,18174,18183],{},[3063,18164,18165,18167,18168,18170,18171,18173],{},[1894,18166,10947],{}," \u002F ",[1894,18169,10950],{}," avec majuscule. ",[1894,18172,645],{}," n'existe pas en Python.",[3063,18175,18176,14723,18178,14727,18181,2097],{},[1894,18177,14722],{},[1894,18179,18180],{},"not (a == b)",[1894,18182,14730],{},[3063,18184,18185,1896,18187,1896,18189,18191,18192,18194,18195,11182,18197,11182,18199,14753],{},[1894,18186,7370],{},[1894,18188,14738],{},[1894,18190,14741],{}," opèrent ",[603,18193,14745],{}," sur des entiers, pas comme ",[1894,18196,13040],{},[1894,18198,13090],{},[1894,18200,13796],{},[594,18202,15959],{"id":15958},[2878,18204,18206],{"className":2880,"code":18205,"language":2882,"meta":2883,"style":2883},"True and False     # False\nTrue or False      # True\nnot True           # False\nA != B             # équivalent xor\nif liste and liste[0] > 5:\n    ...\n",[1894,18207,18208,18219,18230,18239,18252,18272],{"__ignoreMap":2883},[608,18209,18210,18212,18214,18216],{"class":2888,"line":2889},[608,18211,10947],{"class":2892},[608,18213,13191],{"class":3722},[608,18215,13194],{"class":2892},[608,18217,18218],{"class":2915},"     # False\n",[608,18220,18221,18223,18225,18227],{"class":2888,"line":2919},[608,18222,10947],{"class":2892},[608,18224,13209],{"class":3722},[608,18226,13194],{"class":2892},[608,18228,18229],{"class":2915},"      # True\n",[608,18231,18232,18234,18236],{"class":2888,"line":2943},[608,18233,13138],{"class":3722},[608,18235,13227],{"class":2892},[608,18237,18238],{"class":2915},"           # False\n",[608,18240,18241,18244,18246,18249],{"class":2888,"line":3794},[608,18242,18243],{"class":2896},"A ",[608,18245,14008],{"class":3722},[608,18247,18248],{"class":2896}," B             ",[608,18250,18251],{"class":2915},"# équivalent xor\n",[608,18253,18254,18256,18258,18260,18262,18264,18266,18268,18270],{"class":2888,"line":7353},[608,18255,14209],{"class":3722},[608,18257,14360],{"class":2896},[608,18259,13040],{"class":3722},[608,18261,14365],{"class":2896},[608,18263,635],{"class":2892},[608,18265,14370],{"class":2896},[608,18267,14373],{"class":3722},[608,18269,14376],{"class":2892},[608,18271,14379],{"class":2896},[608,18273,18274],{"class":2888,"line":7359},[608,18275,18276],{"class":2892},"    ...\n",[5621,18278,18279],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":2883,"searchDepth":2919,"depth":2919,"links":18281},[18282,18283,18284,18285,18286],{"id":14930,"depth":2919,"text":14931},{"id":14997,"depth":2919,"text":14998},{"id":15486,"depth":2919,"text":15487},{"id":5339,"depth":2919,"text":5340},{"id":15958,"depth":2919,"text":15959},"Mémo de révision sur les opérateurs logiques et les tables de vérité.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbooleens.memo",[],{"title":17839,"description":18287},"1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens.memo","30Sop2HhFOzYiDk72irvpMaoO1upyTiAoHYMVfu9rf4",{"id":18295,"title":18296,"bo":18297,"body":18298,"description":19335,"eval":5636,"extension":5637,"labo":5636,"meta":19336,"navigation":5645,"path":19338,"quizzes":19339,"readingTime":2889,"seo":19362,"stem":19363,"tp":5636,"__hash__":19364},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002Fchapitre.memo.md","Représentation des données — synthèse de chapitre",[589,8827,5654,11704,12961],{"type":591,"value":18299,"toc":19330},[18300,18304,18311,18326,18387,18455,18715,18730,18744,18748,19026,19029,19033,19132,19152,19162,19225,19254,19276,19305],[594,18301,18303],{"id":18302},"vue-densemble","Vue d'ensemble",[599,18305,18306,18307,18310],{},"Ce chapitre vous a appris comment une machine binaire, qui ne manipule que des\n0 et des 1, parvient à représenter ",[603,18308,18309],{},"tout"," ce qu'un programme manipule :\nnombres entiers (positifs et négatifs), nombres à virgule, texte, valeurs\nlogiques.",[599,18312,6190,18313,18316,18317,18319,18320,18322,18323,18325],{},[603,18314,18315],{},"fil rouge"," était le suivant : à chaque type de donnée correspond une\n",[603,18318,11714],{}," qui dit comment des bits doivent être interprétés. Cette\nconvention est invisible quand tout fonctionne, mais elle se révèle dès qu'un\nproblème survient — un nombre négatif mal additionné, un ",[1894,18321,9004],{}," qui ne\ndonne pas ",[1894,18324,9011],{},", un fichier dont les accents deviennent illisibles.",[599,18327,18328,18329,18332,18333,18383,18384,18386],{},"Vous avez d'abord vu les ",[603,18330,18331],{},"bases de numération",". Tout entier positif peut\ns'écrire dans n'importe quelle base ",[608,18334,18336,18353],{"className":18335},[611],[608,18337,18339],{"className":18338},[615],[617,18340,18341],{"xmlns":619},[621,18342,18343,18351],{},[624,18344,18345,18347,18349],{},[876,18346,878],{},[942,18348,944],{},[630,18350,782],{},[637,18352,949],{"encoding":639},[608,18354,18356,18374],{"className":18355,"ariaHidden":645},[644],[608,18357,18359,18362,18365,18368,18371],{"className":18358},[649],[608,18360],{"className":18361,"style":959},[653],[608,18363,878],{"className":18364},[658,894],[608,18366],{"className":18367,"style":967},[966],[608,18369,944],{"className":18370},[971],[608,18372],{"className":18373,"style":967},[966],[608,18375,18377,18380],{"className":18376},[649],[608,18378],{"className":18379,"style":981},[653],[608,18381,782],{"className":18382},[658],", et la machine privilégie la\nbase 2. L'",[603,18385,1948],{}," sert de lecture humaine compacte du binaire (4 bits\npar chiffre hexa). Vous savez désormais convertir un nombre d'une base à\nl'autre, et estimer combien de bits sont nécessaires pour le représenter.",[599,18388,18389,18390,18392,18393,18395,18396,18454],{},"Vous avez ensuite étendu cette écriture aux ",[603,18391,14833],{}," grâce au\n",[603,18394,5882],{},". Ce codage astucieux permet aux processeurs d'utiliser\nles mêmes circuits arithmétiques pour les positifs et les négatifs — l'addition\ndevient une simple addition modulo ",[608,18397,18399,18416],{"className":18398},[611],[608,18400,18402],{"className":18401},[615],[617,18403,18404],{"xmlns":619},[621,18405,18406,18414],{},[624,18407,18408],{},[627,18409,18410,18412],{},[630,18411,782],{},[876,18413,1162],{},[637,18415,4399],{"encoding":639},[608,18417,18419],{"className":18418,"ariaHidden":645},[644],[608,18420,18422,18425],{"className":18421},[649],[608,18423],{"className":18424,"style":4409},[653],[608,18426,18428,18431],{"className":18427},[658],[608,18429,782],{"className":18430},[658],[608,18432,18434],{"className":18433},[669],[608,18435,18437],{"className":18436},[673],[608,18438,18440],{"className":18439},[677],[608,18441,18443],{"className":18442,"style":4409},[681],[608,18444,18445,18448],{"style":684},[608,18446],{"className":18447,"style":689},[688],[608,18449,18451],{"className":18450},[693,694,695,696],[608,18452,1162],{"className":18453},[658,894,696],", sans gestion explicite du signe.",[599,18456,18457,18458,18460,18461,18463,18464,18611,18612,18640,18641,18669,18670,18673,18674,18711,18712,2097],{},"Pour les ",[603,18459,14837],{},", la machine utilise la ",[603,18462,8995],{}," : un\nnombre est codé comme ",[608,18465,18467,18502],{"className":18466},[611],[608,18468,18470],{"className":18469},[615],[617,18471,18472],{"xmlns":619},[621,18473,18474,18500],{},[624,18475,18476,18478,18480,18482,18488,18490,18492,18494],{},[942,18477,1151],{"stretchy":1150},[942,18479,1060],{},[630,18481,659],{},[627,18483,18484,18486],{},[942,18485,1198],{"stretchy":1150},[876,18487,9411],{},[942,18489,1646],{},[876,18491,9416],{},[942,18493,1646],{},[627,18495,18496,18498],{},[630,18497,782],{},[876,18499,9425],{},[637,18501,16987],{"encoding":639},[608,18503,18505,18558,18576],{"className":18504,"ariaHidden":645},[644],[608,18506,18508,18511,18514,18517,18520,18549,18552,18555],{"className":18507},[649],[608,18509],{"className":18510,"style":1213},[653],[608,18512,1151],{"className":18513},[1217],[608,18515,1060],{"className":18516},[658],[608,18518,659],{"className":18519},[658],[608,18521,18523,18526],{"className":18522},[1417],[608,18524,1198],{"className":18525},[1417],[608,18527,18529],{"className":18528},[669],[608,18530,18532],{"className":18531},[673],[608,18533,18535],{"className":18534},[677],[608,18536,18538],{"className":18537,"style":4409},[681],[608,18539,18540,18543],{"style":684},[608,18541],{"className":18542,"style":689},[688],[608,18544,18546],{"className":18545},[693,694,695,696],[608,18547,9411],{"className":18548},[658,894,696],[608,18550],{"className":18551,"style":1082},[966],[608,18553,1646],{"className":18554},[1086],[608,18556],{"className":18557,"style":1082},[966],[608,18559,18561,18564,18567,18570,18573],{"className":18560},[649],[608,18562],{"className":18563,"style":9510},[653],[608,18565,9416],{"className":18566},[658,894],[608,18568],{"className":18569,"style":1082},[966],[608,18571,1646],{"className":18572},[1086],[608,18574],{"className":18575,"style":1082},[966],[608,18577,18579,18582],{"className":18578},[649],[608,18580],{"className":18581,"style":4409},[653],[608,18583,18585,18588],{"className":18584},[658],[608,18586,782],{"className":18587},[658],[608,18589,18591],{"className":18590},[669],[608,18592,18594],{"className":18593},[673],[608,18595,18597],{"className":18596},[677],[608,18598,18600],{"className":18599,"style":4409},[681],[608,18601,18602,18605],{"style":684},[608,18603],{"className":18604,"style":689},[688],[608,18606,18608],{"className":18607},[693,694,695,696],[608,18609,9425],{"className":18610},[658,894,696],", avec une mantisse ",[608,18613,18615,18628],{"className":18614},[611],[608,18616,18618],{"className":18617},[615],[617,18619,18620],{"xmlns":619},[621,18621,18622,18626],{},[624,18623,18624],{},[876,18625,9416],{},[637,18627,9416],{"encoding":639},[608,18629,18631],{"className":18630,"ariaHidden":645},[644],[608,18632,18634,18637],{"className":18633},[649],[608,18635],{"className":18636,"style":4374},[653],[608,18638,9416],{"className":18639},[658,894]," et un\nexposant ",[608,18642,18644,18657],{"className":18643},[611],[608,18645,18647],{"className":18646},[615],[617,18648,18649],{"xmlns":619},[621,18650,18651,18655],{},[624,18652,18653],{},[876,18654,9425],{},[637,18656,9425],{"encoding":639},[608,18658,18660],{"className":18659,"ariaHidden":645},[644],[608,18661,18663,18666],{"className":18662},[649],[608,18664],{"className":18665,"style":4374},[653],[608,18667,9425],{"className":18668},[658,894]," stockés sur un nombre fixe de bits. Vous avez vu que cette\nreprésentation ",[603,18671,18672],{},"approxime"," la plupart des réels — en particulier les\ndécimaux qui n'ont pas d'écriture finie en binaire, comme ",[608,18675,18677,18690],{"className":18676},[611],[608,18678,18680],{"className":18679},[615],[617,18681,18682],{"xmlns":619},[621,18683,18684,18688],{},[624,18685,18686],{},[630,18687,9952],{},[637,18689,9955],{"encoding":639},[608,18691,18693],{"className":18692,"ariaHidden":645},[644],[608,18694,18696,18699,18702,18708],{"className":18695},[649],[608,18697],{"className":18698,"style":5047},[653],[608,18700,635],{"className":18701},[658],[608,18703,18705],{"className":18704},[658],[608,18706,5058],{"className":18707},[5057],[608,18709,659],{"className":18710},[658],". D'où la\nrègle d'or : ",[603,18713,18714],{},"ne jamais tester l'égalité stricte de deux flottants",[599,18716,6190,18717,18720,18721,18723,18724,18726,18727,18729],{},[603,18718,18719],{},"texte"," suit la même logique : chaque caractère reçoit un numéro, et un\n",[603,18722,12330],{}," dit comment ce numéro devient une suite d'octets. ASCII a couvert\nl'anglais, ISO-8859-1 l'Europe de l'Ouest, et ",[603,18725,12211],{}," ambitionne de couvrir\ntoutes les langues du monde. ",[603,18728,12340],{}," est aujourd'hui l'encodage dominant,\nrétrocompatible avec ASCII et économe en octets pour les langues européennes.",[599,18731,18732,18733,18736,18737,1896,18739,1896,18741,18743],{},"Enfin, vous avez formalisé le calcul ",[603,18734,18735],{},"booléen"," : deux valeurs (vrai et\nfaux), trois opérateurs (",[1894,18738,13040],{},[1894,18740,13090],{},[1894,18742,13138],{},"), une logique de\ncourt-circuit, et des tables de vérité comme outil de référence. C'est cette\nlogique qui structure toute prise de décision dans un programme — et qui,\ngravée dans le silicium, fait fonctionner les processeurs.",[594,18745,18747],{"id":18746},"carte-des-notions","Carte des notions",[1859,18749,18750,18766],{},[1862,18751,18752],{},[1865,18753,18754,18757,18760,18763],{},[1868,18755,18756],{},"Cours",[1868,18758,18759],{},"Item BO",[1868,18761,18762],{},"Notion-clé",[1868,18764,18765],{},"À savoir faire",[1881,18767,18768,18911,18985,19000,19013],{},[1865,18769,18770,18773,18775,18908],{},[1886,18771,18772],{},"1. Bases",[1886,18774,589],{},[1886,18776,18777,18778],{},"Notation positionnelle, ",[608,18779,18781,18809],{"className":18780},[611],[608,18782,18784],{"className":18783},[615],[617,18785,18786],{"xmlns":619},[621,18787,18788,18806],{},[624,18789,18790,18792,18798,18800],{},[942,18791,1621],{},[1153,18793,18794,18796],{},[876,18795,878],{},[876,18797,1504],{},[942,18799,1646],{},[627,18801,18802,18804],{},[876,18803,878],{},[876,18805,1504],{},[637,18807,18808],{"encoding":639},"\\sum b_i \\cdot b^i",[608,18810,18812,18873],{"className":18811,"ariaHidden":645},[644],[608,18813,18815,18818,18821,18824,18864,18867,18870],{"className":18814},[649],[608,18816],{"className":18817,"style":1213},[653],[608,18819,1621],{"className":18820,"style":15126},[1689,1734,15125],[608,18822],{"className":18823,"style":1325},[966],[608,18825,18827,18830],{"className":18826},[658],[608,18828,878],{"className":18829},[658,894],[608,18831,18833],{"className":18832},[669],[608,18834,18836,18856],{"className":18835},[673,1230],[608,18837,18839,18853],{"className":18838},[677],[608,18840,18842],{"className":18841,"style":1536},[681],[608,18843,18844,18847],{"style":1240},[608,18845],{"className":18846,"style":689},[688],[608,18848,18850],{"className":18849},[693,694,695,696],[608,18851,1504],{"className":18852},[658,894,696],[608,18854,1263],{"className":18855},[1262],[608,18857,18859],{"className":18858},[677],[608,18860,18862],{"className":18861,"style":1371},[681],[608,18863],{},[608,18865],{"className":18866,"style":1082},[966],[608,18868,1646],{"className":18869},[1086],[608,18871],{"className":18872,"style":1082},[966],[608,18874,18876,18879],{"className":18875},[649],[608,18877],{"className":18878,"style":15252},[653],[608,18880,18882,18885],{"className":18881},[658],[608,18883,878],{"className":18884},[658,894],[608,18886,18888],{"className":18887},[669],[608,18889,18891],{"className":18890},[673],[608,18892,18894],{"className":18893},[677],[608,18895,18897],{"className":18896,"style":15252},[681],[608,18898,18899,18902],{"style":684},[608,18900],{"className":18901,"style":689},[688],[608,18903,18905],{"className":18904},[693,694,695,696],[608,18906,1504],{"className":18907},[658,894,696],[1886,18909,18910],{},"Convertir binaire\u002Fdécimal\u002Fhexa, estimer nombre de bits",[1865,18912,18913,18916,18918,18921],{},[1886,18914,18915],{},"2. Entiers relatifs",[1886,18917,5654],{},[1886,18919,18920],{},"Complément à deux",[1886,18922,16523,18923,7412,18956,18984],{},[608,18924,18926,18941],{"className":18925},[611],[608,18927,18929],{"className":18928},[615],[617,18930,18931],{"xmlns":619},[621,18932,18933,18939],{},[624,18934,18935,18937],{},[942,18936,1060],{},[876,18938,5898],{},[637,18940,6376],{"encoding":639},[608,18942,18944],{"className":18943,"ariaHidden":645},[644],[608,18945,18947,18950,18953],{"className":18946},[649],[608,18948],{"className":18949,"style":6386},[653],[608,18951,1060],{"className":18952},[658],[608,18954,5898],{"className":18955},[658,894],[608,18957,18959,18972],{"className":18958},[611],[608,18960,18962],{"className":18961},[615],[617,18963,18964],{"xmlns":619},[621,18965,18966,18970],{},[624,18967,18968],{},[876,18969,1162],{},[637,18971,1162],{"encoding":639},[608,18973,18975],{"className":18974,"ariaHidden":645},[644],[608,18976,18978,18981],{"className":18977},[649],[608,18979],{"className":18980,"style":4374},[653],[608,18982,1162],{"className":18983},[658,894]," bits, lire un négatif",[1865,18986,18987,18990,18992,18995],{},[1886,18988,18989],{},"3. Flottants",[1886,18991,8827],{},[1886,18993,18994],{},"Virgule flottante IEEE 754",[1886,18996,18997,18998],{},"Comprendre l'approximation, comparer avec ",[1894,18999,11127],{},[1865,19001,19002,19005,19007,19010],{},[1886,19003,19004],{},"4. Encodages",[1886,19006,11704],{},[1886,19008,19009],{},"ASCII, ISO-8859-1, Unicode\u002FUTF-8",[1886,19011,19012],{},"Encoder\u002Fdécoder, identifier un bug d'encodage",[1865,19014,19015,19018,19020,19023],{},[1886,19016,19017],{},"5. Booléens",[1886,19019,12961],{},[1886,19021,19022],{},"Opérateurs logiques, table de vérité",[1886,19024,19025],{},"Dresser une table, comprendre le court-circuit",[599,19027,19028],{},"Les notions s'enchaînent naturellement : les bases (cours 1) sont le fondement\ndes entiers signés (cours 2) qui constituent la mantisse et l'exposant des\nflottants (cours 3). Les encodages de texte (cours 4) sont eux-mêmes des\ntables d'entiers. Les booléens (cours 5) sont à la fois un type à part et le\nfondement du fonctionnement matériel — ils interviennent d'ailleurs déjà dans\nl'arithmétique binaire vue en cours 2.",[594,19030,19032],{"id":19031},"auto-évaluation","Auto-évaluation",[2966,19034,19036,19119],{"correct":782,"slug":19035},"pb-types-valeurs-1-q1",[2970,19037,19038],{"v-slot:question":2883},[599,19039,2974,19040,3056],{},[608,19041,19043,19064],{"className":19042},[611],[608,19044,19046],{"className":19045},[615],[617,19047,19048],{"xmlns":619},[621,19049,19050,19062],{},[624,19051,19052,19054,19056],{},[942,19053,1151],{"stretchy":1150},[630,19055,2991],{},[1153,19057,19058,19060],{},[942,19059,1198],{"stretchy":1150},[630,19061,782],{},[637,19063,3000],{"encoding":639},[608,19065,19067],{"className":19066,"ariaHidden":645},[644],[608,19068,19070,19073,19076,19079],{"className":19069},[649],[608,19071],{"className":19072,"style":1213},[653],[608,19074,1151],{"className":19075},[1217],[608,19077,2991],{"className":19078},[658],[608,19080,19082,19085],{"className":19081},[1417],[608,19083,1198],{"className":19084},[1417],[608,19086,19088],{"className":19087},[669],[608,19089,19091,19111],{"className":19090},[673,1230],[608,19092,19094,19108],{"className":19093},[677],[608,19095,19097],{"className":19096,"style":1237},[681],[608,19098,19099,19102],{"style":1240},[608,19100],{"className":19101,"style":689},[688],[608,19103,19105],{"className":19104},[693,694,695,696],[608,19106,782],{"className":19107},[658,696],[608,19109,1263],{"className":19110},[1262],[608,19112,19114],{"className":19113},[677],[608,19115,19117],{"className":19116,"style":1371},[681],[608,19118],{},[2970,19120,19121],{"v-slot:choices":2883},[3060,19122,19123,19125,19128,19130],{},[3063,19124,2051],{},[3063,19126,19127],{},"12",[3063,19129,3067],{},[3063,19131,3070],{},[2966,19133,19135,19140],{"correct":2413,"slug":19134},"pb-types-valeurs-1-q2",[2970,19136,19137],{"v-slot:question":2883},[599,19138,19139],{},"Avec 8 bits non signés, combien de valeurs distinctes peut-on représenter ?",[2970,19141,19142],{"v-slot:choices":2883},[3060,19143,19144,19146,19148,19150],{},[3063,19145,7756],{},[3063,19147,7719],{},[3063,19149,4789],{},[3063,19151,4713],{},[19153,19154,19157],"self-eval-free-text",{"answer":19155,"case-sensitive":1150,"slug":19156},"2F","pb-types-valeurs-1-q3",[2970,19158,19159],{"v-slot:question":2883},[599,19160,19161],{},"Quelle est l'écriture hexadécimale (sans préfixe) de l'entier 47 ?",[2966,19163,19165,19204],{"correct":659,"slug":19164},"pb-types-valeurs-1-q4",[2970,19166,19167],{"v-slot:question":2883},[599,19168,19169,19170,19203],{},"Sur 8 bits en complément à deux, ",[608,19171,19173,19188],{"className":19172},[611],[608,19174,19176],{"className":19175},[615],[617,19177,19178],{"xmlns":619},[621,19179,19180,19186],{},[624,19181,19182,19184],{},[942,19183,1060],{},[630,19185,659],{},[637,19187,8674],{"encoding":639},[608,19189,19191],{"className":19190,"ariaHidden":645},[644],[608,19192,19194,19197,19200],{"className":19193},[649],[608,19195],{"className":19196,"style":2794},[653],[608,19198,1060],{"className":19199},[658],[608,19201,659],{"className":19202},[658]," s'écrit :",[2970,19205,19206],{"v-slot:choices":2883},[3060,19207,19208,19212,19216,19220],{},[3063,19209,19210],{},[1894,19211,8697],{},[3063,19213,19214],{},[1894,19215,8706],{},[3063,19217,19218],{},[1894,19219,8700],{},[3063,19221,19222],{},[1894,19223,19224],{},"1000 0000",[2966,19226,19228,19235],{"correct":659,"slug":19227},"pb-types-valeurs-1-q5",[2970,19229,19230],{"v-slot:question":2883},[599,19231,19232,19233,4329],{},"En Python, que renvoie ",[1894,19234,10891],{},[2970,19236,19237],{"v-slot:choices":2883},[3060,19238,19239,19243,19247,19249],{},[3063,19240,19241],{},[1894,19242,10947],{},[3063,19244,19245],{},[1894,19246,10950],{},[3063,19248,10953],{},[3063,19250,19251],{},[1894,19252,19253],{},"None",[2966,19255,19257,19262],{"correct":782,"slug":19256},"pb-types-valeurs-1-q6",[2970,19258,19259],{"v-slot:question":2883},[599,19260,19261],{},"Quel est aujourd'hui l'encodage Unicode dominant sur le Web ?",[2970,19263,19264],{"v-slot:choices":2883},[3060,19265,19266,19268,19270,19273],{},[3063,19267,11733],{},[3063,19269,12340],{},[3063,19271,19272],{},"Latin-1 (ISO-8859-1)",[3063,19274,19275],{},"UTF-16",[2966,19277,19279,19289],{"correct":782,"slug":19278},"pb-types-valeurs-1-q7",[2970,19280,19281],{"v-slot:question":2883},[599,19282,19283,19284,19286,19287,5279],{},"La table de vérité de ",[1894,19285,14321],{}," retourne ",[603,19288,12982],{},[2970,19290,19291],{"v-slot:choices":2883},[3060,19292,19293,19296,19299,19302],{},[3063,19294,19295],{},"quand A et B sont tous les deux vrais",[3063,19297,19298],{},"quand au moins un des deux est vrai",[3063,19300,19301],{},"quand A et B sont tous les deux faux",[3063,19303,19304],{},"jamais",[19306,19307,19309,19314],"self-eval-ordering",{"slug":19308},"pb-types-valeurs-1-q8",[2970,19310,19311],{"v-slot:question":2883},[599,19312,19313],{},"Ordonnez ces encodages dans l'ordre chronologique d'apparition :",[2970,19315,19316],{"v-slot:items":2883},[3060,19317,19318,19321,19324,19327],{},[3063,19319,19320],{},"ASCII (1963)",[3063,19322,19323],{},"ISO-8859-1 (1985)",[3063,19325,19326],{},"Unicode 1.0 (1991)",[3063,19328,19329],{},"UTF-8 (1993)",{"title":2883,"searchDepth":2919,"depth":2919,"links":19331},[19332,19333,19334],{"id":18302,"depth":2919,"text":18303},{"id":18746,"depth":2919,"text":18747},{"id":19031,"depth":2919,"text":19032},"Vue d'ensemble du chapitre, points-clés et auto-évaluation finale.",{"self_eval_slug":19337},"nsi-1ere-pb-types-valeurs","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fchapitre.memo",[19340,19342,19344,19348,19351,19354,19356,19359],{"slug":19035,"kind":5642,"question":5643,"choices":19341,"correct":782,"multi":5645,"explanation":2883},[2051,19127,3067,3070],{"slug":19134,"kind":5642,"question":19139,"choices":19343,"correct":2413,"multi":5645,"explanation":2883},[7756,7719,4789,4713],{"slug":19156,"kind":19345,"question":19161,"answer":19155,"caseSensitive":5645,"tolerance":19346,"inputType":19347,"explanation":2883},"free_text",0,"string",{"slug":19164,"kind":5642,"question":19349,"choices":19350,"correct":659,"multi":5645,"explanation":2883},"Sur 8 bits en complément à deux, −1-1−1 s'écrit :",[8697,8706,8700,19224],{"slug":19227,"kind":5642,"question":19352,"choices":19353,"correct":659,"multi":5645,"explanation":2883},"En Python, que renvoie 0.1 + 0.2 == 0.3 ?",[10947,10950,10953,19253],{"slug":19256,"kind":5642,"question":19261,"choices":19355,"correct":782,"multi":5645,"explanation":2883},[11733,12340,19272,19275],{"slug":19278,"kind":5642,"question":19357,"choices":19358,"correct":782,"multi":5645,"explanation":2883},"La table de vérité de A or B retourne faux :",[19295,19298,19301,19304],{"slug":19308,"kind":19360,"question":19313,"items":19361,"explanation":2883},"ordering",[19320,19323,19326,19329],{"title":18296,"description":19335},"1.nsi-1ere\u002Fpb-types-valeurs\u002Fchapitre.memo","Qtq8roq3cJrn0J9Ue_gqN1_9ExHpAMe4CQof_wwZHyA",1788035578939]