{"id":25823,"date":"2021-05-20T00:00:13","date_gmt":"2021-05-20T00:00:13","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25823"},"modified":"2025-03-02T19:58:58","modified_gmt":"2025-03-02T19:58:58","slug":"%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%be%e0%a4%95%e0%a5%83%e0%a4%a4%e0%a4%bf%e0%a4%95-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a5%87-%e0%a4%b8%e0%a4%be","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%be%e0%a4%95%e0%a5%83%e0%a4%a4%e0%a4%bf%e0%a4%95-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a5%87-%e0%a4%b8%e0%a4%be\/","title":{"rendered":"\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u091a\u093e\u0932\u0928 \u0914\u0930 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u091a\u093e\u0932\u0928<\/h1>\n<p style=\"text-align:center;\"><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0939\u092e \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0909\u0928\u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u092a\u0930 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902\u0917\u0947, \u092a\u0940\u0928\u094b \u0915\u0947 \u0905\u0915\u094d\u0937\u093e\u0902\u0936\u094b\u0902 \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u091c\u094b\u0921\u093c, \u0917\u0941\u0923\u0928 \u0914\u0930 \u0936\u0915\u094d\u0924\u093f \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f \u0914\u0930 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0938\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f\u0964 \u0939\u092e \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u0947 \u092e\u0941\u0916\u094d\u092f \u0917\u0941\u0923\u094b\u0902 \u091c\u0948\u0938\u0947 \u0915\u093f \u0938\u093e\u092e\u094d\u092f\u0924\u093e, \u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e, \u0935\u093f\u0924\u0930\u0923 \u0914\u0930 \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u090f\u0935\u0902 \u0909\u0932\u091f\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0915\u0947 \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u093e \u092a\u0930\u0940\u0915\u094d\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0917\u0923\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u0923 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0939\u092e \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0914\u0930 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u093e \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u0928 \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0907\u0938\u0915\u0947 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924, \u0939\u092e \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u093e \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947, \u091c\u093f\u0938\u092e\u0947\u0902 \u0924\u094d\u0930\u093f\u0917\u0941\u0923\u093f\u0924\u093e \u0915\u093e \u0928\u093f\u092f\u092e \u0914\u0930 \u092a\u093e\u0930\u0917\u092e\u094d\u092f\u0924\u093e \u0914\u0930 \u0938\u094d\u0925\u093e\u092f\u093f\u0924\u094d\u0935 \u0915\u0947 \u0917\u0941\u0923 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902, \u0907\u0928 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u094b \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925\u0964 \u0905\u0902\u0924 \u092e\u0947\u0902, \u0939\u092e \u0909\u0932\u091f\u0947 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 (\u0918\u091f\u093e\u0935 \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928) \u0915\u093e \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902\u0917\u0947 \u0914\u0930 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0936\u0915\u094d\u0924\u093f \u0914\u0930 \u0909\u0928\u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u092a\u0924\u093e \u0932\u0917\u093e\u090f\u0902\u0917\u0947\u0964\n<\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>\u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0938\u092e\u093e\u092a\u094d\u0924 \u0939\u094b\u0928\u0947 \u092a\u0930 \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947:\n<\/p>\n<p style=\"text-align:justify;\">\n<ol>\n<li><strong>\u0938\u092e\u091d\u0947\u0902<\/strong> \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f \u0914\u0930 \u0917\u0941\u0923\u0964<\/li>\n<li><strong>\u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902<\/strong> \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923, \u091c\u0948\u0938\u0947 \u0915\u093f \u0938\u093e\u092e\u094d\u092f\u0924\u093e, \u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e, \u0935\u093f\u0924\u0930\u0923, \u0914\u0930 \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u090f\u0935\u0902 \u0909\u0932\u091f\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0915\u0947 \u0928\u093f\u092f\u092e\u0964<\/li>\n<li><strong>\u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902<\/strong> \u0917\u0923\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u0923 \u0915\u094b \u0938\u0930\u0932 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0914\u0930 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u0928 \u0915\u0947 \u0932\u093f\u090f\u0964<\/li>\n<li><strong>\u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0930\u0947\u0902<\/strong> \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0915\u094d\u0930\u092e \u0917\u0941\u0923, \u091c\u0948\u0938\u0947 \u0924\u094d\u0930\u093f\u0917\u0941\u0923\u093f\u0924\u093e \u0915\u093e \u0928\u093f\u092f\u092e \u0914\u0930 \u092a\u093e\u0930\u0917\u092e\u094d\u092f\u0924\u093e \u0914\u0930 \u0938\u094d\u0925\u093e\u092f\u093f\u0924\u094d\u0935 \u0915\u0947 \u0917\u0941\u0923\u0964<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u0935\u093f\u0937\u092f \u0938\u0942\u091a\u0940<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f<\/a><br \/>\n<a href=\"#2\">\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u094d\u0930\u092e<\/a><br \/>\n<a href=\"#3\">\u0909\u0932\u091f\u0947 \u0938\u0902\u091a\u093e\u0932\u0928: \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0918\u091f\u093e\u0935 \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928<\/a><br \/>\n<a href=\"#4\">\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0936\u0915\u094d\u0924\u093f<\/a><br \/>\n<a href=\"#5\">\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924 \u0914\u0930 \u0939\u0932 \u0915\u093f\u090f \u0917\u090f \u0938\u092e\u0938\u094d\u092f\u093e\u090f\u0902<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/jKD71TjMC4s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">\u0939\u093e\u0932\u093e\u0902\u0915\u093f \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u091a\u093e\u0932\u0928 \u091c\u094d\u091e\u093e\u0924 \u0939\u0948\u0902, \u0907\u0938 \u091c\u094d\u091e\u093e\u0928 \u0915\u094b \u00ab\u0925\u094b\u0921\u093c\u0947 \u0905\u0927\u093f\u0915 \u0917\u0923\u093f\u0924\u0940\u092f \u0924\u0930\u0940\u0915\u0947 \u0938\u0947\u00bb \u0938\u0902\u0915\u094d\u0937\u0947\u092a\u093f\u0924 \u0915\u0930\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u0939\u092e \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u091c\u094b\u0921\u093c, \u0917\u0941\u0923\u0928 \u0914\u0930 \u0936\u0915\u094d\u0924\u093f \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0914\u0930 \u0909\u0928\u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u093e \u092e\u0942\u0932<\/h2>\n<h3>\u091c\u094b\u0921\u093c \u0915\u093e \u0938\u0902\u091a\u093e\u0932\u0928<\/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>\u091c\u094b\u0921\u093c \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0915\u093e \u092a\u094d\u0930\u093e\u0930\u0902\u092d \u0939\u092e\u0928\u0947<\/strong><\/span><\/a> <a href=\"http:\/\/toposuranos.com\/material\/hi\/%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%be%e0%a4%95%e0%a5%83%e0%a4%a4%e0%a4%bf%e0%a4%95-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%8f%e0%a4%82-%e0%a4%94%e0%a4%b0-%e0%a4%aa%e0%a5%80\/\" rel=\"noopener\" target=\"_blank\"><strong>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u092a\u0940\u0928\u094b \u0915\u0947 \u0905\u0915\u094d\u0937\u093e\u0902\u0936\u094b\u0902 \u092a\u0930 \u0935\u0930\u094d\u0917 \u092e\u0947\u0902 \u0926\u0947\u0916\u093e,<\/strong><\/a> \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u090f\u0915 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0915\u093e \u0909\u0924\u094d\u0924\u0930\u093e\u0927\u093f\u0915\u093e\u0930\u0940 \u092d\u0940 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/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;\">\u091c\u0948\u0938\u093e \u0915\u093f \u0939\u092e\u0928\u0947 \u0915\u0939\u093e \u0925\u093e <span class=\"katex-eq\" data-katex-display=\"false\">2=S(1), 3=S(2), 4=S(3), \\cdots <\/span><\/bdi> \u0914\u0930 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0906\u0917\u0947, \u0924\u094b \u0939\u092e \u091c\u094b\u0921\u093c \u0915\u094b \u0909\u0924\u094d\u0924\u0930\u093e\u0927\u093f\u0915\u093e\u0930\u0940 \u0938\u0902\u091a\u093e\u0932\u0928 \u0915\u0947 \u0932\u0917\u093e\u0924\u093e\u0930 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0938\u092e\u091d \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/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;\">\u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903:<\/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\\;\u092c\u093e\u0930} n)\\cdots)) <\/span><\/bdi><\/p>\n<h4>\u091c\u094b\u0921\u093c \u0915\u0947 \u0917\u0941\u0923<\/h4>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <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;\">\u0924\u094b \u0907\u0938\u0938\u0947 \u0939\u092e \u091c\u094b\u0921\u093c \u0915\u0947 \u0935\u094b \u0938\u092d\u0940 \u0917\u0941\u0923 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0939\u092e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948\u0902:<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>\u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f\u0924\u093e<\/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>\u0938\u0939\u092f\u094b\u0917\u093f\u0924\u093e<\/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>\u0938\u0930\u0932\u0940\u0915\u0930\u0923<\/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;\">\u092f\u0947 \u0938\u092d\u0940 \u0917\u0941\u0923 \u0906\u0935\u0930\u094d\u0924\u0940 \u092a\u094d\u0930\u092e\u093e\u0923\u0940\u0915\u0930\u0923 \u0926\u094d\u0935\u093e\u0930\u093e \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u092a\u0930 \u0939\u092e \u092f\u0939 \u0915\u093e\u092e \u091b\u094b\u0921\u093c \u0926\u0947\u0902\u0917\u0947\u0964 \u092b\u093f\u0930 \u092d\u0940, \u092e\u0948\u0902 \u0924\u0941\u092e\u094d\u0939\u0947\u0902 \u0907\u0938\u0947 \u0906\u0935\u0930\u094d\u0924\u0940 \u0924\u0915\u0928\u0940\u0915 \u0915\u093e \u0905\u092d\u094d\u092f\u093e\u0938 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0915\u094b\u0936\u093f\u0936 \u0915\u0930\u0928\u0947 \u0915\u0940 \u092a\u094d\u0930\u0947\u0930\u0923\u093e \u0926\u0947\u0924\u093e \u0939\u0942\u0901\u0964<\/p>\n<h3>\u0917\u0941\u0923\u0928 \u0915\u093e \u0938\u0902\u091a\u093e\u0932\u0928<\/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;\">\u0907\u0938\u0940 \u0924\u0930\u0939,<\/span> <\/strong><\/a>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0917\u0941\u0923\u0928 \u092f\u094b\u0917 \u0915\u0947 \u0932\u0917\u093e\u0924\u093e\u0930 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938<\/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\\;\u092c\u093e\u0930}<\/span><\/bdi><\/p>\n<h4>\u0917\u0941\u0923\u0928 \u0915\u0947 \u0917\u0941\u0923<\/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;\">\u0914\u0930 \u0907\u0938\u0940 \u0924\u0930\u0939<\/span><\/strong><\/a> \u0907\u0938\u0915\u0947 \u0917\u0941\u0923 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>\u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f\u0924\u093e<\/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>\u0938\u0939\u092f\u094b\u0917\u093f\u0924\u093e<\/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>\u0938\u0930\u0932\u0940\u0915\u0930\u0923<\/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;\">\u0914\u0930 \u0907\u0938\u0915\u0947 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924, \u0917\u0941\u0923\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947 \u0939\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u00ab1\u00bb \u0915\u094b <strong>\u0907\u0915\u093e\u0908<\/strong> \u0915\u093e \u0917\u0941\u0923 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>\u0907\u0915\u093e\u0908<\/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>\u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u093e \u0938\u0902\u092f\u0941\u0915\u094d\u0924<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u091c\u092c \u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0938\u0902\u092f\u0941\u0915\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092f\u094b\u0917 \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0917\u0941\u0923\u0928 \u0915\u093e \u0935\u093f\u0924\u0930\u0923 \u0917\u0941\u0923 \u092e\u093f\u0932\u0924\u093e \u0939\u0948<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>\u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e<\/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>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u094d\u0930\u092e<\/h2>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0938\u0902\u091a\u093e\u0932\u0928 \u0938\u0947, \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u090f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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> <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0938\u0947 \u0915\u092e \u0939\u0948<\/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> <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0938\u0947 \u0905\u0927\u093f\u0915 \u0939\u0948<\/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>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u0915\u094d\u0930\u092e \u0915\u0947 \u0917\u0941\u0923<\/h3>\n<h4>\u0924\u094d\u0930\u093f\u0915\u094b\u091f\u092e\u0940 \u0915\u093e \u0928\u093f\u092f\u092e<\/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>\u0907\u0938\u0938\u0947 \u092f\u0939 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948 \u0915\u093f \u0915\u0947\u0935\u0932<\/strong><\/span><\/a> \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0940\u0928 \u092a\u0930\u093f\u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u0939\u0940 \u0938\u0902\u092d\u0935 \u0939\u0948:<\/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;\">\u092f\u0926\u093f \u0910\u0938\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f, \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0938\u0947 \u0915\u092e \u0928\u0939\u0940\u0902 \u0939\u0948, \u0924\u094b \u092f\u0939 \u0926\u094b \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f: \u092f\u093e \u0924\u094b <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a=b<\/span><\/bdi>, \u092f\u093e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, \u0905\u0930\u094d\u0925\u093e\u0924 \u092c\u0921\u093c\u093e \u092f\u093e \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0914\u0930 \u0907\u0938\u0947 \u0932\u093f\u0916\u093e \u091c\u093e\u090f\u0917\u093e: <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\geq b.<\/span><\/bdi> \u0914\u0930 \u0907\u0938\u0940 \u0924\u0930\u0939 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> \u091c\u092c \u092f\u0939 \u0915\u092e \u092f\u093e \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4>\u0938\u0902\u0915\u094d\u0930\u092e\u0923\u0940\u092f \u0917\u0941\u0923<\/h4>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span> \u0914\u0930 <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>\u0915\u094b\u0908 \u092d\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0902 \u0939\u094b\u0902, \u0924\u094b \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u0948 \u0915\u093f:<\/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;\">\u0914\u0930 \u0907\u0938\u0940 \u0924\u0930\u0939:<\/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>\u092e\u094b\u0928\u094b\u091f\u094b\u0928\u0940 \u0915\u093e \u0917\u0941\u0923<\/h4>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0926\u094b\u0928\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092e\u094b\u0928\u094b\u091f\u094b\u0928\u0940 \u0917\u0941\u0923 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>\u092f\u094b\u0917 \u0915\u0940 \u092e\u094b\u0928\u094b\u091f\u094b\u0928\u0940<\/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>\u0917\u0941\u0923\u0928 \u0915\u0940 \u092e\u094b\u0928\u094b\u091f\u094b\u0928\u0940<\/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>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0935\u093f\u092a\u0930\u0940\u0924 \u0938\u0902\u091a\u093e\u0932\u0928: \u0918\u091f\u093e\u0935 \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928<\/h2>\n<h3>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0918\u091f\u093e\u0935<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <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;\">\u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0905\u0902\u0924\u0930 (\u0909\u0938 \u0915\u094d\u0930\u092e \u092e\u0947\u0902), \u091c\u093f\u0938\u0947 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b<\/span><\/bdi> \u0932\u093f\u0916\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948<\/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;\">\u091c\u0948\u0938\u093e \u0915\u093f \u0939\u092e \u0926\u0947\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u0915\u0947\u0935\u0932 \u0938\u0924\u094d\u092f \u0939\u094b\u0917\u093e \u092f\u0926\u093f <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0926\u093f <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> \u0939\u094b, \u0924\u094b \u0910\u0938\u093e \u0915\u094b\u0908 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\in \\mathbb{N}<\/span><\/bdi> \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u093f\u0938\u0915\u0947 \u0938\u093e\u0925 \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u092a\u0942\u0930\u093e \u0939\u094b \u0938\u0915\u0947\u0964<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0918\u091f\u093e\u0935 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0928\u093f\u092f\u092e \u0939\u0948 \u00ab\u091c\u094b \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0947 \u090f\u0915 \u0924\u0930\u092b \u091c\u0941\u0921\u093c \u0930\u0939\u093e \u0939\u0948, \u0935\u0939 \u0926\u0942\u0938\u0930\u0940 \u0924\u0930\u092b \u0938\u0947 \u0918\u091f\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0935\u093f\u092a\u0930\u0940\u0924\u00bb\u0964<\/p>\n<h3>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0935\u093f\u092d\u093e\u091c\u0928<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <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>\u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f<\/strong><\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0915\u0947 \u092c\u0940\u091a \u0935\u093f\u092d\u093e\u091c\u0928 (\u0909\u0938 \u0915\u094d\u0930\u092e \u092e\u0947\u0902), \u091c\u093f\u0938\u0947 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi> \u0932\u093f\u0916\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948<\/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;\">\u0935\u093f\u092d\u093e\u091c\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0928\u093f\u092f\u092e \u0939\u0948 \u00ab\u091c\u094b \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0947 \u090f\u0915 \u0924\u0930\u092b \u0917\u0941\u0923\u093e \u0915\u0930 \u0930\u0939\u093e \u0939\u0948, \u0935\u0939 \u0926\u0942\u0938\u0930\u0940 \u0924\u0930\u092b \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0935\u093f\u092a\u0930\u0940\u0924\u00bb\u0964<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u091c\u0948\u0938\u0947 \u0915\u093f \u0918\u091f\u093e\u0935 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a - b<\/span><\/bdi> \u0915\u0947 \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u0915\u0947 \u0932\u093f\u090f \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948 \u0915\u093f <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi> \u0939\u094b, \u0935\u093f\u092d\u093e\u091c\u0928 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi> \u0915\u0947 \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u0915\u0947 \u0932\u093f\u090f \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0926\u094d\u0935\u093e\u0930\u093e \u00ab\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u00bb \u0939\u094b\u0964 \u0907\u0938\u0947 \u0939\u092e \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0926\u094d\u0935\u093e\u0930\u093e \u0935\u093f\u092d\u093e\u091c\u094d\u092f \u0939\u0948 <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>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0936\u0915\u094d\u0924\u093f\u092f\u093e\u0902<\/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>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0939\u092e \u0936\u0915\u094d\u0924\u093f\u092f\u093e\u0902 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong> <\/span><\/a>\u090f\u0915 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> \u0915\u094b, \u091c\u093f\u0938\u0947 \u0939\u092e \u0906\u0927\u093e\u0930 \u0915\u0939\u0924\u0947 \u0939\u0948\u0902, \u0926\u0942\u0938\u0930\u0947 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">n,<\/span> \u0915\u094b, \u091c\u093f\u0938\u0947 \u0939\u092e \u0918\u093e\u0924\u093e\u0902\u0915 \u0915\u0939\u0924\u0947 \u0939\u0948\u0902, \u0909\u0924\u094d\u0925\u093e\u0928 \u0915\u0930\u0928\u093e \u0905\u0930\u094d\u0925\u093e\u0924 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u092c\u093e\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0915\u093e \u0917\u0941\u0923\u093e \u0915\u0930\u0928\u093e \u0939\u0948\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930<\/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\\;\u092c\u093e\u0930}<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n,m\\in\\mathbb{N},<\/span><\/bdi> \u0924\u094b \u0926\u094b\u0939\u0930\u0947 \u0906\u0935\u0930\u094d\u0924\u0928 (\u0907\u0902\u0921\u0915\u094d\u0936\u0928) \u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923 \u0926\u093f\u0916\u093e\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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> \u091c\u092c \u0924\u0915 \u0915\u093f <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>\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924 \u0914\u0930 \u0939\u0932 \u0915\u093f\u090f \u0917\u090f \u0938\u092e\u0938\u094d\u092f\u093e\u090f\u0902<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\u092f\u0939\u093e\u0901 \u0926\u093f\u0916\u093e\u0908 \u0917\u0908 \u0938\u092d\u0940 \u0938\u0902\u092a\u0924\u094d\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u0917\u0923\u093f\u0924\u0940\u092f \u0906\u0935\u0930\u094d\u0924\u0940 (\u0938\u093e\u0927\u093e\u0930\u0923 \u092f\u093e \u0926\u094b\u0939\u0930\u0940) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u092e\u0948\u0902\u0928\u0947 \u0909\u0928\u094d\u0939\u0947\u0902 \u0935\u093f\u0915\u0938\u093f\u0924 \u0928\u0939\u0940\u0902 \u0915\u093f\u092f\u093e \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092a\u094d\u0930\u092e\u093e\u0923 \u0907\u0928 \u0905\u0902\u0924\u0930\u094d\u091c\u094d\u091e\u093e\u0928\u093f\u0915 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0905\u0928\u093e\u0935\u0936\u094d\u092f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0932\u0902\u092c\u093e \u0939\u094b\u0924\u093e\u0964 \u0939\u093e\u0932\u093e\u0902\u0915\u093f, \u091c\u094b \u0907\u0928 \u0915\u0915\u094d\u0937\u093e\u0913\u0902 \u0915\u093e \u0905\u0928\u0941\u0938\u0930\u0923 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0935\u0947 \u0907\u0928 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u094b \u0905\u092d\u094d\u092f\u093e\u0938 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0915\u094b\u0936\u093f\u0936 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 <strong>[\u0915\u0947\u0935\u0932 \u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924]<\/strong><\/li>\n<li>\u0915\u094d\u092f\u093e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{n^m}<\/span><\/bdi> (\u091c\u094b <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{(n^m)})<\/span><\/bdi> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948) \u0914\u0930 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(b^n)^m<\/span><\/bdi> \u0938\u092e\u093e\u0928 \u0939\u0948\u0902? <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1298s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li>\u0926\u0947\u0916\u0940 \u0917\u0908 \u0938\u0902\u092a\u0924\u094d\u0924\u093f\u092f\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u0940 \u091c\u093e\u0902\u091a \u0915\u0930\u0947\u0902:<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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>; \u092f\u0926\u093f <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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>; \u092f\u0926\u093f <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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/strong><\/span><\/a>&nbsp;<\/li>\n<li>\u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f <\/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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^2 = a^2 - 2ab + b^2<\/span><\/bdi>; \u092f\u0926\u093f <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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/strong><\/span><\/a> <\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(a-b) = a^2-b^2<\/span>; \u092f\u0926\u093f <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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>; \u092f\u0926\u093f <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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/strong><\/span><\/a><\/p>\n<p>&nbsp;<\/li>\n<li>\u092a\u0942\u0930\u094d\u0923 \u0906\u0935\u0930\u094d\u0924\u0940 \u0926\u094d\u0935\u093e\u0930\u093e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092a\u0924\u094d\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902:<\/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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/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>[\u0938\u092e\u093e\u0927\u093e\u0928]<\/strong><\/span><\/a>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u091a\u093e\u0932\u0928 \u0938\u093e\u0930\u093e\u0902\u0936: \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0939\u092e \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0909\u0928\u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u092a\u0930 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902\u0917\u0947, \u092a\u0940\u0928\u094b \u0915\u0947 \u0905\u0915\u094d\u0937\u093e\u0902\u0936\u094b\u0902 \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u091c\u094b\u0921\u093c, \u0917\u0941\u0923\u0928 \u0914\u0930 \u0936\u0915\u094d\u0924\u093f \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f \u0914\u0930 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0938\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f\u0964 \u0939\u092e \u0938\u0902\u091a\u093e\u0932\u0928\u094b\u0902 \u0915\u0947 \u092e\u0941\u0916\u094d\u092f \u0917\u0941\u0923\u094b\u0902 \u091c\u0948\u0938\u0947 \u0915\u093f \u0938\u093e\u092e\u094d\u092f\u0924\u093e, \u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e, \u0935\u093f\u0924\u0930\u0923 \u0914\u0930 \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u090f\u0935\u0902 [&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":0,"footnotes":""},"categories":[577,593,1039],"tags":[],"class_list":["post-25823","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-593","category-1039"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 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