{"id":33634,"date":"2021-03-20T00:00:53","date_gmt":"2021-03-20T00:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33634"},"modified":"2025-07-29T07:20:57","modified_gmt":"2025-07-29T07:20:57","slug":"operationes-cum-numeris-naturalibus-et-relationes-ordinis","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\/","title":{"rendered":"Operationes cum Numeris Naturalibus et Relationes Ordinis"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>Operationes cum Numeris Naturalibus et Relationes Ordinis<\/h1>\n<p style=\"text-align:center;\"><strong>Summarium:<\/strong><br \/>\nIn hac lectione numeros naturales et eorum operationes fundamentales altius perscrutabimur, ex incunabulis et proprietatibus additionis, multiplicationis atque potentiationis ex axiomatibus Peano derivatis. Investigabimus proprietates praecipuas ut commutativitatem, associativitatem, distributivitatem ac regulas simplificationis et inversionis. Inductionem mathematicam adhibebimus ad theoremas et proprietates demonstrandas. Praeterea relationem ordinis inter numeros naturales explorabimus, legem trichotomiae et proprietates transitivitatis ac monotoniae complectentem, cum exercitiis practicis ad hos conceptos applicandos. Postremo operationes inversas (subtractio et divisio) tractabimus et potentiationem numerorum naturalium eiusque proprietates explorabimus.\n<\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIVA DISCENDI:<\/strong><br \/>\nPost hanc lectionem discipulus poterit:\n<\/p>\n<p style=\"text-align:justify;\">\n<ol>\n<li><strong>Intellegere<\/strong> originem et proprietates operationum fundamentalium numerorum naturalium.<\/li>\n<li><strong>Applicare<\/strong> proprietates operationum cum numeris naturalibus, ut commutativitatem, associativitatem, distributivitatem atque regulas simplificationis et operationis inversae.<\/li>\n<li><strong>Adhibere<\/strong> inductionem mathematicam ad demonstrationem proprietatum et theoremata simplicia.<\/li>\n<li><strong>Analyzare<\/strong> proprietates ordinis in numeris naturalibus, ut legem trichotomiae et proprietates transitivitatis et monotoniae.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">Origo Operationum Fundamentalium Numerorum Naturalium<\/a><br \/>\n<a href=\"#2\">Ordo ab Operationibus Numerorum Naturalium Inductus<\/a><br \/>\n<a href=\"#3\">Operationes Inversae: Subtractio et Divisio Numerorum Naturalium<\/a><br \/>\n<a href=\"#4\">Potentiae Numerorum Naturalium<\/a><br \/>\n<a href=\"#5\">Problemata Proposita et Soluta<\/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;\">Quamvis operationes cum numeris naturalibus notae sint, necesse est hoc scientiam modis paulo magis mathematicis recensere. Propter hoc revisionem faciemus operationum additionis, multiplicationis et potentiationis numerorum naturalium earumque proprietatum.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Origo Operationum Fundamentalium Numerorum Naturalium<\/h2>\n<h3>Operatio Additionis<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=49s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Germen operationis additionis recensuimus in lectione de<\/strong><\/span><\/a>  <a href=\"http:\/\/toposuranos.com\/material\/es\/los-numeros-naturales-y-los-axiomas-de-peano\/\" rel=\"noopener\" target=\"_blank\"><strong>Numeris Naturalibus et Axiomatibus Peano,<\/strong><\/a> quia successor naturalis etiam sic exhiberi potest:<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">S(n) = n+1<\/span><\/bdi><\/p>\n<p style=\"text-align: justify;\">Cum diximus <span class=\"katex-eq\" data-katex-display=\"false\">2=S(1), 3=S(2), 4=S(3), \\cdots <\/span><\/bdi> et sic porro, tunc summam interpretari possumus ut applicationem iteratam operationis successionis.<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+1 =S(n),<\/span><\/bdi><\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+2 =S(S(n)),<\/span><\/bdi><\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+3 =S(S(S(n))),<\/span><\/bdi><\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/bdi><\/p>\n<p style=\"text-align: justify;\">Et generatim:<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+m = \\underbrace{S(S(\\cdots S(}_{m\\;veces} n)\\cdots)) <\/span><\/bdi><\/p>\n<h4>Proprietates Additionis<\/h4>\n<p style=\"text-align: justify;\">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;\">tunc ex hoc consequi possumus proprietates additionis quas omnes novimus:<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td><strong>Commutativitas<\/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;\">\n<td><strong>Associativitas<\/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;\">\n<td><strong>Simplificatio<\/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;\">Omnes hae proprietates per inductionem demonstrari possunt sed hoc laborem praetermittemus. Attamen te hortor ut id coneris tamquam modum exercendi technicam inductionis.<\/p>\n<h3>Operatio Multiplicationis<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Simili modo, productus numerorum<\/span> <\/strong><\/a>naturalium tamquam applicatio successiva additionis definitur. Habemus igitur<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n\\cdot m = \\underbrace{n+ n+ \\cdots + n}_{m\\;veces}<\/span><\/bdi><\/p>\n<h4>Proprietates Multiplicationis<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=251s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Et similiter<\/span><\/strong><\/a> proprietates eius elici possunt<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td><strong>Commutativitas<\/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;\">\n<td><strong>Associativitas<\/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;\">\n<td><strong>Simplificatio<\/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;\">Et praeterea, ex definitione multiplicationis fit ut \u00ab1\u00bb numerorum naturalium proprium obtineat quod eum in <strong>unitatem<\/strong> convertat:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td><strong>Unitas<\/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>Suma y Producto Combinados<\/h3>\n<p style=\"text-align: justify;\">Cum operationes additionis et multiplicationis combinantur, proprietatem distributionis additionis respectu multiplicationis obtinetur<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td><strong>Distributivitas<\/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>Ordo ab Operationibus Numerorum Naturalium Inductus<\/h2>\n<p style=\"text-align: justify;\">Ex operibus additionis et multiplicationis quae recensuimus inducitur in numeris naturalibus relatio ordinis per sequentes definitiones:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> minor est quam <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;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> maior est quam <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>Proprietates Ordinis in Numeris Naturalibus<\/h3>\n<h4>Lex Trichotomiae<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=513s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Ex hoc sequitur quod una et sola ex tribus sequentibus condicionibus accidere potest:<\/strong><\/span><\/a><\/p>\n<ol style=\"text-align: justify;\">\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;\">Si accideret, exempli gratia, <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> non esse minorem quam <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, tunc una ex duabus contingere deberet: vel <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a=b<\/span><\/bdi>, vel <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, id est maior sive aequalis, quod scribitur: <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\geq b.<\/span><\/bdi>. Eodem modo scribitur <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> cum minor sive aequalis est.<\/p>\n<h4>Proprietas Transitivitatis<\/h4>\n<p style=\"text-align: justify;\">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>numeri naturales quilibet sunt, tunc valet:<\/strong><\/span><\/a><\/p>\n<p style=\"text-align: center;\"><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;\">Et similiter:<\/p>\n<p style=\"text-align: center;\"><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>Proprietas Monotoniae<\/h4>\n<p style=\"text-align: justify;\">Exstat proprietas monotoniae tam pro additione quam pro multiplicatione, quae est:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Monotonia Additionis<\/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;\"><strong>Monotonia Multiplicationis<\/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>Operationes Inversae: Subtractio et Divisio Numerorum Naturalium<\/h2>\n<h3>Subtractio Numerorum Naturalium<\/h3>\n<p style=\"text-align: justify;\">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;\">dicimus differentiam inter<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (hoc ordine), scriptam <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b<\/span><\/bdi>, definiri per relationem<\/p>\n<p style=\"text-align: center;\"><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;\">Ut videre licet, talis relatio vera erit solum si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, quia nullus <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\in \\mathbb{N}<\/span><\/bdi> est quo hac relatione satisfieri possit, si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi><\/p>\n<p style=\"text-align: justify;\">Per definitionem subtractionis habemus notam regulam de \u00abquod ad unam partem aequationis additur, ad alteram transire potest subtractum, et vice versa\u00bb.<\/p>\n<h3>Divisio Numerorum Naturalium<\/h3>\n<p style=\"text-align: justify;\">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>dicimus quod divisio inter<\/strong><\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (hoc ordine), scripta <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi>, definiri per relationem<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b=c \\leftrightarrow a= bc<\/span><\/bdi><\/p>\n<p style=\"text-align: justify;\">Ex definitione divisionis habemus regulam, \u00abquod ad unam partem aequationis multiplicatur, ad alteram transire potest dividendo, et converso\u00bb.<\/p>\n<p style=\"text-align: justify;\">Sicut, ut subtractio <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a - b<\/span><\/bdi> exsistat, requiritur ut <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, sic etiam ut divisio <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi> exsistat, necesse est <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/bdi> \u00abdivisibilis\u00bb esse per <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b.<\/span><\/bdi> Hoc ita repraesentamus scribendo<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> divisibilis est per <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>Potentiae Numerorum Naturalium<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1020s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Cum numeris naturalibus potentiae definiri possunt.<\/strong> <\/span><\/a>Elevare naturalem <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> quem basim appellamus, ad alium naturalem <span class=\"katex-eq\" data-katex-display=\"false\">n,<\/span> quem exponentem dicimus, significat <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> vicibus <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> multiplicare. Ita<\/p>\n<p style=\"text-align: center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^n = \\underbrace{bb\\cdots b}_{n\\;veces}<\/span><\/bdi><\/p>\n<p style=\"text-align: justify;\">Si <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n,m\\in\\mathbb{N},<\/span><\/bdi> per inductionem (duplicem) demonstrare possumus sequentes proprietates:<\/p>\n<ol style=\"text-align: justify;\">\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> dum <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>Problemata Proposita et Soluta<\/h2>\n<ol style=\"text-align: justify;\">\n<li>Omnes proprietates quae hic ostensae sunt per inductionem mathematicam (simplicem sive duplicem) demonstrari possunt, sed eas non evolvi quia demonstratio resultans nimis longa est pro talibus resultatibus intuitive. Tamen, qui has lectiones sequitur, conari potest tales demonstrationes facere ut exercitium. <strong>[Propositum tantum]<\/strong><\/li>\n<li>Num idem est <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{n^m}<\/span><\/bdi> (quod definitur ut <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{(n^m)}<\/span><\/bdi>) ac <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;\">[Solutio]<\/span><\/strong><\/a><\/li>\n<li>Proprietatibus visis utendo, aequationes has verificare:<br \/>\n a) <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>[Solutio]<\/strong><\/span><\/a><\/br><br \/>\n b) <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>[Solutio]<\/strong><\/span><\/a> <\/br><br \/>\n c)<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>[Solutio]<\/strong><\/span><\/a>&nbsp;<\/li>\n<li>Demonstrandum est <\/br><br \/>\n a) <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>[Solutio]<\/strong><\/span><\/a> <\/br><br \/>\n b) <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>[Solutio]<\/strong><\/span><\/a> <\/br><br \/>\n c) <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>[Solutio]<\/strong><\/span><\/a> <\/br><br \/>\n d) <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>[Solutio]<\/strong><\/span><\/a><\/br><br \/>\n e) <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>[Solutio]<\/strong><\/span><\/a><\/p>\n<p>&nbsp;<\/li>\n<li>Pro inductione completa haec proprietates probare:<\/br><br \/>\n a) <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>[Solutio]<\/strong><\/span><\/a> <\/br><br \/>\n b) <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>[Solutio]<\/strong><\/span><\/a><\/br><br \/>\n c) <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>[Solutio]<\/strong><\/span><\/a>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Operationes cum Numeris Naturalibus et Relationes Ordinis Summarium: In hac lectione numeros naturales et eorum operationes fundamentales altius perscrutabimur, ex incunabulis et proprietatibus additionis, multiplicationis atque potentiationis ex axiomatibus Peano derivatis. Investigabimus proprietates praecipuas ut commutativitatem, associativitatem, distributivitatem ac regulas simplificationis et inversionis. Inductionem mathematicam adhibebimus ad theoremas et proprietates demonstrandas. Praeterea relationem ordinis inter [&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":4,"footnotes":""},"categories":[1304,1310,1298],"tags":[],"class_list":["post-33634","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-algebra-generalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Operationes cum Numeris Naturalibus et Relationes Ordinis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Omnia de operationibus et relationibus ordinis Numerorum Naturalium, Axiomatibus Peano, Inductione, trichotomia et monotonia\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Operationes cum Numeris Naturalibus et Relationes Ordinis\" \/>\n<meta property=\"og:description\" content=\"Omnia de operationibus et relationibus ordinis Numerorum Naturalium, Axiomatibus Peano, Inductione, trichotomia et monotonia\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-20T00:00:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-29T07:20:57+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/OPERACIONES-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Operationes cum Numeris Naturalibus et Relationes Ordinis\" \/>\n<meta name=\"twitter:description\" content=\"Omnia de operationibus et relationibus ordinis Numerorum Naturalium, Axiomatibus Peano, Inductione, trichotomia et monotonia\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/OPERACIONES.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Operationes cum Numeris Naturalibus et Relationes Ordinis\",\"datePublished\":\"2021-03-20T00:00:53+00:00\",\"dateModified\":\"2025-07-29T07:20:57+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/\"},\"wordCount\":1293,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/12\\\/OPERACIONES.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Algebra Generalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/operationes-cum-numeris-naturalibus-et-relationes-ordinis\\\/\",\"name\":\"Operationes cum Numeris Naturalibus et Relationes Ordinis - 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