\u041f\u0435\u0440\u0432\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u043a \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430\u043c – ToposUranos.com<\/title> <\/head> <body> <\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u041f\u0435\u0440\u0432\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u043a \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430\u043c: \u043e\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0434\u043e \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0445<\/h1>\n<\/center><\/p>\n\u0420\u0435\u0437\u044e\u043c\u0435:<\/strong><\/br>\u0412 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c, \u043a\u0430\u043a \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u043c\u043e\u0433\u0443\u0442 \u0441\u043b\u0443\u0436\u0438\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0434\u043b\u044f \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u044f \u0434\u0440\u0443\u0433\u0438\u0445 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0445 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0434\u043e\u043b\u0435\u043d\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u0445 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0439. \u041c\u044b \u043d\u0430\u0447\u043d\u0435\u043c \u0441 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u044e\u0442 \u043d\u0430\u043c \u0448\u0438\u0440\u043e\u043a\u043e \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u0442\u044c \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435. \u0417\u0430\u0442\u0435\u043c \u043c\u044b \u043f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u0430\u043c, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u043b\u043d\u043e\u0441\u0442\u044c\u044e \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0442 \u043d\u0430\u043c \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0434\u0435\u043b\u0435\u043d\u0438\u044f. \u0417\u0430\u0442\u0435\u043c \u043c\u044b \u043f\u043e\u0433\u0440\u0443\u0437\u0438\u043c\u0441\u044f \u0432 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430, \u0447\u0442\u043e\u0431\u044b \u0440\u0430\u0431\u043e\u0442\u0430\u0442\u044c \u0441 \u043a\u043e\u0440\u043d\u044f\u043c\u0438 n-\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0438 \u0443\u043f\u043e\u043c\u044f\u043d\u0435\u043c, \u043a\u0430\u043a \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0432\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u044b\u0445 \u0441\u0446\u0435\u043d\u0430\u0440\u0438\u0435\u0432 \u0441 \u043a\u043e\u0440\u043d\u044f\u043c\u0438 n-\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438. \u0427\u0435\u0440\u0435\u0437 \u044d\u0442\u0438 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0431\u0443\u0434\u0435\u0442 \u043f\u043e\u043d\u044f\u0442\u043d\u043e, \u043a\u0430\u043a \u043a\u0430\u0436\u0434\u043e\u0435 \u043d\u043e\u0432\u043e\u0435 \u0447\u0438\u0441\u043b\u043e\u0432\u043e\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c, \u043f\u0440\u0438\u0441\u0443\u0449\u0438\u0445 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u043c\u0443.<\/em><\/p>\n\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f<\/u>:<\/strong> \u041f\u043e \u043e\u043a\u043e\u043d\u0447\u0430\u043d\u0438\u0438 \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442:<\/p>\n<ol>\n<li>\u0418\u0434\u0435\u043d\u0442\u0438\u0444\u0438\u0446\u0438\u0440\u043e\u0432\u0430\u0442\u044c<\/strong> \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445, \u0446\u0435\u043b\u044b\u0445 \u0438 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b.<\/li>\n<li>\u0418\u043d\u0442\u0435\u0440\u043f\u0440\u0435\u0442\u0438\u0440\u043e\u0432\u0430\u0442\u044c<\/strong> \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043d\u0430\u0441\u043b\u0435\u0434\u0443\u044e\u0442\u0441\u044f \u0438\u043b\u0438 \u0438\u0437\u043c\u0435\u043d\u044f\u044e\u0442\u0441\u044f \u043f\u0440\u0438 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0435 \u043e\u0442 \u043e\u0434\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u043e\u0432\u043e\u0433\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043a \u0434\u0440\u0443\u0433\u043e\u043c\u0443.<\/li>\n<li>\u0421\u0440\u0430\u0432\u043d\u0438\u0442\u044c<\/strong> \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0445 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0438 \u0442\u043e, \u043a\u0430\u043a \u043e\u043d\u0438 \u0441\u0432\u044f\u0437\u0430\u043d\u044b \u0434\u0440\u0443\u0433 \u0441 \u0434\u0440\u0443\u0433\u043e\u043c.<\/li>\n<\/ol>\n\u0421\u041e\u0414\u0415\u0420\u0416\u0410\u041d\u0418\u0415<\/u><\/strong> \n<a href=\"#1\">\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/a> \n<a href=\"#2\">\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/a> \n<a href=\"#3\">\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043e\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u043a \u0446\u0435\u043b\u044b\u043c<\/a> \n<a href=\"#4\">\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u0430\u043c<\/a> \n<a href=\"#5\">\u0412\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0438 \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430<\/a> \n<a href=\"#6\">\u041a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430: \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0437\u0430\u043c\u044b\u043a\u0430\u0442\u0435\u043b\u044c \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/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>\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/h2>\n<div class=\"content\">\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=96s\" target=\"_blank\" rel=\"noopener\">\u0412\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430, \u0432\u043c\u0435\u0441\u0442\u0435 \u0441 \u0434\u0440\u0443\u0433\u0438\u043c\u0438 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u043c\u0438 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430\u043c\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0432 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435,<\/strong><\/span><\/a> \u0432\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u043f\u0443\u0442\u0435\u043c \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u044f \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b. \u0421\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0442\u0430\u043a, \u0447\u0442\u043e \u0441 \u0434\u0432\u0443\u043c\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u044b\u043c\u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438 \u043d\u0435 \u0432\u0441\u0435\u0433\u0434\u0430 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e \u043f\u0440\u043e\u0432\u0435\u0441\u0442\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f \u0438\u043b\u0438 \u0434\u0435\u043b\u0435\u043d\u0438\u044f, \u0438 \u044d\u0442\u0438 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u044f \u043f\u0440\u0438\u0437\u0432\u0430\u043d\u044b \u0440\u0435\u0448\u0438\u0442\u044c \u044d\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443.<\/p>\n\u0412 \u0442\u0435\u0447\u0435\u043d\u0438\u0435 \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c <a href=\"https:\/\/toposuranos.com\/operaciones-con-numeros-naturales\/\" rel=\"noopener\" target=\"_blank\">\u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b,<\/strong><\/a> \u0438 \u043d\u0430 \u044d\u0442\u043e\u0439 \u043e\u0441\u043d\u043e\u0432\u0435 \u043c\u044b \u043f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u043a \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u044e \u0432\u0441\u0435\u0445 \u043e\u0441\u0442\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0445 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432, \u0434\u043e\u0441\u0442\u0438\u0433\u0430\u044f \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u0438 \u0434\u0430\u043b\u044c\u0448\u0435.<\/p>\n<a name=\"2\"><\/a><\/p>\n<h2>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=214s\" target=\"_blank\" rel=\"noopener\">\u0420\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0441 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438,<\/span><\/strong><\/a> \u043c\u044b \u0432 \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u043c \u0438\u043c\u0435\u0435\u043c \u0432 \u0432\u0438\u0434\u0443 \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435, \u0432\u043c\u0435\u0441\u0442\u0435 \u0441 \u0438\u0445 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u043c\u0438 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f\u043c\u0438. \u041d\u0438\u0436\u0435 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u044d\u0442\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430:<\/p>\n\u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 a,b,c\\in\\mathbb{N},<\/span> \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0430\u0435\u0442\u0441\u044f, \u0447\u0442\u043e:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td>1. <\/td>\n<td>a + b = b + a<\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left; color: #000000;\">\n<td>2. <\/td>\n<td>a \\pm (b \\pm c) = (a\\pm b)\\pm c <\/span> (\u0432 \u0441\u043b\u0443\u0447\u0430\u0435 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f \u044d\u0442\u043e \u0432\u0435\u0440\u043d\u043e \u0432\u0441\u0435\u0433\u0434\u0430, \u043a\u043e\u0433\u0434\u0430 \u044d\u0442\u043e \u0445\u043e\u0440\u043e\u0448\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e)\n<\/td>\n<\/tr>\n<tr style=\"text-align: left; color: #000000;\">\n<td>3. <\/td>\n<td>a\\cdot b = b \\cdot a <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: left; color: #000000;\">\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; color: #000000;\">\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; color: #000000;\">\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; color: #000000;\">\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>\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043e\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u043a \u0446\u0435\u043b\u044b\u043c<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=418s\" target=\"_blank\" rel=\"noopener\">\u041f\u0435\u0440\u0432\u043e\u0435, \u0447\u0442\u043e \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u043e\u0442\u043c\u0435\u0442\u0438\u0442\u044c, – \u044d\u0442\u043e \u0442\u043e, \u0447\u0442\u043e \u0432 \u0441\u043b\u0443\u0447\u0430\u0435 \u0441\u0443\u043c\u043c:<\/span><\/strong><\/a> (\\forall a,b\\in\\mathbb{N})(a+b\\in\\mathbb{N})<\/span>, \u0432 \u0442\u043e \u0432\u0440\u0435\u043c\u044f \u043a\u0430\u043a \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f: (\\forall a,b\\in\\mathbb{N})(a+b\\in\\mathbb{N} \\leftrightarrow a\\gt b)<\/span>. \u041f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442, \u043a\u043e\u0433\u0434\u0430 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b a<\/span> \u0438 b<\/span> \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0441\u043c\u044b\u0441\u043b\u0430, \u0435\u0441\u043b\u0438 a\\leq b<\/span>; \u0447\u0442\u043e\u0431\u044b \u0440\u0435\u0448\u0438\u0442\u044c \u044d\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443, \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0440\u0430\u0441\u0448\u0438\u0440\u044f\u044e\u0442\u0441\u044f \u0434\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u0433\u0434\u0435 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f \u044d\u0442\u043e\u0433\u043e \u0442\u0438\u043f\u0430 \u043f\u0440\u0438\u043e\u0431\u0440\u0435\u0442\u0430\u044e\u0442 \u0445\u043e\u0440\u043e\u0448\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435. \u041c\u044b \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0430\u0435\u043c \u044d\u0442\u043e \u043d\u043e\u0432\u043e\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/strong> \u0431\u0443\u043a\u0432\u043e\u0439 \\mathbb{Z}<\/span>, \u0438 \u043e\u043d\u043e \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0438\u0437 \u0432\u0441\u0435\u0445 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u0438\u0445 \u0430\u0434\u0434\u0438\u0442\u0438\u0432\u043d\u044b\u0445 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0445 \u0438 \u043d\u0443\u043b\u044f.<\/p>\n\\mathbb{Z} = \\{\\cdots, -3,-2,-1,0,1,2,3,\\cdots \\}<\/span>\n\u0426\u0435\u043b\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430\u0441\u043b\u0435\u0434\u0443\u044e\u0442 \u0432\u0441\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u0441 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u0435\u043c \u043d\u0430 \u0432\u0442\u043e\u0440\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e, \u0438 \u0432\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u043f\u043e\u043d\u044f\u0442\u0438\u044f \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0433\u043e \u0438 \u043d\u0435\u0439\u0442\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0430\u0434\u0434\u0438\u0442\u0438\u0432\u0430.<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: left; color: #000000;\">\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; color: #000000;\">\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; color: #000000;\">\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>\n\u042d\u043b\u0435\u043c\u0435\u043d\u0442 b=-a<\/span> – \u044d\u0442\u043e \u0442\u043e, \u0447\u0442\u043e \u043c\u044b \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u043c \u0430\u0434\u0434\u0438\u0442\u0438\u0432\u043d\u044b\u043c \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c<\/strong> a<\/span>.<\/p>\n<a name=\"4\"><\/a><\/p>\n<h2>\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u0430\u043c<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=755s\" target=\"_blank\" rel=\"noopener\">\u041d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0443 \u043d\u0430\u0441 \u0435\u0449\u0435 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430, – \u044d\u0442\u043e \u0434\u0435\u043b\u0435\u043d\u0438\u0435.<\/span><\/strong><\/a> \u0427\u0442\u043e\u0431\u044b \u0440\u0435\u0448\u0438\u0442\u044c \u044d\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443, \u043c\u044b \u0440\u0430\u0441\u0448\u0438\u0440\u0438\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u0434\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0431\u0443\u0434\u0435\u0442 \u0437\u0430\u0434\u0430\u043d\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c:<\/p>\n\\mathbb{Q}=\\left\\{a= \\displaystyle\\frac{n}{m}\\;|\\;n,m\\in\\mathbb{Z}\\wedge m\\neq 0 \\right\\}<\/span>\n\u0421 \u044d\u0442\u0438\u043c \u043f\u0440\u0438\u043e\u0431\u0440\u0435\u0442\u0430\u0435\u0442\u0441\u044f \u043d\u043e\u0432\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\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; color: #000000;\">\n<td colspan=\"2\">\u0412\u0441\u0435 \u043d\u0435 \u0440\u0430\u0432\u043d\u044b\u0435 \u043d\u0443\u043b\u044e \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0438\u043c\u0435\u044e\u0442 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e. \u041e\u0431\u0440\u0430\u0442\u043d\u043e\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e a<\/span> – \u044d\u0442\u043e a^{-1}<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u0421 \u044d\u0442\u0438\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438, \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f\u043c\u0438 \u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u043c\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0442\u0441\u044f \u043d\u043e\u0432\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0441\u043e \u0441\u0432\u043e\u0438\u043c\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u043c\u0438. \u0412 \u043d\u0438\u0445 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442\u0441\u044f n-\u0430\u044f \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 q<\/span> \u0447\u0435\u0440\u0435\u0437<\/p>\nq^n = \\underbrace{q\\cdot q \\cdot \\cdots \\cdot q}_{n\\;\u0440\u0430\u0437};<\/span> \u0441 n\\in\\mathbb{N}<\/span>\nq^{-n}= \\displaystyle \\frac{1}{q^n}<\/span>\n\u041e\u0431\u0440\u0430\u0442\u0438\u0442\u0435 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435, \u0447\u0442\u043e, \u043d\u0430\u0447\u0438\u043d\u0430\u044f \u0441 \u044d\u0442\u043e\u0433\u043e, \u0438 \u0432\u0441\u0435\u0433\u0434\u0430 \u043a\u043e\u0433\u0434\u0430 q\\neq 0<\/span>, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e<\/p>\nq^0 = 1<\/span>\n\u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u0441\u044f\u043a\u0438\u0439 \u0440\u0430\u0437, \u043a\u043e\u0433\u0434\u0430 \u043f\u043e\u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043d\u0430 \u043d\u043e\u043b\u044c, \u0434\u0430\u043d\u043d\u044b\u0435 \u0434\u0432\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430 a,b<\/span> , \u0438 \u0434\u0432\u0430 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u043b\u0430 n,m<\/span> \u0431\u0443\u0434\u0443\u0442 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0442\u044c\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\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; color: #000000;\">\n<td>12.<\/td>\n<td>(a^n)^m = a^{n\\cdot m} <\/span>\n<\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\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; color: #000000;\">\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; color: #000000;\">\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>\u0412\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0438 \u0418\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1031s\" target=\"_blank\" rel=\"noopener\">\u0422\u0430\u043a \u0436\u0435 \u043a\u0430\u043a \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f (\u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0435 \u043a \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u044e) \u0438 \u0434\u0435\u043b\u0435\u043d\u0438\u044f<\/span><\/strong><\/a> (\u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0435 \u043a \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044e) \u0441\u0434\u0435\u043b\u0430\u043b\u0438 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u043c \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u0435 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u0434\u043e \u0446\u0435\u043b\u044b\u0445 \u0438 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043d\u043d\u043e, \u0447\u0442\u043e\u0431\u044b \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0445\u043e\u0440\u043e\u0448\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438, \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u044b\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0441 \u0441\u0442\u0435\u043f\u0435\u043d\u044f\u043c\u0438. \u041e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043a n<\/span>-\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f n<\/span>-\u043e\u0439 \u043a\u043e\u0440\u043d\u0435\u043c.<\/p>\n<h3>\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043a\u043e\u0440\u043d\u044f<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1071s\" target=\"_blank\" rel=\"noopener\">\u041f\u0443\u0441\u0442\u044c n<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0446\u0435\u043b\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c \u0431\u043e\u043b\u044c\u0448\u0435 1<\/span><\/strong><\/a> \u0438 p,q<\/span> \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u044b\u043c\u0438 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442\u0441\u044f n<\/span>-\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c \u0438\u0437 q<\/span>, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0445 \u043f\u0440\u0430\u0432\u0438\u043b:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td>16.<\/td>\n<td>q=0 \\rightarrow \\sqrt[n]{q} = 0<\/span><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\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; color: #000000;\">\n<td>18.<\/td>\n<td> \\left[ q \\lt 0 \\wedge n {\\;is\\;odd} \\right]\\rightarrow \\left[ \\sqrt[n]{q} = p \\leftrightarrow p^n = q \\right]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u0412 \u0438\u0442\u043e\u0433\u0435, n<\/span>-\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c \u0438\u0437 q<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0447\u0438\u0441\u043b\u043e\u043c p<\/span>, \u0442\u0430\u043a\u0438\u043c \u0447\u0442\u043e, \u0431\u0443\u0434\u0443\u0447\u0438 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u043d\u044b\u043c \u0432 n<\/span>, \u043e\u043d \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0432\u0430\u043c \u0447\u0438\u0441\u043b\u043e q<\/span>. \u0412 \u044d\u0442\u0438\u0445 \u0441\u043b\u0443\u0447\u0430\u044f\u0445, \u043a\u043e\u0433\u0434\u0430 n=2<\/span>, \u0432\u043c\u0435\u0441\u0442\u043e \u0442\u043e\u0433\u043e \u0447\u0442\u043e\u0431\u044b \u043f\u0438\u0441\u0430\u0442\u044c \\sqrt[2]{q}<\/span>, \u043c\u044b \u043f\u0438\u0448\u0435\u043c \\sqrt{q}<\/span> \u0434\u043b\u044f \u043f\u0440\u043e\u0441\u0442\u043e\u0442\u044b.<\/p>\n<h3>\u041f\u043e\u044f\u0432\u043b\u0435\u043d\u0438\u0435 \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1216s\" target=\"_blank\" rel=\"noopener\">\u041d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435 \u043c\u044b \u0437\u0430\u0434\u0430\u0435\u043c\u0441\u044f \u0432\u043e\u043f\u0440\u043e\u0441\u043e\u043c<\/span><\/strong><\/a> \u0425\u043e\u0440\u043e\u0448\u043e \u043b\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d n-\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c \u0434\u043b\u044f \u0432\u0441\u0435\u0445 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \\mathbb{Q}<\/span>? \u041f\u0440\u0430\u0432\u0434\u0430 \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e, \u0445\u043e\u0442\u044f \u044d\u0442\u043e \u0438 \u043d\u0435 \u0442\u0430\u043a \u043e\u0447\u0435\u0432\u0438\u0434\u043d\u043e (\u043f\u043e \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e \u0441 \u0442\u0435\u043c, \u0447\u0442\u043e \u043c\u044b \u0432\u0438\u0434\u0435\u043b\u0438 \u0441 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435\u043c \u0438 \u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c), \u0435\u0441\u0442\u044c \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043d\u0435 \u0438\u043c\u0435\u044e\u0442 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0433\u043e n-\u043e\u0433\u043e \u043a\u043e\u0440\u043d\u044f. \u0427\u0442\u043e\u0431\u044b \u0443\u0432\u0438\u0434\u0435\u0442\u044c \u044d\u0442\u043e, \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440:<\/p>\n\\sqrt{2}<\/span> \u043d\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c.<\/strong><\/em><\/p>\n\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e<\/strong><\/p>\n\u041c\u044b \u0434\u043e\u043a\u0430\u0436\u0435\u043c \u044d\u0442\u043e \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u043e\u0442 \u043f\u0440\u043e\u0442\u0438\u0432\u043d\u043e\u0433\u043e.<\/p>\n\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \\sqrt{2}<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c, \u0442\u043e \u0435\u0441\u0442\u044c \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 p,q\\in\\mathbb{Z}<\/span>, \u0441 q\\neq 0,<\/span> \u0442\u0430\u043a\u0438\u0435, \u0447\u0442\u043e \\sqrt{2}=p\/q,<\/span> \u0438 \u0447\u0442\u043e \u043a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u044d\u0442\u043e \u0431\u044b\u043b\u043e \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u043e \u0434\u043e \u043d\u0435\u0441\u043e\u043a\u0440\u0430\u0442\u0438\u043c\u043e\u0433\u043e \u0432\u0438\u0434\u0430. \u0415\u0441\u043b\u0438 \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u044d\u0442\u043e, \u0442\u043e\u0433\u0434\u0430 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e<\/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>\n\u041d\u043e \u044d\u0442\u043e \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u0440\u0435\u0447\u0438\u0442 \u0442\u043e\u043c\u0443 \u0444\u0430\u043a\u0442\u0443, \u0447\u0442\u043e p\/q<\/span> \u0431\u044b\u043b\u043e \u0437\u0430\u043f\u0438\u0441\u0430\u043d\u043e \u0432 \u043d\u0435\u0441\u043e\u043a\u0440\u0430\u0442\u0438\u043c\u043e\u0439 \u0444\u043e\u0440\u043c\u0435 (\u0442\u0435\u043f\u0435\u0440\u044c \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0447\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u0443\u043f\u0440\u043e\u0441\u0442\u0438\u0442\u044c (p\/q)^2<\/span> \u0438 \u0435\u0433\u043e \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u043c \u0431\u0443\u0434\u0435\u0442 2). \u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435, \u0447\u0442\u043e \\sqrt{2}<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c, \u043f\u0440\u0438\u0432\u043e\u0434\u0438\u0442 \u043a \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u0440\u0435\u0447\u0438\u044e, \u044d\u0442\u043e \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c, \u0438, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043c\u044b \u0433\u043e\u0432\u043e\u0440\u0438\u043c, \u0447\u0442\u043e \u044d\u0442\u043e \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e.<\/p>\n<h3>\u0420\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u0435 \u0434\u043e \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/h3>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1514s\" target=\"_blank\" rel=\"noopener\">\u042d\u0442\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u0443\u044e\u0442 \u0442\u043e\u0442 \u0444\u0430\u043a\u0442, \u0447\u0442\u043e,<\/span> <\/strong><\/a>\u0447\u0442\u043e\u0431\u044b \u043f\u0440\u0430\u0432\u0438\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c n-\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c, \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0440\u0430\u0441\u0448\u0438\u0440\u0438\u0442\u044c \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0434\u043e \u043d\u043e\u0432\u043e\u0433\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430, \u044d\u0442\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u044b \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0430\u0435\u043c \u043a\u0430\u043a \\mathbb{R}<\/span> \u0438 \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u0442 \u043a\u0430\u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435, \u0442\u0430\u043a \u0438 \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430<\/p>\n\\mathbb{R}= \\mathbb{Q}\\cup \\mathbb{Q}^*<\/span>\n<a name=\"6\"><\/a><\/p>\n<h2>\u041a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430: \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0434\u043e\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b<\/h2>\n<a href=\"https:\/\/www.youtube.com\/watch?v=PfK-pIlyCj4&t=1532s\" target=\"_blank\" rel=\"noopener\">\u041d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435 \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u043e\u0442\u043c\u0435\u0442\u0438\u0442\u044c \u0434\u0432\u0435 \u0432\u0435\u0449\u0438:<\/span><\/strong><\/a> (1) \u043a\u043e\u0433\u0434\u0430 n<\/span> \u0447\u0435\u0442\u043d\u043e\u0435, n-\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f \u043c\u043d\u043e\u0433\u043e\u0437\u043d\u0430\u0447\u043d\u044b\u043c \u0438, (2) \u0435\u0441\u043b\u0438 \u043a \u0442\u043e\u043c\u0443 \u0436\u0435 \u043f\u044b\u0442\u0430\u0442\u044c\u0441\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \\sqrt[n]{q}<\/span> \u0441 q\\lt 0,<\/span> \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c, \u0447\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c.<\/p>\n\u041f\u0435\u0440\u0432\u0430\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u0440\u0435\u0448\u0430\u0435\u0442\u0441\u044f \u043f\u0443\u0442\u0435\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0433\u043b\u0430\u0432\u043d\u043e\u0433\u043e \u043a\u043e\u0440\u043d\u044f<\/strong>, \u0432\u043d\u043e\u0441\u044f \u043d\u0435\u0431\u043e\u043b\u044c\u0448\u043e\u0435 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0432 \u043f\u0443\u043d\u043a\u0442 (17), \u043a\u0430\u0441\u0430\u044e\u0449\u0438\u0439\u0441\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043a\u043e\u0440\u043d\u044f, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0442\u0435\u043f\u0435\u0440\u044c \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\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>\n\u0412\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u0440\u0435\u0448\u0430\u0435\u0442\u0441\u044f \u043f\u0443\u0442\u0435\u043c \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u0434\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \\mathbb{C},<\/span> \u043d\u043e \u044d\u0442\u043e \u0441\u0442\u0440\u043e\u0435\u043d\u0438\u0435 \u043e\u0441\u0442\u0430\u043d\u0435\u0442\u0441\u044f \u0434\u043b\u044f \u0431\u0443\u0434\u0443\u0449\u0435\u0433\u043e.<\/p>\n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"\u041f\u0435\u0440\u0432\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u043a \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430\u043c – ToposUranos.com \u041f\u0435\u0440\u0432\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u043a \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430\u043c: \u043e\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0434\u043e \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0445 \u0420\u0435\u0437\u044e\u043c\u0435:\u0412 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c, \u043a\u0430\u043a \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u043c\u043e\u0433\u0443\u0442 \u0441\u043b\u0443\u0436\u0438\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0434\u043b\u044f \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u044f \u0434\u0440\u0443\u0433\u0438\u0445 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0445 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0434\u043e\u043b\u0435\u043d\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u0445 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0439. \u041c\u044b \u043d\u0430\u0447\u043d\u0435\u043c \u0441 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u044e\u0442 \u043d\u0430\u043c \u0448\u0438\u0440\u043e\u043a\u043e \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u0442\u044c \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435. \u0417\u0430\u0442\u0435\u043c \u043c\u044b \u043f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u0430\u043c, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 […]<\/p>\n","protected":false},"author":1,"featured_media":25040,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":26,"footnotes":""},"categories":[589,573,1041],"tags":[],"class_list":["post-24916","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-589","category-573","category-1041"],"yoast_head":"\n<title>\u0427\u0438\u0441\u043b\u043e\u0432\u044b\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430: \u043e\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0434\u043e \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0445 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u041f\u043e\u043b\u0443\u0447\u0438\u0442\u0435 \u043f\u0435\u0440\u0432\u043e\u0435 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