{"id":25854,"date":"2021-03-20T00:00:12","date_gmt":"2021-03-20T00:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25854"},"modified":"2025-03-02T19:59:22","modified_gmt":"2025-03-02T19:59:22","slug":"operations-avec-des-nombres-naturels-et-relations-dordre","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/operations-avec-des-nombres-naturels-et-relations-dordre\/","title":{"rendered":"Op\u00e9rations avec des Nombres Naturels et Relations d&#8217;Ordre"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>Op\u00e9rations avec des Nombres Naturels et Relations d&#8217;Ordre<\/h1>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, nous approfondirons les nombres naturels et leurs op\u00e9rations de base, en commen\u00e7ant par l&#8217;origine et les propri\u00e9t\u00e9s de l&#8217;addition, de la multiplication et de la puissance, en relation avec les Axiomes de Peano. Nous examinerons des propri\u00e9t\u00e9s cl\u00e9s telles que la commutativit\u00e9, l&#8217;associativit\u00e9, la distributivit\u00e9 et les r\u00e8gles de simplification et d&#8217;inversion. Nous utiliserons l&#8217;induction math\u00e9matique pour d\u00e9montrer des th\u00e9or\u00e8mes et des propri\u00e9t\u00e9s. De plus, nous analyserons la relation d&#8217;ordre entre les nombres naturels, y compris la loi de trichotomie et les propri\u00e9t\u00e9s de transitivit\u00e9 et de monotonie, avec des exercices pratiques pour appliquer ces concepts. Enfin, nous aborderons les op\u00e9rations inverses (soustraction et division) et explorerons la puissance des nombres naturels et leurs propri\u00e9t\u00e9s.\n<\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<p style=\"text-align:justify;\">\n<ol>\n<li><strong>Comprendre<\/strong> l&#8217;origine et les propri\u00e9t\u00e9s des op\u00e9rations de base des nombres naturels.<\/li>\n<li><strong>Appliquer<\/strong> les propri\u00e9t\u00e9s des op\u00e9rations avec les nombres naturels, comme la commutativit\u00e9, l&#8217;associativit\u00e9, la distributivit\u00e9, et les r\u00e8gles pour la simplification et l&#8217;op\u00e9ration inverse.<\/li>\n<li><strong>Appliquer<\/strong> l&#8217;induction math\u00e9matique pour la d\u00e9monstration de propri\u00e9t\u00e9s et th\u00e9or\u00e8mes simples.<\/li>\n<li><strong>Analyser<\/strong> les propri\u00e9t\u00e9s de l&#8217;ordre dans les nombres naturels, comme la loi de trichotomie et les propri\u00e9t\u00e9s de transitivit\u00e9 et de monotonie.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX DES CONTENUS<\/u>:<\/strong><br \/>\n<a href=\"#1\">L&#8217;origine des Op\u00e9rations de Base des Nombres Naturels<\/a><br \/>\n<a href=\"#2\">L&#8217;ordre Induit par les Op\u00e9rations des Nombres Naturels<\/a><br \/>\n<a href=\"#3\">Op\u00e9rations Inverses : Soustraction et Division des Nombres Naturels<\/a><br \/>\n<a href=\"#4\">Puissances des Nombres Naturels<\/a><br \/>\n<a href=\"#5\">Probl\u00e8mes Propos\u00e9s et R\u00e9solus<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/jKD71TjMC4s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Bien que les op\u00e9rations avec les nombres naturels soient connues, il est n\u00e9cessaire de synth\u00e9tiser cette connaissance en utilisant une approche \u00abun peu plus math\u00e9matique\u00bb. Pour cette raison, nous r\u00e9viserons les op\u00e9rations d&#8217;addition, de multiplication et de puissance des nombres naturels et leurs propri\u00e9t\u00e9s.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>L&#8217;origine des Op\u00e9rations de Base des Nombres Naturels<\/h2>\n<h3>Op\u00e9ration d&#8217;Addition<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=49s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Le germe de l&#8217;op\u00e9ration d&#8217;addition a \u00e9t\u00e9 examin\u00e9 dans le cours sur<\/strong><\/span><\/a>  <a href=\"http:\/\/toposuranos.com\/material\/fr\/les-nombres-naturels-et-les-axiomes-de-peano\/\" rel=\"noopener\" target=\"_blank\"><strong>Les Nombres Naturels et les Axiomes de Peano,<\/strong><\/a> car le successeur d&#8217;un naturel peut \u00e9galement \u00eatre pr\u00e9sent\u00e9 ainsi :<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">S(n) = n+1<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Comme nous l&#8217;avons dit, <span class=\"katex-eq\" data-katex-display=\"false\">2=S(1), 3=S(2), 4=S(3), \\cdots <\/span><\/bdi> et ainsi de suite, alors nous pouvons interpr\u00e9ter l&#8217;addition comme l&#8217;application successive de l&#8217;op\u00e9ration de succession.<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+1 =S(n),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+2 =S(S(n)),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+3 =S(S(S(n))),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et en g\u00e9n\u00e9ral :<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+m = \\underbrace{S(S(\\cdots S(}_{m\\;fois} n)\\cdots)) <\/span><\/bdi><\/p>\n<h4>Propri\u00e9t\u00e9s de l&#8217;Addition<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N},<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=131s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">alors \u00e0 partir de cela nous pouvons obtenir les propri\u00e9t\u00e9s de l&#8217;addition que nous connaissons tous :<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Commutativit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b=b+a<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Associativit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b+c=(a+b)+c=a+(b+c)<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Simplification<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\leftrightarrow b=c <\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Toutes ces propri\u00e9t\u00e9s peuvent \u00eatre d\u00e9montr\u00e9es par induction mais nous allons sauter ce travail. Cependant, je vous encourage \u00e0 essayer cela comme un moyen de pratiquer la technique d&#8217;induction.<\/p>\n<h3>Op\u00e9ration de Multiplication<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">De mani\u00e8re similaire, le produit de nombres<\/span> <\/strong><\/a>naturels est d\u00e9fini comme une application successive de l&#8217;addition. Nous avons donc<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n\\cdot m = \\underbrace{n+ n+ \\cdots + n}_{m\\;fois}<\/span><\/bdi><\/p>\n<h4>Propri\u00e9t\u00e9s du Produit<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=251s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Et de mani\u00e8re analogue<\/span><\/strong><\/a> on peut obtenir ses propri\u00e9t\u00e9s<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Commutativit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">ab=ba<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Associativit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">abc=(ab)c=a(bc)<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Simplification<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">ab=ac \\leftrightarrow b=c <\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Et en outre, \u00e0 partir de la d\u00e9finition du produit, le \u00ab1\u00bb des naturels acquiert la qualit\u00e9 qui le transforme en <strong>unit\u00e9 :<\/strong><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Unit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">1a=a=a1<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Somme et Produit Combin\u00e9s<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Lorsque les op\u00e9rations de somme et de produit sont combin\u00e9es, on obtient la propri\u00e9t\u00e9 de distribution de la somme par rapport \u00e0 la multiplication<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Distributivit\u00e9<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a(b+c)=ab+ac<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>L&#8217;ordre Induit par les Op\u00e9rations des Nombres Naturels<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Depuis les op\u00e9rations d&#8217;addition et de multiplication que nous avons examin\u00e9es, une relation d&#8217;ordre est induite chez les naturels \u00e0 travers les d\u00e9finitions suivantes :<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> est inf\u00e9rieur \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b := (\\exists k \\in \\mathbb{N}) (a + k = b)<\/span><\/bdi><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> est sup\u00e9rieur \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b := (\\exists k \\in \\mathbb{N}) (a = b + k)<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Propri\u00e9t\u00e9s de l&#8217;Ordre dans les Nombres Naturels<\/h3>\n<h4>Loi de Trichotomie<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=513s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>\u00c0 partir de cela, il se trouve que seulement<\/strong><\/span><\/a> l&#8217;une des trois situations suivantes peut se produire :<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a = b<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">Si cela arrivait que, par exemple <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> n&#8217;est pas inf\u00e9rieur \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, alors il faudrait qu&#8217;une des deux se produise : soit <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a=b<\/span><\/bdi>, soit <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, c&#8217;est-\u00e0-dire plus grand ou \u00e9gal, et on \u00e9crirait : <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\geq b.<\/span><\/bdi> Et de mani\u00e8re analogue on \u00e9crit <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> lorsque c&#8217;est inf\u00e9rieur ou \u00e9gal.<\/p>\n<h4>Propri\u00e9t\u00e9 Transitive<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=625s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>sont des naturels quelconques, alors il est vrai que :<\/strong><\/span><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">[(a\\lt b) \\wedge (b\\lt c)] \\rightarrow (a\\lt c)<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et de mani\u00e8re analogue :<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">[(a\\gt b) \\wedge (b\\gt c)] \\rightarrow (a\\gt c)<\/span><\/bdi><\/p>\n<h4>Propri\u00e9t\u00e9 de Monotonie<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Il existe une propri\u00e9t\u00e9 de monotonie \u00e0 la fois pour l&#8217;addition et pour la multiplication, elle est la suivante :<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>Monotonie de l&#8217;addition<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\lt b) \\leftrightarrow (a+c \\lt b+c) <\/span><\/bdi><bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\gt b) \\leftrightarrow (a+c \\gt b+c)<\/span><\/bdi><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>Monotonie de la multiplication<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\lt b) \\leftrightarrow (a c \\lt b c) <\/span><\/bdi><bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\gt b) \\leftrightarrow (a c \\gt b c)<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Op\u00e9rations Inverses : Soustraction et Division des Nombres Naturels<\/h2>\n<h3>Soustraction des Nombres Naturels<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N}<\/span><\/bdi>, <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=782s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">nous disons que la diff\u00e9rence entre<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (dans cet ordre), \u00e9crite <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b<\/span><\/bdi>, est d\u00e9finie par la relation<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b=c \\leftrightarrow a= b+c<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Comme nous pouvons le voir, cette relation sera vraie seulement si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, car il n&#8217;existe pas de <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\in \\mathbb{N}<\/span><\/bdi> avec lequel cette relation puisse \u00eatre satisfaite si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 travers la d\u00e9finition de la soustraction, nous avons la r\u00e8gle connue de \u00abce qui est ajout\u00e9 d&#8217;un c\u00f4t\u00e9 de l&#8217;\u00e9galit\u00e9 peut passer de l&#8217;autre c\u00f4t\u00e9 en soustrayant, et vice versa\u00bb.<\/p>\n<h3>Division des Nombres Naturels<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N}<\/span><\/bdi>, <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=917s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>nous disons que la division entre<\/strong><\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (dans cet ordre), \u00e9crite <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi>, est d\u00e9finie par la relation<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b=c \\leftrightarrow a= bc<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">De la d\u00e9finition de la division, nous avons la r\u00e8gle de \u00abce qui multiplie d&#8217;un c\u00f4t\u00e9 de l&#8217;\u00e9galit\u00e9 peut passer de l&#8217;autre c\u00f4t\u00e9 en divisant, et vice versa\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Tout comme pour que la soustraction <bidi><span class=\"katex-eq\" data-katex-display=\"false\">a - b<\/span><\/bidi> existe, il faut que <bidi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bidi>, pour que la division <bidi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bidi> existe, il est n\u00e9cessaire que <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> soit \u00abdivisible\u00bb par <span class=\"katex-eq\" data-katex-display=\"false\">b.<\/span> Cela est repr\u00e9sent\u00e9 par l&#8217;\u00e9criture<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> est divisible par <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\; :=a|b \\; := \\; (\\exists k \\in \\mathbb{N})(a = kb)<\/span><\/bdi><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Puissances des Nombres Naturels<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1020s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Avec les nombres naturels, on peut d\u00e9finir les puissances.<\/strong> <\/span><\/a>\u00c9lever un naturel <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> que nous appelons la base, \u00e0 un autre naturel <span class=\"katex-eq\" data-katex-display=\"false\">n,<\/span> que nous appelons l&#8217;exposant, signifie multiplier <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> par lui-m\u00eame <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> fois. Ainsi<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^n = \\underbrace{bb\\cdots b}_{n\\;fois}<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n,m\\in\\mathbb{N},<\/span><\/bdi> par induction (double) on peut d\u00e9montrer les propri\u00e9t\u00e9s suivantes :<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b^nb^m=b^{n+m}<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{b^n}{b^m} = b^{n-m},<\/span><\/bdi> \u00e0 condition que <bdi><span class=\"katex-eq\" data-katex-display=\"false\">n\\lt m<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (ab)^n=a^nb^n<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{a}{b}\\right)^n = \\frac{a^n}{b^n}<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (b^n)^m=b^{nm}<\/span><\/bdi><\/li>\n<\/ol>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Probl\u00e8mes Propos\u00e9s et R\u00e9solus<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Toutes les propri\u00e9t\u00e9s pr\u00e9sent\u00e9es ici peuvent \u00eatre d\u00e9montr\u00e9es en utilisant l&#8217;induction math\u00e9matique (simple ou double), mais je ne les ai pas d\u00e9velopp\u00e9es car la d\u00e9monstration r\u00e9sultante serait inutilement longue pour ces r\u00e9sultats si intuitifs. Cependant, ceux qui suivent ces cours peuvent essayer de r\u00e9aliser ces d\u00e9monstrations comme exercice. <strong>[Uniquement propos\u00e9]<\/strong><\/li>\n<li>Est-ce la m\u00eame chose <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{n^m}<\/span><\/bdi> (qui est d\u00e9fini comme <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{(n^m)})<\/span><\/bdi> que <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(b^n)^m<\/span><\/bdi>? <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1298s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[Solution]<\/span><\/strong><\/a><\/li>\n<li>En utilisant les propri\u00e9t\u00e9s vues, v\u00e9rifiez les \u00e9galit\u00e9s :<br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(c+d) = ac+ad+bc+bd<\/span><\/bdi><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1556s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(c-d) = ac-ad+bc-bd,<\/span><\/bdi>; si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1660s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)(c-d) = ac-ad-bc+bd,<\/span><\/bdi>; si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1730s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a>&nbsp;<\/li>\n<li>D\u00e9montrez que <\/br><br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)^2 = a^2 + 2ab + b^2<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1903s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^2 = a^2 - 2ab + b^2<\/span><\/bdi>; si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1953s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(a-b) = a^2-b^2<\/span>; si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1978s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nd) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)^3 = a^3 + 3a^2b+3ab^2+b^3<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2008s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\ne) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^3 = a^3 - 3a^2b+3ab^2-b^3<\/span><\/bdi>; si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2124s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[Solution]<\/strong><\/span><\/a><\/p>\n<p>&nbsp;<\/li>\n<li>D\u00e9montrez par induction compl\u00e8te les propri\u00e9t\u00e9s suivantes :<\/br><br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1+2+3+4+\\cdots+n = \\displaystyle \\frac{n(n+1)}{2}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2328s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1^2+2^2+3^2+4^2+\\cdots+n^2 = \\displaystyle \\frac{n(n+1)(2n+1)}{6}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2505s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1^3+2^3+3^3+4^3+\\cdots+n^3 = \\displaystyle \\frac{n^2(n+1)^2}{4}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2972s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Op\u00e9rations avec des Nombres Naturels et Relations d&#8217;Ordre R\u00e9sum\u00e9 : Dans ce cours, nous approfondirons les nombres naturels et leurs op\u00e9rations de base, en commen\u00e7ant par l&#8217;origine et les propri\u00e9t\u00e9s de l&#8217;addition, de la multiplication et de la puissance, en relation avec les Axiomes de Peano. Nous examinerons des propri\u00e9t\u00e9s cl\u00e9s telles que la commutativit\u00e9, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25803,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[585,1043,569],"tags":[],"class_list":["post-25854","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebre-et-geometrie","category-algebre-generale","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Op\u00e9rations avec des Nombres Naturels et Relations d&#039;Ordre - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Tout sur les op\u00e9rations et les relations d&#039;ordre des Nombres Naturels, les Axiomes de Peano, l&#039;Induction, la trichotomie et la monotonie\" \/>\n<meta name=\"robots\" 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