Prima appropinquatio ad Coniuncta Numerica – ToposUranos.com<\/title> <\/head> <body> <\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Prima Appropinquatio ad Coniuncta Numerica: A Naturalibus ad Complexos<\/h1>\n<\/center><\/p>\nSummarium:<\/strong><\/br>In hac lectione perscrutabimur quomodo numeri naturales uti possint fundamentum ad aliorum coniunctuum numerorum constructionem ut certas limitationes operationum superare valeant. Incipiemus a numeris integris, qui nobis permittunt subtractiones late perficere. Deinde progrediemur ad numeros rationales, qui instrumentum divisionis plene praebent. Postea in numeros reales penitus ingrediemur ut radicibus n-nesimis operari possimus, atque commemorabimus quomodo numeri complexi introducantur ad specifica n-nesimarum radicum casus tractandos. Per hos progressus intelligetur quomodo unumquodque novum coniunctum numericum oritur ad quaestiones prioris intrinsecas solvendas.<\/em><\/p>\nObjectiva Discendi<\/u>:<\/strong> Lectione hac confecta discipulus poterit:<\/p>\n<ol>\n<li>Identificare<\/strong> proprietates fundamentales numerorum naturalium, integrorum et rationalium.<\/li>\n<li>Interpretari<\/strong> proprietates et operationes fundamentales quae heredantur vel mutantur dum ab uno coniuncto numerorum ad aliud transitur.<\/li>\n<li>Comparare<\/strong> proprietates diversorum coniunctuum numerorum et quomodo inter se connexae sint.<\/li>\n<\/ol>\nIndex Contentorum<\/u><\/strong> \n<a href=\"#1\">Introductio<\/a> \n<a href=\"#2\">Proprietates Numerorum Naturalium<\/a> \n<a href=\"#3\">Transitus e Numeris Naturalibus ad Integros<\/a> \n<a href=\"#4\">Saltus ad Numeros Rationales<\/a> \n<a href=\"#5\">Numeri Reales et Irrationales<\/a> \n<a href=\"#6\">Complexi: Clausus Algebraicus Numerorum Realium<\/a>\n<\/p>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PfK-pIlyCj4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<a name=\"1\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<div class=\"content\">\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=96s\" target=\"_blank\" rel=\"noopener\">Numeri reales, una cum aliis coniunctis numerorum quae in hac lectione explorabimus,<\/strong><\/span><\/a> introducuntur per extensionem numerorum naturalium. Contingit enim ut, cum duobus numeris naturalibus quibuscumque, non semper fieri possit operationes subtractionis aut divisionis efficere, et hae ampliationes ad hoc incommodum removendum destinatae sunt.<\/p>\nIn hac lectione recensibimus <a href=\"https:\/\/toposuranos.com\/operaciones-con-numeros-naturales\/\" rel=\"noopener\" target=\"_blank\">operationes et proprietates numerorum naturalium,<\/strong><\/a> et hac in fundamentali, procedemus ad structuram omnium reliquorum coniunctuum numerorum, usque ad numeros reales et ultra.<\/p>\n<a name=\"2\"><\/a><\/p>\n<h2>Proprietates Numerorum Naturalium<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=214s\" target=\"_blank\" rel=\"noopener\">Cum operationes cum numeris naturalibus tractamus,<\/span><\/strong><\/a> maxime ad summam et productum cum suis respectivis operationibus inversis referimus. Infra hae proprietates breviantur:<\/p>\nCum a,b,c\\in\\mathbb{N},<\/span> verificatur:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>1. <\/td>\n<td>a + b = b + a<\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>2. <\/td>\n<td>a \\pm (b \\pm c) = (a\\pm b)\\pm c <\/span> (en el caso de la resta, es v\u00e1lida siempre que est\u00e9 bien definida)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>3. <\/td>\n<td>a\\cdot b = b \\cdot a <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>4. <\/td>\n<td>a\\cdot(b\\cdot c)= (a\\cdot b)\\cdot c <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>5.\\;\\;\\;\\;<\/span><\/td>\n<td>a\\cdot b = a \\leftrightarrow b=1 <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>6.<\/td>\n<td>\\displaystyle \\frac{a}{b}\\in\\mathbb{N} \\leftrightarrow (\\exists k\\in\\mathbb{N})(a=b\\cdot k) <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>7.<\/td>\n<td>a\\cdot(b+c)=a\\cdot b + a \\cdot c <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<a name=\"3\"><\/a><\/p>\n<h2>Transitus e Numeris Naturalibus ad Integros<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=418s\" target=\"_blank\" rel=\"noopener\">Primum quod animadvertendum est est in casu additionum:<\/span><\/strong><\/a> (\\forall a,b\\in\\mathbb{N})(a+b\\in\\mathbb{N})<\/span>, dum pro subtractionibus: (\\forall a,b\\in\\mathbb{N})(a+b\\in\\mathbb{N} \\leftrightarrow a\\gt b)<\/span>. Incommodum oritur cum subtractio inter duos numeros naturales a<\/span> et b<\/span> sensum non habet si a\\leq b<\/span>; ad hanc rem emendandam, numeri naturales extenduntur ad coniunctum numerorum integrorum, ubi huiusmodi subtractiones valor bene definitus accipiunt. Hoc novum coniunctum numerorum integrorum littera \\mathbb{Z}<\/span> denotamus, quod ex omnibus numeris naturalibus, eorum inversis additivis atque zero constat.<\/p>\n\\mathbb{Z} = \\{\\cdots, -3,-2,-1,0,1,2,3,\\cdots \\}<\/span>\nNumeri integri omnes proprietates et operationes numerorum naturalium heredant, cum extensione super secundam proprietatem, et notiones inversi et elementi neutri additivi introducuntur.<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: left;\">\n<td>2*.<\/td>\n<td>a \\pm (b \\pm c) = (a\\pm b) \\pm c <\/span><\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>8.<\/td>\n<td>(\\forall a\\in\\mathbb{Z})(\\exists ! b\\in\\mathbb{Z})(a+b=0 \\leftrightarrow b=-a)<\/span><\/td>\n<\/tr>\n<tr style=\"text-align: left;\">\n<td>9.<\/td>\n<td>(\\forall a\\in\\mathbb{Z})(\\exists ! b\\in\\mathbb{Z})(a+b=a \\leftrightarrow b=0)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nElementum b=-a<\/span> illud vocamus inversum additivum<\/strong> a<\/span>.<\/p>\n<a name=\"4\"><\/a><\/p>\n<h2>Saltus ad Numeros Rationales<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=755s\" target=\"_blank\" rel=\"noopener\">Hoc loco sola operatio quae recte definire nondum possumus est divisio.<\/span><\/strong><\/a> Ad hoc solvendum expansionem perficemus de coniuncto numerorum integrorum ad coniunctum numerorum rationalium, quod hisce definietur:<\/p>\n\\mathbb{Q}=\\left\\{a= \\displaystyle\\frac{n}{m}\\;|\\;n,m\\in\\mathbb{Z}\\wedge m\\neq 0 \\right\\}<\/span>\nHoc fit ut nova proprietas accedat<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>10.<\/td>\n<td>(\\forall a \\in \\mathbb{Q}\\setminus\\{0\\})(\\exists ! b \\in \\mathbb{Q})<\/span> \\left[(a\\cdot b = 1) \\leftrightarrow \\left( b = \\displaystyle \\frac{1}{a} = a^{-1} \\right)\\right]<\/span><\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td colspan=\"2\">Omnis rationalis non nullus inversum multiplicativum habet. Inversum multiplicativum a<\/span> est a^{-1}<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nCum his numeris, operationibus et proprietatibus novae operationes cum proprietatibus definiuntur. In his definitur potestas n-nesima rationis q<\/span> per<\/p>\nq^n = \\underbrace{q\\cdot q \\cdot \\cdots \\cdot q}_{n\\;veces};<\/span> cum n\\in\\mathbb{N}<\/span>\nq^{-n}= \\displaystyle \\frac{1}{q^n}<\/span>\nAnimadvertamus quod, ex hoc, et quoties q\\neq 0<\/span>, dicere possumus quod<\/p>\nq^0 = 1<\/span>\nPraeterea, quoties divisiones per nullum occurrant, datis duobus rationalibus quibusvis a,b<\/span>, et duobus integris n,m<\/span>, sequentia verificabuntur:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>11.<\/td>\n<td>a^n \\cdot a^m = a^{n+m} <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>12.<\/td>\n<td>(a^n)^m = a^{n\\cdot m} <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>13.<\/td>\n<td>(a\\cdot b)^n = a^{n} \\cdot a^{m} <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>14.<\/td>\n<td>\\left(\\displaystyle \\frac{a}{a}\\right)^n = \\frac{a^n}{a^n} <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>15.<\/td>\n<td>\\displaystyle \\frac{a^n}{a^m} = a^{n-m} = \\frac{1}{a^{m-n}} <\/span>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<a name=\"5\"><\/a><\/p>\n<h2>Numeri Reales et Irrationales<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1031s\" target=\"_blank\" rel=\"noopener\">Sicut operatio subtractionis (inversa additionis) et divisio<\/span><\/strong><\/a> (inversa producti) necessarium reddiderunt naturales ad integros et rationales, respective, extendere ut operationes bene definiretur, similiter accidit cum potentiis. Operatio inversa n<\/span>-esimae potestatis est radix n<\/span>-esima.<\/p>\n<h3>Definitio Radicis<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1071s\" target=\"_blank\" rel=\"noopener\">Sit n<\/span> integer maior quam 1<\/span><\/strong><\/a> et p,q<\/span> numeri rationales quilibet, definitur radix n<\/span>-esima q<\/span>, quam per sequentes regulas repraesentamus:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>16.<\/td>\n<td>q=0 \\rightarrow \\sqrt[n]{q} = 0<\/span><\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>17.<\/td>\n<td>q \\gt 0 \\rightarrow \\left[ \\sqrt[n]{q} = p \\leftrightarrow p^n = q \\right]<\/span><\/td>\n<\/tr>\n<tr style=\"text-align: justify;\">\n<td>18.<\/td>\n<td> \\left[ q \\lt 0 \\wedge n {\\;es\\;impar} \\right]\\rightarrow \\left[ \\sqrt[n]{q} = p \\leftrightarrow p^n = q \\right]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSummatim, n<\/span>-esima radix q<\/span> est numerus p<\/span> talis ut, cum ad n<\/span> elevatur, numerum q<\/span> reddat. His in casibus, cum n=2<\/span>, loco \\sqrt[2]{q}<\/span> scribendi, simpliciter scribimus \\sqrt{q}.<\/span>\n<h3>Apparitio Numerorum Irrationalium<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1216s\" target=\"_blank\" rel=\"noopener\">Huc usque progressi nos interrogamus<\/span><\/strong><\/a> Numne radix n-nesima omnibus elementis \\mathbb{Q}<\/span> bene definita erit? Verum est, quamvis non tam evidens (comparatum cum eo quod in subtractione et divisione visum est), exsistere rationales qui radicem n-nesimam rationalem non habent. Ut hoc videamus sufficit sequentem exemplum considerare:<\/p>\n\\sqrt{2}<\/span> non est numerus rationalis.<\/strong><\/em><\/p>\nDEMONSTRATIO<\/strong><\/p>\nHoc per reductionem ad absurdum probabimus.<\/p>\nSupponamus \\sqrt{2}<\/span> numerum rationalem esse, id est, existere p,q\\in\\mathbb{Z}<\/span>, cum q\\neq 0,<\/span> tales ut \\sqrt{2}=p\/q,<\/span> et praeterea fractionem esse ad formam irreducibilem redactam. Si sic facimus tunc dicere possumus quod<\/p>\n2 = \\left(\\sqrt{2} \\right)^2 =\\displaystyle \\frac{p^2}{q^2} = <\/span> \\left(\\displaystyle \\frac{p}{q}\\right)^2<\/span>\n<\/span><\/p>\nSed hoc in contradictionem incidit cum eo quod p\/q<\/span> in forma irreducibili scriptum erat (nunc enim constat (p\/q)^2<\/span> simplificari posse et eius exitum esse 2). Quia supponere \\sqrt{2}<\/span> rationalem esse contradictionem gignit, ergo hic numerus rationalis esse non potest et, consequenter, irrationale est.<\/p>\n<h3>Expansio ad Numeros Reales<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1514s\" target=\"_blank\" rel=\"noopener\">Haec eventa in lucem ponunt quod,<\/span> <\/strong><\/a>ut radicem n-nesimam recte definiamus necesse est rationales ad novum coniunctum extendere, hoc est coniunctum numerorum realium, quod \\mathbb{R}<\/span> denotamus et quod tam rationales quam irrationales continet<\/p>\n\\mathbb{R}= \\mathbb{Q}\\cup \\mathbb{Q}^*<\/span>\n<a name=\"6\"><\/a><\/p>\n<h2>Complexi: Clausula Algebraica Numerorum Realium<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1532s\" target=\"_blank\" rel=\"noopener\">Hoc loco duo animadvertenda sunt:<\/span><\/strong><\/a> (1) cum n<\/span> par est, radix n-nesima multiplicem valorem habet et, (2) si insuper conamur computare \\sqrt[n]{q}<\/span> cum q\\lt 0,<\/span>, videbimus talem numerum numerum realem esse non posse.<\/p>\nPrimum solvitur definiendo la radicem principalem<\/strong> levem mutationem in articulo (17) de definitione radicis applicando, ita ut res sic se habeat:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>17*.<\/td>\n<td>q\\gt 0 \\rightarrow \\left[ 0\\lt p=\\sqrt[n]{q} \\leftrightarrow p^n=q \\right]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSecundum obtinetur ampliando coniunctum realium ad coniunctum numerorum complexorum \\mathbb{C},<\/span> sed haec constructio in posterum reservabitur.<\/p>\n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"Prima appropinquatio ad Coniuncta Numerica – ToposUranos.com Prima Appropinquatio ad Coniuncta Numerica: A Naturalibus ad Complexos Summarium:In hac lectione perscrutabimur quomodo numeri naturales uti possint fundamentum ad aliorum coniunctuum numerorum constructionem ut certas limitationes operationum superare valeant. Incipiemus a numeris integris, qui nobis permittunt subtractiones late perficere. Deinde progrediemur ad numeros rationales, qui instrumentum divisionis […]<\/p>\n","protected":false},"author":1,"featured_media":25027,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1304,1310,1298],"tags":[],"class_list":["post-33613","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-algebra-generalis","category-mathematica"],"yoast_head":"\n<title>Coniuncta Numerica: A Naturalibus ad Complexos - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Obtine primam cognitionem quomodo construantur coniuncta numerica, a numeris naturalibus usque ad complexos.\" \/>\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\/coniuncta-numerica-a-naturalibus-ad-complexos\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Coniuncta Numerica: A Naturalibus ad Complexos\" \/>\n<meta property=\"og:description\" content=\"Obtine primam cognitionem quomodo construantur coniuncta numerica, a numeris naturalibus usque ad complexos.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/coniuncta-numerica-a-naturalibus-ad-complexos\/\" \/>\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=\"2021-03-16T00:00:35+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-29T07:39:58+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/naturales-captura-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Coniuncta Numerica: A Naturalibus ad Complexos\" \/>\n<meta name=\"twitter:description\" content=\"Obtine primam cognitionem quomodo construantur coniuncta numerica, a numeris naturalibus usque ad complexos.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/naturales-captura.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=\"5 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Coniuncta Numerica: A Naturalibus ad Complexos\",\"datePublished\":\"2021-03-16T00:00:35+00:00\",\"dateModified\":\"2025-07-29T07:39:58+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/\"},\"wordCount\":1303,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/conjuntosnumericos.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Algebra Generalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/coniuncta-numerica-a-naturalibus-ad-complexos\\\/\",\"name\":\"Coniuncta Numerica: A Naturalibus ad Complexos - 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