{"id":25810,"date":"2021-03-20T00:00:10","date_gmt":"2021-03-20T00:00:10","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25810"},"modified":"2025-03-02T19:58:13","modified_gmt":"2025-03-02T19:58:13","slug":"operacoes-com-numeros-naturais-e-relacoes-de-ordem","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/operacoes-com-numeros-naturais-e-relacoes-de-ordem\/","title":{"rendered":"Opera\u00e7\u00f5es com N\u00fameros Naturais e Rela\u00e7\u00f5es de Ordem"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>Opera\u00e7\u00f5es com N\u00fameros Naturais e Rela\u00e7\u00f5es de Ordem<\/h1>\n<p style=\"text-align:center;\"><strong>Resumo:<\/strong><br \/>\nNesta aula, aprofundaremos nos n\u00fameros naturais e suas opera\u00e7\u00f5es b\u00e1sicas, come\u00e7ando pela origem e propriedades da soma, multiplica\u00e7\u00e3o e potencia\u00e7\u00e3o, em rela\u00e7\u00e3o aos Axiomas de Peano. Examinaremos propriedades-chave como comutatividade, associatividade, distributividade e regras de simplifica\u00e7\u00e3o e invers\u00e3o. Utilizaremos a indu\u00e7\u00e3o matem\u00e1tica para demonstrar teoremas e propriedades. Al\u00e9m disso, analisaremos a rela\u00e7\u00e3o de ordem entre os n\u00fameros naturais, incluindo a lei de tricotomia e as propriedades de transitividade e monotonicidade, com exerc\u00edcios pr\u00e1ticos para aplicar esses conceitos. Finalmente, abordaremos as opera\u00e7\u00f5es inversas (subtra\u00e7\u00e3o e divis\u00e3o) e exploraremos a potencia\u00e7\u00e3o de n\u00fameros naturais e suas propriedades.\n<\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo finalizar esta aula, o estudante ser\u00e1 capaz de:\n<\/p>\n<p style=\"text-align:justify;\">\n<ol>\n<li><strong>Compreender<\/strong> a origem e as propriedades das opera\u00e7\u00f5es b\u00e1sicas dos n\u00fameros naturais.<\/li>\n<li><strong>Aplicar<\/strong> as propriedades das opera\u00e7\u00f5es com n\u00fameros naturais, como comutatividade, associatividade, distributividade e as regras para simplifica\u00e7\u00e3o e opera\u00e7\u00e3o inversa.<\/li>\n<li><strong>Aplicar<\/strong> a indu\u00e7\u00e3o matem\u00e1tica para demonstra\u00e7\u00e3o de propriedades e teoremas simples.<\/li>\n<li><strong>Analisar<\/strong> as propriedades da ordem nos n\u00fameros naturais, como a lei de tricotomia e as propriedades de transitividade e monotonicidade.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">A origem das Opera\u00e7\u00f5es B\u00e1sicas dos N\u00fameros Naturais<\/a><br \/>\n<a href=\"#2\">A ordem Induzida pelas Opera\u00e7\u00f5es dos N\u00fameros Naturais<\/a><br \/>\n<a href=\"#3\">Opera\u00e7\u00f5es Inversas: Subtra\u00e7\u00e3o e Divis\u00e3o de N\u00fameros Naturais<\/a><br \/>\n<a href=\"#4\">Pot\u00eancias de N\u00fameros Naturais<\/a><br \/>\n<a href=\"#5\">Problemas Propostos e Resolvidos<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/jKD71TjMC4s\" title=\"Reprodutor de v\u00eddeo do YouTube\" 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;\">Embora as opera\u00e7\u00f5es com os n\u00fameros naturais sejam conhecidas, \u00e9 necess\u00e1rio sintetizar este conhecimento usando um \u00abmodo um pouco mais matem\u00e1tico\u00bb. Por este motivo, faremos uma revis\u00e3o das opera\u00e7\u00f5es de soma, multiplica\u00e7\u00e3o e pot\u00eancia de n\u00fameros naturais e suas propriedades.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>A origem das Opera\u00e7\u00f5es B\u00e1sicas com N\u00fameros Naturais<\/h2>\n<h3>Opera\u00e7\u00e3o de Soma<\/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>O germe da opera\u00e7\u00e3o de soma foi revisado na aula sobre<\/strong><\/span><\/a> <a href=\"http:\/\/toposuranos.com\/material\/pt\/os-numeros-naturais-e-os-axiomas-de-peano\/\" rel=\"noopener\" target=\"_blank\"><strong>Os N\u00fameros Naturais e os Axiomas de Peano,<\/strong><\/a> porque o sucessor de um n\u00famero natural tamb\u00e9m pode ser apresentado assim:<\/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;\">Como dissemos que <span class=\"katex-eq\" data-katex-display=\"false\">2=S(1), 3=S(2), 4=S(3), \\cdots <\/span><\/bdi> e assim sucessivamente, ent\u00e3o podemos interpretar a soma como a aplica\u00e7\u00e3o sucessiva da opera\u00e7\u00e3o de sucess\u00e3o.<\/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;\">E em geral:<\/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\\;vezes} n)\\cdots)) <\/span><\/bdi><\/p>\n<h4>Propriedades da Soma<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Se <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;\">ent\u00e3o a partir disso podemos obter as propriedades da soma que todos conhecemos:<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Comutatividade<\/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>Associatividade<\/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>Simplifica\u00e7\u00e3o<\/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;\">Todas estas propriedades podem ser demonstradas por indu\u00e7\u00e3o, mas vamos pular esse trabalho. No entanto, encorajo que tente isso como uma forma de praticar a t\u00e9cnica de indu\u00e7\u00e3o.<\/p>\n<h3>Opera\u00e7\u00e3o de Produto<\/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 forma similar, define-se o produto de n\u00fameros<\/span> <\/strong><\/a>naturais como uma aplica\u00e7\u00e3o sucessiva da soma. Portanto, temos que<\/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\\;vezes}<\/span><\/bdi><\/p>\n<h4>Propriedades do Produto<\/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;\">E de forma an\u00e1loga<\/span><\/strong><\/a> podem-se obter suas propriedades<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Comutatividade<\/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>Associatividade<\/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>Simplifica\u00e7\u00e3o<\/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;\">Al\u00e9m disso, a partir da defini\u00e7\u00e3o do produto, o \u00ab1\u00bb dos naturais adquire a qualidade que o transforma em <strong>unidade:<\/strong><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Unidade<\/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>Soma e Produto Combinados<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Quando as opera\u00e7\u00f5es de soma e produto s\u00e3o combinadas, obt\u00e9m-se a propriedade de distribui\u00e7\u00e3o da soma em rela\u00e7\u00e3o \u00e0 multiplica\u00e7\u00e3o<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Distributividade<\/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>A Ordem Induzida pelas Opera\u00e7\u00f5es dos N\u00fameros Naturais<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Desde as opera\u00e7\u00f5es de soma e produto que revisamos, uma rela\u00e7\u00e3o de ordem \u00e9 induzida nos n\u00fameros naturais atrav\u00e9s das seguintes defini\u00e7\u00f5es:<\/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> \u00e9 menor que <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> \u00e9 maior que <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>Propriedades da Ordem nos N\u00fameros Naturais<\/h3>\n<h4>Lei da Tricotomia<\/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>A partir disso, tem-se que apenas<\/strong><\/span><\/a> pode ocorrer uma e apenas uma das seguintes tr\u00eas situa\u00e7\u00f5es:<\/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;\">Se ocorresse que, por exemplo, <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> n\u00e3o \u00e9 menor que <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, ent\u00e3o teria que ocorrer uma das duas: ou <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a=b<\/span><\/bdi>, ou <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, ou seja, maior ou igual e se escreveria: <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\geq b.<\/span><\/bdi> E de forma an\u00e1loga se escreve <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> quando for menor ou igual.<\/p>\n<h4>Propriedade Transitiva<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span> e <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>s\u00e3o n\u00fameros naturais quaisquer, ent\u00e3o cumpre-se 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;\">E analogamente:<\/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>Propriedade de Monotonicidade<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Existe uma propriedade de monotonicidade tanto para a soma quanto para o produto, que \u00e9:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>Monotonicidade da soma<\/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>Monotonicidade do produto<\/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>Opera\u00e7\u00f5es Inversas: Subtra\u00e7\u00e3o e Divis\u00e3o de N\u00fameros Naturais<\/h2>\n<h3>Subtra\u00e7\u00e3o de N\u00fameros Naturais<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Se <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;\">dizemos que a diferen\u00e7a entre<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (nessa ordem), escrita como <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b<\/span><\/bdi>, \u00e9 definida atrav\u00e9s da rela\u00e7\u00e3o<\/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;\">Como podemos ver, essa rela\u00e7\u00e3o ser\u00e1 verdadeira apenas se <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, pois n\u00e3o existe um <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\in \\mathbb{N}<\/span><\/bdi> com o qual se possa satisfazer esta rela\u00e7\u00e3o se <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Pela defini\u00e7\u00e3o de subtra\u00e7\u00e3o, temos a regra conhecida de \\\u00bbo que est\u00e1 somando de um lado da igualdade pode passar para o outro lado subtraindo, e vice-versa\\\u00bb.<\/p>\n<h3>Divis\u00e3o de N\u00fameros Naturais<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Se <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>dizemos que a divis\u00e3o entre<\/strong><\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (nessa ordem), escrita como <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi>, \u00e9 definida atrav\u00e9s da rela\u00e7\u00e3o<\/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;\">Pela defini\u00e7\u00e3o de divis\u00e3o, temos a regra de \\\u00bbo que est\u00e1 multiplicando de um lado da igualdade pode passar para o outro lado dividindo, e vice-versa\\\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Assim como para que a subtra\u00e7\u00e3o <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a - b<\/span><\/bdi> exista, deve cumprir-se que <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, para que exista a divis\u00e3o <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi> \u00e9 necess\u00e1rio que <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> seja \\\u00bbdivis\u00edvel\\\u00bb por <span class=\"katex-eq\" data-katex-display=\"false\">b.<\/span> Isso \u00e9 representado escrevendo<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u00e9 divis\u00edvel por <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>Pot\u00eancias de N\u00fameros Naturais<\/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>Com os n\u00fameros naturais, podem-se definir pot\u00eancias.<\/strong> <\/span><\/a>Elevar um natural <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> que chamamos de base, a outro natural <span class=\"katex-eq\" data-katex-display=\"false\">n,<\/span> que chamamos de expoente, significa multiplicar <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> vezes <span class=\"katex-eq\" data-katex-display=\"false\">n.<\/span> Assim sendo<\/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\\;vezes}<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n,m\\in\\mathbb{N},<\/span><\/bdi> por indu\u00e7\u00e3o (dupla) podem-se demonstrar as seguintes propriedades:<\/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> desde 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>Problemas Propostos e Resolvidos<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Todas as propriedades que foram mostradas aqui podem ser demonstradas utilizando indu\u00e7\u00e3o matem\u00e1tica (simples ou dupla), mas n\u00e3o as desenvolvi porque a demonstra\u00e7\u00e3o resultante \u00e9 desnecessariamente longa para estes resultados t\u00e3o intuitivos. No entanto, quem seguir estas aulas pode tentar realizar essas demonstra\u00e7\u00f5es como exerc\u00edcio. <strong>[Apenas proposto]<\/strong><\/li>\n<li>\u00c9 o mesmo <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{n^m}<\/span><\/bdi> (que se define como <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;\">[Solu\u00e7\u00e3o]<\/span><\/strong><\/a><\/li>\n<li>Utilizando as propriedades vistas, verifique as igualdades:<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>[Solu\u00e7\u00e3o]<\/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>; se <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>[Solu\u00e7\u00e3o]<\/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>; se <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>[Solu\u00e7\u00e3o]<\/strong><\/span><\/a>&nbsp;<\/li>\n<li>Demonstre 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>[Solu\u00e7\u00e3o]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^2 = a^2 - 2ab + b^2<\/span><\/bidi>; se <bidi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bidi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1953s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solu\u00e7\u00e3o]<\/strong><\/span><\/a> <\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(a-b) = a^2-b^2<\/span><\/bidi>; se <bidi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bidi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1978s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solu\u00e7\u00e3o]<\/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>[Solu\u00e7\u00e3o]<\/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>; se <bidi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bidi><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2124s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[Solu\u00e7\u00e3o]<\/strong><\/span><\/a><\/p>\n<p>&nbsp;<\/li>\n<li>Prove por indu\u00e7\u00e3o completa as seguintes propriedades:<\/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>[Solu\u00e7\u00e3o]<\/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>[Solu\u00e7\u00e3o]<\/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>[Solu\u00e7\u00e3o]<\/strong><\/span><\/a>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Opera\u00e7\u00f5es com N\u00fameros Naturais e Rela\u00e7\u00f5es de Ordem Resumo: Nesta aula, aprofundaremos nos n\u00fameros naturais e suas opera\u00e7\u00f5es b\u00e1sicas, come\u00e7ando pela origem e propriedades da soma, multiplica\u00e7\u00e3o e potencia\u00e7\u00e3o, em rela\u00e7\u00e3o aos Axiomas de Peano. Examinaremos propriedades-chave como comutatividade, associatividade, distributividade e regras de simplifica\u00e7\u00e3o e invers\u00e3o. Utilizaremos a indu\u00e7\u00e3o matem\u00e1tica para demonstrar teoremas e [&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":13,"footnotes":""},"categories":[587,1033,571],"tags":[],"class_list":["post-25810","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-e-geometria","category-algebra-geral","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Opera\u00e7\u00f5es com N\u00fameros Naturais e Rela\u00e7\u00f5es de Ordem - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Tudo sobre opera\u00e7\u00f5es e rela\u00e7\u00f5es de ordem dos N\u00fameros Naturais, Axiomas de Peano, Indu\u00e7\u00e3o, tricotomia e monotonia\" \/>\n<meta name=\"robots\" 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