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tables","\u002Fsnt-2nde\u002Fsd-donnees-structurees\u002Ftraitement-tables","0.snt-2nde\u002Fsd-donnees-structurees\u002F4.traitement-tables",{"title":115,"path":116,"stem":117},"Le 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 OpenStreetMap","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fcartes-numeriques","0.snt-2nde\u002Fse-localisation-cartographie\u002F3.cartes-numeriques",{"title":137,"path":138,"stem":139},"Itinéraires et graphes","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fitineraires-graphes","0.snt-2nde\u002Fse-localisation-cartographie\u002F4.itineraires-graphes",{"title":141,"path":142,"stem":143},"Confidentialité de la position","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fconfidentialite-position","0.snt-2nde\u002Fse-localisation-cartographie\u002F5.confidentialite-position",{"title":145,"path":146,"stem":147,"children":148},"Informatique embarquée et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee","0.snt-2nde\u002Fsf-info-embarquee\u002F0.index",[149,150,154,158],{"title":145,"path":146,"stem":147},{"title":151,"path":152,"stem":153},"Systèmes informatiques embarqués","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fsystemes-embarques","0.snt-2nde\u002Fsf-info-embarquee\u002F1.systemes-embarques",{"title":155,"path":156,"stem":157},"Capteurs et actionneurs","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fcapteurs-actionneurs","0.snt-2nde\u002Fsf-info-embarquee\u002F2.capteurs-actionneurs",{"title":159,"path":160,"stem":161},"IHM et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fihm-objet-connecte","0.snt-2nde\u002Fsf-info-embarquee\u002F3.ihm-objet-connecte",{"title":163,"path":164,"stem":165,"children":166},"Photographie numérique","\u002Fsnt-2nde\u002Fsg-photo-numerique","0.snt-2nde\u002Fsg-photo-numerique\u002F0.index",[167,168,172,176,180],{"title":163,"path":164,"stem":165},{"title":169,"path":170,"stem":171},"Photosites, pixels et résolution","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fpixels-resolution","0.snt-2nde\u002Fsg-photo-numerique\u002F1.pixels-resolution",{"title":173,"path":174,"stem":175},"Les algorithmes de 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 l'informatique","\u002Fnsi-1ere\u002Fpa-histoire-info","1.nsi-1ere\u002Fpa-histoire-info\u002F0.index",[239,240,244,248,252],{"title":235,"path":236,"stem":237},{"title":241,"path":242,"stem":243},"De la préantiquité aux machines à calculer","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fpreantiquite-aux-machines","1.nsi-1ere\u002Fpa-histoire-info\u002F1.preantiquite-aux-machines",{"title":245,"path":246,"stem":247},"La machine universelle (1930-1950)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fmachine-universelle","1.nsi-1ere\u002Fpa-histoire-info\u002F2.machine-universelle",{"title":249,"path":250,"stem":251},"Les ordinateurs modernes (1947-1990)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fordinateurs-modernes","1.nsi-1ere\u002Fpa-histoire-info\u002F3.ordinateurs-modernes",{"title":253,"path":254,"stem":255},"Réseaux, Web et IA (1969 à 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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compréhension","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fcomprehension","1.nsi-1ere\u002Fpc-types-construits\u002F2.comprehension",{"title":297,"path":298,"stem":299},"Tuples et p-uplets nommés","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftuples","1.nsi-1ere\u002Fpc-types-construits\u002F3.tuples",{"title":301,"path":302,"stem":303},"Dictionnaires","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fdictionnaires","1.nsi-1ere\u002Fpc-types-construits\u002F4.dictionnaires",{"title":305,"path":306,"stem":307,"children":308},"Traitement de données en tables","\u002Fnsi-1ere\u002Fpd-tables-donnees","1.nsi-1ere\u002Fpd-tables-donnees\u002F0.index",[309,310,314,318,322],{"title":305,"path":306,"stem":307},{"title":311,"path":312,"stem":313},"Tables et CSV","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftables-csv","1.nsi-1ere\u002Fpd-tables-donnees\u002F1.tables-csv",{"title":315,"path":316,"stem":317},"Recherche et 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",{"id":586,"title":267,"bo":587,"body":589,"description":3893,"eval":3894,"extension":3895,"labo":3894,"meta":3896,"navigation":3667,"path":268,"quizzes":3897,"readingTime":2356,"seo":3905,"stem":269,"tp":3894,"__hash__":3906},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.md",[588],"PB03",{"type":590,"value":591,"toc":3881},"minimark",[592,597,667,674,678,694,697,781,788,808,818,822,934,970,1050,1204,1222,1288,1594,1790,1793,2123,2239,2341,2497,2500,2508,2512,2543,2742,3241,3248,3272,3276,3279,3622,3627,3631,3637,3719,3722,3781,3785,3866,3870,3877],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,605,606,659,660,663,664],"p",{},"Vous savez désormais représenter un entier positif en binaire. Mais comment\nfait-on pour coder un nombre ",[602,603,604],"strong",{},"négatif"," comme ",[607,608,611,638],"span",{"className":609},[610],"katex",[607,612,615],{"className":613},[614],"katex-mathml",[616,617,619],"math",{"xmlns":618},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[620,621,622,633],"semantics",{},[623,624,625,629],"mrow",{},[626,627,628],"mo",{},"−",[630,631,632],"mn",{},"5",[634,635,637],"annotation",{"encoding":636},"application\u002Fx-tex","-5",[607,639,643],{"className":640,"ariaHidden":642},[641],"katex-html","true",[607,644,647,652,656],{"className":645},[646],"base",[607,648],{"className":649,"style":651},[650],"strut","height:0.7278em;vertical-align:-0.0833em;",[607,653,628],{"className":654},[655],"mord",[607,657,632],{"className":658},[655]," ? 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 : ",[602,661,662],{},"le complément à deux",". ",[665,666],"bo-ref",{"code":588},[598,668,669,670,673],{},"Ce codage a une propriété quasi magique : il permet d'",[602,671,672],{},"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.",[593,675,677],{"id":676},"le-bit-de-signe-première-idée","Le bit de signe — première idée",[598,679,680,681,684,685,689,690,693],{},"L'idée la plus naïve consiste à réserver le ",[602,682,683],{},"bit de poids fort"," comme bit de\nsigne : ",[686,687,688],"code",{},"0"," pour le positif, ",[686,691,692],{},"1"," pour le négatif. Les autres bits codent la\nvaleur absolue.",[598,695,696],{},"Sur 4 bits :",[698,699,700,743],"ul",{},[701,702,703,706,707,742],"li",{},[686,704,705],{},"0101"," représenterait ",[607,708,710,727],{"className":709},[610],[607,711,713],{"className":712},[614],[616,714,715],{"xmlns":618},[620,716,717,724],{},[623,718,719,722],{},[626,720,721],{},"+",[630,723,632],{},[634,725,726],{"encoding":636},"+5",[607,728,730],{"className":729,"ariaHidden":642},[641],[607,731,733,736,739],{"className":732},[646],[607,734],{"className":735,"style":651},[650],[607,737,721],{"className":738},[655],[607,740,632],{"className":741},[655],".",[701,744,745,706,748,742],{},[686,746,747],{},"1101",[607,749,751,766],{"className":750},[610],[607,752,754],{"className":753},[614],[616,755,756],{"xmlns":618},[620,757,758,764],{},[623,759,760,762],{},[626,761,628],{},[630,763,632],{},[634,765,637],{"encoding":636},[607,767,769],{"className":768,"ariaHidden":642},[641],[607,770,772,775,778],{"className":771},[646],[607,773],{"className":774,"style":651},[650],[607,776,628],{"className":777},[655],[607,779,632],{"className":780},[655],[598,782,783,784,787],{},"Cette approche, appelée ",[602,785,786],{},"signe et valeur absolue",", semble naturelle… mais\nelle souffre de défauts rédhibitoires :",[698,789,790,805],{},[701,791,792,793,796,797,800,801,804],{},"Il existe ",[602,794,795],{},"deux zéros"," : ",[686,798,799],{},"0000"," et ",[686,802,803],{},"1000",", ce qui complique les\ncomparaisons.",[701,806,807],{},"L'addition d'un positif et d'un négatif ne fonctionne plus de façon\nélémentaire : il faut un circuit dédié pour gérer le signe.",[809,810,811],"warning",{},[598,812,813,814,817],{},"Ce n'est ",[602,815,816],{},"pas"," ce codage qui est utilisé en pratique. Il est présenté ici\nuniquement pour comprendre pourquoi le complément à deux a été inventé.",[593,819,821],{"id":820},"le-complément-à-deux","Le complément à deux",[598,823,824,825,857,858,861,862,891,892,920,921,924,925,927,928,930,931,742],{},"Sur ",[607,826,828,843],{"className":827},[610],[607,829,831],{"className":830},[614],[616,832,833],{"xmlns":618},[620,834,835,841],{},[623,836,837],{},[838,839,840],"mi",{},"n",[634,842,840],{"encoding":636},[607,844,846],{"className":845,"ariaHidden":642},[641],[607,847,849,853],{"className":848},[646],[607,850],{"className":851,"style":852},[650],"height:0.4306em;",[607,854,840],{"className":855},[655,856],"mathnormal"," bits, le ",[602,859,860],{},"complément à deux"," d'un entier ",[607,863,865,879],{"className":864},[610],[607,866,868],{"className":867},[614],[616,869,870],{"xmlns":618},[620,871,872,877],{},[623,873,874],{},[838,875,876],{},"x",[634,878,876],{"encoding":636},[607,880,882],{"className":881,"ariaHidden":642},[641],[607,883,885,888],{"className":884},[646],[607,886],{"className":887,"style":852},[650],[607,889,876],{"className":890},[655,856]," se définit ainsi : on\nécrit ",[607,893,895,908],{"className":894},[610],[607,896,898],{"className":897},[614],[616,899,900],{"xmlns":618},[620,901,902,906],{},[623,903,904],{},[838,905,876],{},[634,907,876],{"encoding":636},[607,909,911],{"className":910,"ariaHidden":642},[641],[607,912,914,917],{"className":913},[646],[607,915],{"className":916,"style":852},[650],[607,918,876],{"className":919},[655,856]," comme un entier non signé, on ",[602,922,923],{},"inverse tous les bits"," (les ",[686,926,688],{},"\ndeviennent ",[686,929,692],{}," et inversement), puis on ",[602,932,933],{},"ajoute 1",[598,935,936,937,742],{},"Exemple sur 8 bits : représenter ",[607,938,940,955],{"className":939},[610],[607,941,943],{"className":942},[614],[616,944,945],{"xmlns":618},[620,946,947,953],{},[623,948,949,951],{},[626,950,628],{},[630,952,632],{},[634,954,637],{"encoding":636},[607,956,958],{"className":957,"ariaHidden":642},[641],[607,959,961,964,967],{"className":960},[646],[607,962],{"className":963,"style":651},[650],[607,965,628],{"className":966},[655],[607,968,632],{"className":969},[655],[971,972,973,986],"table",{},[974,975,976],"thead",{},[977,978,979,983],"tr",{},[980,981,982],"th",{},"Étape",[980,984,985],{},"Valeur",[987,988,989,1030,1040],"tbody",{},[977,990,991,1025],{},[992,993,994,995,1024],"td",{},"1. Écrire ",[607,996,998,1011],{"className":997},[610],[607,999,1001],{"className":1000},[614],[616,1002,1003],{"xmlns":618},[620,1004,1005,1009],{},[623,1006,1007],{},[630,1008,632],{},[634,1010,632],{"encoding":636},[607,1012,1014],{"className":1013,"ariaHidden":642},[641],[607,1015,1017,1021],{"className":1016},[646],[607,1018],{"className":1019,"style":1020},[650],"height:0.6444em;",[607,1022,632],{"className":1023},[655]," sur 8 bits",[992,1026,1027],{},[686,1028,1029],{},"0000 0101",[977,1031,1032,1035],{},[992,1033,1034],{},"2. Inverser tous les bits",[992,1036,1037],{},[686,1038,1039],{},"1111 1010",[977,1041,1042,1045],{},[992,1043,1044],{},"3. Ajouter 1",[992,1046,1047],{},[686,1048,1049],{},"1111 1011",[598,1051,1052,1053,1203],{},"Donc ",[607,1054,1056,1098],{"className":1055},[610],[607,1057,1059],{"className":1058},[614],[616,1060,1061],{"xmlns":618},[620,1062,1063,1095],{},[623,1064,1065,1067,1069,1072,1076,1079,1083,1086],{},[626,1066,628],{},[630,1068,632],{},[626,1070,1071],{},"=",[626,1073,1075],{"stretchy":1074},"false","(",[630,1077,1078],{},"1111",[1080,1081,1082],"mtext",{}," ",[630,1084,1085],{},"1011",[1087,1088,1089,1092],"msub",{},[626,1090,1091],{"stretchy":1074},")",[630,1093,1094],{},"2",[634,1096,1097],{"encoding":636},"-5 = (1111\\,1011)_2",[607,1099,1101,1125],{"className":1100,"ariaHidden":642},[641],[607,1102,1104,1107,1110,1113,1118,1122],{"className":1103},[646],[607,1105],{"className":1106,"style":651},[650],[607,1108,628],{"className":1109},[655],[607,1111,632],{"className":1112},[655],[607,1114],{"className":1115,"style":1117},[1116],"mspace","margin-right:0.2778em;",[607,1119,1071],{"className":1120},[1121],"mrel",[607,1123],{"className":1124,"style":1117},[1116],[607,1126,1128,1132,1136,1139,1143,1146],{"className":1127},[646],[607,1129],{"className":1130,"style":1131},[650],"height:1em;vertical-align:-0.25em;",[607,1133,1075],{"className":1134},[1135],"mopen",[607,1137,1078],{"className":1138},[655],[607,1140],{"className":1141,"style":1142},[1116],"margin-right:0.1667em;",[607,1144,1085],{"className":1145},[655],[607,1147,1150,1153],{"className":1148},[1149],"mclose",[607,1151,1091],{"className":1152},[1149],[607,1154,1157],{"className":1155},[1156],"msupsub",[607,1158,1162,1194],{"className":1159},[1160,1161],"vlist-t","vlist-t2",[607,1163,1166,1189],{"className":1164},[1165],"vlist-r",[607,1167,1171],{"className":1168,"style":1170},[1169],"vlist","height:0.3011em;",[607,1172,1174,1179],{"style":1173},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[607,1175],{"className":1176,"style":1178},[1177],"pstrut","height:2.7em;",[607,1180,1186],{"className":1181},[1182,1183,1184,1185],"sizing","reset-size6","size3","mtight",[607,1187,1094],{"className":1188},[655,1185],[607,1190,1193],{"className":1191},[1192],"vlist-s","​",[607,1195,1197],{"className":1196},[1165],[607,1198,1201],{"className":1199,"style":1200},[1169],"height:0.15em;",[607,1202],{}," en complément à deux sur 8 bits.",[1205,1206,1207],"note",{},[598,1208,1209,1210,1212,1213,1215,1216,1218,1219,1221],{},"Le ",[602,1211,683],{}," indique toujours le signe : ",[686,1214,688],{}," pour les positifs,\n",[686,1217,692],{}," pour les négatifs. Mais contrairement au codage signe\u002Fvaleur absolue, les\nautres bits ne sont ",[602,1220,816],{}," la valeur absolue — ils résultent de la double\nopération « inversion + 1 ».",[593,1223,1225,1226],{"id":1224},"pourquoi-ça-marche-laddition-modulo-2n2n2n","Pourquoi ça marche — l'addition modulo ",[607,1227,1229,1248],{"className":1228},[610],[607,1230,1232],{"className":1231},[614],[616,1233,1234],{"xmlns":618},[620,1235,1236,1245],{},[623,1237,1238],{},[1239,1240,1241,1243],"msup",{},[630,1242,1094],{},[838,1244,840],{},[634,1246,1247],{"encoding":636},"2^n",[607,1249,1251],{"className":1250,"ariaHidden":642},[641],[607,1252,1254,1258],{"className":1253},[646],[607,1255],{"className":1256,"style":1257},[650],"height:0.6644em;",[607,1259,1261,1264],{"className":1260},[655],[607,1262,1094],{"className":1263},[655],[607,1265,1267],{"className":1266},[1156],[607,1268,1270],{"className":1269},[1160],[607,1271,1273],{"className":1272},[1165],[607,1274,1276],{"className":1275,"style":1257},[1169],[607,1277,1279,1282],{"style":1278},"top:-3.063em;margin-right:0.05em;",[607,1280],{"className":1281,"style":1178},[1177],[607,1283,1285],{"className":1284},[1182,1183,1184,1185],[607,1286,840],{"className":1287},[655,856,1185],[598,1289,824,1290,1318,1319,1380,1381,1416,1417,1501,1502,1560,1561,742],{},[607,1291,1293,1306],{"className":1292},[610],[607,1294,1296],{"className":1295},[614],[616,1297,1298],{"xmlns":618},[620,1299,1300,1304],{},[623,1301,1302],{},[838,1303,840],{},[634,1305,840],{"encoding":636},[607,1307,1309],{"className":1308,"ariaHidden":642},[641],[607,1310,1312,1315],{"className":1311},[646],[607,1313],{"className":1314,"style":852},[650],[607,1316,840],{"className":1317},[655,856]," bits, l'arithmétique se fait ",[602,1320,1321,1322],{},"modulo ",[607,1323,1325,1342],{"className":1324},[610],[607,1326,1328],{"className":1327},[614],[616,1329,1330],{"xmlns":618},[620,1331,1332,1340],{},[623,1333,1334],{},[1239,1335,1336,1338],{},[630,1337,1094],{},[838,1339,840],{},[634,1341,1247],{"encoding":636},[607,1343,1345],{"className":1344,"ariaHidden":642},[641],[607,1346,1348,1351],{"className":1347},[646],[607,1349],{"className":1350,"style":1257},[650],[607,1352,1354,1357],{"className":1353},[655],[607,1355,1094],{"className":1356},[655],[607,1358,1360],{"className":1359},[1156],[607,1361,1363],{"className":1362},[1160],[607,1364,1366],{"className":1365},[1165],[607,1367,1369],{"className":1368,"style":1257},[1169],[607,1370,1371,1374],{"style":1278},[607,1372],{"className":1373,"style":1178},[1177],[607,1375,1377],{"className":1376},[1182,1183,1184,1185],[607,1378,840],{"className":1379},[655,856,1185],". Coder un entier négatif\n",[607,1382,1384,1400],{"className":1383},[610],[607,1385,1387],{"className":1386},[614],[616,1388,1389],{"xmlns":618},[620,1390,1391,1397],{},[623,1392,1393,1395],{},[626,1394,628],{},[838,1396,876],{},[634,1398,1399],{"encoding":636},"-x",[607,1401,1403],{"className":1402,"ariaHidden":642},[641],[607,1404,1406,1410,1413],{"className":1405},[646],[607,1407],{"className":1408,"style":1409},[650],"height:0.6667em;vertical-align:-0.0833em;",[607,1411,628],{"className":1412},[655],[607,1414,876],{"className":1415},[655,856]," revient à utiliser la valeur positive ",[607,1418,1420,1442],{"className":1419},[610],[607,1421,1423],{"className":1422},[614],[616,1424,1425],{"xmlns":618},[620,1426,1427,1439],{},[623,1428,1429,1435,1437],{},[1239,1430,1431,1433],{},[630,1432,1094],{},[838,1434,840],{},[626,1436,628],{},[838,1438,876],{},[634,1440,1441],{"encoding":636},"2^n - x",[607,1443,1445,1492],{"className":1444,"ariaHidden":642},[641],[607,1446,1448,1452,1481,1485,1489],{"className":1447},[646],[607,1449],{"className":1450,"style":1451},[650],"height:0.7477em;vertical-align:-0.0833em;",[607,1453,1455,1458],{"className":1454},[655],[607,1456,1094],{"className":1457},[655],[607,1459,1461],{"className":1460},[1156],[607,1462,1464],{"className":1463},[1160],[607,1465,1467],{"className":1466},[1165],[607,1468,1470],{"className":1469,"style":1257},[1169],[607,1471,1472,1475],{"style":1278},[607,1473],{"className":1474,"style":1178},[1177],[607,1476,1478],{"className":1477},[1182,1183,1184,1185],[607,1479,840],{"className":1480},[655,856,1185],[607,1482],{"className":1483,"style":1484},[1116],"margin-right:0.2222em;",[607,1486,628],{"className":1487},[1488],"mbin",[607,1490],{"className":1491,"style":1484},[1116],[607,1493,1495,1498],{"className":1494},[646],[607,1496],{"className":1497,"style":852},[650],[607,1499,876],{"className":1500},[655,856]," qui, modulo ",[607,1503,1505,1522],{"className":1504},[610],[607,1506,1508],{"className":1507},[614],[616,1509,1510],{"xmlns":618},[620,1511,1512,1520],{},[623,1513,1514],{},[1239,1515,1516,1518],{},[630,1517,1094],{},[838,1519,840],{},[634,1521,1247],{"encoding":636},[607,1523,1525],{"className":1524,"ariaHidden":642},[641],[607,1526,1528,1531],{"className":1527},[646],[607,1529],{"className":1530,"style":1257},[650],[607,1532,1534,1537],{"className":1533},[655],[607,1535,1094],{"className":1536},[655],[607,1538,1540],{"className":1539},[1156],[607,1541,1543],{"className":1542},[1160],[607,1544,1546],{"className":1545},[1165],[607,1547,1549],{"className":1548,"style":1257},[1169],[607,1550,1551,1554],{"style":1278},[607,1552],{"className":1553,"style":1178},[1177],[607,1555,1557],{"className":1556},[1182,1183,1184,1185],[607,1558,840],{"className":1559},[655,856,1185],", se\ncomporte exactement comme ",[607,1562,1564,1579],{"className":1563},[610],[607,1565,1567],{"className":1566},[614],[616,1568,1569],{"xmlns":618},[620,1570,1571,1577],{},[623,1572,1573,1575],{},[626,1574,628],{},[838,1576,876],{},[634,1578,1399],{"encoding":636},[607,1580,1582],{"className":1581,"ariaHidden":642},[641],[607,1583,1585,1588,1591],{"className":1584},[646],[607,1586],{"className":1587,"style":1409},[650],[607,1589,628],{"className":1590},[655],[607,1592,876],{"className":1593},[655,856],[598,1595,1596,1597,1630,1631,1789],{},"Vérification : sur 8 bits, ",[607,1598,1600,1615],{"className":1599},[610],[607,1601,1603],{"className":1602},[614],[616,1604,1605],{"xmlns":618},[620,1606,1607,1613],{},[623,1608,1609,1611],{},[626,1610,628],{},[630,1612,632],{},[634,1614,637],{"encoding":636},[607,1616,1618],{"className":1617,"ariaHidden":642},[641],[607,1619,1621,1624,1627],{"className":1620},[646],[607,1622],{"className":1623,"style":651},[650],[607,1625,628],{"className":1626},[655],[607,1628,632],{"className":1629},[655]," correspond à ",[607,1632,1634,1674],{"className":1633},[610],[607,1635,1637],{"className":1636},[614],[616,1638,1639],{"xmlns":618},[620,1640,1641,1671],{},[623,1642,1643,1646,1648,1650,1652,1655,1657,1659,1661,1663,1665],{},[630,1644,1645],{},"256",[626,1647,628],{},[630,1649,632],{},[626,1651,1071],{},[630,1653,1654],{},"251",[626,1656,1071],{},[626,1658,1075],{"stretchy":1074},[630,1660,1078],{},[1080,1662,1082],{},[630,1664,1085],{},[1087,1666,1667,1669],{},[626,1668,1091],{"stretchy":1074},[630,1670,1094],{},[634,1672,1673],{"encoding":636},"256 - 5 = 251 = (1111\\,1011)_2",[607,1675,1677,1695,1713,1731],{"className":1676,"ariaHidden":642},[641],[607,1678,1680,1683,1686,1689,1692],{"className":1679},[646],[607,1681],{"className":1682,"style":651},[650],[607,1684,1645],{"className":1685},[655],[607,1687],{"className":1688,"style":1484},[1116],[607,1690,628],{"className":1691},[1488],[607,1693],{"className":1694,"style":1484},[1116],[607,1696,1698,1701,1704,1707,1710],{"className":1697},[646],[607,1699],{"className":1700,"style":1020},[650],[607,1702,632],{"className":1703},[655],[607,1705],{"className":1706,"style":1117},[1116],[607,1708,1071],{"className":1709},[1121],[607,1711],{"className":1712,"style":1117},[1116],[607,1714,1716,1719,1722,1725,1728],{"className":1715},[646],[607,1717],{"className":1718,"style":1020},[650],[607,1720,1654],{"className":1721},[655],[607,1723],{"className":1724,"style":1117},[1116],[607,1726,1071],{"className":1727},[1121],[607,1729],{"className":1730,"style":1117},[1116],[607,1732,1734,1737,1740,1743,1746,1749],{"className":1733},[646],[607,1735],{"className":1736,"style":1131},[650],[607,1738,1075],{"className":1739},[1135],[607,1741,1078],{"className":1742},[655],[607,1744],{"className":1745,"style":1142},[1116],[607,1747,1085],{"className":1748},[655],[607,1750,1752,1755],{"className":1751},[1149],[607,1753,1091],{"className":1754},[1149],[607,1756,1758],{"className":1757},[1156],[607,1759,1761,1781],{"className":1760},[1160,1161],[607,1762,1764,1778],{"className":1763},[1165],[607,1765,1767],{"className":1766,"style":1170},[1169],[607,1768,1769,1772],{"style":1173},[607,1770],{"className":1771,"style":1178},[1177],[607,1773,1775],{"className":1774},[1182,1183,1184,1185],[607,1776,1094],{"className":1777},[655,1185],[607,1779,1193],{"className":1780},[1192],[607,1782,1784],{"className":1783},[1165],[607,1785,1787],{"className":1786,"style":1200},[1169],[607,1788],{},".\nC'est bien le résultat trouvé plus haut.",[598,1791,1792],{},"Et l'addition fonctionne :",[607,1794,1797],{"className":1795},[1796],"katex-display",[607,1798,1800,1877],{"className":1799},[610],[607,1801,1803],{"className":1802},[614],[616,1804,1806],{"xmlns":618,"display":1805},"block",[620,1807,1808,1874],{},[623,1809,1810,1812,1814,1816,1818,1820,1822,1824,1826,1828,1830,1832,1838,1840,1842,1844,1846,1848,1854,1856,1858,1860,1862,1864,1866,1868],{},[630,1811,632],{},[626,1813,721],{},[626,1815,1075],{"stretchy":1074},[626,1817,628],{},[630,1819,632],{},[626,1821,1091],{"stretchy":1074},[626,1823,1071],{},[626,1825,1075],{"stretchy":1074},[630,1827,799],{},[1080,1829,1082],{},[630,1831,705],{},[1087,1833,1834,1836],{},[626,1835,1091],{"stretchy":1074},[630,1837,1094],{},[626,1839,721],{},[626,1841,1075],{"stretchy":1074},[630,1843,1078],{},[1080,1845,1082],{},[630,1847,1085],{},[1087,1849,1850,1852],{},[626,1851,1091],{"stretchy":1074},[630,1853,1094],{},[626,1855,1071],{},[626,1857,1075],{"stretchy":1074},[630,1859,692],{},[1080,1861,1082],{},[630,1863,799],{},[1080,1865,1082],{},[630,1867,799],{},[1087,1869,1870,1872],{},[626,1871,1091],{"stretchy":1074},[630,1873,1094],{},[634,1875,1876],{"encoding":636},"5 + (-5) = (0000\\,0101)_2 + (1111\\,1011)_2 = (1\\,0000\\,0000)_2",[607,1878,1880,1898,1925,1992,2059],{"className":1879,"ariaHidden":642},[641],[607,1881,1883,1886,1889,1892,1895],{"className":1882},[646],[607,1884],{"className":1885,"style":651},[650],[607,1887,632],{"className":1888},[655],[607,1890],{"className":1891,"style":1484},[1116],[607,1893,721],{"className":1894},[1488],[607,1896],{"className":1897,"style":1484},[1116],[607,1899,1901,1904,1907,1910,1913,1916,1919,1922],{"className":1900},[646],[607,1902],{"className":1903,"style":1131},[650],[607,1905,1075],{"className":1906},[1135],[607,1908,628],{"className":1909},[655],[607,1911,632],{"className":1912},[655],[607,1914,1091],{"className":1915},[1149],[607,1917],{"className":1918,"style":1117},[1116],[607,1920,1071],{"className":1921},[1121],[607,1923],{"className":1924,"style":1117},[1116],[607,1926,1928,1931,1934,1937,1940,1943,1983,1986,1989],{"className":1927},[646],[607,1929],{"className":1930,"style":1131},[650],[607,1932,1075],{"className":1933},[1135],[607,1935,799],{"className":1936},[655],[607,1938],{"className":1939,"style":1142},[1116],[607,1941,705],{"className":1942},[655],[607,1944,1946,1949],{"className":1945},[1149],[607,1947,1091],{"className":1948},[1149],[607,1950,1952],{"className":1951},[1156],[607,1953,1955,1975],{"className":1954},[1160,1161],[607,1956,1958,1972],{"className":1957},[1165],[607,1959,1961],{"className":1960,"style":1170},[1169],[607,1962,1963,1966],{"style":1173},[607,1964],{"className":1965,"style":1178},[1177],[607,1967,1969],{"className":1968},[1182,1183,1184,1185],[607,1970,1094],{"className":1971},[655,1185],[607,1973,1193],{"className":1974},[1192],[607,1976,1978],{"className":1977},[1165],[607,1979,1981],{"className":1980,"style":1200},[1169],[607,1982],{},[607,1984],{"className":1985,"style":1484},[1116],[607,1987,721],{"className":1988},[1488],[607,1990],{"className":1991,"style":1484},[1116],[607,1993,1995,1998,2001,2004,2007,2010,2050,2053,2056],{"className":1994},[646],[607,1996],{"className":1997,"style":1131},[650],[607,1999,1075],{"className":2000},[1135],[607,2002,1078],{"className":2003},[655],[607,2005],{"className":2006,"style":1142},[1116],[607,2008,1085],{"className":2009},[655],[607,2011,2013,2016],{"className":2012},[1149],[607,2014,1091],{"className":2015},[1149],[607,2017,2019],{"className":2018},[1156],[607,2020,2022,2042],{"className":2021},[1160,1161],[607,2023,2025,2039],{"className":2024},[1165],[607,2026,2028],{"className":2027,"style":1170},[1169],[607,2029,2030,2033],{"style":1173},[607,2031],{"className":2032,"style":1178},[1177],[607,2034,2036],{"className":2035},[1182,1183,1184,1185],[607,2037,1094],{"className":2038},[655,1185],[607,2040,1193],{"className":2041},[1192],[607,2043,2045],{"className":2044},[1165],[607,2046,2048],{"className":2047,"style":1200},[1169],[607,2049],{},[607,2051],{"className":2052,"style":1117},[1116],[607,2054,1071],{"className":2055},[1121],[607,2057],{"className":2058,"style":1117},[1116],[607,2060,2062,2065,2068,2071,2074,2077,2080,2083],{"className":2061},[646],[607,2063],{"className":2064,"style":1131},[650],[607,2066,1075],{"className":2067},[1135],[607,2069,692],{"className":2070},[655],[607,2072],{"className":2073,"style":1142},[1116],[607,2075,799],{"className":2076},[655],[607,2078],{"className":2079,"style":1142},[1116],[607,2081,799],{"className":2082},[655],[607,2084,2086,2089],{"className":2085},[1149],[607,2087,1091],{"className":2088},[1149],[607,2090,2092],{"className":2091},[1156],[607,2093,2095,2115],{"className":2094},[1160,1161],[607,2096,2098,2112],{"className":2097},[1165],[607,2099,2101],{"className":2100,"style":1170},[1169],[607,2102,2103,2106],{"style":1173},[607,2104],{"className":2105,"style":1178},[1177],[607,2107,2109],{"className":2108},[1182,1183,1184,1185],[607,2110,1094],{"className":2111},[655,1185],[607,2113,1193],{"className":2114},[1192],[607,2116,2118],{"className":2117},[1165],[607,2119,2121],{"className":2120,"style":1200},[1169],[607,2122],{},[598,2124,2125,2126,2238],{},"Le 9ᵉ bit déborde de l'octet et est ignoré : il reste ",[607,2127,2129,2159],{"className":2128},[610],[607,2130,2132],{"className":2131},[614],[616,2133,2134],{"xmlns":618},[620,2135,2136,2156],{},[623,2137,2138,2140,2142,2144,2146,2152,2154],{},[626,2139,1075],{"stretchy":1074},[630,2141,799],{},[1080,2143,1082],{},[630,2145,799],{},[1087,2147,2148,2150],{},[626,2149,1091],{"stretchy":1074},[630,2151,1094],{},[626,2153,1071],{},[630,2155,688],{},[634,2157,2158],{"encoding":636},"(0000\\,0000)_2 = 0",[607,2160,2162,2229],{"className":2161,"ariaHidden":642},[641],[607,2163,2165,2168,2171,2174,2177,2180,2220,2223,2226],{"className":2164},[646],[607,2166],{"className":2167,"style":1131},[650],[607,2169,1075],{"className":2170},[1135],[607,2172,799],{"className":2173},[655],[607,2175],{"className":2176,"style":1142},[1116],[607,2178,799],{"className":2179},[655],[607,2181,2183,2186],{"className":2182},[1149],[607,2184,1091],{"className":2185},[1149],[607,2187,2189],{"className":2188},[1156],[607,2190,2192,2212],{"className":2191},[1160,1161],[607,2193,2195,2209],{"className":2194},[1165],[607,2196,2198],{"className":2197,"style":1170},[1169],[607,2199,2200,2203],{"style":1173},[607,2201],{"className":2202,"style":1178},[1177],[607,2204,2206],{"className":2205},[1182,1183,1184,1185],[607,2207,1094],{"className":2208},[655,1185],[607,2210,1193],{"className":2211},[1192],[607,2213,2215],{"className":2214},[1165],[607,2216,2218],{"className":2217,"style":1200},[1169],[607,2219],{},[607,2221],{"className":2222,"style":1117},[1116],[607,2224,1071],{"className":2225},[1121],[607,2227],{"className":2228,"style":1117},[1116],[607,2230,2232,2235],{"className":2231},[646],[607,2233],{"className":2234,"style":1020},[650],[607,2236,688],{"className":2237},[655],".\nL'opération donne bien zéro.",[2240,2241,2242],"tip",{},[598,2243,2244,2247,2248,2251,2252,2255,2256,2258,2259,2317,2318,2321,2322,2325,2326,800,2328,2331,2332,2335,2336,2338,2339,742],{},[602,2245,2246],{},"Deux opérateurs Python utilisés ci-dessous"," :\n",[686,2249,2250],{},"a \u003C\u003C n"," décale les bits de ",[686,2253,2254],{},"a"," de ",[686,2257,840],{}," rangs vers la gauche (équivaut à\nmultiplier par ",[607,2260,2262,2279],{"className":2261},[610],[607,2263,2265],{"className":2264},[614],[616,2266,2267],{"xmlns":618},[620,2268,2269,2277],{},[623,2270,2271],{},[1239,2272,2273,2275],{},[630,2274,1094],{},[838,2276,840],{},[634,2278,1247],{"encoding":636},[607,2280,2282],{"className":2281,"ariaHidden":642},[641],[607,2283,2285,2288],{"className":2284},[646],[607,2286],{"className":2287,"style":1257},[650],[607,2289,2291,2294],{"className":2290},[655],[607,2292,1094],{"className":2293},[655],[607,2295,2297],{"className":2296},[1156],[607,2298,2300],{"className":2299},[1160],[607,2301,2303],{"className":2302},[1165],[607,2304,2306],{"className":2305,"style":1257},[1169],[607,2307,2308,2311],{"style":1278},[607,2309],{"className":2310,"style":1178},[1177],[607,2312,2314],{"className":2313},[1182,1183,1184,1185],[607,2315,840],{"className":2316},[655,856,1185],") ; ",[686,2319,2320],{},"a & b"," réalise un ",[602,2323,2324],{},"ET bit-à-bit"," (chaque bit du\nrésultat vaut 1 seulement si les deux bits correspondants de ",[686,2327,2254],{},[686,2329,2330],{},"b","\nvalent 1). Combinés, ",[686,2333,2334],{},"x & ((1 \u003C\u003C n) - 1)"," ne garde que les ",[686,2337,840],{}," bits de poids\nfaible de ",[686,2340,876],{},[2342,2343,2344],"python-snippet",{},[2345,2346,2351],"pre",{"className":2347,"code":2348,"language":2349,"meta":2350,"style":2350},"language-python shiki shiki-themes github-light github-light github-dark","# 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","python","",[686,2352,2353,2361,2367,2381,2395,2401,2438],{"__ignoreMap":2350},[607,2354,2357],{"class":2355,"line":2356},"line",1,[607,2358,2360],{"class":2359},"sCsY4","# Python gère les entiers signés en complément à deux conceptuellement\n",[607,2362,2364],{"class":2355,"line":2363},2,[607,2365,2366],{"class":2359},"# mais sans largeur fixe. On simule un complément à deux sur 8 bits.\n",[607,2368,2370,2374,2377],{"class":2355,"line":2369},3,[607,2371,2373],{"class":2372},"sxrX7","n_bits ",[607,2375,1071],{"class":2376},"s8jYJ",[607,2378,2380],{"class":2379},"sBjJW"," 8\n",[607,2382,2384,2387,2389,2392],{"class":2355,"line":2383},4,[607,2385,2386],{"class":2372},"x ",[607,2388,1071],{"class":2376},[607,2390,2391],{"class":2376}," -",[607,2393,2394],{"class":2379},"5\n",[607,2396,2398],{"class":2355,"line":2397},5,[607,2399,2400],{"class":2359},"# Conversion en complément à deux sur 8 bits :\n",[607,2402,2404,2407,2409,2412,2415,2418,2420,2423,2426,2429,2432,2435],{"class":2355,"line":2403},6,[607,2405,2406],{"class":2372},"code ",[607,2408,1071],{"class":2376},[607,2410,2411],{"class":2372}," x ",[607,2413,2414],{"class":2376},"&",[607,2416,2417],{"class":2372}," ((",[607,2419,692],{"class":2379},[607,2421,2422],{"class":2376}," \u003C\u003C",[607,2424,2425],{"class":2372}," n_bits) ",[607,2427,2428],{"class":2376},"-",[607,2430,2431],{"class":2379}," 1",[607,2433,2434],{"class":2372},")  ",[607,2436,2437],{"class":2359},"# masque les 8 bits de poids faible\n",[607,2439,2441,2444,2446,2449,2453,2456,2458,2461,2464,2466,2469,2471,2474,2477,2480,2482,2485,2487,2489,2491,2494],{"class":2355,"line":2440},7,[607,2442,2443],{"class":2379},"print",[607,2445,1075],{"class":2372},[607,2447,2448],{"class":2376},"f",[607,2450,2452],{"class":2451},"sIIMD","\"",[607,2454,2455],{"class":2379},"{",[607,2457,876],{"class":2372},[607,2459,2460],{"class":2379},"}",[607,2462,2463],{"class":2451}," sur ",[607,2465,2455],{"class":2379},[607,2467,2468],{"class":2372},"n_bits",[607,2470,2460],{"class":2379},[607,2472,2473],{"class":2451}," bits : ",[607,2475,2476],{"class":2379},"{bin",[607,2478,2479],{"class":2372},"(code)",[607,2481,2460],{"class":2379},[607,2483,2484],{"class":2451}," (",[607,2486,2455],{"class":2379},[607,2488,686],{"class":2372},[607,2490,2460],{"class":2379},[607,2492,2493],{"class":2451}," non signé)\"",[607,2495,2496],{"class":2372},")\n",[598,2498,2499],{},"Sortie attendue :",[2345,2501,2506],{"className":2502,"code":2504,"language":2505},[2503],"language-text","-5 sur 8 bits : 0b11111011 (251 non signé)\n","text",[686,2507,2504],{"__ignoreMap":2350},[593,2509,2511],{"id":2510},"étendue-représentable","Étendue représentable",[598,2513,824,2514,2542],{},[607,2515,2517,2530],{"className":2516},[610],[607,2518,2520],{"className":2519},[614],[616,2521,2522],{"xmlns":618},[620,2523,2524,2528],{},[623,2525,2526],{},[838,2527,840],{},[634,2529,840],{"encoding":636},[607,2531,2533],{"className":2532,"ariaHidden":642},[641],[607,2534,2536,2539],{"className":2535},[646],[607,2537],{"className":2538,"style":852},[650],[607,2540,840],{"className":2541},[655,856]," bits en complément à deux, on couvre l'intervalle :",[607,2544,2546],{"className":2545},[1796],[607,2547,2549,2598],{"className":2548},[610],[607,2550,2552],{"className":2551},[614],[616,2553,2554],{"xmlns":618,"display":1805},[620,2555,2556,2595],{},[623,2557,2558,2560,2572,2575,2577,2579,2591,2593],{},[626,2559,628],{},[1239,2561,2562,2564],{},[630,2563,1094],{},[623,2565,2566,2568,2570],{},[838,2567,840],{},[626,2569,628],{},[630,2571,692],{},[626,2573,2574],{},"≤",[838,2576,876],{},[626,2578,2574],{},[1239,2580,2581,2583],{},[630,2582,1094],{},[623,2584,2585,2587,2589],{},[838,2586,840],{},[626,2588,628],{},[630,2590,692],{},[626,2592,628],{},[630,2594,692],{},[634,2596,2597],{"encoding":636},"-2^{n-1} \\leq x \\leq 2^{n-1} - 1",[607,2599,2601,2660,2679,2733],{"className":2600,"ariaHidden":642},[641],[607,2602,2604,2608,2611,2651,2654,2657],{"className":2603},[646],[607,2605],{"className":2606,"style":2607},[650],"height:1.0001em;vertical-align:-0.136em;",[607,2609,628],{"className":2610},[655],[607,2612,2614,2617],{"className":2613},[655],[607,2615,1094],{"className":2616},[655],[607,2618,2620],{"className":2619},[1156],[607,2621,2623],{"className":2622},[1160],[607,2624,2626],{"className":2625},[1165],[607,2627,2630],{"className":2628,"style":2629},[1169],"height:0.8641em;",[607,2631,2633,2636],{"style":2632},"top:-3.113em;margin-right:0.05em;",[607,2634],{"className":2635,"style":1178},[1177],[607,2637,2639],{"className":2638},[1182,1183,1184,1185],[607,2640,2642,2645,2648],{"className":2641},[655,1185],[607,2643,840],{"className":2644},[655,856,1185],[607,2646,628],{"className":2647},[1488,1185],[607,2649,692],{"className":2650},[655,1185],[607,2652],{"className":2653,"style":1117},[1116],[607,2655,2574],{"className":2656},[1121],[607,2658],{"className":2659,"style":1117},[1116],[607,2661,2663,2667,2670,2673,2676],{"className":2662},[646],[607,2664],{"className":2665,"style":2666},[650],"height:0.7719em;vertical-align:-0.136em;",[607,2668,876],{"className":2669},[655,856],[607,2671],{"className":2672,"style":1117},[1116],[607,2674,2574],{"className":2675},[1121],[607,2677],{"className":2678,"style":1117},[1116],[607,2680,2682,2686,2724,2727,2730],{"className":2681},[646],[607,2683],{"className":2684,"style":2685},[650],"height:0.9474em;vertical-align:-0.0833em;",[607,2687,2689,2692],{"className":2688},[655],[607,2690,1094],{"className":2691},[655],[607,2693,2695],{"className":2694},[1156],[607,2696,2698],{"className":2697},[1160],[607,2699,2701],{"className":2700},[1165],[607,2702,2704],{"className":2703,"style":2629},[1169],[607,2705,2706,2709],{"style":2632},[607,2707],{"className":2708,"style":1178},[1177],[607,2710,2712],{"className":2711},[1182,1183,1184,1185],[607,2713,2715,2718,2721],{"className":2714},[655,1185],[607,2716,840],{"className":2717},[655,856,1185],[607,2719,628],{"className":2720},[1488,1185],[607,2722,692],{"className":2723},[655,1185],[607,2725],{"className":2726,"style":1484},[1116],[607,2728,628],{"className":2729},[1488],[607,2731],{"className":2732,"style":1484},[1116],[607,2734,2736,2739],{"className":2735},[646],[607,2737],{"className":2738,"style":1020},[650],[607,2740,692],{"className":2741},[655],[971,2743,2744,2757],{},[974,2745,2746],{},[977,2747,2748,2751,2754],{},[980,2749,2750],{},"Bits",[980,2752,2753],{},"Minimum",[980,2755,2756],{},"Maximum",[987,2758,2759,2838,2938,3078],{},[977,2760,2761,2764,2801],{},[992,2762,2763],{},"8",[992,2765,2766],{},[607,2767,2769,2786],{"className":2768},[610],[607,2770,2772],{"className":2771},[614],[616,2773,2774],{"xmlns":618},[620,2775,2776,2783],{},[623,2777,2778,2780],{},[626,2779,628],{},[630,2781,2782],{},"128",[634,2784,2785],{"encoding":636},"-128",[607,2787,2789],{"className":2788,"ariaHidden":642},[641],[607,2790,2792,2795,2798],{"className":2791},[646],[607,2793],{"className":2794,"style":651},[650],[607,2796,628],{"className":2797},[655],[607,2799,2782],{"className":2800},[655],[992,2802,2803],{},[607,2804,2806,2823],{"className":2805},[610],[607,2807,2809],{"className":2808},[614],[616,2810,2811],{"xmlns":618},[620,2812,2813,2820],{},[623,2814,2815,2817],{},[626,2816,721],{},[630,2818,2819],{},"127",[634,2821,2822],{"encoding":636},"+127",[607,2824,2826],{"className":2825,"ariaHidden":642},[641],[607,2827,2829,2832,2835],{"className":2828},[646],[607,2830],{"className":2831,"style":651},[650],[607,2833,721],{"className":2834},[655],[607,2836,2819],{"className":2837},[655],[977,2839,2840,2843,2891],{},[992,2841,2842],{},"16",[992,2844,2845],{},[607,2846,2848,2870],{"className":2847},[610],[607,2849,2851],{"className":2850},[614],[616,2852,2853],{"xmlns":618},[620,2854,2855,2867],{},[623,2856,2857,2859,2862,2864],{},[626,2858,628],{},[630,2860,2861],{},"32",[1080,2863,1082],{},[630,2865,2866],{},"768",[634,2868,2869],{"encoding":636},"-32\\,768",[607,2871,2873],{"className":2872,"ariaHidden":642},[641],[607,2874,2876,2879,2882,2885,2888],{"className":2875},[646],[607,2877],{"className":2878,"style":651},[650],[607,2880,628],{"className":2881},[655],[607,2883,2861],{"className":2884},[655],[607,2886],{"className":2887,"style":1142},[1116],[607,2889,2866],{"className":2890},[655],[992,2892,2893],{},[607,2894,2896,2917],{"className":2895},[610],[607,2897,2899],{"className":2898},[614],[616,2900,2901],{"xmlns":618},[620,2902,2903,2914],{},[623,2904,2905,2907,2909,2911],{},[626,2906,721],{},[630,2908,2861],{},[1080,2910,1082],{},[630,2912,2913],{},"767",[634,2915,2916],{"encoding":636},"+32\\,767",[607,2918,2920],{"className":2919,"ariaHidden":642},[641],[607,2921,2923,2926,2929,2932,2935],{"className":2922},[646],[607,2924],{"className":2925,"style":651},[650],[607,2927,721],{"className":2928},[655],[607,2930,2861],{"className":2931},[655],[607,2933],{"className":2934,"style":1142},[1116],[607,2936,2913],{"className":2937},[655],[977,2939,2940,2942,3011],{},[992,2941,2861],{},[992,2943,2944],{},[607,2945,2947,2978],{"className":2946},[610],[607,2948,2950],{"className":2949},[614],[616,2951,2952],{"xmlns":618},[620,2953,2954,2975],{},[623,2955,2956,2958,2960,2962,2965,2967,2970,2972],{},[626,2957,628],{},[630,2959,1094],{},[1080,2961,1082],{},[630,2963,2964],{},"147",[1080,2966,1082],{},[630,2968,2969],{},"483",[1080,2971,1082],{},[630,2973,2974],{},"648",[634,2976,2977],{"encoding":636},"-2\\,147\\,483\\,648",[607,2979,2981],{"className":2980,"ariaHidden":642},[641],[607,2982,2984,2987,2990,2993,2996,2999,3002,3005,3008],{"className":2983},[646],[607,2985],{"className":2986,"style":651},[650],[607,2988,628],{"className":2989},[655],[607,2991,1094],{"className":2992},[655],[607,2994],{"className":2995,"style":1142},[1116],[607,2997,2964],{"className":2998},[655],[607,3000],{"className":3001,"style":1142},[1116],[607,3003,2969],{"className":3004},[655],[607,3006],{"className":3007,"style":1142},[1116],[607,3009,2974],{"className":3010},[655],[992,3012,3013],{},[607,3014,3016,3045],{"className":3015},[610],[607,3017,3019],{"className":3018},[614],[616,3020,3021],{"xmlns":618},[620,3022,3023,3042],{},[623,3024,3025,3027,3029,3031,3033,3035,3037,3039],{},[626,3026,721],{},[630,3028,1094],{},[1080,3030,1082],{},[630,3032,2964],{},[1080,3034,1082],{},[630,3036,2969],{},[1080,3038,1082],{},[630,3040,3041],{},"647",[634,3043,3044],{"encoding":636},"+2\\,147\\,483\\,647",[607,3046,3048],{"className":3047,"ariaHidden":642},[641],[607,3049,3051,3054,3057,3060,3063,3066,3069,3072,3075],{"className":3050},[646],[607,3052],{"className":3053,"style":651},[650],[607,3055,721],{"className":3056},[655],[607,3058,1094],{"className":3059},[655],[607,3061],{"className":3062,"style":1142},[1116],[607,3064,2964],{"className":3065},[655],[607,3067],{"className":3068,"style":1142},[1116],[607,3070,2969],{"className":3071},[655],[607,3073],{"className":3074,"style":1142},[1116],[607,3076,3041],{"className":3077},[655],[977,3079,3080,3083,3155],{},[992,3081,3082],{},"64",[992,3084,3085],{},[607,3086,3088,3109],{"className":3087},[610],[607,3089,3091],{"className":3090},[614],[616,3092,3093],{"xmlns":618},[620,3094,3095,3106],{},[623,3096,3097,3099],{},[626,3098,628],{},[1239,3100,3101,3103],{},[630,3102,1094],{},[630,3104,3105],{},"63",[634,3107,3108],{"encoding":636},"-2^{63}",[607,3110,3112],{"className":3111,"ariaHidden":642},[641],[607,3113,3115,3119,3122],{"className":3114},[646],[607,3116],{"className":3117,"style":3118},[650],"height:0.8974em;vertical-align:-0.0833em;",[607,3120,628],{"className":3121},[655],[607,3123,3125,3128],{"className":3124},[655],[607,3126,1094],{"className":3127},[655],[607,3129,3131],{"className":3130},[1156],[607,3132,3134],{"className":3133},[1160],[607,3135,3137],{"className":3136},[1165],[607,3138,3141],{"className":3139,"style":3140},[1169],"height:0.8141em;",[607,3142,3143,3146],{"style":1278},[607,3144],{"className":3145,"style":1178},[1177],[607,3147,3149],{"className":3148},[1182,1183,1184,1185],[607,3150,3152],{"className":3151},[655,1185],[607,3153,3105],{"className":3154},[655,1185],[992,3156,3157],{},[607,3158,3160,3182],{"className":3159},[610],[607,3161,3163],{"className":3162},[614],[616,3164,3165],{"xmlns":618},[620,3166,3167,3179],{},[623,3168,3169,3175,3177],{},[1239,3170,3171,3173],{},[630,3172,1094],{},[630,3174,3105],{},[626,3176,628],{},[630,3178,692],{},[634,3180,3181],{"encoding":636},"2^{63} - 1",[607,3183,3185,3232],{"className":3184,"ariaHidden":642},[641],[607,3186,3188,3191,3223,3226,3229],{"className":3187},[646],[607,3189],{"className":3190,"style":3118},[650],[607,3192,3194,3197],{"className":3193},[655],[607,3195,1094],{"className":3196},[655],[607,3198,3200],{"className":3199},[1156],[607,3201,3203],{"className":3202},[1160],[607,3204,3206],{"className":3205},[1165],[607,3207,3209],{"className":3208,"style":3140},[1169],[607,3210,3211,3214],{"style":1278},[607,3212],{"className":3213,"style":1178},[1177],[607,3215,3217],{"className":3216},[1182,1183,1184,1185],[607,3218,3220],{"className":3219},[655,1185],[607,3221,3105],{"className":3222},[655,1185],[607,3224],{"className":3225,"style":1484},[1116],[607,3227,628],{"className":3228},[1488],[607,3230],{"className":3231,"style":1484},[1116],[607,3233,3235,3238],{"className":3234},[646],[607,3236],{"className":3237,"style":1020},[650],[607,3239,692],{"className":3240},[655],[598,3242,3243,3244,3247],{},"On remarque que l'intervalle n'est ",[602,3245,3246],{},"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).",[3249,3250,3253,3259],"self-eval-qcm",{"correct":3251,"slug":3252},"3","entiers-relatifs-c1",[3254,3255,3256],"template",{"v-slot:question":2350},[598,3257,3258],{},"Sur 8 bits en complément à deux, combien d'entiers distincts peut-on représenter ?",[3254,3260,3261],{"v-slot:choices":2350},[698,3262,3263,3265,3267,3270],{},[701,3264,2819],{},[701,3266,2782],{},[701,3268,3269],{},"255",[701,3271,1645],{},[593,3273,3275],{"id":3274},"bits-nécessaires-pour-une-somme-ou-un-produit","Bits nécessaires pour une somme ou un produit",[598,3277,3278],{},"Combien de bits faut-il prévoir si l'on additionne ou si l'on multiplie deux\nentiers ?",[698,3280,3281,3368],{},[701,3282,3283,3286,3287,3315,3316,3367],{},[602,3284,3285],{},"Somme"," de deux entiers de ",[607,3288,3290,3303],{"className":3289},[610],[607,3291,3293],{"className":3292},[614],[616,3294,3295],{"xmlns":618},[620,3296,3297,3301],{},[623,3298,3299],{},[838,3300,840],{},[634,3302,840],{"encoding":636},[607,3304,3306],{"className":3305,"ariaHidden":642},[641],[607,3307,3309,3312],{"className":3308},[646],[607,3310],{"className":3311,"style":852},[650],[607,3313,840],{"className":3314},[655,856]," bits : il peut y avoir une retenue qui\najoute 1 bit. Le résultat tient sur ",[607,3317,3319,3337],{"className":3318},[610],[607,3320,3322],{"className":3321},[614],[616,3323,3324],{"xmlns":618},[620,3325,3326,3334],{},[623,3327,3328,3330,3332],{},[838,3329,840],{},[626,3331,721],{},[630,3333,692],{},[634,3335,3336],{"encoding":636},"n+1",[607,3338,3340,3358],{"className":3339,"ariaHidden":642},[641],[607,3341,3343,3346,3349,3352,3355],{"className":3342},[646],[607,3344],{"className":3345,"style":1409},[650],[607,3347,840],{"className":3348},[655,856],[607,3350],{"className":3351,"style":1484},[1116],[607,3353,721],{"className":3354},[1488],[607,3356],{"className":3357,"style":1484},[1116],[607,3359,3361,3364],{"className":3360},[646],[607,3362],{"className":3363,"style":1020},[650],[607,3365,692],{"className":3366},[655]," bits.",[701,3369,3370,3286,3373,3401,3402,3587,3588,3367],{},[602,3371,3372],{},"Produit",[607,3374,3376,3389],{"className":3375},[610],[607,3377,3379],{"className":3378},[614],[616,3380,3381],{"xmlns":618},[620,3382,3383,3387],{},[623,3384,3385],{},[838,3386,840],{},[634,3388,840],{"encoding":636},[607,3390,3392],{"className":3391,"ariaHidden":642},[641],[607,3393,3395,3398],{"className":3394},[646],[607,3396],{"className":3397,"style":852},[650],[607,3399,840],{"className":3400},[655,856]," bits : le résultat peut atteindre\n",[607,3403,3405,3448],{"className":3404},[610],[607,3406,3408],{"className":3407},[614],[616,3409,3410],{"xmlns":618},[620,3411,3412,3445],{},[623,3413,3414,3416,3422,3424,3426,3432,3435],{},[626,3415,1075],{"stretchy":1074},[1239,3417,3418,3420],{},[630,3419,1094],{},[838,3421,840],{},[626,3423,628],{},[630,3425,692],{},[1239,3427,3428,3430],{},[626,3429,1091],{"stretchy":1074},[630,3431,1094],{},[626,3433,3434],{},"≈",[1239,3436,3437,3439],{},[630,3438,1094],{},[623,3440,3441,3443],{},[630,3442,1094],{},[838,3444,840],{},[634,3446,3447],{"encoding":636},"(2^n - 1)^2 \\approx 2^{2n}",[607,3449,3451,3498,3546],{"className":3450,"ariaHidden":642},[641],[607,3452,3454,3457,3460,3489,3492,3495],{"className":3453},[646],[607,3455],{"className":3456,"style":1131},[650],[607,3458,1075],{"className":3459},[1135],[607,3461,3463,3466],{"className":3462},[655],[607,3464,1094],{"className":3465},[655],[607,3467,3469],{"className":3468},[1156],[607,3470,3472],{"className":3471},[1160],[607,3473,3475],{"className":3474},[1165],[607,3476,3478],{"className":3477,"style":1257},[1169],[607,3479,3480,3483],{"style":1278},[607,3481],{"className":3482,"style":1178},[1177],[607,3484,3486],{"className":3485},[1182,1183,1184,1185],[607,3487,840],{"className":3488},[655,856,1185],[607,3490],{"className":3491,"style":1484},[1116],[607,3493,628],{"className":3494},[1488],[607,3496],{"className":3497,"style":1484},[1116],[607,3499,3501,3505,3508,3537,3540,3543],{"className":3500},[646],[607,3502],{"className":3503,"style":3504},[650],"height:1.0641em;vertical-align:-0.25em;",[607,3506,692],{"className":3507},[655],[607,3509,3511,3514],{"className":3510},[1149],[607,3512,1091],{"className":3513},[1149],[607,3515,3517],{"className":3516},[1156],[607,3518,3520],{"className":3519},[1160],[607,3521,3523],{"className":3522},[1165],[607,3524,3526],{"className":3525,"style":3140},[1169],[607,3527,3528,3531],{"style":1278},[607,3529],{"className":3530,"style":1178},[1177],[607,3532,3534],{"className":3533},[1182,1183,1184,1185],[607,3535,1094],{"className":3536},[655,1185],[607,3538],{"className":3539,"style":1117},[1116],[607,3541,3434],{"className":3542},[1121],[607,3544],{"className":3545,"style":1117},[1116],[607,3547,3549,3552],{"className":3548},[646],[607,3550],{"className":3551,"style":3140},[650],[607,3553,3555,3558],{"className":3554},[655],[607,3556,1094],{"className":3557},[655],[607,3559,3561],{"className":3560},[1156],[607,3562,3564],{"className":3563},[1160],[607,3565,3567],{"className":3566},[1165],[607,3568,3570],{"className":3569,"style":3140},[1169],[607,3571,3572,3575],{"style":1278},[607,3573],{"className":3574,"style":1178},[1177],[607,3576,3578],{"className":3577},[1182,1183,1184,1185],[607,3579,3581,3584],{"className":3580},[655,1185],[607,3582,1094],{"className":3583},[655,1185],[607,3585,840],{"className":3586},[655,856,1185],", donc il faut prévoir ",[607,3589,3591,3607],{"className":3590},[610],[607,3592,3594],{"className":3593},[614],[616,3595,3596],{"xmlns":618},[620,3597,3598,3604],{},[623,3599,3600,3602],{},[630,3601,1094],{},[838,3603,840],{},[634,3605,3606],{"encoding":636},"2n",[607,3608,3610],{"className":3609,"ariaHidden":642},[641],[607,3611,3613,3616,3619],{"className":3612},[646],[607,3614],{"className":3615,"style":1020},[650],[607,3617,1094],{"className":3618},[655],[607,3620,840],{"className":3621},[655,856],[2240,3623,3624],{},[598,3625,3626],{},"En pratique, sur un processeur 64 bits, multiplier deux entiers de 32 bits\ntient encore largement dans 64 bits, donc sans débordement.",[593,3628,3630],{"id":3629},"le-cas-python-entiers-de-taille-arbitraire","Le cas Python — entiers de taille arbitraire",[598,3632,3633,3634,3636],{},"Python (à partir de la version 3) ne limite ",[602,3635,816],{}," la taille des entiers : ils\npeuvent croître autant que la mémoire le permet.",[2345,3638,3640],{"className":2347,"code":3639,"language":2349,"meta":2350,"style":2350},"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",[686,3641,3642,3658,3663,3669,3686,3691,3695,3713],{"__ignoreMap":2350},[607,3643,3644,3646,3648,3650,3653,3656],{"class":2355,"line":2356},[607,3645,2443],{"class":2379},[607,3647,1075],{"class":2372},[607,3649,1094],{"class":2379},[607,3651,3652],{"class":2376}," **",[607,3654,3655],{"class":2379}," 100",[607,3657,2496],{"class":2372},[607,3659,3660],{"class":2355,"line":2363},[607,3661,3662],{"class":2359},"# 1267650600228229401496703205376\n",[607,3664,3665],{"class":2355,"line":2369},[607,3666,3668],{"emptyLinePlaceholder":3667},true,"\n",[607,3670,3671,3673,3675,3677,3679,3681,3684],{"class":2355,"line":2383},[607,3672,2443],{"class":2379},[607,3674,1075],{"class":2372},[607,3676,2428],{"class":2376},[607,3678,692],{"class":2379},[607,3680,2422],{"class":2376},[607,3682,3683],{"class":2379}," 8",[607,3685,2496],{"class":2372},[607,3687,3688],{"class":2355,"line":2397},[607,3689,3690],{"class":2359},"# -256\n",[607,3692,3693],{"class":2355,"line":2403},[607,3694,3668],{"emptyLinePlaceholder":3667},[607,3696,3697,3699,3701,3704,3706,3708,3710],{"class":2355,"line":2440},[607,3698,2443],{"class":2379},[607,3700,1075],{"class":2372},[607,3702,3703],{"class":2379},"bin",[607,3705,1075],{"class":2372},[607,3707,2428],{"class":2376},[607,3709,632],{"class":2379},[607,3711,3712],{"class":2372},"))\n",[607,3714,3716],{"class":2355,"line":3715},8,[607,3717,3718],{"class":2359},"# -0b101  — Python affiche le signe, pas le complément à deux\n",[598,3720,3721],{},"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.",[3249,3723,3725,3765],{"correct":3251,"slug":3724},"entiers-relatifs-c2",[3254,3726,3727],{"v-slot:question":2350},[598,3728,3729,3730,3764],{},"Que vaut ",[607,3731,3733,3749],{"className":3732},[610],[607,3734,3736],{"className":3735},[614],[616,3737,3738],{"xmlns":618},[620,3739,3740,3746],{},[623,3741,3742,3744],{},[626,3743,628],{},[630,3745,692],{},[634,3747,3748],{"encoding":636},"-1",[607,3750,3752],{"className":3751,"ariaHidden":642},[641],[607,3753,3755,3758,3761],{"className":3754},[646],[607,3756],{"className":3757,"style":651},[650],[607,3759,628],{"className":3760},[655],[607,3762,692],{"className":3763},[655]," en complément à deux sur 8 bits ?",[3254,3766,3767],{"v-slot:choices":2350},[698,3768,3769,3772,3775,3778],{},[701,3770,3771],{},"0000 0001",[701,3773,3774],{},"1000 0001",[701,3776,3777],{},"1111 1110",[701,3779,3780],{},"1111 1111",[593,3782,3784],{"id":3783},"pièges-courants","Pièges courants",[809,3786,3787],{},[698,3788,3789,3798,3805],{},[701,3790,1209,3791,3794,3795,3797],{},[602,3792,3793],{},"bit de poids fort vaut 1"," pour tout négatif, mais ce n'est ",[602,3796,816],{},"\nl'opposé en valeur absolue : il faut vraiment refaire « inverser + 1 ».",[701,3799,3800,3801,3804],{},"Sur un type entier de taille fixe (C, Java), additionner deux grands\npositifs peut donner un négatif à cause du ",[602,3802,3803],{},"débordement"," (overflow). Python\nvous protège de cela.",[701,3806,3807,3808,742],{},"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",[607,3809,3811,3828],{"className":3810},[610],[607,3812,3814],{"className":3813},[614],[616,3815,3816],{"xmlns":618},[620,3817,3818,3826],{},[623,3819,3820],{},[1239,3821,3822,3824],{},[630,3823,1094],{},[838,3825,840],{},[634,3827,1247],{"encoding":636},[607,3829,3831],{"className":3830,"ariaHidden":642},[641],[607,3832,3834,3837],{"className":3833},[646],[607,3835],{"className":3836,"style":1257},[650],[607,3838,3840,3843],{"className":3839},[655],[607,3841,1094],{"className":3842},[655],[607,3844,3846],{"className":3845},[1156],[607,3847,3849],{"className":3848},[1160],[607,3850,3852],{"className":3851},[1165],[607,3853,3855],{"className":3854,"style":1257},[1169],[607,3856,3857,3860],{"style":1278},[607,3858],{"className":3859,"style":1178},[1177],[607,3861,3863],{"className":3862},[1182,1183,1184,1185],[607,3864,840],{"className":3865},[655,856,1185],[593,3867,3869],{"id":3868},"pour-aller-plus-loin","Pour aller plus loin",[598,3871,3872,3873,3876],{},"Le complément à deux explique aussi pourquoi le décalage à droite d'un nombre\nnégatif est subtil : il faut ",[602,3874,3875],{},"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.",[3878,3879,3880],"style",{},"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":2350,"searchDepth":2363,"depth":2363,"links":3882},[3883,3884,3885,3886,3888,3889,3890,3891,3892],{"id":595,"depth":2363,"text":596},{"id":676,"depth":2363,"text":677},{"id":820,"depth":2363,"text":821},{"id":1224,"depth":2363,"text":3887},"Pourquoi ça marche — l'addition modulo 2n2^n2n",{"id":2510,"depth":2363,"text":2511},{"id":3274,"depth":2363,"text":3275},{"id":3629,"depth":2363,"text":3630},{"id":3783,"depth":2363,"text":3784},{"id":3868,"depth":2363,"text":3869},"Comment représenter un entier négatif en binaire et pourquoi le complément à deux s'est imposé.",null,"md",{},[3898,3902],{"slug":3252,"kind":3899,"question":3258,"choices":3900,"correct":3251,"multi":3901,"explanation":2350},"qcm",[2819,2782,3269,1645],false,{"slug":3724,"kind":3899,"question":3903,"choices":3904,"correct":3251,"multi":3901,"explanation":2350},"Que vaut −1-1−1 en complément à deux sur 8 bits ?",[3771,3774,3777,3780],{"title":267,"description":3893},"hrB49T9jmSbyZSG7hdTQsZO1Znz_hHVLK_SUDFpArlc",{"id":3908,"title":3909,"bo":3910,"body":3911,"description":4789,"eval":3894,"extension":3895,"labo":3894,"meta":4790,"navigation":3901,"path":4791,"quizzes":4792,"readingTime":2356,"seo":4793,"stem":4794,"tp":3894,"__hash__":4795},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.memo.md","Entiers relatifs et complément à deux — l'essentiel",[588],{"type":590,"value":3912,"toc":4782},[3913,3917,4065,4069,4361,4365,4675,4677,4731,4735,4779],[593,3914,3916],{"id":3915},"en-une-phrase","En une phrase",[598,3918,1209,3919,3921,3922,2463,3955,3983,3984,4064],{},[602,3920,860],{}," code un entier négatif ",[607,3923,3925,3940],{"className":3924},[610],[607,3926,3928],{"className":3927},[614],[616,3929,3930],{"xmlns":618},[620,3931,3932,3938],{},[623,3933,3934,3936],{},[626,3935,628],{},[838,3937,876],{},[634,3939,1399],{"encoding":636},[607,3941,3943],{"className":3942,"ariaHidden":642},[641],[607,3944,3946,3949,3952],{"className":3945},[646],[607,3947],{"className":3948,"style":1409},[650],[607,3950,628],{"className":3951},[655],[607,3953,876],{"className":3954},[655,856],[607,3956,3958,3971],{"className":3957},[610],[607,3959,3961],{"className":3960},[614],[616,3962,3963],{"xmlns":618},[620,3964,3965,3969],{},[623,3966,3967],{},[838,3968,840],{},[634,3970,840],{"encoding":636},[607,3972,3974],{"className":3973,"ariaHidden":642},[641],[607,3975,3977,3980],{"className":3976},[646],[607,3978],{"className":3979,"style":852},[650],[607,3981,840],{"className":3982},[655,856]," bits comme\n",[607,3985,3987,4008],{"className":3986},[610],[607,3988,3990],{"className":3989},[614],[616,3991,3992],{"xmlns":618},[620,3993,3994,4006],{},[623,3995,3996,4002,4004],{},[1239,3997,3998,4000],{},[630,3999,1094],{},[838,4001,840],{},[626,4003,628],{},[838,4005,876],{},[634,4007,1441],{"encoding":636},[607,4009,4011,4055],{"className":4010,"ariaHidden":642},[641],[607,4012,4014,4017,4046,4049,4052],{"className":4013},[646],[607,4015],{"className":4016,"style":1451},[650],[607,4018,4020,4023],{"className":4019},[655],[607,4021,1094],{"className":4022},[655],[607,4024,4026],{"className":4025},[1156],[607,4027,4029],{"className":4028},[1160],[607,4030,4032],{"className":4031},[1165],[607,4033,4035],{"className":4034,"style":1257},[1169],[607,4036,4037,4040],{"style":1278},[607,4038],{"className":4039,"style":1178},[1177],[607,4041,4043],{"className":4042},[1182,1183,1184,1185],[607,4044,840],{"className":4045},[655,856,1185],[607,4047],{"className":4048,"style":1484},[1116],[607,4050,628],{"className":4051},[1488],[607,4053],{"className":4054,"style":1484},[1116],[607,4056,4058,4061],{"className":4057},[646],[607,4059],{"className":4060,"style":852},[650],[607,4062,876],{"className":4063},[655,856]," — ce qui permet de faire des additions sans circuit dédié au signe.",[593,4066,4068],{"id":4067},"à-mémoriser","À mémoriser",[698,4070,4071,4281,4349,4358],{},[701,4072,824,4073,4101,4102,742],{},[607,4074,4076,4089],{"className":4075},[610],[607,4077,4079],{"className":4078},[614],[616,4080,4081],{"xmlns":618},[620,4082,4083,4087],{},[623,4084,4085],{},[838,4086,840],{},[634,4088,840],{"encoding":636},[607,4090,4092],{"className":4091,"ariaHidden":642},[641],[607,4093,4095,4098],{"className":4094},[646],[607,4096],{"className":4097,"style":852},[650],[607,4099,840],{"className":4100},[655,856]," bits en complément à deux : intervalle ",[607,4103,4105,4159],{"className":4104},[610],[607,4106,4108],{"className":4107},[614],[616,4109,4110],{"xmlns":618},[620,4111,4112,4156],{},[623,4113,4114,4117,4119,4131,4134,4137,4149,4151,4153],{},[626,4115,4116],{"stretchy":1074},"[",[626,4118,628],{},[1239,4120,4121,4123],{},[630,4122,1094],{},[623,4124,4125,4127,4129],{},[838,4126,840],{},[626,4128,628],{},[630,4130,692],{},[626,4132,4133],{"separator":642},",",[1080,4135,4136],{}," ",[1239,4138,4139,4141],{},[630,4140,1094],{},[623,4142,4143,4145,4147],{},[838,4144,840],{},[626,4146,628],{},[630,4148,692],{},[626,4150,628],{},[630,4152,692],{},[626,4154,4155],{"stretchy":1074},"]",[634,4157,4158],{"encoding":636},"[-2^{n-1},\\ 2^{n-1} - 1]",[607,4160,4162,4269],{"className":4161,"ariaHidden":642},[641],[607,4163,4165,4168,4171,4174,4212,4216,4219,4222,4260,4263,4266],{"className":4164},[646],[607,4166],{"className":4167,"style":3504},[650],[607,4169,4116],{"className":4170},[1135],[607,4172,628],{"className":4173},[655],[607,4175,4177,4180],{"className":4176},[655],[607,4178,1094],{"className":4179},[655],[607,4181,4183],{"className":4182},[1156],[607,4184,4186],{"className":4185},[1160],[607,4187,4189],{"className":4188},[1165],[607,4190,4192],{"className":4191,"style":3140},[1169],[607,4193,4194,4197],{"style":1278},[607,4195],{"className":4196,"style":1178},[1177],[607,4198,4200],{"className":4199},[1182,1183,1184,1185],[607,4201,4203,4206,4209],{"className":4202},[655,1185],[607,4204,840],{"className":4205},[655,856,1185],[607,4207,628],{"className":4208},[1488,1185],[607,4210,692],{"className":4211},[655,1185],[607,4213,4133],{"className":4214},[4215],"mpunct",[607,4217,4136],{"className":4218},[1116],[607,4220],{"className":4221,"style":1142},[1116],[607,4223,4225,4228],{"className":4224},[655],[607,4226,1094],{"className":4227},[655],[607,4229,4231],{"className":4230},[1156],[607,4232,4234],{"className":4233},[1160],[607,4235,4237],{"className":4236},[1165],[607,4238,4240],{"className":4239,"style":3140},[1169],[607,4241,4242,4245],{"style":1278},[607,4243],{"className":4244,"style":1178},[1177],[607,4246,4248],{"className":4247},[1182,1183,1184,1185],[607,4249,4251,4254,4257],{"className":4250},[655,1185],[607,4252,840],{"className":4253},[655,856,1185],[607,4255,628],{"className":4256},[1488,1185],[607,4258,692],{"className":4259},[655,1185],[607,4261],{"className":4262,"style":1484},[1116],[607,4264,628],{"className":4265},[1488],[607,4267],{"className":4268,"style":1484},[1116],[607,4270,4272,4275,4278],{"className":4271},[646],[607,4273],{"className":4274,"style":1131},[650],[607,4276,692],{"className":4277},[655],[607,4279,4155],{"className":4280},[1149],[701,4282,4283,4319,4320,4348],{},[602,4284,4285,4286],{},"Méthode pour coder ",[607,4287,4289,4304],{"className":4288},[610],[607,4290,4292],{"className":4291},[614],[616,4293,4294],{"xmlns":618},[620,4295,4296,4302],{},[623,4297,4298,4300],{},[626,4299,628],{},[838,4301,876],{},[634,4303,1399],{"encoding":636},[607,4305,4307],{"className":4306,"ariaHidden":642},[641],[607,4308,4310,4313,4316],{"className":4309},[646],[607,4311],{"className":4312,"style":1409},[650],[607,4314,628],{"className":4315},[655],[607,4317,876],{"className":4318},[655,856]," : écrire ",[607,4321,4323,4336],{"className":4322},[610],[607,4324,4326],{"className":4325},[614],[616,4327,4328],{"xmlns":618},[620,4329,4330,4334],{},[623,4331,4332],{},[838,4333,876],{},[634,4335,876],{"encoding":636},[607,4337,4339],{"className":4338,"ariaHidden":642},[641],[607,4340,4342,4345],{"className":4341},[646],[607,4343],{"className":4344,"style":852},[650],[607,4346,876],{"className":4347},[655,856]," en binaire, inverser tous les bits,\najouter 1.",[701,4350,4351,4352,4354,4355,4357],{},"Le bit de poids fort = bit de signe (",[686,4353,688],{}," positif, ",[686,4356,692],{}," négatif).",[701,4359,4360],{},"Tailles courantes : 8, 16, 32, 64 bits.",[593,4362,4364],{"id":4363},"à-savoir-reconnaître","À savoir reconnaître",[971,4366,4367,4377],{},[974,4368,4369],{},[977,4370,4371,4374],{},[980,4372,4373],{},"Situation",[980,4375,4376],{},"Action à mener",[987,4378,4379,4455,4522,4607],{},[977,4380,4381,4446],{},[992,4382,4383,4384,2463,4417,4445],{},"Coder ",[607,4385,4387,4402],{"className":4386},[610],[607,4388,4390],{"className":4389},[614],[616,4391,4392],{"xmlns":618},[620,4393,4394,4400],{},[623,4395,4396,4398],{},[626,4397,628],{},[838,4399,876],{},[634,4401,1399],{"encoding":636},[607,4403,4405],{"className":4404,"ariaHidden":642},[641],[607,4406,4408,4411,4414],{"className":4407},[646],[607,4409],{"className":4410,"style":1409},[650],[607,4412,628],{"className":4413},[655],[607,4415,876],{"className":4416},[655,856],[607,4418,4420,4433],{"className":4419},[610],[607,4421,4423],{"className":4422},[614],[616,4424,4425],{"xmlns":618},[620,4426,4427,4431],{},[623,4428,4429],{},[838,4430,840],{},[634,4432,840],{"encoding":636},[607,4434,4436],{"className":4435,"ariaHidden":642},[641],[607,4437,4439,4442],{"className":4438},[646],[607,4440],{"className":4441,"style":852},[650],[607,4443,840],{"className":4444},[655,856]," bits",[992,4447,4448,4451,4452,742],{},[686,4449,4450],{},"not(x)"," bit à bit puis ",[686,4453,4454],{},"+1",[977,4456,4457,4460],{},[992,4458,4459],{},"Lire un négatif",[992,4461,4462,4463,4521],{},"Bit de poids fort à 1, soustraire ",[607,4464,4466,4483],{"className":4465},[610],[607,4467,4469],{"className":4468},[614],[616,4470,4471],{"xmlns":618},[620,4472,4473,4481],{},[623,4474,4475],{},[1239,4476,4477,4479],{},[630,4478,1094],{},[838,4480,840],{},[634,4482,1247],{"encoding":636},[607,4484,4486],{"className":4485,"ariaHidden":642},[641],[607,4487,4489,4492],{"className":4488},[646],[607,4490],{"className":4491,"style":1257},[650],[607,4493,4495,4498],{"className":4494},[655],[607,4496,1094],{"className":4497},[655],[607,4499,4501],{"className":4500},[1156],[607,4502,4504],{"className":4503},[1160],[607,4505,4507],{"className":4506},[1165],[607,4508,4510],{"className":4509,"style":1257},[1169],[607,4511,4512,4515],{"style":1278},[607,4513],{"className":4514,"style":1178},[1177],[607,4516,4518],{"className":4517},[1182,1183,1184,1185],[607,4519,840],{"className":4520},[655,856,1185]," à la valeur lue.",[977,4523,4524,4555],{},[992,4525,4526,4527,4445],{},"Estimer la taille d'une somme de ",[607,4528,4530,4543],{"className":4529},[610],[607,4531,4533],{"className":4532},[614],[616,4534,4535],{"xmlns":618},[620,4536,4537,4541],{},[623,4538,4539],{},[838,4540,840],{},[634,4542,840],{"encoding":636},[607,4544,4546],{"className":4545,"ariaHidden":642},[641],[607,4547,4549,4552],{"className":4548},[646],[607,4550],{"className":4551,"style":852},[650],[607,4553,840],{"className":4554},[655,856],[992,4556,4557,3367],{},[607,4558,4560,4577],{"className":4559},[610],[607,4561,4563],{"className":4562},[614],[616,4564,4565],{"xmlns":618},[620,4566,4567,4575],{},[623,4568,4569,4571,4573],{},[838,4570,840],{},[626,4572,721],{},[630,4574,692],{},[634,4576,3336],{"encoding":636},[607,4578,4580,4598],{"className":4579,"ariaHidden":642},[641],[607,4581,4583,4586,4589,4592,4595],{"className":4582},[646],[607,4584],{"className":4585,"style":1409},[650],[607,4587,840],{"className":4588},[655,856],[607,4590],{"className":4591,"style":1484},[1116],[607,4593,721],{"className":4594},[1488],[607,4596],{"className":4597,"style":1484},[1116],[607,4599,4601,4604],{"className":4600},[646],[607,4602],{"className":4603,"style":1020},[650],[607,4605,692],{"className":4606},[655],[977,4608,4609,4640],{},[992,4610,4611,4612,4445],{},"Estimer la taille d'un produit de ",[607,4613,4615,4628],{"className":4614},[610],[607,4616,4618],{"className":4617},[614],[616,4619,4620],{"xmlns":618},[620,4621,4622,4626],{},[623,4623,4624],{},[838,4625,840],{},[634,4627,840],{"encoding":636},[607,4629,4631],{"className":4630,"ariaHidden":642},[641],[607,4632,4634,4637],{"className":4633},[646],[607,4635],{"className":4636,"style":852},[650],[607,4638,840],{"className":4639},[655,856],[992,4641,4642,3367],{},[607,4643,4645,4660],{"className":4644},[610],[607,4646,4648],{"className":4647},[614],[616,4649,4650],{"xmlns":618},[620,4651,4652,4658],{},[623,4653,4654,4656],{},[630,4655,1094],{},[838,4657,840],{},[634,4659,3606],{"encoding":636},[607,4661,4663],{"className":4662,"ariaHidden":642},[641],[607,4664,4666,4669,4672],{"className":4665},[646],[607,4667],{"className":4668,"style":1020},[650],[607,4670,1094],{"className":4671},[655],[607,4673,840],{"className":4674},[655,856],[593,4676,3784],{"id":3783},[698,4678,4679,4685,4725],{},[701,4680,4681,4682,4684],{},"L'intervalle n'est ",[602,4683,3246],{}," : un négatif de plus que de positifs.",[701,4686,4687,4720,4721,4724],{},[607,4688,4690,4705],{"className":4689},[610],[607,4691,4693],{"className":4692},[614],[616,4694,4695],{"xmlns":618},[620,4696,4697,4703],{},[623,4698,4699,4701],{},[626,4700,628],{},[630,4702,692],{},[634,4704,3748],{"encoding":636},[607,4706,4708],{"className":4707,"ariaHidden":642},[641],[607,4709,4711,4714,4717],{"className":4710},[646],[607,4712],{"className":4713,"style":651},[650],[607,4715,628],{"className":4716},[655],[607,4718,692],{"className":4719},[655]," en complément à deux vaut ",[686,4722,4723],{},"1111 ... 1111"," (tous les bits à 1).",[701,4726,4727,4728,4730],{},"Python n'a ",[602,4729,816],{}," de taille fixe pour ses entiers : pas de débordement.",[593,4732,4734],{"id":4733},"syntaxe-python","Syntaxe Python",[2345,4736,4738],{"className":2347,"code":4737,"language":2349,"meta":2350,"style":2350},"x = -5\ncode = x & 0xFF      # complément à deux sur 8 bits → 251\nbin(code)            # '0b11111011'\n",[686,4739,4740,4750,4769],{"__ignoreMap":2350},[607,4741,4742,4744,4746,4748],{"class":2355,"line":2356},[607,4743,2386],{"class":2372},[607,4745,1071],{"class":2376},[607,4747,2391],{"class":2376},[607,4749,2394],{"class":2379},[607,4751,4752,4754,4756,4758,4760,4763,4766],{"class":2355,"line":2363},[607,4753,2406],{"class":2372},[607,4755,1071],{"class":2376},[607,4757,2411],{"class":2372},[607,4759,2414],{"class":2376},[607,4761,4762],{"class":2376}," 0x",[607,4764,4765],{"class":2379},"FF",[607,4767,4768],{"class":2359},"      # complément à deux sur 8 bits → 251\n",[607,4770,4771,4773,4776],{"class":2355,"line":2369},[607,4772,3703],{"class":2379},[607,4774,4775],{"class":2372},"(code)            ",[607,4777,4778],{"class":2359},"# '0b11111011'\n",[3878,4780,4781],{},"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":2350,"searchDepth":2363,"depth":2363,"links":4783},[4784,4785,4786,4787,4788],{"id":3915,"depth":2363,"text":3916},{"id":4067,"depth":2363,"text":4068},{"id":4363,"depth":2363,"text":4364},{"id":3783,"depth":2363,"text":3784},{"id":4733,"depth":2363,"text":4734},"Mémo de révision sur la représentation binaire des entiers signés.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs.memo",[],{"title":3909,"description":4789},"1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.memo","_qSVkfWrJe7L9M8vAcZQh6-D5nu1dmDQ2Y70e_mtBlo",{"id":4797,"title":4798,"body":4799,"description":6056,"extension":3895,"meta":6057,"navigation":3901,"path":6058,"seo":6059,"stem":6060,"__hash__":6061},"coursSlides\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs.slides.md","Entiers relatifs et complément à deux — diapositives",{"type":590,"value":4800,"toc":6054},[4801,4806,4843,4846,4850,4924,4927,4940,4942,4944,4985,5024,5026,5030,5200,5277,5279,5281,5484,5819,5821,5825,5973,5975,5979,6027,6029,6033,6040,6042,6046,6051],[4802,4803,4805],"h1",{"id":4804},"entiers-relatifs","Entiers relatifs",[598,4807,4808,4809,4842],{},"Comment représenter ",[607,4810,4812,4827],{"className":4811},[610],[607,4813,4815],{"className":4814},[614],[616,4816,4817],{"xmlns":618},[620,4818,4819,4825],{},[623,4820,4821,4823],{},[626,4822,628],{},[630,4824,632],{},[634,4826,637],{"encoding":636},[607,4828,4830],{"className":4829,"ariaHidden":642},[641],[607,4831,4833,4836,4839],{"className":4832},[646],[607,4834],{"className":4835,"style":651},[650],[607,4837,628],{"className":4838},[655],[607,4840,632],{"className":4841},[655]," en binaire ?",[4844,4845],"hr",{},[4802,4847,4849],{"id":4848},"première-idée-bit-de-signe","Première idée — bit de signe",[598,4851,4852,4854,4855,4888,4889,4854,4891,742],{},[686,4853,705],{}," → ",[607,4856,4858,4873],{"className":4857},[610],[607,4859,4861],{"className":4860},[614],[616,4862,4863],{"xmlns":618},[620,4864,4865,4871],{},[623,4866,4867,4869],{},[626,4868,721],{},[630,4870,632],{},[634,4872,726],{"encoding":636},[607,4874,4876],{"className":4875,"ariaHidden":642},[641],[607,4877,4879,4882,4885],{"className":4878},[646],[607,4880],{"className":4881,"style":651},[650],[607,4883,721],{"className":4884},[655],[607,4886,632],{"className":4887},[655],", ",[686,4890,747],{},[607,4892,4894,4909],{"className":4893},[610],[607,4895,4897],{"className":4896},[614],[616,4898,4899],{"xmlns":618},[620,4900,4901,4907],{},[623,4902,4903,4905],{},[626,4904,628],{},[630,4906,632],{},[634,4908,637],{"encoding":636},[607,4910,4912],{"className":4911,"ariaHidden":642},[641],[607,4913,4915,4918,4921],{"className":4914},[646],[607,4916],{"className":4917,"style":651},[650],[607,4919,628],{"className":4920},[655],[607,4922,632],{"className":4923},[655],[598,4925,4926],{},"Problèmes :",[698,4928,4929,4937],{},[701,4930,4931,4932,800,4934,4936],{},"Deux zéros (",[686,4933,799],{},[686,4935,803],{},").",[701,4938,4939],{},"L'addition nécessite un circuit dédié.",[4844,4941],{},[4802,4943,821],{"id":820},[4945,4946,4947,4979,4982],"ol",{},[701,4948,4949,4950,4978],{},"Écrire ",[607,4951,4953,4966],{"className":4952},[610],[607,4954,4956],{"className":4955},[614],[616,4957,4958],{"xmlns":618},[620,4959,4960,4964],{},[623,4961,4962],{},[838,4963,876],{},[634,4965,876],{"encoding":636},[607,4967,4969],{"className":4968,"ariaHidden":642},[641],[607,4970,4972,4975],{"className":4971},[646],[607,4973],{"className":4974,"style":852},[650],[607,4976,876],{"className":4977},[655,856]," en binaire.",[701,4980,4981],{},"Inverser tous les bits.",[701,4983,4984],{},"Ajouter 1.",[598,4986,4987,4988,5021,5022,742],{},"Sur 8 bits : ",[607,4989,4991,5006],{"className":4990},[610],[607,4992,4994],{"className":4993},[614],[616,4995,4996],{"xmlns":618},[620,4997,4998,5004],{},[623,4999,5000,5002],{},[626,5001,628],{},[630,5003,632],{},[634,5005,637],{"encoding":636},[607,5007,5009],{"className":5008,"ariaHidden":642},[641],[607,5010,5012,5015,5018],{"className":5011},[646],[607,5013],{"className":5014,"style":651},[650],[607,5016,628],{"className":5017},[655],[607,5019,632],{"className":5020},[655]," = ",[686,5023,1049],{},[4844,5025],{},[4802,5027,5029],{"id":5028},"pourquoi-ça-marche","Pourquoi ça marche",[598,5031,5032,5033,796,5091,742],{},"Arithmétique modulo ",[607,5034,5036,5053],{"className":5035},[610],[607,5037,5039],{"className":5038},[614],[616,5040,5041],{"xmlns":618},[620,5042,5043,5051],{},[623,5044,5045],{},[1239,5046,5047,5049],{},[630,5048,1094],{},[838,5050,840],{},[634,5052,1247],{"encoding":636},[607,5054,5056],{"className":5055,"ariaHidden":642},[641],[607,5057,5059,5062],{"className":5058},[646],[607,5060],{"className":5061,"style":1257},[650],[607,5063,5065,5068],{"className":5064},[655],[607,5066,1094],{"className":5067},[655],[607,5069,5071],{"className":5070},[1156],[607,5072,5074],{"className":5073},[1160],[607,5075,5077],{"className":5076},[1165],[607,5078,5080],{"className":5079,"style":1257},[1169],[607,5081,5082,5085],{"style":1278},[607,5083],{"className":5084,"style":1178},[1177],[607,5086,5088],{"className":5087},[1182,1183,1184,1185],[607,5089,840],{"className":5090},[655,856,1185],[607,5092,5094,5123],{"className":5093},[610],[607,5095,5097],{"className":5096},[614],[616,5098,5099],{"xmlns":618},[620,5100,5101,5120],{},[623,5102,5103,5105,5107,5110,5116,5118],{},[626,5104,628],{},[838,5106,876],{},[626,5108,5109],{},"≡",[1239,5111,5112,5114],{},[630,5113,1094],{},[838,5115,840],{},[626,5117,628],{},[838,5119,876],{},[634,5121,5122],{"encoding":636},"-x \\equiv 2^n - x",[607,5124,5126,5147,5191],{"className":5125,"ariaHidden":642},[641],[607,5127,5129,5132,5135,5138,5141,5144],{"className":5128},[646],[607,5130],{"className":5131,"style":1409},[650],[607,5133,628],{"className":5134},[655],[607,5136,876],{"className":5137},[655,856],[607,5139],{"className":5140,"style":1117},[1116],[607,5142,5109],{"className":5143},[1121],[607,5145],{"className":5146,"style":1117},[1116],[607,5148,5150,5153,5182,5185,5188],{"className":5149},[646],[607,5151],{"className":5152,"style":1451},[650],[607,5154,5156,5159],{"className":5155},[655],[607,5157,1094],{"className":5158},[655],[607,5160,5162],{"className":5161},[1156],[607,5163,5165],{"className":5164},[1160],[607,5166,5168],{"className":5167},[1165],[607,5169,5171],{"className":5170,"style":1257},[1169],[607,5172,5173,5176],{"style":1278},[607,5174],{"className":5175,"style":1178},[1177],[607,5177,5179],{"className":5178},[1182,1183,1184,1185],[607,5180,840],{"className":5181},[655,856,1185],[607,5183],{"className":5184,"style":1484},[1116],[607,5186,628],{"className":5187},[1488],[607,5189],{"className":5190,"style":1484},[1116],[607,5192,5194,5197],{"className":5193},[646],[607,5195],{"className":5196,"style":852},[650],[607,5198,876],{"className":5199},[655,856],[598,5201,5202,5021,5268,5270,5271,5021,5273,5276],{},[607,5203,5205,5229],{"className":5204},[610],[607,5206,5208],{"className":5207},[614],[616,5209,5210],{"xmlns":618},[620,5211,5212,5226],{},[623,5213,5214,5216,5218,5220,5222,5224],{},[630,5215,632],{},[626,5217,721],{},[626,5219,1075],{"stretchy":1074},[626,5221,628],{},[630,5223,632],{},[626,5225,1091],{"stretchy":1074},[634,5227,5228],{"encoding":636},"5 + (-5)",[607,5230,5232,5250],{"className":5231,"ariaHidden":642},[641],[607,5233,5235,5238,5241,5244,5247],{"className":5234},[646],[607,5236],{"className":5237,"style":651},[650],[607,5239,632],{"className":5240},[655],[607,5242],{"className":5243,"style":1484},[1116],[607,5245,721],{"className":5246},[1488],[607,5248],{"className":5249,"style":1484},[1116],[607,5251,5253,5256,5259,5262,5265],{"className":5252},[646],[607,5254],{"className":5255,"style":1131},[650],[607,5257,1075],{"className":5258},[1135],[607,5260,628],{"className":5261},[655],[607,5263,632],{"className":5264},[655],[607,5266,1091],{"className":5267},[1149],[686,5269,1029],{}," + ",[686,5272,1049],{},[686,5274,5275],{},"1 0000 0000"," → 0 après débordement du 9ᵉ bit.",[4844,5278],{},[4802,5280,2511],{"id":2510},[598,5282,824,5283,2473,5311,742],{},[607,5284,5286,5299],{"className":5285},[610],[607,5287,5289],{"className":5288},[614],[616,5290,5291],{"xmlns":618},[620,5292,5293,5297],{},[623,5294,5295],{},[838,5296,840],{},[634,5298,840],{"encoding":636},[607,5300,5302],{"className":5301,"ariaHidden":642},[641],[607,5303,5305,5308],{"className":5304},[646],[607,5306],{"className":5307,"style":852},[650],[607,5309,840],{"className":5310},[655,856],[607,5312,5314,5363],{"className":5313},[610],[607,5315,5317],{"className":5316},[614],[616,5318,5319],{"xmlns":618},[620,5320,5321,5361],{},[623,5322,5323,5325,5327,5339,5341,5343,5355,5357,5359],{},[626,5324,4116],{"stretchy":1074},[626,5326,628],{},[1239,5328,5329,5331],{},[630,5330,1094],{},[623,5332,5333,5335,5337],{},[838,5334,840],{},[626,5336,628],{},[630,5338,692],{},[626,5340,4133],{"separator":642},[1080,5342,4136],{},[1239,5344,5345,5347],{},[630,5346,1094],{},[623,5348,5349,5351,5353],{},[838,5350,840],{},[626,5352,628],{},[630,5354,692],{},[626,5356,628],{},[630,5358,692],{},[626,5360,4155],{"stretchy":1074},[634,5362,4158],{"encoding":636},[607,5364,5366,5472],{"className":5365,"ariaHidden":642},[641],[607,5367,5369,5372,5375,5378,5416,5419,5422,5425,5463,5466,5469],{"className":5368},[646],[607,5370],{"className":5371,"style":3504},[650],[607,5373,4116],{"className":5374},[1135],[607,5376,628],{"className":5377},[655],[607,5379,5381,5384],{"className":5380},[655],[607,5382,1094],{"className":5383},[655],[607,5385,5387],{"className":5386},[1156],[607,5388,5390],{"className":5389},[1160],[607,5391,5393],{"className":5392},[1165],[607,5394,5396],{"className":5395,"style":3140},[1169],[607,5397,5398,5401],{"style":1278},[607,5399],{"className":5400,"style":1178},[1177],[607,5402,5404],{"className":5403},[1182,1183,1184,1185],[607,5405,5407,5410,5413],{"className":5406},[655,1185],[607,5408,840],{"className":5409},[655,856,1185],[607,5411,628],{"className":5412},[1488,1185],[607,5414,692],{"className":5415},[655,1185],[607,5417,4133],{"className":5418},[4215],[607,5420,4136],{"className":5421},[1116],[607,5423],{"className":5424,"style":1142},[1116],[607,5426,5428,5431],{"className":5427},[655],[607,5429,1094],{"className":5430},[655],[607,5432,5434],{"className":5433},[1156],[607,5435,5437],{"className":5436},[1160],[607,5438,5440],{"className":5439},[1165],[607,5441,5443],{"className":5442,"style":3140},[1169],[607,5444,5445,5448],{"style":1278},[607,5446],{"className":5447,"style":1178},[1177],[607,5449,5451],{"className":5450},[1182,1183,1184,1185],[607,5452,5454,5457,5460],{"className":5453},[655,1185],[607,5455,840],{"className":5456},[655,856,1185],[607,5458,628],{"className":5459},[1488,1185],[607,5461,692],{"className":5462},[655,1185],[607,5464],{"className":5465,"style":1484},[1116],[607,5467,628],{"className":5468},[1488],[607,5470],{"className":5471,"style":1484},[1116],[607,5473,5475,5478,5481],{"className":5474},[646],[607,5476],{"className":5477,"style":1131},[650],[607,5479,692],{"className":5480},[655],[607,5482,4155],{"className":5483},[1149],[971,5485,5486,5498],{},[974,5487,5488],{},[977,5489,5490,5492,5495],{},[980,5491,2750],{},[980,5493,5494],{},"Min",[980,5496,5497],{},"Max",[987,5499,5500,5574],{},[977,5501,5502,5504,5539],{},[992,5503,2763],{},[992,5505,5506],{},[607,5507,5509,5524],{"className":5508},[610],[607,5510,5512],{"className":5511},[614],[616,5513,5514],{"xmlns":618},[620,5515,5516,5522],{},[623,5517,5518,5520],{},[626,5519,628],{},[630,5521,2782],{},[634,5523,2785],{"encoding":636},[607,5525,5527],{"className":5526,"ariaHidden":642},[641],[607,5528,5530,5533,5536],{"className":5529},[646],[607,5531],{"className":5532,"style":651},[650],[607,5534,628],{"className":5535},[655],[607,5537,2782],{"className":5538},[655],[992,5540,5541],{},[607,5542,5544,5559],{"className":5543},[610],[607,5545,5547],{"className":5546},[614],[616,5548,5549],{"xmlns":618},[620,5550,5551,5557],{},[623,5552,5553,5555],{},[626,5554,721],{},[630,5556,2819],{},[634,5558,2822],{"encoding":636},[607,5560,5562],{"className":5561,"ariaHidden":642},[641],[607,5563,5565,5568,5571],{"className":5564},[646],[607,5566],{"className":5567,"style":651},[650],[607,5569,721],{"className":5570},[655],[607,5572,2819],{"className":5573},[655],[977,5575,5576,5578,5701],{},[992,5577,2861],{},[992,5579,5580],{},[607,5581,5583,5616],{"className":5582},[610],[607,5584,5586],{"className":5585},[614],[616,5587,5588],{"xmlns":618},[620,5589,5590,5613],{},[623,5591,5592,5594,5596,5598,5600,5602,5605],{},[626,5593,3434],{},[626,5595,628],{},[630,5597,1094],{},[626,5599,4133],{"separator":642},[630,5601,692],{},[626,5603,5604],{},"⋅",[1239,5606,5607,5610],{},[630,5608,5609],{},"10",[630,5611,5612],{},"9",[634,5614,5615],{"encoding":636},"\\approx -2,1 \\cdot 10^9",[607,5617,5619,5632,5663],{"className":5618,"ariaHidden":642},[641],[607,5620,5622,5626,5629],{"className":5621},[646],[607,5623],{"className":5624,"style":5625},[650],"height:0.4831em;",[607,5627,3434],{"className":5628},[1121],[607,5630],{"className":5631,"style":1117},[1116],[607,5633,5635,5639,5642,5645,5648,5651,5654,5657,5660],{"className":5634},[646],[607,5636],{"className":5637,"style":5638},[650],"height:0.8389em;vertical-align:-0.1944em;",[607,5640,628],{"className":5641},[655],[607,5643,1094],{"className":5644},[655],[607,5646,4133],{"className":5647},[4215],[607,5649],{"className":5650,"style":1142},[1116],[607,5652,692],{"className":5653},[655],[607,5655],{"className":5656,"style":1484},[1116],[607,5658,5604],{"className":5659},[1488],[607,5661],{"className":5662,"style":1484},[1116],[607,5664,5666,5669,5672],{"className":5665},[646],[607,5667],{"className":5668,"style":3140},[650],[607,5670,692],{"className":5671},[655],[607,5673,5675,5678],{"className":5674},[655],[607,5676,688],{"className":5677},[655],[607,5679,5681],{"className":5680},[1156],[607,5682,5684],{"className":5683},[1160],[607,5685,5687],{"className":5686},[1165],[607,5688,5690],{"className":5689,"style":3140},[1169],[607,5691,5692,5695],{"style":1278},[607,5693],{"className":5694,"style":1178},[1177],[607,5696,5698],{"className":5697},[1182,1183,1184,1185],[607,5699,5612],{"className":5700},[655,1185],[992,5702,5703],{},[607,5704,5706,5736],{"className":5705},[610],[607,5707,5709],{"className":5708},[614],[616,5710,5711],{"xmlns":618},[620,5712,5713,5733],{},[623,5714,5715,5717,5719,5721,5723,5725,5727],{},[626,5716,3434],{},[626,5718,721],{},[630,5720,1094],{},[626,5722,4133],{"separator":642},[630,5724,692],{},[626,5726,5604],{},[1239,5728,5729,5731],{},[630,5730,5609],{},[630,5732,5612],{},[634,5734,5735],{"encoding":636},"\\approx +2,1 \\cdot 10^9",[607,5737,5739,5751,5781],{"className":5738,"ariaHidden":642},[641],[607,5740,5742,5745,5748],{"className":5741},[646],[607,5743],{"className":5744,"style":5625},[650],[607,5746,3434],{"className":5747},[1121],[607,5749],{"className":5750,"style":1117},[1116],[607,5752,5754,5757,5760,5763,5766,5769,5772,5775,5778],{"className":5753},[646],[607,5755],{"className":5756,"style":5638},[650],[607,5758,721],{"className":5759},[655],[607,5761,1094],{"className":5762},[655],[607,5764,4133],{"className":5765},[4215],[607,5767],{"className":5768,"style":1142},[1116],[607,5770,692],{"className":5771},[655],[607,5773],{"className":5774,"style":1484},[1116],[607,5776,5604],{"className":5777},[1488],[607,5779],{"className":5780,"style":1484},[1116],[607,5782,5784,5787,5790],{"className":5783},[646],[607,5785],{"className":5786,"style":3140},[650],[607,5788,692],{"className":5789},[655],[607,5791,5793,5796],{"className":5792},[655],[607,5794,688],{"className":5795},[655],[607,5797,5799],{"className":5798},[1156],[607,5800,5802],{"className":5801},[1160],[607,5803,5805],{"className":5804},[1165],[607,5806,5808],{"className":5807,"style":3140},[1169],[607,5809,5810,5813],{"style":1278},[607,5811],{"className":5812,"style":1178},[1177],[607,5814,5816],{"className":5815},[1182,1183,1184,1185],[607,5817,5612],{"className":5818},[655,1185],[4844,5820],{},[4802,5822,5824],{"id":5823},"taille-du-résultat","Taille du résultat",[698,5826,5827,5909],{},[701,5828,5829,5830,2473,5858,3367],{},"Somme de ",[607,5831,5833,5846],{"className":5832},[610],[607,5834,5836],{"className":5835},[614],[616,5837,5838],{"xmlns":618},[620,5839,5840,5844],{},[623,5841,5842],{},[838,5843,840],{},[634,5845,840],{"encoding":636},[607,5847,5849],{"className":5848,"ariaHidden":642},[641],[607,5850,5852,5855],{"className":5851},[646],[607,5853],{"className":5854,"style":852},[650],[607,5856,840],{"className":5857},[655,856],[607,5859,5861,5879],{"className":5860},[610],[607,5862,5864],{"className":5863},[614],[616,5865,5866],{"xmlns":618},[620,5867,5868,5876],{},[623,5869,5870,5872,5874],{},[838,5871,840],{},[626,5873,721],{},[630,5875,692],{},[634,5877,5878],{"encoding":636},"n + 1",[607,5880,5882,5900],{"className":5881,"ariaHidden":642},[641],[607,5883,5885,5888,5891,5894,5897],{"className":5884},[646],[607,5886],{"className":5887,"style":1409},[650],[607,5889,840],{"className":5890},[655,856],[607,5892],{"className":5893,"style":1484},[1116],[607,5895,721],{"className":5896},[1488],[607,5898],{"className":5899,"style":1484},[1116],[607,5901,5903,5906],{"className":5902},[646],[607,5904],{"className":5905,"style":1020},[650],[607,5907,692],{"className":5908},[655],[701,5910,5911,5912,2473,5940,3367],{},"Produit de ",[607,5913,5915,5928],{"className":5914},[610],[607,5916,5918],{"className":5917},[614],[616,5919,5920],{"xmlns":618},[620,5921,5922,5926],{},[623,5923,5924],{},[838,5925,840],{},[634,5927,840],{"encoding":636},[607,5929,5931],{"className":5930,"ariaHidden":642},[641],[607,5932,5934,5937],{"className":5933},[646],[607,5935],{"className":5936,"style":852},[650],[607,5938,840],{"className":5939},[655,856],[607,5941,5943,5958],{"className":5942},[610],[607,5944,5946],{"className":5945},[614],[616,5947,5948],{"xmlns":618},[620,5949,5950,5956],{},[623,5951,5952,5954],{},[630,5953,1094],{},[838,5955,840],{},[634,5957,3606],{"encoding":636},[607,5959,5961],{"className":5960,"ariaHidden":642},[641],[607,5962,5964,5967,5970],{"className":5963},[646],[607,5965],{"className":5966,"style":1020},[650],[607,5968,1094],{"className":5969},[655],[607,5971,840],{"className":5972},[655,856],[4844,5974],{},[4802,5976,5978],{"id":5977},"en-python","En Python",[2345,5980,5982],{"className":2347,"code":5981,"language":2349,"meta":2350,"style":2350},"2 ** 100   # OK, taille arbitraire\n-5 & 0xFF  # 251 — encodage sur 8 bits\nbin(-5)    # '-0b101' (notation Python)\n",[686,5983,5984,5995,6011],{"__ignoreMap":2350},[607,5985,5986,5988,5990,5992],{"class":2355,"line":2356},[607,5987,1094],{"class":2379},[607,5989,3652],{"class":2376},[607,5991,3655],{"class":2379},[607,5993,5994],{"class":2359},"   # OK, taille arbitraire\n",[607,5996,5997,5999,6001,6004,6006,6008],{"class":2355,"line":2363},[607,5998,2428],{"class":2376},[607,6000,632],{"class":2379},[607,6002,6003],{"class":2376}," &",[607,6005,4762],{"class":2376},[607,6007,4765],{"class":2379},[607,6009,6010],{"class":2359},"  # 251 — encodage sur 8 bits\n",[607,6012,6013,6015,6017,6019,6021,6024],{"class":2355,"line":2369},[607,6014,3703],{"class":2379},[607,6016,1075],{"class":2372},[607,6018,2428],{"class":2376},[607,6020,632],{"class":2379},[607,6022,6023],{"class":2372},")    ",[607,6025,6026],{"class":2359},"# '-0b101' (notation Python)\n",[4844,6028],{},[4802,6030,6032],{"id":6031},"schéma-à-dessiner-au-tableau","Schéma à dessiner au tableau",[6034,6035],"illustration",{"alt":6036,"kind":6037,"src":6038,"svg":6039},"Cercle modulaire 8 bits : 0 en haut, +127 à droite, -128 en bas, -1 à gauche, montrant la correspondance circulaire entre complément à deux et entier signé","schema","\u002Fillustrations\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002Fcercle-modulaire.svg","\u003Csvg class=\"illustration-svg\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 400 400\" role=\"img\" aria-label=\"Cercle modulaire sur 8 bits : 0 en haut (00000000), +127 à droite (01111111), -128 en bas (10000000), -1 à gauche (11111111). Les valeurs positives 0 à 127 occupent la moitié droite, les négatives -128 à -1 la moitié gauche ; le passage de +127 à -128 illustre la correspondance circulaire du complément à deux.\">\n  \u003Ctitle>Cercle modulaire 8 bits — complément à deux\u003C\u002Ftitle>\n  \u003Cstyle>\n    .cm-svg   { font-family: Geist, Inter, system-ui, sans-serif; }\n    .cm-ring  { fill: none; stroke: var(--diagram-border); stroke-width: 2; }\n    .cm-arc-pos { fill: none; stroke: var(--diagram-node-border); stroke-width: 4; stroke-linecap: round; }\n    .cm-arc-neg { fill: none; stroke: var(--diagram-line); stroke-width: 4; stroke-linecap: round; stroke-dasharray: 2 7; }\n    .cm-node  { fill: var(--diagram-node-bg); stroke: var(--diagram-node-border); stroke-width: 1.5; }\n    .cm-val   { fill: var(--diagram-text); font-size: 18px; font-weight: 600; }\n    .cm-bits  { fill: var(--diagram-line); font-family: 'JetBrains Mono', monospace; font-size: 11px; }\n    .cm-side  { fill: var(--diagram-line); font-size: 12px; letter-spacing: 1px; }\n    .cm-seam  { fill: var(--diagram-text); font-size: 12px; font-style: italic; }\n  \u003C\u002Fstyle>\n\n  \u003Cg class=\"cm-svg\">\n    \u003C!-- anneau de base -->\n    \u003Ccircle cx=\"200\" cy=\"200\" r=\"120\" class=\"cm-ring\"\u002F>\n\n    \u003C!-- moitié droite : valeurs positives 0 → +127 (haut vers droite vers bas, sens horaire) -->\n    \u003Cpath class=\"cm-arc-pos\" d=\"M 200 80 A 120 120 0 0 1 200 320\"\u002F>\n    \u003C!-- moitié gauche : valeurs négatives -128 → -1 (bas vers gauche vers haut, sens horaire) -->\n    \u003Cpath class=\"cm-arc-neg\" d=\"M 200 320 A 120 120 0 0 1 200 80\"\u002F>\n\n    \u003C!-- étiquettes des demi-cercles -->\n    \u003Ctext x=\"312\" y=\"204\" text-anchor=\"middle\" class=\"cm-side\">positifs\u003C\u002Ftext>\n    \u003Ctext x=\"88\"  y=\"204\" text-anchor=\"middle\" class=\"cm-side\">négatifs\u003C\u002Ftext>\n\n    \u003C!-- couture +127 \u002F -128 : le saut circulaire du complément à deux,\n         placé dans le quadrant bas-droit entre les deux nœuds concernés -->\n    \u003Ctext x=\"304\" y=\"300\" text-anchor=\"middle\" class=\"cm-seam\">+127 → -128\u003C\u002Ftext>\n    \u003Ctext x=\"304\" y=\"316\" text-anchor=\"middle\" class=\"cm-seam\">(débordement)\u003C\u002Ftext>\n\n    \u003C!-- 0 (haut) -->\n    \u003Ccircle cx=\"200\" cy=\"80\" r=\"22\" class=\"cm-node\"\u002F>\n    \u003Ctext x=\"200\" y=\"86\" text-anchor=\"middle\" class=\"cm-val\">0\u003C\u002Ftext>\n    \u003Ctext x=\"200\" y=\"46\" text-anchor=\"middle\" class=\"cm-bits\">00000000\u003C\u002Ftext>\n\n    \u003C!-- +127 (droite) -->\n    \u003Ccircle cx=\"320\" cy=\"200\" r=\"22\" class=\"cm-node\"\u002F>\n    \u003Ctext x=\"320\" y=\"205\" text-anchor=\"middle\" class=\"cm-val\">+127\u003C\u002Ftext>\n    \u003Ctext x=\"320\" y=\"246\" text-anchor=\"middle\" class=\"cm-bits\">01111111\u003C\u002Ftext>\n\n    \u003C!-- -128 (bas) -->\n    \u003Ccircle cx=\"200\" cy=\"320\" r=\"22\" class=\"cm-node\"\u002F>\n    \u003Ctext x=\"200\" y=\"325\" text-anchor=\"middle\" class=\"cm-val\">-128\u003C\u002Ftext>\n    \u003Ctext x=\"200\" y=\"358\" text-anchor=\"middle\" class=\"cm-bits\">10000000\u003C\u002Ftext>\n\n    \u003C!-- -1 (gauche) -->\n    \u003Ccircle cx=\"80\" cy=\"200\" r=\"22\" class=\"cm-node\"\u002F>\n    \u003Ctext x=\"80\" y=\"205\" text-anchor=\"middle\" class=\"cm-val\">-1\u003C\u002Ftext>\n    \u003Ctext x=\"80\" y=\"246\" text-anchor=\"middle\" class=\"cm-bits\">11111111\u003C\u002Ftext>\n  \u003C\u002Fg>\n\u003C\u002Fsvg>\n",[4844,6041],{},[4802,6043,6045],{"id":6044},"à-retenir","À retenir",[598,6047,6048],{},[602,6049,6050],{},"Le complément à deux n'est pas qu'une convention de notation — c'est ce qui\npermet aux processeurs d'additionner des entiers signés avec le même circuit\nque des non signés.",[3878,6052,6053],{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html 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