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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 flottante","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants","1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants",{"title":275,"path":276,"stem":277},"Textes et encodages — ASCII, ISO-8859-1, Unicode","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Ftextes-encodages","1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages",{"title":279,"path":280,"stem":281},"Valeurs booléennes et opérateurs logiques","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbooleens","1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens",{"title":283,"path":284,"stem":285,"children":286},"Représentation des données : types construits","\u002Fnsi-1ere\u002Fpc-types-construits","1.nsi-1ere\u002Fpc-types-construits\u002F0.index",[287,288,292,296,300],{"title":283,"path":284,"stem":285},{"title":289,"path":290,"stem":291},"Tableaux et listes","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftableaux-listes","1.nsi-1ere\u002Fpc-types-construits\u002F1.tableaux-listes",{"title":293,"path":294,"stem":295},"Listes en 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 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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":271,"bo":587,"body":589,"description":3594,"eval":3595,"extension":3596,"labo":3595,"meta":3597,"navigation":2940,"path":272,"quizzes":3598,"readingTime":2486,"seo":3606,"stem":273,"tp":3595,"__hash__":3607},"cours\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.md",[588],"PB02",{"type":590,"value":591,"toc":3581},"minimark",[592,597,801,817,821,824,1031,1034,1198,1205,1395,1485,1489,1496,1634,1645,1771,1781,1825,2036,2235,2317,2470,2474,2536,2622,2696,2699,2707,2739,2842,2846,2888,2966,2973,3006,3020,3033,3036,3194,3265,3294,3318,3322,3396,3400,3403,3523,3571,3577],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,660,661,784,785,789,790,793,794,797,798],"p",{},"Les entiers, c'est bien — mais comment représente-t-on ",[602,603,606,629],"span",{"className":604},[605],"katex",[602,607,610],{"className":608},[609],"katex-mathml",[611,612,614],"math",{"xmlns":613},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[615,616,617,624],"semantics",{},[618,619,620],"mrow",{},[621,622,623],"mn",{},"3,14",[625,626,628],"annotation",{"encoding":627},"application\u002Fx-tex","3{,}14",[602,630,634],{"className":631,"ariaHidden":633},[632],"katex-html","true",[602,635,638,643,648,656],{"className":636},[637],"base",[602,639],{"className":640,"style":642},[641],"strut","height:0.8389em;vertical-align:-0.1944em;",[602,644,647],{"className":645},[646],"mord","3",[602,649,651],{"className":650},[646],[602,652,655],{"className":653},[654],"mpunct",",",[602,657,659],{"className":658},[646],"14"," ou\n",[602,662,664,684],{"className":663},[605],[602,665,667],{"className":666},[609],[611,668,669],{"xmlns":613},[615,670,671,681],{},[618,672,673],{},[674,675,676,679],"mfrac",{},[621,677,678],{},"1",[621,680,647],{},[625,682,683],{"encoding":627},"\\frac{1}{3}",[602,685,687],{"className":686,"ariaHidden":633},[632],[602,688,690,694],{"className":689},[637],[602,691],{"className":692,"style":693},[641],"height:1.1901em;vertical-align:-0.345em;",[602,695,697,702,780],{"className":696},[646],[602,698],{"className":699},[700,701],"mopen","nulldelimiter",[602,703,705],{"className":704},[674],[602,706,710,771],{"className":707},[708,709],"vlist-t","vlist-t2",[602,711,714,766],{"className":712},[713],"vlist-r",[602,715,719,740,751],{"className":716,"style":718},[717],"vlist","height:0.8451em;",[602,720,722,727],{"style":721},"top:-2.655em;",[602,723],{"className":724,"style":726},[725],"pstrut","height:3em;",[602,728,734],{"className":729},[730,731,732,733],"sizing","reset-size6","size3","mtight",[602,735,737],{"className":736},[646,733],[602,738,647],{"className":739},[646,733],[602,741,743,746],{"style":742},"top:-3.23em;",[602,744],{"className":745,"style":726},[725],[602,747],{"className":748,"style":750},[749],"frac-line","border-bottom-width:0.04em;",[602,752,754,757],{"style":753},"top:-3.394em;",[602,755],{"className":756,"style":726},[725],[602,758,760],{"className":759},[730,731,732,733],[602,761,763],{"className":762},[646,733],[602,764,678],{"className":765},[646,733],[602,767,770],{"className":768},[769],"vlist-s","​",[602,772,774],{"className":773},[713],[602,775,778],{"className":776,"style":777},[717],"height:0.345em;",[602,779],{},[602,781],{"className":782},[783,701],"mclose"," ? Les nombres réels posent un problème fondamental : il en existe\nune ",[786,787,788],"strong",{},"infinité"," entre deux entiers consécutifs, alors qu'une machine ne\ndispose que d'un nombre fini de bits. Il faudra donc ",[786,791,792],{},"approximer",". C'est ce\nque fait la représentation en ",[786,795,796],{},"virgule flottante",", normalisée par le standard\nIEEE 754 et utilisée par tous les processeurs modernes. ",[799,800],"bo-ref",{"code":588},[598,802,803,804,808,809,812,813,816],{},"Cette approximation a un effet déroutant : en Python, ",[805,806,807],"code",{},"0.1 + 0.2"," ne vaut\n",[786,810,811],{},"pas"," exactement ",[805,814,815],{},"0.3",". Comprendre pourquoi vous évitera bien des bugs.",[593,818,820],{"id":819},"lanalogie-avec-la-notation-scientifique","L'analogie avec la notation scientifique",[598,822,823],{},"En mathématiques, on note souvent un grand ou un petit nombre en notation\nscientifique :",[602,825,828],{"className":826},[827],"katex-display",[602,829,831,881],{"className":830},[605],[602,832,834],{"className":833},[609],[611,835,837],{"xmlns":613,"display":836},"block",[615,838,839,878],{},[618,840,841,844,848,857,861,864,866],{},[621,842,843],{},"6,022",[845,846,847],"mo",{},"⋅",[849,850,851,854],"msup",{},[621,852,853],{},"10",[621,855,856],{},"23",[858,859],"mspace",{"width":860},"2em",[621,862,863],{},"1,6",[845,865,847],{},[849,867,868,870],{},[621,869,853],{},[618,871,872,875],{},[845,873,874],{},"−",[621,876,877],{},"19",[625,879,880],{"encoding":627},"6{,}022 \\cdot 10^{23} \\qquad 1{,}6 \\cdot 10^{-19}",[602,882,884,915,987],{"className":883,"ariaHidden":633},[632],[602,885,887,890,894,900,904,908,912],{"className":886},[637],[602,888],{"className":889,"style":642},[641],[602,891,893],{"className":892},[646],"6",[602,895,897],{"className":896},[646],[602,898,655],{"className":899},[654],[602,901,903],{"className":902},[646],"022",[602,905],{"className":906,"style":907},[858],"margin-right:0.2222em;",[602,909,847],{"className":910},[911],"mbin",[602,913],{"className":914,"style":907},[858],[602,916,918,922,925,962,966,969,975,978,981,984],{"className":917},[637],[602,919],{"className":920,"style":921},[641],"height:1.0585em;vertical-align:-0.1944em;",[602,923,678],{"className":924},[646],[602,926,928,932],{"className":927},[646],[602,929,931],{"className":930},[646],"0",[602,933,936],{"className":934},[935],"msupsub",[602,937,939],{"className":938},[708],[602,940,942],{"className":941},[713],[602,943,946],{"className":944,"style":945},[717],"height:0.8641em;",[602,947,949,953],{"style":948},"top:-3.113em;margin-right:0.05em;",[602,950],{"className":951,"style":952},[725],"height:2.7em;",[602,954,956],{"className":955},[730,731,732,733],[602,957,959],{"className":958},[646,733],[602,960,856],{"className":961},[646,733],[602,963],{"className":964,"style":965},[858],"margin-right:2em;",[602,967,678],{"className":968},[646],[602,970,972],{"className":971},[646],[602,973,655],{"className":974},[654],[602,976,893],{"className":977},[646],[602,979],{"className":980,"style":907},[858],[602,982,847],{"className":983},[911],[602,985],{"className":986,"style":907},[858],[602,988,990,993,996],{"className":989},[637],[602,991],{"className":992,"style":945},[641],[602,994,678],{"className":995},[646],[602,997,999,1002],{"className":998},[646],[602,1000,931],{"className":1001},[646],[602,1003,1005],{"className":1004},[935],[602,1006,1008],{"className":1007},[708],[602,1009,1011],{"className":1010},[713],[602,1012,1014],{"className":1013,"style":945},[717],[602,1015,1016,1019],{"style":948},[602,1017],{"className":1018,"style":952},[725],[602,1020,1022],{"className":1021},[730,731,732,733],[602,1023,1025,1028],{"className":1024},[646,733],[602,1026,874],{"className":1027},[646,733],[602,1029,877],{"className":1030},[646,733],[598,1032,1033],{},"Trois éléments structurent cette écriture :",[1035,1036,1037,1122,1127],"ul",{},[1038,1039,1040,1041,1044,1045,1083,1084,655],"li",{},"une ",[786,1042,1043],{},"mantisse"," : ici ",[602,1046,1048,1062],{"className":1047},[605],[602,1049,1051],{"className":1050},[609],[611,1052,1053],{"xmlns":613},[615,1054,1055,1059],{},[618,1056,1057],{},[621,1058,843],{},[625,1060,1061],{"encoding":627},"6{,}022",[602,1063,1065],{"className":1064,"ariaHidden":633},[632],[602,1066,1068,1071,1074,1080],{"className":1067},[637],[602,1069],{"className":1070,"style":642},[641],[602,1072,893],{"className":1073},[646],[602,1075,1077],{"className":1076},[646],[602,1078,655],{"className":1079},[654],[602,1081,903],{"className":1082},[646]," ou ",[602,1085,1087,1101],{"className":1086},[605],[602,1088,1090],{"className":1089},[609],[611,1091,1092],{"xmlns":613},[615,1093,1094,1098],{},[618,1095,1096],{},[621,1097,863],{},[625,1099,1100],{"encoding":627},"1{,}6",[602,1102,1104],{"className":1103,"ariaHidden":633},[632],[602,1105,1107,1110,1113,1119],{"className":1106},[637],[602,1108],{"className":1109,"style":642},[641],[602,1111,678],{"className":1112},[646],[602,1114,1116],{"className":1115},[646],[602,1117,655],{"className":1118},[654],[602,1120,893],{"className":1121},[646],[1038,1123,1040,1124,1126],{},[786,1125,637],{}," : ici 10,",[1038,1128,1129,1130,1044,1133,1083,1162,1197],{},"un ",[786,1131,1132],{},"exposant",[602,1134,1136,1149],{"className":1135},[605],[602,1137,1139],{"className":1138},[609],[611,1140,1141],{"xmlns":613},[615,1142,1143,1147],{},[618,1144,1145],{},[621,1146,856],{},[625,1148,856],{"encoding":627},[602,1150,1152],{"className":1151,"ariaHidden":633},[632],[602,1153,1155,1159],{"className":1154},[637],[602,1156],{"className":1157,"style":1158},[641],"height:0.6444em;",[602,1160,856],{"className":1161},[646],[602,1163,1165,1181],{"className":1164},[605],[602,1166,1168],{"className":1167},[609],[611,1169,1170],{"xmlns":613},[615,1171,1172,1178],{},[618,1173,1174,1176],{},[845,1175,874],{},[621,1177,877],{},[625,1179,1180],{"encoding":627},"-19",[602,1182,1184],{"className":1183,"ariaHidden":633},[632],[602,1185,1187,1191,1194],{"className":1186},[637],[602,1188],{"className":1189,"style":1190},[641],"height:0.7278em;vertical-align:-0.0833em;",[602,1192,874],{"className":1193},[646],[602,1195,877],{"className":1196},[646],".",[598,1199,1200,1201,1204],{},"La virgule flottante reprend exactement cette idée, ",[786,1202,1203],{},"en base 2",". Un flottant\ns'écrit alors comme :",[602,1206,1208],{"className":1207},[827],[602,1209,1211,1261],{"className":1210},[605],[602,1212,1214],{"className":1213},[609],[611,1215,1216],{"xmlns":613,"display":836},[615,1217,1218,1258],{},[618,1219,1220,1224,1227,1231,1233,1235,1243,1245,1248,1250],{},[1221,1222,1223],"mi",{},"x",[845,1225,1226],{},"=",[845,1228,1230],{"stretchy":1229},"false","(",[845,1232,874],{},[621,1234,678],{},[849,1236,1237,1240],{},[845,1238,1239],{"stretchy":1229},")",[1221,1241,1242],{},"s",[845,1244,847],{},[1221,1246,1247],{},"m",[845,1249,847],{},[849,1251,1252,1255],{},[621,1253,1254],{},"2",[1221,1256,1257],{},"e",[625,1259,1260],{"encoding":627},"x = (-1)^s \\cdot m \\cdot 2^e",[602,1262,1264,1286,1341,1360],{"className":1263,"ariaHidden":633},[632],[602,1265,1267,1271,1275,1279,1283],{"className":1266},[637],[602,1268],{"className":1269,"style":1270},[641],"height:0.4306em;",[602,1272,1223],{"className":1273},[646,1274],"mathnormal",[602,1276],{"className":1277,"style":1278},[858],"margin-right:0.2778em;",[602,1280,1226],{"className":1281},[1282],"mrel",[602,1284],{"className":1285,"style":1278},[858],[602,1287,1289,1293,1296,1299,1302,1332,1335,1338],{"className":1288},[637],[602,1290],{"className":1291,"style":1292},[641],"height:1em;vertical-align:-0.25em;",[602,1294,1230],{"className":1295},[700],[602,1297,874],{"className":1298},[646],[602,1300,678],{"className":1301},[646],[602,1303,1305,1308],{"className":1304},[783],[602,1306,1239],{"className":1307},[783],[602,1309,1311],{"className":1310},[935],[602,1312,1314],{"className":1313},[708],[602,1315,1317],{"className":1316},[713],[602,1318,1321],{"className":1319,"style":1320},[717],"height:0.7144em;",[602,1322,1323,1326],{"style":948},[602,1324],{"className":1325,"style":952},[725],[602,1327,1329],{"className":1328},[730,731,732,733],[602,1330,1242],{"className":1331},[646,1274,733],[602,1333],{"className":1334,"style":907},[858],[602,1336,847],{"className":1337},[911],[602,1339],{"className":1340,"style":907},[858],[602,1342,1344,1348,1351,1354,1357],{"className":1343},[637],[602,1345],{"className":1346,"style":1347},[641],"height:0.4445em;",[602,1349,1247],{"className":1350},[646,1274],[602,1352],{"className":1353,"style":907},[858],[602,1355,847],{"className":1356},[911],[602,1358],{"className":1359,"style":907},[858],[602,1361,1363,1366],{"className":1362},[637],[602,1364],{"className":1365,"style":1320},[641],[602,1367,1369,1372],{"className":1368},[646],[602,1370,1254],{"className":1371},[646],[602,1373,1375],{"className":1374},[935],[602,1376,1378],{"className":1377},[708],[602,1379,1381],{"className":1380},[713],[602,1382,1384],{"className":1383,"style":1320},[717],[602,1385,1386,1389],{"style":948},[602,1387],{"className":1388,"style":952},[725],[602,1390,1392],{"className":1391},[730,731,732,733],[602,1393,1257],{"className":1394},[646,1274,733],[598,1396,1397,1398,1426,1427,1455,1456,1484],{},"où ",[602,1399,1401,1414],{"className":1400},[605],[602,1402,1404],{"className":1403},[609],[611,1405,1406],{"xmlns":613},[615,1407,1408,1412],{},[618,1409,1410],{},[1221,1411,1242],{},[625,1413,1242],{"encoding":627},[602,1415,1417],{"className":1416,"ariaHidden":633},[632],[602,1418,1420,1423],{"className":1419},[637],[602,1421],{"className":1422,"style":1270},[641],[602,1424,1242],{"className":1425},[646,1274]," est le bit de signe, ",[602,1428,1430,1443],{"className":1429},[605],[602,1431,1433],{"className":1432},[609],[611,1434,1435],{"xmlns":613},[615,1436,1437,1441],{},[618,1438,1439],{},[1221,1440,1247],{},[625,1442,1247],{"encoding":627},[602,1444,1446],{"className":1445,"ariaHidden":633},[632],[602,1447,1449,1452],{"className":1448},[637],[602,1450],{"className":1451,"style":1270},[641],[602,1453,1247],{"className":1454},[646,1274]," la mantisse (en binaire, entre 1 et 2 environ),\net ",[602,1457,1459,1472],{"className":1458},[605],[602,1460,1462],{"className":1461},[609],[611,1463,1464],{"xmlns":613},[615,1465,1466,1470],{},[618,1467,1468],{},[1221,1469,1257],{},[625,1471,1257],{"encoding":627},[602,1473,1475],{"className":1474,"ariaHidden":633},[632],[602,1476,1478,1481],{"className":1477},[637],[602,1479],{"className":1480,"style":1270},[641],[602,1482,1257],{"className":1483},[646,1274]," l'exposant (un entier signé).",[593,1486,1488],{"id":1487},"la-norme-ieee-754-vue-simplifiée","La norme IEEE 754, vue simplifiée",[598,1490,1491,1492,1495],{},"La norme IEEE 754 fixe la répartition des bits. En ",[786,1493,1494],{},"simple précision"," (32\nbits) :",[1497,1498,1499,1515],"table",{},[1500,1501,1502],"thead",{},[1503,1504,1505,1509,1512],"tr",{},[1506,1507,1508],"th",{},"Champ",[1506,1510,1511],{},"Bits",[1506,1513,1514],{},"Rôle",[1516,1517,1518,1557,1596],"tbody",{},[1503,1519,1520,1552,1554],{},[1521,1522,1523,1524],"td",{},"Signe ",[602,1525,1527,1540],{"className":1526},[605],[602,1528,1530],{"className":1529},[609],[611,1531,1532],{"xmlns":613},[615,1533,1534,1538],{},[618,1535,1536],{},[1221,1537,1242],{},[625,1539,1242],{"encoding":627},[602,1541,1543],{"className":1542,"ariaHidden":633},[632],[602,1544,1546,1549],{"className":1545},[637],[602,1547],{"className":1548,"style":1270},[641],[602,1550,1242],{"className":1551},[646,1274],[1521,1553,678],{},[1521,1555,1556],{},"0 = positif, 1 = négatif",[1503,1558,1559,1590,1593],{},[1521,1560,1561,1562],{},"Exposant ",[602,1563,1565,1578],{"className":1564},[605],[602,1566,1568],{"className":1567},[609],[611,1569,1570],{"xmlns":613},[615,1571,1572,1576],{},[618,1573,1574],{},[1221,1575,1257],{},[625,1577,1257],{"encoding":627},[602,1579,1581],{"className":1580,"ariaHidden":633},[632],[602,1582,1584,1587],{"className":1583},[637],[602,1585],{"className":1586,"style":1270},[641],[602,1588,1257],{"className":1589},[646,1274],[1521,1591,1592],{},"8",[1521,1594,1595],{},"Codé avec un biais de 127",[1503,1597,1598,1629,1631],{},[1521,1599,1600,1601],{},"Mantisse ",[602,1602,1604,1617],{"className":1603},[605],[602,1605,1607],{"className":1606},[609],[611,1608,1609],{"xmlns":613},[615,1610,1611,1615],{},[618,1612,1613],{},[1221,1614,1247],{},[625,1616,1247],{"encoding":627},[602,1618,1620],{"className":1619,"ariaHidden":633},[632],[602,1621,1623,1626],{"className":1622},[637],[602,1624],{"className":1625,"style":1270},[641],[602,1627,1247],{"className":1628},[646,1274],[1521,1630,856],{},[1521,1632,1633],{},"Partie fractionnaire après le « 1, » implicite",[598,1635,1636,1637,1640,1641,1644],{},"En ",[786,1638,1639],{},"double précision"," (64 bits, le ",[805,1642,1643],{},"float"," Python par défaut) :",[1497,1646,1647,1657],{},[1500,1648,1649],{},[1503,1650,1651,1653,1655],{},[1506,1652,1508],{},[1506,1654,1511],{},[1506,1656,1514],{},[1516,1658,1659,1695,1733],{},[1503,1660,1661,1691,1693],{},[1521,1662,1523,1663],{},[602,1664,1666,1679],{"className":1665},[605],[602,1667,1669],{"className":1668},[609],[611,1670,1671],{"xmlns":613},[615,1672,1673,1677],{},[618,1674,1675],{},[1221,1676,1242],{},[625,1678,1242],{"encoding":627},[602,1680,1682],{"className":1681,"ariaHidden":633},[632],[602,1683,1685,1688],{"className":1684},[637],[602,1686],{"className":1687,"style":1270},[641],[602,1689,1242],{"className":1690},[646,1274],[1521,1692,678],{},[1521,1694,1556],{},[1503,1696,1697,1727,1730],{},[1521,1698,1561,1699],{},[602,1700,1702,1715],{"className":1701},[605],[602,1703,1705],{"className":1704},[609],[611,1706,1707],{"xmlns":613},[615,1708,1709,1713],{},[618,1710,1711],{},[1221,1712,1257],{},[625,1714,1257],{"encoding":627},[602,1716,1718],{"className":1717,"ariaHidden":633},[632],[602,1719,1721,1724],{"className":1720},[637],[602,1722],{"className":1723,"style":1270},[641],[602,1725,1257],{"className":1726},[646,1274],[1521,1728,1729],{},"11",[1521,1731,1732],{},"Biais de 1023",[1503,1734,1735,1765,1768],{},[1521,1736,1600,1737],{},[602,1738,1740,1753],{"className":1739},[605],[602,1741,1743],{"className":1742},[609],[611,1744,1745],{"xmlns":613},[615,1746,1747,1751],{},[618,1748,1749],{},[1221,1750,1247],{},[625,1752,1247],{"encoding":627},[602,1754,1756],{"className":1755,"ariaHidden":633},[632],[602,1757,1759,1762],{"className":1758},[637],[602,1760],{"className":1761,"style":1270},[641],[602,1763,1247],{"className":1764},[646,1274],[1521,1766,1767],{},"52",[1521,1769,1770],{},"Partie fractionnaire",[1772,1773,1774],"note",{},[598,1775,1776,1777,1780],{},"Le programme officiel ne vous demande ",[786,1778,1779],{},"aucune connaissance précise"," des bits\nde la norme IEEE 754. Ce qui compte, c'est de retenir le principe : signe,\nmantisse, exposant — et les conséquences sur la précision.",[593,1782,1784,1785,1824],{"id":1783},"pourquoi-010101-pose-problème","Pourquoi ",[602,1786,1788,1803],{"className":1787},[605],[602,1789,1791],{"className":1790},[609],[611,1792,1793],{"xmlns":613},[615,1794,1795,1800],{},[618,1796,1797],{},[621,1798,1799],{},"0,1",[625,1801,1802],{"encoding":627},"0{,}1",[602,1804,1806],{"className":1805,"ariaHidden":633},[632],[602,1807,1809,1812,1815,1821],{"className":1808},[637],[602,1810],{"className":1811,"style":642},[641],[602,1813,931],{"className":1814},[646],[602,1816,1818],{"className":1817},[646],[602,1819,655],{"className":1820},[654],[602,1822,678],{"className":1823},[646]," pose problème",[598,1826,1827,1828,1926,1927,1929,1930,1933,1934,2031,2032,2035],{},"En base 10, ",[602,1829,1831,1849],{"className":1830},[605],[602,1832,1834],{"className":1833},[609],[611,1835,1836],{"xmlns":613},[615,1837,1838,1846],{},[618,1839,1840],{},[674,1841,1842,1844],{},[621,1843,678],{},[621,1845,853],{},[625,1847,1848],{"encoding":627},"\\frac{1}{10}",[602,1850,1852],{"className":1851,"ariaHidden":633},[632],[602,1853,1855,1858],{"className":1854},[637],[602,1856],{"className":1857,"style":693},[641],[602,1859,1861,1864,1923],{"className":1860},[646],[602,1862],{"className":1863},[700,701],[602,1865,1867],{"className":1866},[674],[602,1868,1870,1915],{"className":1869},[708,709],[602,1871,1873,1912],{"className":1872},[713],[602,1874,1876,1890,1898],{"className":1875,"style":718},[717],[602,1877,1878,1881],{"style":721},[602,1879],{"className":1880,"style":726},[725],[602,1882,1884],{"className":1883},[730,731,732,733],[602,1885,1887],{"className":1886},[646,733],[602,1888,853],{"className":1889},[646,733],[602,1891,1892,1895],{"style":742},[602,1893],{"className":1894,"style":726},[725],[602,1896],{"className":1897,"style":750},[749],[602,1899,1900,1903],{"style":753},[602,1901],{"className":1902,"style":726},[725],[602,1904,1906],{"className":1905},[730,731,732,733],[602,1907,1909],{"className":1908},[646,733],[602,1910,678],{"className":1911},[646,733],[602,1913,770],{"className":1914},[769],[602,1916,1918],{"className":1917},[713],[602,1919,1921],{"className":1920,"style":777},[717],[602,1922],{},[602,1924],{"className":1925},[783,701]," s'écrit simplement ",[805,1928,1799],{},". Mais en ",[786,1931,1932],{},"base 2",",\n",[602,1935,1937,1954],{"className":1936},[605],[602,1938,1940],{"className":1939},[609],[611,1941,1942],{"xmlns":613},[615,1943,1944,1952],{},[618,1945,1946],{},[674,1947,1948,1950],{},[621,1949,678],{},[621,1951,853],{},[625,1953,1848],{"encoding":627},[602,1955,1957],{"className":1956,"ariaHidden":633},[632],[602,1958,1960,1963],{"className":1959},[637],[602,1961],{"className":1962,"style":693},[641],[602,1964,1966,1969,2028],{"className":1965},[646],[602,1967],{"className":1968},[700,701],[602,1970,1972],{"className":1971},[674],[602,1973,1975,2020],{"className":1974},[708,709],[602,1976,1978,2017],{"className":1977},[713],[602,1979,1981,1995,2003],{"className":1980,"style":718},[717],[602,1982,1983,1986],{"style":721},[602,1984],{"className":1985,"style":726},[725],[602,1987,1989],{"className":1988},[730,731,732,733],[602,1990,1992],{"className":1991},[646,733],[602,1993,853],{"className":1994},[646,733],[602,1996,1997,2000],{"style":742},[602,1998],{"className":1999,"style":726},[725],[602,2001],{"className":2002,"style":750},[749],[602,2004,2005,2008],{"style":753},[602,2006],{"className":2007,"style":726},[725],[602,2009,2011],{"className":2010},[730,731,732,733],[602,2012,2014],{"className":2013},[646,733],[602,2015,678],{"className":2016},[646,733],[602,2018,770],{"className":2019},[769],[602,2021,2023],{"className":2022},[713],[602,2024,2026],{"className":2025,"style":777},[717],[602,2027],{},[602,2029],{"className":2030},[783,701]," a un développement ",[786,2033,2034],{},"infini périodique"," :",[602,2037,2039],{"className":2038},[827],[602,2040,2042,2077],{"className":2041},[605],[602,2043,2045],{"className":2044},[609],[611,2046,2047],{"xmlns":613,"display":836},[615,2048,2049,2074],{},[618,2050,2051,2057,2059,2061,2064,2067],{},[674,2052,2053,2055],{},[621,2054,678],{},[621,2056,853],{},[845,2058,1226],{},[845,2060,1230],{"stretchy":1229},[621,2062,2063],{},"0,0001100110011",[845,2065,2066],{},"…",[2068,2069,2070,2072],"msub",{},[845,2071,1239],{"stretchy":1229},[621,2073,1254],{},[625,2075,2076],{"encoding":627},"\\frac{1}{10} = (0{,}0001100110011\\ldots)_2",[602,2078,2080,2162],{"className":2079,"ariaHidden":633},[632],[602,2081,2083,2087,2153,2156,2159],{"className":2082},[637],[602,2084],{"className":2085,"style":2086},[641],"height:2.0074em;vertical-align:-0.686em;",[602,2088,2090,2093,2150],{"className":2089},[646],[602,2091],{"className":2092},[700,701],[602,2094,2096],{"className":2095},[674],[602,2097,2099,2141],{"className":2098},[708,709],[602,2100,2102,2138],{"className":2101},[713],[602,2103,2106,2118,2126],{"className":2104,"style":2105},[717],"height:1.3214em;",[602,2107,2109,2112],{"style":2108},"top:-2.314em;",[602,2110],{"className":2111,"style":726},[725],[602,2113,2115],{"className":2114},[646],[602,2116,853],{"className":2117},[646],[602,2119,2120,2123],{"style":742},[602,2121],{"className":2122,"style":726},[725],[602,2124],{"className":2125,"style":750},[749],[602,2127,2129,2132],{"style":2128},"top:-3.677em;",[602,2130],{"className":2131,"style":726},[725],[602,2133,2135],{"className":2134},[646],[602,2136,678],{"className":2137},[646],[602,2139,770],{"className":2140},[769],[602,2142,2144],{"className":2143},[713],[602,2145,2148],{"className":2146,"style":2147},[717],"height:0.686em;",[602,2149],{},[602,2151],{"className":2152},[783,701],[602,2154],{"className":2155,"style":1278},[858],[602,2157,1226],{"className":2158},[1282],[602,2160],{"className":2161,"style":1278},[858],[602,2163,2165,2168,2171,2174,2180,2184,2188,2192],{"className":2164},[637],[602,2166],{"className":2167,"style":1292},[641],[602,2169,1230],{"className":2170},[700],[602,2172,931],{"className":2173},[646],[602,2175,2177],{"className":2176},[646],[602,2178,655],{"className":2179},[654],[602,2181,2183],{"className":2182},[646],"0001100110011",[602,2185],{"className":2186,"style":2187},[858],"margin-right:0.1667em;",[602,2189,2066],{"className":2190},[2191],"minner",[602,2193,2195,2198],{"className":2194},[783],[602,2196,1239],{"className":2197},[783],[602,2199,2201],{"className":2200},[935],[602,2202,2204,2226],{"className":2203},[708,709],[602,2205,2207,2223],{"className":2206},[713],[602,2208,2211],{"className":2209,"style":2210},[717],"height:0.3011em;",[602,2212,2214,2217],{"style":2213},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[602,2215],{"className":2216,"style":952},[725],[602,2218,2220],{"className":2219},[730,731,732,733],[602,2221,1254],{"className":2222},[646,733],[602,2224,770],{"className":2225},[769],[602,2227,2229],{"className":2228},[713],[602,2230,2233],{"className":2231,"style":2232},[717],"height:0.15em;",[602,2234],{},[598,2236,2237,2238,2241,2242,2279,2280,1197],{},"Comme la mantisse a une taille bornée (52 bits en double précision), la machine\ntronque ce développement infini. Le résultat est une ",[786,2239,2240],{},"approximation très\nproche"," de ",[602,2243,2245,2258],{"className":2244},[605],[602,2246,2248],{"className":2247},[609],[611,2249,2250],{"xmlns":613},[615,2251,2252,2256],{},[618,2253,2254],{},[621,2255,1799],{},[625,2257,1802],{"encoding":627},[602,2259,2261],{"className":2260,"ariaHidden":633},[632],[602,2262,2264,2267,2270,2276],{"className":2263},[637],[602,2265],{"className":2266,"style":642},[641],[602,2268,931],{"className":2269},[646],[602,2271,2273],{"className":2272},[646],[602,2274,655],{"className":2275},[654],[602,2277,678],{"className":2278},[646],", mais pas exactement ",[602,2281,2283,2296],{"className":2282},[605],[602,2284,2286],{"className":2285},[609],[611,2287,2288],{"xmlns":613},[615,2289,2290,2294],{},[618,2291,2292],{},[621,2293,1799],{},[625,2295,1802],{"encoding":627},[602,2297,2299],{"className":2298,"ariaHidden":633},[632],[602,2300,2302,2305,2308,2314],{"className":2301},[637],[602,2303],{"className":2304,"style":642},[641],[602,2306,931],{"className":2307},[646],[602,2309,2311],{"className":2310},[646],[602,2312,655],{"className":2313},[654],[602,2315,678],{"className":2316},[646],[2318,2319,2320],"tip",{},[598,2321,2322,2323,2420,2421,2469],{},"La même chose se produit en base 10 avec ",[602,2324,2326,2343],{"className":2325},[605],[602,2327,2329],{"className":2328},[609],[611,2330,2331],{"xmlns":613},[615,2332,2333,2341],{},[618,2334,2335],{},[674,2336,2337,2339],{},[621,2338,678],{},[621,2340,647],{},[625,2342,683],{"encoding":627},[602,2344,2346],{"className":2345,"ariaHidden":633},[632],[602,2347,2349,2352],{"className":2348},[637],[602,2350],{"className":2351,"style":693},[641],[602,2353,2355,2358,2417],{"className":2354},[646],[602,2356],{"className":2357},[700,701],[602,2359,2361],{"className":2360},[674],[602,2362,2364,2409],{"className":2363},[708,709],[602,2365,2367,2406],{"className":2366},[713],[602,2368,2370,2384,2392],{"className":2369,"style":718},[717],[602,2371,2372,2375],{"style":721},[602,2373],{"className":2374,"style":726},[725],[602,2376,2378],{"className":2377},[730,731,732,733],[602,2379,2381],{"className":2380},[646,733],[602,2382,647],{"className":2383},[646,733],[602,2385,2386,2389],{"style":742},[602,2387],{"className":2388,"style":726},[725],[602,2390],{"className":2391,"style":750},[749],[602,2393,2394,2397],{"style":753},[602,2395],{"className":2396,"style":726},[725],[602,2398,2400],{"className":2399},[730,731,732,733],[602,2401,2403],{"className":2402},[646,733],[602,2404,678],{"className":2405},[646,733],[602,2407,770],{"className":2408},[769],[602,2410,2412],{"className":2411},[713],[602,2413,2415],{"className":2414,"style":777},[717],[602,2416],{},[602,2418],{"className":2419},[783,701]," : son écriture\ndécimale ",[602,2422,2424,2441],{"className":2423},[605],[602,2425,2427],{"className":2426},[609],[611,2428,2429],{"xmlns":613},[615,2430,2431,2438],{},[618,2432,2433,2436],{},[621,2434,2435],{},"0,3333",[845,2437,2066],{},[625,2439,2440],{"encoding":627},"0{,}3333\\ldots",[602,2442,2444],{"className":2443,"ariaHidden":633},[632],[602,2445,2447,2450,2453,2459,2463,2466],{"className":2446},[637],[602,2448],{"className":2449,"style":642},[641],[602,2451,931],{"className":2452},[646],[602,2454,2456],{"className":2455},[646],[602,2457,655],{"className":2458},[654],[602,2460,2462],{"className":2461},[646],"3333",[602,2464],{"className":2465,"style":2187},[858],[602,2467,2066],{"className":2468},[2191]," ne s'arrête jamais. Si on devait l'écrire avec un\nnombre fini de chiffres, on obtiendrait une approximation, jamais la valeur\nexacte. La base 2 a le même genre de souci, mais pour d'autres fractions —\nnotamment celles dont le dénominateur n'est pas une puissance de 2.",[593,2471,2473],{"id":2472},"la-preuve-par-python","La preuve par Python",[2475,2476,2481],"pre",{"className":2477,"code":2478,"language":2479,"meta":2480,"style":2480},"language-python shiki shiki-themes github-light github-light github-dark","print(0.1 + 0.2)       # 0.30000000000000004\nprint(0.1 + 0.2 == 0.3)  # False\n","python","",[805,2482,2483,2511],{"__ignoreMap":2480},[602,2484,2487,2491,2494,2497,2501,2504,2507],{"class":2485,"line":2486},"line",1,[602,2488,2490],{"class":2489},"sBjJW","print",[602,2492,1230],{"class":2493},"sxrX7",[602,2495,2496],{"class":2489},"0.1",[602,2498,2500],{"class":2499},"s8jYJ"," +",[602,2502,2503],{"class":2489}," 0.2",[602,2505,2506],{"class":2493},")       ",[602,2508,2510],{"class":2509},"sCsY4","# 0.30000000000000004\n",[602,2512,2514,2516,2518,2520,2522,2524,2527,2530,2533],{"class":2485,"line":2513},2,[602,2515,2490],{"class":2489},[602,2517,1230],{"class":2493},[602,2519,2496],{"class":2489},[602,2521,2500],{"class":2499},[602,2523,2503],{"class":2489},[602,2525,2526],{"class":2499}," ==",[602,2528,2529],{"class":2489}," 0.3",[602,2531,2532],{"class":2493},")  ",[602,2534,2535],{"class":2509},"# False\n",[598,2537,2538,2539,2614,2615,2618,2619,2621],{},"L'écart est minuscule — de l'ordre de ",[602,2540,2542,2565],{"className":2541},[605],[602,2543,2545],{"className":2544},[609],[611,2546,2547],{"xmlns":613},[615,2548,2549,2562],{},[618,2550,2551],{},[849,2552,2553,2555],{},[621,2554,853],{},[618,2556,2557,2559],{},[845,2558,874],{},[621,2560,2561],{},"17",[625,2563,2564],{"encoding":627},"10^{-17}",[602,2566,2568],{"className":2567,"ariaHidden":633},[632],[602,2569,2571,2575,2578],{"className":2570},[637],[602,2572],{"className":2573,"style":2574},[641],"height:0.8141em;",[602,2576,678],{"className":2577},[646],[602,2579,2581,2584],{"className":2580},[646],[602,2582,931],{"className":2583},[646],[602,2585,2587],{"className":2586},[935],[602,2588,2590],{"className":2589},[708],[602,2591,2593],{"className":2592},[713],[602,2594,2596],{"className":2595,"style":2574},[717],[602,2597,2599,2602],{"style":2598},"top:-3.063em;margin-right:0.05em;",[602,2600],{"className":2601,"style":952},[725],[602,2603,2605],{"className":2604},[730,731,732,733],[602,2606,2608,2611],{"className":2607},[646,733],[602,2609,874],{"className":2610},[646,733],[602,2612,2561],{"className":2613},[646,733]," — mais il est ",[786,2616,2617],{},"non nul",", et\nil suffit à faire échouer un test d'égalité strict. Ce comportement n'est\n",[786,2620,811],{}," un bug de Python : c'est la conséquence directe de la représentation\nbinaire des flottants.",[2623,2624,2625],"python-snippet",{},[2475,2626,2628],{"className":2477,"code":2627,"language":2479,"meta":2480,"style":2480},"# Affichage très précis pour voir la \"vraie\" valeur stockée\nfrom decimal import Decimal\nprint(Decimal(0.1))\nprint(Decimal(0.2))\nprint(Decimal(0.1) + Decimal(0.2))\n",[805,2629,2630,2635,2649,2662,2674],{"__ignoreMap":2480},[602,2631,2632],{"class":2485,"line":2486},[602,2633,2634],{"class":2509},"# Affichage très précis pour voir la \"vraie\" valeur stockée\n",[602,2636,2637,2640,2643,2646],{"class":2485,"line":2513},[602,2638,2639],{"class":2499},"from",[602,2641,2642],{"class":2493}," decimal ",[602,2644,2645],{"class":2499},"import",[602,2647,2648],{"class":2493}," Decimal\n",[602,2650,2652,2654,2657,2659],{"class":2485,"line":2651},3,[602,2653,2490],{"class":2489},[602,2655,2656],{"class":2493},"(Decimal(",[602,2658,2496],{"class":2489},[602,2660,2661],{"class":2493},"))\n",[602,2663,2665,2667,2669,2672],{"class":2485,"line":2664},4,[602,2666,2490],{"class":2489},[602,2668,2656],{"class":2493},[602,2670,2671],{"class":2489},"0.2",[602,2673,2661],{"class":2493},[602,2675,2677,2679,2681,2683,2686,2689,2692,2694],{"class":2485,"line":2676},5,[602,2678,2490],{"class":2489},[602,2680,2656],{"class":2493},[602,2682,2496],{"class":2489},[602,2684,2685],{"class":2493},") ",[602,2687,2688],{"class":2499},"+",[602,2690,2691],{"class":2493}," Decimal(",[602,2693,2671],{"class":2489},[602,2695,2661],{"class":2493},[598,2697,2698],{},"Sortie attendue (extrait) :",[2475,2700,2705],{"className":2701,"code":2703,"language":2704},[2702],"language-text","0.1000000000000000055511151231257827021181583404541015625\n0.200000000000000011102230246251565404236316680908203125\n0.3000000000000000166533453694\n","text",[805,2706,2703],{"__ignoreMap":2480},[598,2708,2709,2710,2713,2714,2716,2717,2719,2720,2723,2724,2727,2728,2731,2732,2735,2736,2738],{},"Les deux premières lignes affichent la ",[786,2711,2712],{},"valeur binaire exacte"," des floats\n",[805,2715,2496],{}," et ",[805,2718,2671],{}," (le constructeur ",[805,2721,2722],{},"Decimal(float)"," n'arrondit pas). La 3ᵉ ligne\ncorrespond à l'",[786,2725,2726],{},"addition"," : Python repasse alors par le contexte ",[805,2729,2730],{},"Decimal","\nde précision par défaut, qui arrondit à ",[786,2733,2734],{},"28 chiffres significatifs",". La\nvaleur reste différente de ",[805,2737,815],{}," exact — ce qui confirme l'erreur de\nreprésentation binaire.",[2740,2741,2743,2826],"self-eval-qcm",{"correct":678,"slug":2742},"flottants-c1",[2744,2745,2746],"template",{"v-slot:question":2480},[598,2747,2748,2749,2825],{},"Que renvoie en Python l'expression ",[602,2750,2752,2776],{"className":2751},[605],[602,2753,2755],{"className":2754},[609],[611,2756,2757],{"xmlns":613},[615,2758,2759,2773],{},[618,2760,2761,2763,2765,2767,2769,2771],{},[621,2762,2496],{},[845,2764,2688],{},[621,2766,2671],{},[845,2768,1226],{},[845,2770,1226],{},[621,2772,815],{},[625,2774,2775],{"encoding":627},"0.1 + 0.2 == 0.3",[602,2777,2779,2797,2816],{"className":2778,"ariaHidden":633},[632],[602,2780,2782,2785,2788,2791,2794],{"className":2781},[637],[602,2783],{"className":2784,"style":1190},[641],[602,2786,2496],{"className":2787},[646],[602,2789],{"className":2790,"style":907},[858],[602,2792,2688],{"className":2793},[911],[602,2795],{"className":2796,"style":907},[858],[602,2798,2800,2803,2806,2809,2813],{"className":2799},[637],[602,2801],{"className":2802,"style":1158},[641],[602,2804,2671],{"className":2805},[646],[602,2807],{"className":2808,"style":1278},[858],[602,2810,2812],{"className":2811},[1282],"==",[602,2814],{"className":2815,"style":1278},[858],[602,2817,2819,2822],{"className":2818},[637],[602,2820],{"className":2821,"style":1158},[641],[602,2823,815],{"className":2824},[646]," ?",[2744,2827,2828],{"v-slot:choices":2480},[1035,2829,2830,2833,2836,2839],{},[1038,2831,2832],{},"True",[1038,2834,2835],{},"False",[1038,2837,2838],{},"une erreur",[1038,2840,2841],{},"cela dépend",[593,2843,2845],{"id":2844},"comment-comparer-deux-flottants","Comment comparer deux flottants",[598,2847,2848,2849,2852,2853,2856,2857,2887],{},"La règle d'or est simple : ",[786,2850,2851],{},"on ne teste jamais l'égalité stricte de deux\nflottants",". À la place, on vérifie qu'ils sont ",[786,2854,2855],{},"suffisamment proches",", à une\ntolérance ",[602,2858,2860,2875],{"className":2859},[605],[602,2861,2863],{"className":2862},[609],[611,2864,2865],{"xmlns":613},[615,2866,2867,2872],{},[618,2868,2869],{},[1221,2870,2871],{},"ε",[625,2873,2874],{"encoding":627},"\\varepsilon",[602,2876,2878],{"className":2877,"ariaHidden":633},[632],[602,2879,2881,2884],{"className":2880},[637],[602,2882],{"className":2883,"style":1270},[641],[602,2885,2871],{"className":2886},[646,1274]," près.",[2475,2889,2891],{"className":2477,"code":2890,"language":2479,"meta":2480,"style":2480},"def proche(a, b, eps=1e-9):\n    return abs(a - b) \u003C eps\n\nprint(proche(0.1 + 0.2, 0.3))  # True\n",[805,2892,2893,2913,2936,2942],{"__ignoreMap":2480},[602,2894,2895,2898,2902,2905,2907,2910],{"class":2485,"line":2486},[602,2896,2897],{"class":2499},"def",[602,2899,2901],{"class":2900},"snPdu"," proche",[602,2903,2904],{"class":2493},"(a, b, eps",[602,2906,1226],{"class":2499},[602,2908,2909],{"class":2489},"1e-9",[602,2911,2912],{"class":2493},"):\n",[602,2914,2915,2918,2921,2924,2927,2930,2933],{"class":2485,"line":2513},[602,2916,2917],{"class":2499},"    return",[602,2919,2920],{"class":2489}," abs",[602,2922,2923],{"class":2493},"(a ",[602,2925,2926],{"class":2499},"-",[602,2928,2929],{"class":2493}," b) ",[602,2931,2932],{"class":2499},"\u003C",[602,2934,2935],{"class":2493}," eps\n",[602,2937,2938],{"class":2485,"line":2651},[602,2939,2941],{"emptyLinePlaceholder":2940},true,"\n",[602,2943,2944,2946,2949,2951,2953,2955,2958,2960,2963],{"class":2485,"line":2664},[602,2945,2490],{"class":2489},[602,2947,2948],{"class":2493},"(proche(",[602,2950,2496],{"class":2489},[602,2952,2500],{"class":2499},[602,2954,2503],{"class":2489},[602,2956,2957],{"class":2493},", ",[602,2959,815],{"class":2489},[602,2961,2962],{"class":2493},"))  ",[602,2964,2965],{"class":2509},"# True\n",[598,2967,2968,2969,2972],{},"Python fournit aussi ",[805,2970,2971],{},"math.isclose"," pour cette comparaison, plus robuste :",[2475,2974,2976],{"className":2477,"code":2975,"language":2479,"meta":2480,"style":2480},"import math\nprint(math.isclose(0.1 + 0.2, 0.3))  # True\n",[805,2977,2978,2985],{"__ignoreMap":2480},[602,2979,2980,2982],{"class":2485,"line":2486},[602,2981,2645],{"class":2499},[602,2983,2984],{"class":2493}," math\n",[602,2986,2987,2989,2992,2994,2996,2998,3000,3002,3004],{"class":2485,"line":2513},[602,2988,2490],{"class":2489},[602,2990,2991],{"class":2493},"(math.isclose(",[602,2993,2496],{"class":2489},[602,2995,2500],{"class":2499},[602,2997,2503],{"class":2489},[602,2999,2957],{"class":2493},[602,3001,815],{"class":2489},[602,3003,2962],{"class":2493},[602,3005,2965],{"class":2509},[3007,3008,3009],"warning",{},[598,3010,3011,3012,3015,3016,3019],{},"Cette précaution est ",[786,3013,3014],{},"systématique",". Que vous compariez le résultat d'un\ncalcul de physique, une distance, une fraction… utilisez toujours ",[805,3017,3018],{},"isclose"," ou\nune tolérance explicite quand des flottants sont en jeu.",[593,3021,3023,3024,2957,3027,2957,3030],{"id":3022},"valeurs-spéciales-inf-inf-nan","Valeurs spéciales — ",[805,3025,3026],{},"inf",[805,3028,3029],{},"-inf",[805,3031,3032],{},"nan",[598,3034,3035],{},"IEEE 754 prévoit quelques valeurs particulières :",[1035,3037,3038,3043,3048],{},[1038,3039,3040,3042],{},[805,3041,3026],{}," (infini positif) : résultat d'un dépassement ou d'une division par\nzéro positive.",[1038,3044,3045,3047],{},[805,3046,3029],{}," (infini négatif).",[1038,3049,3050,3052,3053,3057,3058,660,3093,1197],{},[805,3051,3032],{}," (",[3054,3055,3056],"em",{},"Not a Number",") : résultat d'une opération invalide, comme ",[602,3059,3061,3081],{"className":3060},[605],[602,3062,3064],{"className":3063},[609],[611,3065,3066],{"xmlns":613},[615,3067,3068,3078],{},[618,3069,3070,3072,3076],{},[621,3071,931],{},[1221,3073,3075],{"mathvariant":3074},"normal","\u002F",[621,3077,931],{},[625,3079,3080],{"encoding":627},"0\u002F0",[602,3082,3084],{"className":3083,"ariaHidden":633},[632],[602,3085,3087,3090],{"className":3086},[637],[602,3088],{"className":3089,"style":1292},[641],[602,3091,3080],{"className":3092},[646],[602,3094,3096,3117],{"className":3095},[605],[602,3097,3099],{"className":3098},[609],[611,3100,3101],{"xmlns":613},[615,3102,3103,3114],{},[618,3104,3105],{},[3106,3107,3108],"msqrt",{},[618,3109,3110,3112],{},[845,3111,874],{},[621,3113,678],{},[625,3115,3116],{"encoding":627},"\\sqrt{-1}",[602,3118,3120],{"className":3119,"ariaHidden":633},[632],[602,3121,3123,3127],{"className":3122},[637],[602,3124],{"className":3125,"style":3126},[641],"height:1.04em;vertical-align:-0.1744em;",[602,3128,3131],{"className":3129},[646,3130],"sqrt",[602,3132,3134,3185],{"className":3133},[708,709],[602,3135,3137,3182],{"className":3136},[713],[602,3138,3141,3159],{"className":3139,"style":3140},[717],"height:0.8656em;",[602,3142,3146,3149],{"className":3143,"style":3145},[3144],"svg-align","top:-3em;",[602,3147],{"className":3148,"style":726},[725],[602,3150,3153,3156],{"className":3151,"style":3152},[646],"padding-left:0.833em;",[602,3154,874],{"className":3155},[646],[602,3157,678],{"className":3158},[646],[602,3160,3162,3165],{"style":3161},"top:-2.8256em;",[602,3163],{"className":3164,"style":726},[725],[602,3166,3170],{"className":3167,"style":3169},[3168],"hide-tail","min-width:0.853em;height:1.08em;",[3171,3172,3178],"svg",{"xmlns":3173,"width":3174,"height":3175,"viewBox":3176,"preserveAspectRatio":3177},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[3179,3180],"path",{"d":3181},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[602,3183,770],{"className":3184},[769],[602,3186,3188],{"className":3187},[713],[602,3189,3192],{"className":3190,"style":3191},[717],"height:0.1744em;",[602,3193],{},[2475,3195,3197],{"className":2477,"code":3196,"language":2479,"meta":2480,"style":2480},"import math\nprint(math.inf)              # inf — constante du module math\nprint(1 \u002F 0.0)               # ZeroDivisionError en Python pur (et non inf)\nprint(float('nan') == float('nan'))  # False — un nan n'est jamais égal à lui-même !\n",[805,3198,3199,3205,3215,3235],{"__ignoreMap":2480},[602,3200,3201,3203],{"class":2485,"line":2486},[602,3202,2645],{"class":2499},[602,3204,2984],{"class":2493},[602,3206,3207,3209,3212],{"class":2485,"line":2513},[602,3208,2490],{"class":2489},[602,3210,3211],{"class":2493},"(math.inf)              ",[602,3213,3214],{"class":2509},"# inf — constante du module math\n",[602,3216,3217,3219,3221,3223,3226,3229,3232],{"class":2485,"line":2651},[602,3218,2490],{"class":2489},[602,3220,1230],{"class":2493},[602,3222,678],{"class":2489},[602,3224,3225],{"class":2499}," \u002F",[602,3227,3228],{"class":2489}," 0.0",[602,3230,3231],{"class":2493},")               ",[602,3233,3234],{"class":2509},"# ZeroDivisionError en Python pur (et non inf)\n",[602,3236,3237,3239,3241,3243,3245,3249,3251,3253,3256,3258,3260,3262],{"class":2485,"line":2664},[602,3238,2490],{"class":2489},[602,3240,1230],{"class":2493},[602,3242,1643],{"class":2489},[602,3244,1230],{"class":2493},[602,3246,3248],{"class":3247},"sIIMD","'nan'",[602,3250,2685],{"class":2493},[602,3252,2812],{"class":2499},[602,3254,3255],{"class":2489}," float",[602,3257,1230],{"class":2493},[602,3259,3248],{"class":3247},[602,3261,2962],{"class":2493},[602,3263,3264],{"class":2509},"# False — un nan n'est jamais égal à lui-même !\n",[1772,3266,3267],{},[598,3268,3269,3270,3273,3274,3280,3281,3283,3284,3287,3288,3290,3291,3293],{},"En Python pur, ",[805,3271,3272],{},"1 \u002F 0.0"," ",[786,3275,3276,3277],{},"lève une ",[805,3278,3279],{},"ZeroDivisionError"," — l'infini ne sort\npas naturellement de la division. Pour obtenir ",[805,3282,3026],{},", on passe par la\nconstante ",[805,3285,3286],{},"math.inf"," (ou par des bibliothèques comme NumPy, où ",[805,3289,3272],{},"\nrenvoie effectivement ",[805,3292,3026],{},").",[2740,3295,3297,3302],{"correct":1254,"slug":3296},"flottants-c2",[2744,3298,3299],{"v-slot:question":2480},[598,3300,3301],{},"Combien de bits utilise un float en double précision ?",[2744,3303,3304],{"v-slot:choices":2480},[1035,3305,3306,3309,3312,3315],{},[1038,3307,3308],{},"16",[1038,3310,3311],{},"32",[1038,3313,3314],{},"64",[1038,3316,3317],{},"128",[593,3319,3321],{"id":3320},"pièges-courants","Pièges courants",[3007,3323,3324],{},[1035,3325,3326,3338,3341],{},[1038,3327,3328,3331,3332,3335,3336,1197],{},[786,3329,3330],{},"Ne jamais"," tester ",[805,3333,3334],{},"a == b"," pour des flottants. Utiliser ",[805,3337,2971],{},[1038,3339,3340],{},"Additionner beaucoup de petits flottants peut perdre en précision (les\npetits termes deviennent négligeables face au cumul).",[1038,3342,3343,3344,3347,3348,3052,3350,3353,3354,3357,3358,3395],{},"Convertir une ",[805,3345,3346],{},"str"," en ",[805,3349,1643],{},[805,3351,3352],{},"float(\"0.1\")",") crée la ",[786,3355,3356],{},"même\napproximation"," : il n'y a pas de chemin pour obtenir un ",[602,3359,3361,3374],{"className":3360},[605],[602,3362,3364],{"className":3363},[609],[611,3365,3366],{"xmlns":613},[615,3367,3368,3372],{},[618,3369,3370],{},[621,3371,1799],{},[625,3373,1802],{"encoding":627},[602,3375,3377],{"className":3376,"ariaHidden":633},[632],[602,3378,3380,3383,3386,3392],{"className":3379},[637],[602,3381],{"className":3382,"style":642},[641],[602,3384,931],{"className":3385},[646],[602,3387,3389],{"className":3388},[646],[602,3390,655],{"className":3391},[654],[602,3393,678],{"className":3394},[646]," exact en\nflottant binaire.",[593,3397,3399],{"id":3398},"pour-aller-plus-loin","Pour aller plus loin",[598,3401,3402],{},"Pour des calculs financiers ou scientifiques exigeants, Python fournit deux\nalternatives :",[1035,3404,3405,3412],{},[1038,3406,3407,3408,3411],{},"Le module ",[805,3409,3410],{},"decimal"," (calcul en base 10, à précision arbitraire) — bien adapté\naux montants monétaires.",[1038,3413,3407,3414,3417,3418,3522],{},[805,3415,3416],{},"fractions"," (rationnels exacts ",[602,3419,3421,3440],{"className":3420},[605],[602,3422,3424],{"className":3423},[609],[611,3425,3426],{"xmlns":613},[615,3427,3428,3437],{},[618,3429,3430],{},[674,3431,3432,3434],{},[1221,3433,598],{},[1221,3435,3436],{},"q",[625,3438,3439],{"encoding":627},"\\frac{p}{q}",[602,3441,3443],{"className":3442,"ariaHidden":633},[632],[602,3444,3446,3450],{"className":3445},[637],[602,3447],{"className":3448,"style":3449},[641],"height:1.2286em;vertical-align:-0.4811em;",[602,3451,3453,3456,3519],{"className":3452},[646],[602,3454],{"className":3455},[700,701],[602,3457,3459],{"className":3458},[674],[602,3460,3462,3510],{"className":3461},[708,709],[602,3463,3465,3507],{"className":3464},[713],[602,3466,3469,3484,3492],{"className":3467,"style":3468},[717],"height:0.7475em;",[602,3470,3471,3474],{"style":721},[602,3472],{"className":3473,"style":726},[725],[602,3475,3477],{"className":3476},[730,731,732,733],[602,3478,3480],{"className":3479},[646,733],[602,3481,3436],{"className":3482,"style":3483},[646,1274,733],"margin-right:0.0359em;",[602,3485,3486,3489],{"style":742},[602,3487],{"className":3488,"style":726},[725],[602,3490],{"className":3491,"style":750},[749],[602,3493,3495,3498],{"style":3494},"top:-3.4461em;",[602,3496],{"className":3497,"style":726},[725],[602,3499,3501],{"className":3500},[730,731,732,733],[602,3502,3504],{"className":3503},[646,733],[602,3505,598],{"className":3506},[646,1274,733],[602,3508,770],{"className":3509},[769],[602,3511,3513],{"className":3512},[713],[602,3514,3517],{"className":3515,"style":3516},[717],"height:0.4811em;",[602,3518],{},[602,3520],{"className":3521},[783,701],") — utile pour les\ncalculs symboliques où les approximations sont interdites.",[2475,3524,3526],{"className":2477,"code":3525,"language":2479,"meta":2480,"style":2480},"from fractions import Fraction\nprint(Fraction(1, 10) + Fraction(2, 10))  # 3\u002F10\n",[805,3527,3528,3540],{"__ignoreMap":2480},[602,3529,3530,3532,3535,3537],{"class":2485,"line":2486},[602,3531,2639],{"class":2499},[602,3533,3534],{"class":2493}," fractions ",[602,3536,2645],{"class":2499},[602,3538,3539],{"class":2493}," Fraction\n",[602,3541,3542,3544,3547,3549,3551,3553,3555,3557,3560,3562,3564,3566,3568],{"class":2485,"line":2513},[602,3543,2490],{"class":2489},[602,3545,3546],{"class":2493},"(Fraction(",[602,3548,678],{"class":2489},[602,3550,2957],{"class":2493},[602,3552,853],{"class":2489},[602,3554,2685],{"class":2493},[602,3556,2688],{"class":2499},[602,3558,3559],{"class":2493}," Fraction(",[602,3561,1254],{"class":2489},[602,3563,2957],{"class":2493},[602,3565,853],{"class":2489},[602,3567,2962],{"class":2493},[602,3569,3570],{"class":2509},"# 3\u002F10\n",[598,3572,3573,3574,1197],{},"Ces alternatives sont plus lentes, mais elles fournissent le résultat\n",[786,3575,3576],{},"exact",[3578,3579,3580],"style",{},"html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":2480,"searchDepth":2513,"depth":2513,"links":3582},[3583,3584,3585,3586,3588,3589,3590,3592,3593],{"id":595,"depth":2513,"text":596},{"id":819,"depth":2513,"text":820},{"id":1487,"depth":2513,"text":1488},{"id":1783,"depth":2513,"text":3587},"Pourquoi 0,10{,}10,1 pose problème",{"id":2472,"depth":2513,"text":2473},{"id":2844,"depth":2513,"text":2845},{"id":3022,"depth":2513,"text":3591},"Valeurs spéciales — inf, -inf, nan",{"id":3320,"depth":2513,"text":3321},{"id":3398,"depth":2513,"text":3399},"Comment la machine approxime les nombres réels et pourquoi 0.1 + 0.2 ne vaut pas exactement 0.3.",null,"md",{},[3599,3604],{"slug":2742,"kind":3600,"question":3601,"choices":3602,"correct":678,"multi":3603,"explanation":2480},"qcm","Que renvoie en Python l'expression 0.1+0.2==0.30.1 + 0.2 == 0.30.1+0.2==0.3 ?",[2832,2835,2838,2841],false,{"slug":3296,"kind":3600,"question":3301,"choices":3605,"correct":1254,"multi":3603,"explanation":2480},[3308,3311,3314,3317],{"title":271,"description":3594},"xlRY1HXwAAnvG_w8rV44fLO50aNDFDiEibupS_Wxhio",{"id":3609,"title":3610,"bo":3611,"body":3612,"description":4242,"eval":3595,"extension":3596,"labo":3595,"meta":4243,"navigation":3603,"path":4244,"quizzes":4245,"readingTime":2486,"seo":4246,"stem":4247,"tp":3595,"__hash__":4248},"coursMemo\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.memo.md","Nombres flottants — l'essentiel",[588],{"type":590,"value":3613,"toc":4235},[3614,3618,3778,3782,4012,4016,4074,4076,4135,4139,4232],[593,3615,3617],{"id":3616},"en-une-phrase","En une phrase",[598,3619,3620,3621,3624,3625,3774,3775,1197],{},"Un ",[786,3622,3623],{},"flottant"," approxime un nombre réel par ",[602,3626,3628,3664],{"className":3627},[605],[602,3629,3631],{"className":3630},[609],[611,3632,3633],{"xmlns":613},[615,3634,3635,3661],{},[618,3636,3637,3639,3641,3643,3649,3651,3653,3655],{},[845,3638,1230],{"stretchy":1229},[845,3640,874],{},[621,3642,678],{},[849,3644,3645,3647],{},[845,3646,1239],{"stretchy":1229},[1221,3648,1242],{},[845,3650,847],{},[1221,3652,1247],{},[845,3654,847],{},[849,3656,3657,3659],{},[621,3658,1254],{},[1221,3660,1257],{},[625,3662,3663],{"encoding":627},"(-1)^s \\cdot m \\cdot 2^e",[602,3665,3667,3721,3739],{"className":3666,"ariaHidden":633},[632],[602,3668,3670,3673,3676,3679,3682,3712,3715,3718],{"className":3669},[637],[602,3671],{"className":3672,"style":1292},[641],[602,3674,1230],{"className":3675},[700],[602,3677,874],{"className":3678},[646],[602,3680,678],{"className":3681},[646],[602,3683,3685,3688],{"className":3684},[783],[602,3686,1239],{"className":3687},[783],[602,3689,3691],{"className":3690},[935],[602,3692,3694],{"className":3693},[708],[602,3695,3697],{"className":3696},[713],[602,3698,3701],{"className":3699,"style":3700},[717],"height:0.6644em;",[602,3702,3703,3706],{"style":2598},[602,3704],{"className":3705,"style":952},[725],[602,3707,3709],{"className":3708},[730,731,732,733],[602,3710,1242],{"className":3711},[646,1274,733],[602,3713],{"className":3714,"style":907},[858],[602,3716,847],{"className":3717},[911],[602,3719],{"className":3720,"style":907},[858],[602,3722,3724,3727,3730,3733,3736],{"className":3723},[637],[602,3725],{"className":3726,"style":1347},[641],[602,3728,1247],{"className":3729},[646,1274],[602,3731],{"className":3732,"style":907},[858],[602,3734,847],{"className":3735},[911],[602,3737],{"className":3738,"style":907},[858],[602,3740,3742,3745],{"className":3741},[637],[602,3743],{"className":3744,"style":3700},[641],[602,3746,3748,3751],{"className":3747},[646],[602,3749,1254],{"className":3750},[646],[602,3752,3754],{"className":3753},[935],[602,3755,3757],{"className":3756},[708],[602,3758,3760],{"className":3759},[713],[602,3761,3763],{"className":3762,"style":3700},[717],[602,3764,3765,3768],{"style":2598},[602,3766],{"className":3767,"style":952},[725],[602,3769,3771],{"className":3770},[730,731,732,733],[602,3772,1257],{"className":3773},[646,1274,733]," — d'où\ndes erreurs d'arrondi inévitables, dont la plus célèbre est\n",[805,3776,3777],{},"0.1 + 0.2 != 0.3",[593,3779,3781],{"id":3780},"à-mémoriser","À mémoriser",[1035,3783,3784,3881,3886,3989,3999],{},[1038,3785,3786,3787,2957,3818,2957,3849,3880],{},"Décomposition : ",[786,3788,3789,3790],{},"signe ",[602,3791,3793,3806],{"className":3792},[605],[602,3794,3796],{"className":3795},[609],[611,3797,3798],{"xmlns":613},[615,3799,3800,3804],{},[618,3801,3802],{},[1221,3803,1242],{},[625,3805,1242],{"encoding":627},[602,3807,3809],{"className":3808,"ariaHidden":633},[632],[602,3810,3812,3815],{"className":3811},[637],[602,3813],{"className":3814,"style":1270},[641],[602,3816,1242],{"className":3817},[646,1274],[786,3819,3820,3821],{},"mantisse ",[602,3822,3824,3837],{"className":3823},[605],[602,3825,3827],{"className":3826},[609],[611,3828,3829],{"xmlns":613},[615,3830,3831,3835],{},[618,3832,3833],{},[1221,3834,1247],{},[625,3836,1247],{"encoding":627},[602,3838,3840],{"className":3839,"ariaHidden":633},[632],[602,3841,3843,3846],{"className":3842},[637],[602,3844],{"className":3845,"style":1270},[641],[602,3847,1247],{"className":3848},[646,1274],[786,3850,3851,3852],{},"exposant ",[602,3853,3855,3868],{"className":3854},[605],[602,3856,3858],{"className":3857},[609],[611,3859,3860],{"xmlns":613},[615,3861,3862,3866],{},[618,3863,3864],{},[1221,3865,1257],{},[625,3867,1257],{"encoding":627},[602,3869,3871],{"className":3870,"ariaHidden":633},[632],[602,3872,3874,3877],{"className":3873},[637],[602,3875],{"className":3876,"style":1270},[641],[602,3878,1257],{"className":3879},[646,1274],", base 2.",[1038,3882,3883,3884,3293],{},"Tailles IEEE 754 : simple précision (32 bits), double précision (64 bits — Python ",[805,3885,1643],{},[1038,3887,3888,3985,3986,3988],{},[602,3889,3891,3908],{"className":3890},[605],[602,3892,3894],{"className":3893},[609],[611,3895,3896],{"xmlns":613},[615,3897,3898,3906],{},[618,3899,3900],{},[674,3901,3902,3904],{},[621,3903,678],{},[621,3905,853],{},[625,3907,1848],{"encoding":627},[602,3909,3911],{"className":3910,"ariaHidden":633},[632],[602,3912,3914,3917],{"className":3913},[637],[602,3915],{"className":3916,"style":693},[641],[602,3918,3920,3923,3982],{"className":3919},[646],[602,3921],{"className":3922},[700,701],[602,3924,3926],{"className":3925},[674],[602,3927,3929,3974],{"className":3928},[708,709],[602,3930,3932,3971],{"className":3931},[713],[602,3933,3935,3949,3957],{"className":3934,"style":718},[717],[602,3936,3937,3940],{"style":721},[602,3938],{"className":3939,"style":726},[725],[602,3941,3943],{"className":3942},[730,731,732,733],[602,3944,3946],{"className":3945},[646,733],[602,3947,853],{"className":3948},[646,733],[602,3950,3951,3954],{"style":742},[602,3952],{"className":3953,"style":726},[725],[602,3955],{"className":3956,"style":750},[749],[602,3958,3959,3962],{"style":753},[602,3960],{"className":3961,"style":726},[725],[602,3963,3965],{"className":3964},[730,731,732,733],[602,3966,3968],{"className":3967},[646,733],[602,3969,678],{"className":3970},[646,733],[602,3972,770],{"className":3973},[769],[602,3975,3977],{"className":3976},[713],[602,3978,3980],{"className":3979,"style":777},[717],[602,3981],{},[602,3983],{"className":3984},[783,701]," n'a ",[786,3987,811],{}," d'écriture finie en binaire.",[1038,3990,3991,3331,3993,3995,3996,3998],{},[786,3992,3330],{},[805,3994,3334],{}," pour deux flottants : utiliser ",[805,3997,2971],{}," ou une tolérance.",[1038,4000,4001,4002,2957,4004,2957,4006,4008,4009,3293],{},"Valeurs spéciales : ",[805,4003,3026],{},[805,4005,3029],{},[805,4007,3032],{}," (et ",[805,4010,4011],{},"nan != nan",[593,4013,4015],{"id":4014},"à-savoir-reconnaître","À savoir reconnaître",[1497,4017,4018,4028],{},[1500,4019,4020],{},[1503,4021,4022,4025],{},[1506,4023,4024],{},"Situation",[1506,4026,4027],{},"Action à mener",[1516,4029,4030,4043,4054,4064],{},[1503,4031,4032,4035],{},[1521,4033,4034],{},"Comparer deux flottants",[1521,4036,4037,1083,4040,1197],{},[805,4038,4039],{},"math.isclose(a, b)",[805,4041,4042],{},"abs(a - b) \u003C eps",[1503,4044,4045,4048],{},[1521,4046,4047],{},"Calcul monétaire exact",[1521,4049,4050,4051,1197],{},"Utiliser ",[805,4052,4053],{},"decimal.Decimal",[1503,4055,4056,4059],{},[1521,4057,4058],{},"Calcul rationnel exact",[1521,4060,4050,4061,1197],{},[805,4062,4063],{},"fractions.Fraction",[1503,4065,4066,4069],{},[1521,4067,4068],{},"Vérifier que c'est bien un nombre",[1521,4070,4071,1197],{},[805,4072,4073],{},"not math.isnan(x) and math.isfinite(x)",[593,4075,3321],{"id":3320},[1035,4077,4078,4085,4088],{},[1038,4079,4080,4082,4083,1197],{},[805,4081,2775],{}," renvoie ",[805,4084,2835],{},[1038,4086,4087],{},"Sommer beaucoup de petits termes perd de la précision (les petits sont écrasés par les grands).",[1038,4089,4090,4091,3347,4094,4096,4097,4134],{},"Convertir ",[805,4092,4093],{},"\"0.1\"",[805,4095,1643],{}," recrée la même approximation — pas de chemin pour avoir un ",[602,4098,4100,4113],{"className":4099},[605],[602,4101,4103],{"className":4102},[609],[611,4104,4105],{"xmlns":613},[615,4106,4107,4111],{},[618,4108,4109],{},[621,4110,1799],{},[625,4112,1802],{"encoding":627},[602,4114,4116],{"className":4115,"ariaHidden":633},[632],[602,4117,4119,4122,4125,4131],{"className":4118},[637],[602,4120],{"className":4121,"style":642},[641],[602,4123,931],{"className":4124},[646],[602,4126,4128],{"className":4127},[646],[602,4129,655],{"className":4130},[654],[602,4132,678],{"className":4133},[646]," exact.",[593,4136,4138],{"id":4137},"syntaxe-python","Syntaxe Python",[2475,4140,4142],{"className":2477,"code":4141,"language":2479,"meta":2480,"style":2480},"import math\nmath.isclose(0.1 + 0.2, 0.3)   # True\nmath.inf                       # infini positif\nmath.isnan(float('nan'))       # True\n\nfrom decimal import Decimal\nDecimal('0.1') + Decimal('0.2')  # Decimal('0.3') — exact\n",[805,4143,4144,4150,4170,4178,4194,4198,4209],{"__ignoreMap":2480},[602,4145,4146,4148],{"class":2485,"line":2486},[602,4147,2645],{"class":2499},[602,4149,2984],{"class":2493},[602,4151,4152,4155,4157,4159,4161,4163,4165,4168],{"class":2485,"line":2513},[602,4153,4154],{"class":2493},"math.isclose(",[602,4156,2496],{"class":2489},[602,4158,2500],{"class":2499},[602,4160,2503],{"class":2489},[602,4162,2957],{"class":2493},[602,4164,815],{"class":2489},[602,4166,4167],{"class":2493},")   ",[602,4169,2965],{"class":2509},[602,4171,4172,4175],{"class":2485,"line":2651},[602,4173,4174],{"class":2493},"math.inf                       ",[602,4176,4177],{"class":2509},"# infini positif\n",[602,4179,4180,4183,4185,4187,4189,4192],{"class":2485,"line":2664},[602,4181,4182],{"class":2493},"math.isnan(",[602,4184,1643],{"class":2489},[602,4186,1230],{"class":2493},[602,4188,3248],{"class":3247},[602,4190,4191],{"class":2493},"))       ",[602,4193,2965],{"class":2509},[602,4195,4196],{"class":2485,"line":2676},[602,4197,2941],{"emptyLinePlaceholder":2940},[602,4199,4201,4203,4205,4207],{"class":2485,"line":4200},6,[602,4202,2639],{"class":2499},[602,4204,2642],{"class":2493},[602,4206,2645],{"class":2499},[602,4208,2648],{"class":2493},[602,4210,4212,4215,4218,4220,4222,4224,4227,4229],{"class":2485,"line":4211},7,[602,4213,4214],{"class":2493},"Decimal(",[602,4216,4217],{"class":3247},"'0.1'",[602,4219,2685],{"class":2493},[602,4221,2688],{"class":2499},[602,4223,2691],{"class":2493},[602,4225,4226],{"class":3247},"'0.2'",[602,4228,2532],{"class":2493},[602,4230,4231],{"class":2509},"# Decimal('0.3') — exact\n",[3578,4233,4234],{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sBjJW, html code.shiki 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.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":2480,"searchDepth":2513,"depth":2513,"links":4236},[4237,4238,4239,4240,4241],{"id":3616,"depth":2513,"text":3617},{"id":3780,"depth":2513,"text":3781},{"id":4014,"depth":2513,"text":4015},{"id":3320,"depth":2513,"text":3321},{"id":4137,"depth":2513,"text":4138},"Mémo de révision sur la représentation IEEE 754 et les approximations.",{},"\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants.memo",[],{"title":3610,"description":4242},"1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.memo","uJ-K2xS15p4OyKBDiBFPTwe4ZXL00h0tJGWzTxjacTs",{"id":4250,"title":4251,"body":4252,"description":5224,"extension":3596,"meta":5225,"navigation":3603,"path":5226,"seo":5227,"stem":5228,"__hash__":5229},"coursSlides\u002F1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants.slides.md","Nombres flottants — diapositives",{"type":590,"value":4253,"toc":5222},[4254,4259,4262,4265,4269,4367,4384,4386,4390,4562,4657,4659,4663,4711,4717,4719,4761,4945,4952,4954,4958,5001,5003,5007,5012,5041,5043,5047,5177,5182,5184,5188,5205,5208,5210,5214,5219],[4255,4256,4258],"h1",{"id":4257},"nombres-flottants","Nombres flottants",[598,4260,4261],{},"Comment représenter un réel sur un nombre fini de bits ?",[4263,4264],"hr",{},[4255,4266,4268],{"id":4267},"la-notation-scientifique","La notation scientifique",[598,4270,4271],{},[602,4272,4274,4296],{"className":4273},[605],[602,4275,4277],{"className":4276},[609],[611,4278,4279],{"xmlns":613},[615,4280,4281,4293],{},[618,4282,4283,4285,4287],{},[621,4284,843],{},[845,4286,847],{},[849,4288,4289,4291],{},[621,4290,853],{},[621,4292,856],{},[625,4294,4295],{"encoding":627},"6{,}022 \\cdot 10^{23}",[602,4297,4299,4326],{"className":4298,"ariaHidden":633},[632],[602,4300,4302,4305,4308,4314,4317,4320,4323],{"className":4301},[637],[602,4303],{"className":4304,"style":642},[641],[602,4306,893],{"className":4307},[646],[602,4309,4311],{"className":4310},[646],[602,4312,655],{"className":4313},[654],[602,4315,903],{"className":4316},[646],[602,4318],{"className":4319,"style":907},[858],[602,4321,847],{"className":4322},[911],[602,4324],{"className":4325,"style":907},[858],[602,4327,4329,4332,4335],{"className":4328},[637],[602,4330],{"className":4331,"style":2574},[641],[602,4333,678],{"className":4334},[646],[602,4336,4338,4341],{"className":4337},[646],[602,4339,931],{"className":4340},[646],[602,4342,4344],{"className":4343},[935],[602,4345,4347],{"className":4346},[708],[602,4348,4350],{"className":4349},[713],[602,4351,4353],{"className":4352,"style":2574},[717],[602,4354,4355,4358],{"style":2598},[602,4356],{"className":4357,"style":952},[725],[602,4359,4361],{"className":4360},[730,731,732,733],[602,4362,4364],{"className":4363},[646,733],[602,4365,856],{"className":4366},[646,733],[1035,4368,4369,4374,4379],{},[1038,4370,4371,4373],{},[786,4372,1043],{}," : 6,022",[1038,4375,4376,4378],{},[786,4377,637],{}," : 10",[1038,4380,4381,4383],{},[786,4382,1132],{}," : 23",[4263,4385],{},[4255,4387,4389],{"id":4388},"adaptation-à-la-machine-base-2","Adaptation à la machine — base 2",[602,4391,4393],{"className":4392},[827],[602,4394,4396,4435],{"className":4395},[605],[602,4397,4399],{"className":4398},[609],[611,4400,4401],{"xmlns":613,"display":836},[615,4402,4403,4433],{},[618,4404,4405,4407,4409,4411,4413,4415,4421,4423,4425,4427],{},[1221,4406,1223],{},[845,4408,1226],{},[845,4410,1230],{"stretchy":1229},[845,4412,874],{},[621,4414,678],{},[849,4416,4417,4419],{},[845,4418,1239],{"stretchy":1229},[1221,4420,1242],{},[845,4422,847],{},[1221,4424,1247],{},[845,4426,847],{},[849,4428,4429,4431],{},[621,4430,1254],{},[1221,4432,1257],{},[625,4434,1260],{"encoding":627},[602,4436,4438,4456,4509,4527],{"className":4437,"ariaHidden":633},[632],[602,4439,4441,4444,4447,4450,4453],{"className":4440},[637],[602,4442],{"className":4443,"style":1270},[641],[602,4445,1223],{"className":4446},[646,1274],[602,4448],{"className":4449,"style":1278},[858],[602,4451,1226],{"className":4452},[1282],[602,4454],{"className":4455,"style":1278},[858],[602,4457,4459,4462,4465,4468,4471,4500,4503,4506],{"className":4458},[637],[602,4460],{"className":4461,"style":1292},[641],[602,4463,1230],{"className":4464},[700],[602,4466,874],{"className":4467},[646],[602,4469,678],{"className":4470},[646],[602,4472,4474,4477],{"className":4473},[783],[602,4475,1239],{"className":4476},[783],[602,4478,4480],{"className":4479},[935],[602,4481,4483],{"className":4482},[708],[602,4484,4486],{"className":4485},[713],[602,4487,4489],{"className":4488,"style":1320},[717],[602,4490,4491,4494],{"style":948},[602,4492],{"className":4493,"style":952},[725],[602,4495,4497],{"className":4496},[730,731,732,733],[602,4498,1242],{"className":4499},[646,1274,733],[602,4501],{"className":4502,"style":907},[858],[602,4504,847],{"className":4505},[911],[602,4507],{"className":4508,"style":907},[858],[602,4510,4512,4515,4518,4521,4524],{"className":4511},[637],[602,4513],{"className":4514,"style":1347},[641],[602,4516,1247],{"className":4517},[646,1274],[602,4519],{"className":4520,"style":907},[858],[602,4522,847],{"className":4523},[911],[602,4525],{"className":4526,"style":907},[858],[602,4528,4530,4533],{"className":4529},[637],[602,4531],{"className":4532,"style":1320},[641],[602,4534,4536,4539],{"className":4535},[646],[602,4537,1254],{"className":4538},[646],[602,4540,4542],{"className":4541},[935],[602,4543,4545],{"className":4544},[708],[602,4546,4548],{"className":4547},[713],[602,4549,4551],{"className":4550,"style":1320},[717],[602,4552,4553,4556],{"style":948},[602,4554],{"className":4555,"style":952},[725],[602,4557,4559],{"className":4558},[730,731,732,733],[602,4560,1257],{"className":4561},[646,1274,733],[1035,4563,4564,4595,4626],{},[1038,4565,4566,4594],{},[602,4567,4569,4582],{"className":4568},[605],[602,4570,4572],{"className":4571},[609],[611,4573,4574],{"xmlns":613},[615,4575,4576,4580],{},[618,4577,4578],{},[1221,4579,1242],{},[625,4581,1242],{"encoding":627},[602,4583,4585],{"className":4584,"ariaHidden":633},[632],[602,4586,4588,4591],{"className":4587},[637],[602,4589],{"className":4590,"style":1270},[641],[602,4592,1242],{"className":4593},[646,1274]," : bit de signe",[1038,4596,4597,4625],{},[602,4598,4600,4613],{"className":4599},[605],[602,4601,4603],{"className":4602},[609],[611,4604,4605],{"xmlns":613},[615,4606,4607,4611],{},[618,4608,4609],{},[1221,4610,1247],{},[625,4612,1247],{"encoding":627},[602,4614,4616],{"className":4615,"ariaHidden":633},[632],[602,4617,4619,4622],{"className":4618},[637],[602,4620],{"className":4621,"style":1270},[641],[602,4623,1247],{"className":4624},[646,1274]," : mantisse en binaire",[1038,4627,4628,4656],{},[602,4629,4631,4644],{"className":4630},[605],[602,4632,4634],{"className":4633},[609],[611,4635,4636],{"xmlns":613},[615,4637,4638,4642],{},[618,4639,4640],{},[1221,4641,1257],{},[625,4643,1257],{"encoding":627},[602,4645,4647],{"className":4646,"ariaHidden":633},[632],[602,4648,4650,4653],{"className":4649},[637],[602,4651],{"className":4652,"style":1270},[641],[602,4654,1257],{"className":4655},[646,1274]," : exposant signé",[4263,4658],{},[4255,4660,4662],{"id":4661},"ieee-754-tailles-courantes","IEEE 754 — tailles courantes",[1497,4664,4665,4683],{},[1500,4666,4667],{},[1503,4668,4669,4672,4674,4677,4680],{},[1506,4670,4671],{},"Précision",[1506,4673,1511],{},[1506,4675,4676],{},"Signe",[1506,4678,4679],{},"Exposant",[1506,4681,4682],{},"Mantisse",[1516,4684,4685,4698],{},[1503,4686,4687,4690,4692,4694,4696],{},[1521,4688,4689],{},"Simple",[1521,4691,3311],{},[1521,4693,678],{},[1521,4695,1592],{},[1521,4697,856],{},[1503,4699,4700,4703,4705,4707,4709],{},[1521,4701,4702],{},"Double",[1521,4704,3314],{},[1521,4706,678],{},[1521,4708,1729],{},[1521,4710,1767],{},[598,4712,4713,4714,4716],{},"Python utilise la ",[786,4715,1639],{}," par défaut.",[4263,4718],{},[4255,4720,4722,4723,4760],{"id":4721},"le-piège-010101-en-binaire","Le piège — ",[602,4724,4726,4739],{"className":4725},[605],[602,4727,4729],{"className":4728},[609],[611,4730,4731],{"xmlns":613},[615,4732,4733,4737],{},[618,4734,4735],{},[621,4736,1799],{},[625,4738,1802],{"encoding":627},[602,4740,4742],{"className":4741,"ariaHidden":633},[632],[602,4743,4745,4748,4751,4757],{"className":4744},[637],[602,4746],{"className":4747,"style":642},[641],[602,4749,931],{"className":4750},[646],[602,4752,4754],{"className":4753},[646],[602,4755,655],{"className":4756},[654],[602,4758,678],{"className":4759},[646]," en binaire",[602,4762,4764],{"className":4763},[827],[602,4765,4767,4798],{"className":4766},[605],[602,4768,4770],{"className":4769},[609],[611,4771,4772],{"xmlns":613,"display":836},[615,4773,4774,4796],{},[618,4775,4776,4782,4784,4786,4788,4790],{},[674,4777,4778,4780],{},[621,4779,678],{},[621,4781,853],{},[845,4783,1226],{},[845,4785,1230],{"stretchy":1229},[621,4787,2063],{},[845,4789,2066],{},[2068,4791,4792,4794],{},[845,4793,1239],{"stretchy":1229},[621,4795,1254],{},[625,4797,2076],{"encoding":627},[602,4799,4801,4878],{"className":4800,"ariaHidden":633},[632],[602,4802,4804,4807,4869,4872,4875],{"className":4803},[637],[602,4805],{"className":4806,"style":2086},[641],[602,4808,4810,4813,4866],{"className":4809},[646],[602,4811],{"className":4812},[700,701],[602,4814,4816],{"className":4815},[674],[602,4817,4819,4858],{"className":4818},[708,709],[602,4820,4822,4855],{"className":4821},[713],[602,4823,4825,4836,4844],{"className":4824,"style":2105},[717],[602,4826,4827,4830],{"style":2108},[602,4828],{"className":4829,"style":726},[725],[602,4831,4833],{"className":4832},[646],[602,4834,853],{"className":4835},[646],[602,4837,4838,4841],{"style":742},[602,4839],{"className":4840,"style":726},[725],[602,4842],{"className":4843,"style":750},[749],[602,4845,4846,4849],{"style":2128},[602,4847],{"className":4848,"style":726},[725],[602,4850,4852],{"className":4851},[646],[602,4853,678],{"className":4854},[646],[602,4856,770],{"className":4857},[769],[602,4859,4861],{"className":4860},[713],[602,4862,4864],{"className":4863,"style":2147},[717],[602,4865],{},[602,4867],{"className":4868},[783,701],[602,4870],{"className":4871,"style":1278},[858],[602,4873,1226],{"className":4874},[1282],[602,4876],{"className":4877,"style":1278},[858],[602,4879,4881,4884,4887,4890,4896,4899,4902,4905],{"className":4880},[637],[602,4882],{"className":4883,"style":1292},[641],[602,4885,1230],{"className":4886},[700],[602,4888,931],{"className":4889},[646],[602,4891,4893],{"className":4892},[646],[602,4894,655],{"className":4895},[654],[602,4897,2183],{"className":4898},[646],[602,4900],{"className":4901,"style":2187},[858],[602,4903,2066],{"className":4904},[2191],[602,4906,4908,4911],{"className":4907},[783],[602,4909,1239],{"className":4910},[783],[602,4912,4914],{"className":4913},[935],[602,4915,4917,4937],{"className":4916},[708,709],[602,4918,4920,4934],{"className":4919},[713],[602,4921,4923],{"className":4922,"style":2210},[717],[602,4924,4925,4928],{"style":2213},[602,4926],{"className":4927,"style":952},[725],[602,4929,4931],{"className":4930},[730,731,732,733],[602,4932,1254],{"className":4933},[646,733],[602,4935,770],{"className":4936},[769],[602,4938,4940],{"className":4939},[713],[602,4941,4943],{"className":4942,"style":2232},[717],[602,4944],{},[598,4946,4947,4948,4951],{},"Développement ",[786,4949,4950],{},"infini"," → la mantisse tronque → approximation.",[4263,4953],{},[4255,4955,4957],{"id":4956},"python-le-révèle","Python le révèle",[2475,4959,4961],{"className":2477,"code":4960,"language":2479,"meta":2480,"style":2480},">>> 0.1 + 0.2\n0.30000000000000004\n>>> 0.1 + 0.2 == 0.3\nFalse\n",[805,4962,4963,4976,4981,4996],{"__ignoreMap":2480},[602,4964,4965,4968,4971,4973],{"class":2485,"line":2486},[602,4966,4967],{"class":2499},">>>",[602,4969,4970],{"class":2489}," 0.1",[602,4972,2500],{"class":2499},[602,4974,4975],{"class":2489}," 0.2\n",[602,4977,4978],{"class":2485,"line":2513},[602,4979,4980],{"class":2489},"0.30000000000000004\n",[602,4982,4983,4985,4987,4989,4991,4993],{"class":2485,"line":2651},[602,4984,4967],{"class":2499},[602,4986,4970],{"class":2489},[602,4988,2500],{"class":2499},[602,4990,2503],{"class":2489},[602,4992,2526],{"class":2499},[602,4994,4995],{"class":2489}," 0.3\n",[602,4997,4998],{"class":2485,"line":2664},[602,4999,5000],{"class":2489},"False\n",[4263,5002],{},[4255,5004,5006],{"id":5005},"la-règle-dor","La règle d'or",[598,5008,5009],{},[786,5010,5011],{},"Ne jamais tester l'égalité stricte de deux flottants.",[2475,5013,5015],{"className":2477,"code":5014,"language":2479,"meta":2480,"style":2480},"import math\nmath.isclose(0.1 + 0.2, 0.3)   # True\n",[805,5016,5017,5023],{"__ignoreMap":2480},[602,5018,5019,5021],{"class":2485,"line":2486},[602,5020,2645],{"class":2499},[602,5022,2984],{"class":2493},[602,5024,5025,5027,5029,5031,5033,5035,5037,5039],{"class":2485,"line":2513},[602,5026,4154],{"class":2493},[602,5028,2496],{"class":2489},[602,5030,2500],{"class":2499},[602,5032,2503],{"class":2489},[602,5034,2957],{"class":2493},[602,5036,815],{"class":2489},[602,5038,4167],{"class":2493},[602,5040,2965],{"class":2509},[4263,5042],{},[4255,5044,5046],{"id":5045},"valeurs-spéciales","Valeurs spéciales",[1035,5048,5049,5054,5059],{},[1038,5050,5051,5053],{},[805,5052,3026],{}," : infini positif.",[1038,5055,5056,5058],{},[805,5057,3029],{}," : infini négatif.",[1038,5060,5061,5063,5064,2957,5096,3293],{},[805,5062,3032],{}," : résultat invalide (",[602,5065,5067,5084],{"className":5066},[605],[602,5068,5070],{"className":5069},[609],[611,5071,5072],{"xmlns":613},[615,5073,5074,5082],{},[618,5075,5076,5078,5080],{},[621,5077,931],{},[1221,5079,3075],{"mathvariant":3074},[621,5081,931],{},[625,5083,3080],{"encoding":627},[602,5085,5087],{"className":5086,"ariaHidden":633},[632],[602,5088,5090,5093],{"className":5089},[637],[602,5091],{"className":5092,"style":1292},[641],[602,5094,3080],{"className":5095},[646],[602,5097,5099,5118],{"className":5098},[605],[602,5100,5102],{"className":5101},[609],[611,5103,5104],{"xmlns":613},[615,5105,5106,5116],{},[618,5107,5108],{},[3106,5109,5110],{},[618,5111,5112,5114],{},[845,5113,874],{},[621,5115,678],{},[625,5117,3116],{"encoding":627},[602,5119,5121],{"className":5120,"ariaHidden":633},[632],[602,5122,5124,5127],{"className":5123},[637],[602,5125],{"className":5126,"style":3126},[641],[602,5128,5130],{"className":5129},[646,3130],[602,5131,5133,5169],{"className":5132},[708,709],[602,5134,5136,5166],{"className":5135},[713],[602,5137,5139,5154],{"className":5138,"style":3140},[717],[602,5140,5142,5145],{"className":5141,"style":3145},[3144],[602,5143],{"className":5144,"style":726},[725],[602,5146,5148,5151],{"className":5147,"style":3152},[646],[602,5149,874],{"className":5150},[646],[602,5152,678],{"className":5153},[646],[602,5155,5156,5159],{"style":3161},[602,5157],{"className":5158,"style":726},[725],[602,5160,5162],{"className":5161,"style":3169},[3168],[3171,5163,5164],{"xmlns":3173,"width":3174,"height":3175,"viewBox":3176,"preserveAspectRatio":3177},[3179,5165],{"d":3181},[602,5167,770],{"className":5168},[769],[602,5170,5172],{"className":5171},[713],[602,5173,5175],{"className":5174,"style":3191},[717],[602,5176],{},[598,5178,5179,5181],{},[805,5180,4011],{}," — surprenant mais voulu par la norme.",[4263,5183],{},[4255,5185,5187],{"id":5186},"quand-lexactitude-est-requise","Quand l'exactitude est requise",[1035,5189,5190,5198],{},[1038,5191,5192,5195,5196,1197],{},[786,5193,5194],{},"Argent"," → ",[805,5197,4053],{},[1038,5199,5200,5195,5203,1197],{},[786,5201,5202],{},"Fractions exactes",[805,5204,4063],{},[598,5206,5207],{},"Plus lents, mais exacts.",[4263,5209],{},[4255,5211,5213],{"id":5212},"à-retenir","À retenir",[598,5215,5216],{},[786,5217,5218],{},"Un flottant est une approximation. Le programmeur doit en être conscient à\nchaque comparaison et à chaque cumul.",[3578,5220,5221],{},"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 .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: 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