Pourquoi 1+1=2 n’est pas si \u00e9vident \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --ocre: #c8850a;\n --ocre-clair: #f0c060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n \/* HERO *\/\n .hero {\n background: var(--noir);\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u2211 \u03c0 \u221e \u221a\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.04;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner {\n max-width: 760px;\n margin: 0 auto;\n }\n .article-rubrique {\n display: inline-block;\n background: var(--ocre);\n color: white;\n font-size: 0.7rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n padding: 4px 12px;\n margin-bottom: 24px;\n font-family: 'Source Serif 4', serif;\n font-weight: 600;\n }\n .hero h1 {\n font-family: 'Playfair Display', serif;\n font-size: clamp(2rem, 5vw, 3.4rem);\n line-height: 1.2;\n margin-bottom: 24px;\n font-weight: 700;\n }\n .hero h1 em {\n color: var(--ocre-clair);\n font-style: italic;\n }\n .hero-intro {\n font-size: 1.15rem;\n opacity: 0.75;\n max-width: 600px;\n font-style: italic;\n border-left: 3px solid var(--ocre);\n padding-left: 20px;\n margin-bottom: 32px;\n }\n .meta {\n font-size: 0.8rem;\n opacity: 0.45;\n letter-spacing: 0.08em;\n text-transform: uppercase;\n }\n\n \/* CONTENU *\/\n .article-body {\n max-width: 760px;\n margin: 0 auto;\n padding: 60px 40px 80px;\n }\n\n .article-body p {\n font-size: 1.05rem;\n margin-bottom: 1.6em;\n color: #2a2520;\n }\n\n h2 {\n font-family: 'Playfair Display', serif;\n font-size: 1.6rem;\n margin: 2.5em 0 0.8em;\n color: var(--noir);\n font-weight: 700;\n }\n h2::before {\n content: \"\u00a7 \";\n color: var(--ocre);\n }\n\n h3 {\n font-family: 'Playfair Display', serif;\n font-size: 1.15rem;\n font-style: italic;\n margin: 2em 0 0.6em;\n color: var(--rouge);\n }\n\n \/* ENCADR\u00c9S *\/\n .encadre {\n border: 1.5px solid var(--ocre);\n background: white;\n padding: 24px 28px;\n margin: 2em 0;\n position: relative;\n }\n .encadre-titre {\n position: absolute;\n top: -12px; left: 20px;\n background: white;\n padding: 0 8px;\n font-size: 0.72rem;\n letter-spacing: 0.15em;\n text-transform: uppercase;\n color: var(--ocre);\n font-family: 'Source Serif 4', serif;\n font-weight: 600;\n }\n .encadre p { margin-bottom: 0.8em; font-size: 0.98rem; }\n .encadre p:last-child { margin-bottom: 0; }\n\n .encadre-histoire {\n background: #1a150f;\n color: var(--creme);\n border: none;\n padding: 32px 36px;\n margin: 2.5em 0;\n }\n .encadre-histoire .encadre-titre {\n background: #1a150f;\n color: var(--ocre-clair);\n }\n .encadre-histoire p { color: rgba(245,240,232,0.85); }\n\n .definition {\n background: var(--gris-clair);\n border-left: 4px solid var(--rouge);\n padding: 20px 24px;\n margin: 1.8em 0;\n font-style: italic;\n }\n .definition strong {\n font-style: normal;\n color: var(--rouge);\n font-family: 'Playfair Display', serif;\n }\n\n \/* MATHS *\/\n .formule {\n text-align: center;\n font-size: 1.5rem;\n font-family: 'Playfair Display', serif;\n margin: 1.5em 0;\n color: var(--noir);\n padding: 16px;\n background: white;\n border-top: 2px solid var(--ocre);\n border-bottom: 2px solid var(--ocre);\n }\n\n .etape {\n background: white;\n border-left: 3px solid var(--ocre);\n padding: 12px 20px;\n margin: 0.6em 0;\n font-size: 0.97rem;\n }\n .etape-numero {\n font-family: 'Playfair Display', serif;\n font-weight: 700;\n color: var(--ocre);\n margin-right: 8px;\n }\n\n \/* EXERCICE *\/\n .exercice {\n background: #0f0e0b;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 {\n color: var(--ocre-clair);\n font-style: normal;\n margin-top: 0;\n margin-bottom: 1em;\n font-size: 1.1rem;\n letter-spacing: 0.05em;\n }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction {\n border-top: 1px solid rgba(200,133,10,0.4);\n margin-top: 1.5em;\n padding-top: 1.2em;\n }\n .exercice .correction-titre {\n font-size: 0.7rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n color: var(--ocre-clair);\n margin-bottom: 0.8em;\n font-weight: 600;\n }\n\n \/* CITATION *\/\n blockquote {\n border-left: none;\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--ocre);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite {\n display: block;\n font-size: 0.82rem;\n font-style: normal;\n letter-spacing: 0.1em;\n text-transform: uppercase;\n color: var(--gris);\n margin-top: 8px;\n }\n\n \/* CONCLUSION *\/\n .conclusion {\n border-top: 2px solid var(--noir);\n margin-top: 3em;\n padding-top: 2em;\n }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n }\n<\/style>\n<\/head>\n<body>\n\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Fondements \u00b7 Article #1<\/div>\n <h1>Pourquoi <em>1+1=2<\/em><br>n’est pas si \u00e9vident<\/h1>\n <p class=\"hero-intro\">Ce que tout le monde croit savoir depuis l’enfance cache l’une des aventures intellectuelles les plus profondes de l’histoire des math\u00e9matiques.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 8 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Il y a quelques ann\u00e9es, un \u00e9l\u00e8ve de 6\u00e8me m’a pos\u00e9 une question qui m’a arr\u00eat\u00e9 net au tableau noir :\n <em>\u00ab\u00a0Monsieur, pourquoi 1+1 fait 2 et pas autre chose ?\u00a0\u00bb<\/em>\n J’ai souri. Et puis j’ai r\u00e9alis\u00e9 que je n’avais pas de r\u00e9ponse simple. Parce qu’il n’y en a pas.\n <\/p>\n\n <p>\n Nous grandissons tous avec cette certitude absolue : 1+1=2. C’est la premi\u00e8re chose qu’on apprend en math\u00e9matiques. C’est tellement \u00e9vident qu’on ne se pose jamais la question. Et pourtant, deux des plus grands math\u00e9maticiens du XX<sup>e<\/sup> si\u00e8cle \u2014 Bertrand Russell et Alfred North Whitehead \u2014 ont consacr\u00e9 des centaines de pages d’un ouvrage monumental rien que pour <em>d\u00e9montrer<\/em> que 1+1=2.\n <\/p>\n\n <p>\n Alors, qu’est-ce qui se cache vraiment derri\u00e8re cette petite \u00e9quation ?\n <\/p>\n\n <h2>L’histoire d’une question ignor\u00e9e<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Anecdote historique<\/span>\n <p>\n Dans les march\u00e9s de nos villes africaines, quand une vendeuse compte ses billets, elle ne se demande pas pourquoi 1+1=2. Elle le <em>sait<\/em>. Elle le <em>vit<\/em>. Mais ce savoir pratique et ce savoir math\u00e9matique rigoureux sont deux choses tr\u00e8s diff\u00e9rentes.\n <\/p>\n <p>\n Pendant des mill\u00e9naires, les math\u00e9maticiens ont utilis\u00e9 les nombres sans vraiment se demander <em>ce qu’ils \u00e9taient<\/em>. Ce n’est qu’au XIX<sup>e<\/sup> si\u00e8cle que des hommes comme Giuseppe Peano ont d\u00e9cid\u00e9 de partir de z\u00e9ro \u2014 absolument de z\u00e9ro \u2014 pour construire les math\u00e9matiques sur des bases solides.\n <\/p>\n <\/div>\n\n <p>\n Le probl\u00e8me est le suivant : si tu veux <em>prouver<\/em> que 1+1=2, tu dois d’abord d\u00e9finir ce qu’est le chiffre 1. Et le chiffre 2. Et l’op\u00e9ration \u00ab\u00a0+\u00a0\u00bb. Et l’\u00e9galit\u00e9 \u00ab\u00a0=\u00a0\u00bb. La question semble sans fin.\n <\/p>\n\n <p>\n C’est le probl\u00e8me des <strong>fondements des math\u00e9matiques<\/strong>. Et il a failli faire exploser toute la discipline.\n <\/p>\n\n <h2>Peano : construire les nombres depuis le vide<\/h2>\n\n <p>\n En 1889, le math\u00e9maticien italien Giuseppe Peano a propos\u00e9 une solution \u00e9l\u00e9gante : d\u00e9finir tous les nombres naturels (0, 1, 2, 3, …) \u00e0 partir de presque rien, avec seulement quelques r\u00e8gles appel\u00e9es <strong>axiomes<\/strong>.\n <\/p>\n\n <div class=\"definition\">\n <strong>Axiome :<\/strong> Une v\u00e9rit\u00e9 que l’on accepte sans d\u00e9monstration, comme point de d\u00e9part absolu. C’est la fondation sur laquelle tout le reste est construit.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcd0 Les axiomes de Peano (version accessible)<\/span>\n <p><strong>Axiome 1 :<\/strong> 0 est un nombre naturel.<\/p>\n <p><strong>Axiome 2 :<\/strong> Tout nombre naturel <em>n<\/em> a un successeur, not\u00e9 <em>S(n)<\/em>.<\/p>\n <p><strong>Axiome 3 :<\/strong> 0 n’est le successeur d’aucun nombre.<\/p>\n <p><strong>Axiome 4 :<\/strong> Si deux nombres ont le m\u00eame successeur, ils sont \u00e9gaux.<\/p>\n <p><strong>Axiome 5 :<\/strong> Si une propri\u00e9t\u00e9 est vraie pour 0, et vraie pour <em>n+1<\/em> quand elle est vraie pour <em>n<\/em>, alors elle est vraie pour tous les nombres. (Principe de r\u00e9currence.)<\/p>\n <\/div>\n\n <p>\n Avec ces cinq r\u00e8gles, on peut <em>d\u00e9finir<\/em> tous les nombres naturels. Le nombre 1 ? C’est S(0), le successeur de z\u00e9ro. Le nombre 2 ? C’est S(S(0)), le successeur du successeur de z\u00e9ro.\n <\/p>\n\n <h2>Alors, 1+1=2 : la d\u00e9monstration<\/h2>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udccc D\u00e9finition de l’addition<\/span>\n <p><strong>R\u00e8gle 1 :<\/strong> n + 0 = n (ajouter z\u00e9ro ne change rien)<\/p>\n <p><strong>R\u00e8gle 2 :<\/strong> n + S(m) = S(n + m) (ajouter un successeur, c’est prendre le successeur du r\u00e9sultat)<\/p>\n <\/div>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> Par d\u00e9finition : 1 = S(0) et 2 = S(S(0)) = S(1)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> Calculons 1 + 1 = 1 + S(0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> Par la R\u00e8gle 2 : 1 + S(0) = S(1 + 0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> Par la R\u00e8gle 1 : 1 + 0 = 1<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2464<\/span> Donc : S(1 + 0) = S(1) = 2<\/div>\n\n <div class=\"formule\">\u2234 1 + 1 = 2 \u2713<\/div>\n\n <h2>Mais alors, 1+1 peut-il faire autre chose que 2 ?<\/h2>\n\n <p>Oui ! Dans certains syst\u00e8mes math\u00e9matiques appel\u00e9s <strong>arithm\u00e9tiques modulaires<\/strong>, les r\u00e8gles changent. Par exemple, en arithm\u00e9tique modulo 2 (celle des ordinateurs) :<\/p>\n\n <div class=\"formule\">1 + 1 = 0 (modulo 2)<\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Exemple du quotidien africain<\/span>\n <p>\n Pense aux jours de la semaine. Si aujourd’hui nous sommes <strong>vendredi (jour 6)<\/strong> et que tu ajoutes <strong>2 jours<\/strong>, tu arrives \u00e0 <strong>dimanche (jour 1)<\/strong>, pas au jour 8 ! C’est de l’arithm\u00e9tique modulaire \u2014 celle du calendrier, des heures, et de tout ce qui tourne en cycle.\n <\/p>\n <\/div>\n\n <h2>Ce que tout cela nous enseigne<\/h2>\n\n <blockquote>\n Les math\u00e9matiques ne sont pas une collection de v\u00e9rit\u00e9s tomb\u00e9es du ciel. Ce sont des constructions humaines, b\u00e2ties avec soin, brique par brique, sur des fondations choisies.\n <cite>\u2014 Le\u00e7on \u00e0 retenir<\/cite>\n <\/blockquote>\n\n <p>\n Les maths ne sont pas une liste de r\u00e8gles \u00e0 apprendre par c\u0153ur. Ce sont des raisonnements vivants, \u00e0 comprendre, \u00e0 questionner, \u00e0 habiter. Comme une belle maison : on peut y vivre sans conna\u00eetre chaque d\u00e9tail de sa construction \u2014 mais quand on les conna\u00eet, on l’aime encore plus.\n <\/p>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>En arithm\u00e9tique modulo 7 (comme les jours de la semaine num\u00e9rot\u00e9s de 0 \u00e0 6), calcule :<\/p>\n <p><strong>a)<\/strong> 5 + 3 = ?<\/p>\n <p><strong>b)<\/strong> 6 + 6 = ?<\/p>\n <p><strong>c)<\/strong> Si aujourd’hui est mercredi (jour 3), quel jour sera-t-on dans 10 jours ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> 5 + 3 = 8. Or 8 = 7 + 1, donc le reste est <strong>1<\/strong> (lundi).<\/p>\n <p><strong>b)<\/strong> 6 + 6 = 12. Or 12 = 7 + 5, donc le reste est <strong>5<\/strong> (samedi).<\/p>\n <p><strong>c)<\/strong> 3 + 10 = 13. Or 13 = 7 + 6, donc le reste est <strong>6<\/strong> \u2192 <strong>Samedi<\/strong>.<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>Pour aller plus loin<\/h2>\n <p>\n Si cette question t’a intrigu\u00e9, cherche sur YouTube <em>\u00ab\u00a0axiomes de Peano\u00a0\u00bb<\/em> ou explore le site <strong>Images des Maths<\/strong> du CNRS, enti\u00e8rement gratuit.\n <\/p>\n <p>\n Dans le prochain article, nous voyagerons en \u00c9gypte ancienne pour d\u00e9couvrir comment Thal\u00e8s a mesur\u00e9 la hauteur des pyramides sans les escalader \u2014 une histoire de triangles, d’ombres et de g\u00e9nie.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi 1+1=2 n’est pas si \u00e9vident \u2014 MathsVivantes Fondements \u00b7 Article #1 Pourquoi 1+1=2n’est pas si \u00e9vident Ce que tout […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-45","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Pourquoi 1+1=2 n'est pas toujours si \u00e9vident - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"pourquoi 1+1=2 n'est pas si \u00e9vident ? 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{"id":45,"date":"2026-04-13T13:09:47","date_gmt":"2026-04-13T12:09:47","guid":{"rendered":"https:\/\/vmi3217629.contaboserver.net\/?p=45"},"modified":"2026-04-19T22:29:00","modified_gmt":"2026-04-19T21:29:00","slug":"pourquoi-1-1-2-nest-pas-si-evident","status":"publish","type":"post","link":"https:\/\/vmi3217629.contaboserver.net\/?p=45","title":{"rendered":"Pourquoi 1+1=2 n’est pas toujours si \u00e9vident"},"content":{"rendered":"\n\n\n\n\n\nPourquoi 1+1=2 n’est pas si \u00e9vident \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --ocre: #c8850a;\n --ocre-clair: #f0c060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n \/* HERO *\/\n .hero {\n background: var(--noir);\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u2211 \u03c0 \u221e \u221a\";\n position: absolute;\n top: 20px; 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}\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n }\n<\/style>\n<\/head>\n<body>\n\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Fondements \u00b7 Article #1<\/div>\n <h1>Pourquoi <em>1+1=2<\/em><br>n’est pas si \u00e9vident<\/h1>\n <p class=\"hero-intro\">Ce que tout le monde croit savoir depuis l’enfance cache l’une des aventures intellectuelles les plus profondes de l’histoire des math\u00e9matiques.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 8 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Il y a quelques ann\u00e9es, un \u00e9l\u00e8ve de 6\u00e8me m’a pos\u00e9 une question qui m’a arr\u00eat\u00e9 net au tableau noir :\n <em>\u00ab\u00a0Monsieur, pourquoi 1+1 fait 2 et pas autre chose ?\u00a0\u00bb<\/em>\n J’ai souri. Et puis j’ai r\u00e9alis\u00e9 que je n’avais pas de r\u00e9ponse simple. Parce qu’il n’y en a pas.\n <\/p>\n\n <p>\n Nous grandissons tous avec cette certitude absolue : 1+1=2. C’est la premi\u00e8re chose qu’on apprend en math\u00e9matiques. C’est tellement \u00e9vident qu’on ne se pose jamais la question. Et pourtant, deux des plus grands math\u00e9maticiens du XX<sup>e<\/sup> si\u00e8cle \u2014 Bertrand Russell et Alfred North Whitehead \u2014 ont consacr\u00e9 des centaines de pages d’un ouvrage monumental rien que pour <em>d\u00e9montrer<\/em> que 1+1=2.\n <\/p>\n\n <p>\n Alors, qu’est-ce qui se cache vraiment derri\u00e8re cette petite \u00e9quation ?\n <\/p>\n\n <h2>L’histoire d’une question ignor\u00e9e<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Anecdote historique<\/span>\n <p>\n Dans les march\u00e9s de nos villes africaines, quand une vendeuse compte ses billets, elle ne se demande pas pourquoi 1+1=2. Elle le <em>sait<\/em>. Elle le <em>vit<\/em>. Mais ce savoir pratique et ce savoir math\u00e9matique rigoureux sont deux choses tr\u00e8s diff\u00e9rentes.\n <\/p>\n <p>\n Pendant des mill\u00e9naires, les math\u00e9maticiens ont utilis\u00e9 les nombres sans vraiment se demander <em>ce qu’ils \u00e9taient<\/em>. Ce n’est qu’au XIX<sup>e<\/sup> si\u00e8cle que des hommes comme Giuseppe Peano ont d\u00e9cid\u00e9 de partir de z\u00e9ro \u2014 absolument de z\u00e9ro \u2014 pour construire les math\u00e9matiques sur des bases solides.\n <\/p>\n <\/div>\n\n <p>\n Le probl\u00e8me est le suivant : si tu veux <em>prouver<\/em> que 1+1=2, tu dois d’abord d\u00e9finir ce qu’est le chiffre 1. Et le chiffre 2. Et l’op\u00e9ration \u00ab\u00a0+\u00a0\u00bb. Et l’\u00e9galit\u00e9 \u00ab\u00a0=\u00a0\u00bb. La question semble sans fin.\n <\/p>\n\n <p>\n C’est le probl\u00e8me des <strong>fondements des math\u00e9matiques<\/strong>. Et il a failli faire exploser toute la discipline.\n <\/p>\n\n <h2>Peano : construire les nombres depuis le vide<\/h2>\n\n <p>\n En 1889, le math\u00e9maticien italien Giuseppe Peano a propos\u00e9 une solution \u00e9l\u00e9gante : d\u00e9finir tous les nombres naturels (0, 1, 2, 3, …) \u00e0 partir de presque rien, avec seulement quelques r\u00e8gles appel\u00e9es <strong>axiomes<\/strong>.\n <\/p>\n\n <div class=\"definition\">\n <strong>Axiome :<\/strong> Une v\u00e9rit\u00e9 que l’on accepte sans d\u00e9monstration, comme point de d\u00e9part absolu. C’est la fondation sur laquelle tout le reste est construit.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcd0 Les axiomes de Peano (version accessible)<\/span>\n <p><strong>Axiome 1 :<\/strong> 0 est un nombre naturel.<\/p>\n <p><strong>Axiome 2 :<\/strong> Tout nombre naturel <em>n<\/em> a un successeur, not\u00e9 <em>S(n)<\/em>.<\/p>\n <p><strong>Axiome 3 :<\/strong> 0 n’est le successeur d’aucun nombre.<\/p>\n <p><strong>Axiome 4 :<\/strong> Si deux nombres ont le m\u00eame successeur, ils sont \u00e9gaux.<\/p>\n <p><strong>Axiome 5 :<\/strong> Si une propri\u00e9t\u00e9 est vraie pour 0, et vraie pour <em>n+1<\/em> quand elle est vraie pour <em>n<\/em>, alors elle est vraie pour tous les nombres. (Principe de r\u00e9currence.)<\/p>\n <\/div>\n\n <p>\n Avec ces cinq r\u00e8gles, on peut <em>d\u00e9finir<\/em> tous les nombres naturels. Le nombre 1 ? C’est S(0), le successeur de z\u00e9ro. Le nombre 2 ? C’est S(S(0)), le successeur du successeur de z\u00e9ro.\n <\/p>\n\n <h2>Alors, 1+1=2 : la d\u00e9monstration<\/h2>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udccc D\u00e9finition de l’addition<\/span>\n <p><strong>R\u00e8gle 1 :<\/strong> n + 0 = n (ajouter z\u00e9ro ne change rien)<\/p>\n <p><strong>R\u00e8gle 2 :<\/strong> n + S(m) = S(n + m) (ajouter un successeur, c’est prendre le successeur du r\u00e9sultat)<\/p>\n <\/div>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> Par d\u00e9finition : 1 = S(0) et 2 = S(S(0)) = S(1)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> Calculons 1 + 1 = 1 + S(0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> Par la R\u00e8gle 2 : 1 + S(0) = S(1 + 0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> Par la R\u00e8gle 1 : 1 + 0 = 1<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2464<\/span> Donc : S(1 + 0) = S(1) = 2<\/div>\n\n <div class=\"formule\">\u2234 1 + 1 = 2 \u2713<\/div>\n\n <h2>Mais alors, 1+1 peut-il faire autre chose que 2 ?<\/h2>\n\n <p>Oui ! Dans certains syst\u00e8mes math\u00e9matiques appel\u00e9s <strong>arithm\u00e9tiques modulaires<\/strong>, les r\u00e8gles changent. Par exemple, en arithm\u00e9tique modulo 2 (celle des ordinateurs) :<\/p>\n\n <div class=\"formule\">1 + 1 = 0 (modulo 2)<\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Exemple du quotidien africain<\/span>\n <p>\n Pense aux jours de la semaine. Si aujourd’hui nous sommes <strong>vendredi (jour 6)<\/strong> et que tu ajoutes <strong>2 jours<\/strong>, tu arrives \u00e0 <strong>dimanche (jour 1)<\/strong>, pas au jour 8 ! C’est de l’arithm\u00e9tique modulaire \u2014 celle du calendrier, des heures, et de tout ce qui tourne en cycle.\n <\/p>\n <\/div>\n\n <h2>Ce que tout cela nous enseigne<\/h2>\n\n <blockquote>\n Les math\u00e9matiques ne sont pas une collection de v\u00e9rit\u00e9s tomb\u00e9es du ciel. Ce sont des constructions humaines, b\u00e2ties avec soin, brique par brique, sur des fondations choisies.\n <cite>\u2014 Le\u00e7on \u00e0 retenir<\/cite>\n <\/blockquote>\n\n <p>\n Les maths ne sont pas une liste de r\u00e8gles \u00e0 apprendre par c\u0153ur. Ce sont des raisonnements vivants, \u00e0 comprendre, \u00e0 questionner, \u00e0 habiter. Comme une belle maison : on peut y vivre sans conna\u00eetre chaque d\u00e9tail de sa construction \u2014 mais quand on les conna\u00eet, on l’aime encore plus.\n <\/p>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>En arithm\u00e9tique modulo 7 (comme les jours de la semaine num\u00e9rot\u00e9s de 0 \u00e0 6), calcule :<\/p>\n <p><strong>a)<\/strong> 5 + 3 = ?<\/p>\n <p><strong>b)<\/strong> 6 + 6 = ?<\/p>\n <p><strong>c)<\/strong> Si aujourd’hui est mercredi (jour 3), quel jour sera-t-on dans 10 jours ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> 5 + 3 = 8. Or 8 = 7 + 1, donc le reste est <strong>1<\/strong> (lundi).<\/p>\n <p><strong>b)<\/strong> 6 + 6 = 12. Or 12 = 7 + 5, donc le reste est <strong>5<\/strong> (samedi).<\/p>\n <p><strong>c)<\/strong> 3 + 10 = 13. Or 13 = 7 + 6, donc le reste est <strong>6<\/strong> \u2192 <strong>Samedi<\/strong>.<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>Pour aller plus loin<\/h2>\n <p>\n Si cette question t’a intrigu\u00e9, cherche sur YouTube <em>\u00ab\u00a0axiomes de Peano\u00a0\u00bb<\/em> ou explore le site <strong>Images des Maths<\/strong> du CNRS, enti\u00e8rement gratuit.\n <\/p>\n <p>\n Dans le prochain article, nous voyagerons en \u00c9gypte ancienne pour d\u00e9couvrir comment Thal\u00e8s a mesur\u00e9 la hauteur des pyramides sans les escalader \u2014 une histoire de triangles, d’ombres et de g\u00e9nie.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi 1+1=2 n’est pas si \u00e9vident \u2014 MathsVivantes Fondements \u00b7 Article #1 Pourquoi 1+1=2n’est pas si \u00e9vident Ce que tout […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-45","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Pourquoi 1+1=2 n'est pas toujours si \u00e9vident - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"pourquoi 1+1=2 n'est pas si \u00e9vident ? 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