{"id":27912,"date":"2026-09-06T22:12:28","date_gmt":"2026-09-06T16:42:28","guid":{"rendered":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/?p=27912"},"modified":"2026-09-06T22:12:28","modified_gmt":"2026-09-06T16:42:28","slug":"%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%a3%e0%a4%be%e0%a4%82%e0%a4%95%e0%a5%8b%e0%a4%82-%e0%a4%95%e0%a5%87-%e0%a4%b8%e0%a4%be%e0%a4%a5-%e0%a4%af%e0%a5%8b%e0%a4%97-%e0%a4%94%e0%a4%b0-%e0%a4%97","status":"publish","type":"post","link":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%a3%e0%a4%be%e0%a4%82%e0%a4%95%e0%a5%8b%e0%a4%82-%e0%a4%95%e0%a5%87-%e0%a4%b8%e0%a4%be%e0%a4%a5-%e0%a4%af%e0%a5%8b%e0%a4%97-%e0%a4%94%e0%a4%b0-%e0%a4%97\/","title":{"rendered":"\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0940 \u092a\u095c\u0924\u093e\u0932"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0940 \u092a\u095c\u0924\u093e\u0932 \u0915\u0940 \u0907\u0938 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u093e \u092f\u0939 \u0924\u0940\u0938\u0930\u093e \u0932\u0947\u0916 \u0939\u0948\u0964 (i) \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 (ii) \u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902<sup>1<\/sup> (Non-negative rational numbers) \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u0907\u0938 \u0932\u0947\u0916 \u092e\u0947\u0902 \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0907\u0938 \u0936\u0943\u0902\u0916\u0932\u093e, \u0907\u0938\u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0914\u0930 \u0907\u0938\u0938\u0947 \u0939\u092e\u093e\u0930\u0940 \u091c\u094b \u0906\u0915\u093e\u0902\u0915\u094d\u0937\u093e\u090f\u0901 \u0939\u0948\u0902 \u0909\u0928 \u092a\u0930 \u091a\u093f\u0928\u094d\u0924\u0928 \u0915\u0930\u0928\u0947 \u0915\u093e \u092f\u0939 \u090f\u0915 \u0905\u091a\u094d\u091b\u093e \u0938\u092e\u092f \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"fs-6 wp-block-paragraph\"><sup>1<\/sup>\u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u092e\u0947\u0902 \u092d\u093f\u0928\u094d\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0906\u0917\u0947 \u0938\u0947 \u0939\u092e \u0907\u0938 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u094b \u092d\u093f\u0928\u094d\u0928 \u0915\u0939\u0947\u0902\u0917\u0947\u0964<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u0939\u092e \u092f\u0939 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u094d\u092f\u094b\u0902 \u0915\u0930 \u0930\u0939\u0947 \u0939\u0948\u0902?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0938 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u093e \u092e\u0942\u0932 \u0935\u093f\u091a\u093e\u0930 <em>\u090f\u091f \u0930\u093e\u0907\u091f \u090f\u0902\u0917\u0932\u094d\u0938<\/em> \u0915\u0947 \u092e\u093e\u0930\u094d\u091a 2013 \u0915\u0947 \u0905\u0902\u0915 \u092e\u0947\u0902 \u2018\u0917\u0941\u0923\u0928\u2019 \u092a\u0930 \u092a\u094d\u0930\u0915\u093e\u0936\u093f\u0924 \u092a\u0941\u0932\u0906\u0909\u091f \u0938\u0947 \u0906\u092f\u093e \u0925\u093e\u0964 \u0909\u0938 \u092a\u0941\u0932\u0906\u0909\u091f \u092e\u0947\u0902 \u092a\u0926\u094d\u092e\u092a\u094d\u0930\u093f\u092f\u093e \u0936\u093f\u0930\u093e\u0932\u0940 \u0928\u0947 \u092c\u0924\u093e\u092f\u093e \u0925\u093e \u0915\u093f \u0917\u0941\u0923\u0928 \u0915\u0947 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f, \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0914\u0930 \u0935\u093f\u0924\u0930\u0923 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u0947 \u0932\u093f\u090f \u0926\u0943\u0936\u094d\u092f \u0930\u0942\u092a \u0938\u0947 (visually) \u0915\u0948\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0928\u094d\u0939\u0947\u0902 \u0938\u0939\u0940 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0940 \u091c\u094d\u093c\u092f\u093e\u0926\u093e \u0932\u094b\u0915\u092a\u094d\u0930\u093f\u092f \u0935\u093f\u0927\u093f \u092e\u0947\u0902 \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0926\u094b (\u092f\u093e \u0924\u0940\u0928) \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0932\u0947\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u091c\u093f\u0938 \u0938\u0902\u0916\u094d\u092f\u093e \u0924\u0925\u094d\u092f (number fact) \u0915\u094b \u0939\u092e \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902 (\u091c\u0948\u0938\u0947 \u0915\u093f 3 \u00d7 7 = 7 \u00d7 3, (5 \u00d7 4) \u00d7 9 = 5 \u00d7 (4 \u00d7 9) \u0906\u0926\u093f) \u0909\u0938\u0915\u0947 \u092c\u093e\u090f\u0901 \u092a\u0915\u094d\u0937 (LHS) \u0914\u0930 \u0926\u093e\u090f\u0901 \u092a\u0915\u094d\u0937 (RHS) \u0926\u094b\u0928\u094b\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u091c\u093e\u0901\u091a\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0935\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948\u0902\u0964 \u092f\u0939 \u0906\u0917\u092e\u0928 \u0935\u093f\u0927\u093f (inductive process) \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u092a\u0926\u094d\u092e\u092a\u094d\u0930\u093f\u092f\u093e \u0915\u0940 \u0935\u093f\u0927\u093f \u0915\u094b \u0926\u094b \u092f\u093e \u0924\u0940\u0928 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0932\u093f\u090f \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u092b\u093f\u0930 \u092d\u0932\u0947 \u0939\u0940 \u0935\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0915\u093f\u0924\u0928\u0940 \u092d\u0940 \u092c\u095c\u0940 \u0915\u094d\u092f\u094b\u0902 \u0928 \u0939\u094b\u0902\u0964 \u092f\u0926\u093f \u0939\u092e \u0907\u0938 \u092c\u093e\u0924 \u0938\u0947 \u0938\u0939\u092e\u0924 \u0939\u0948\u0902 \u0915\u093f \u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0909\u0924\u0928\u0947 \u0939\u0940 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0924\u094b \u0909\u0928\u0915\u0940 \u0935\u093f\u0927\u093f\u092f\u093e\u0901 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0932\u093f\u090f \u0915\u093e\u092e \u0915\u0930\u0947\u0902\u0917\u0940\u0964 \u092c\u0921\u093c\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093e \u0915\u0930\u0928\u093e \u092e\u0941\u0936\u094d\u0915\u093f\u0932 \u092f\u093e \u0905\u0938\u092e\u094d\u092d\u0935 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0930\u0942\u092a \u0938\u0947 \u0907\u0938\u0915\u0940 \u0915\u0932\u094d\u092a\u0928\u093e \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u0909\u0928\u0915\u0940 \u0935\u093f\u0927\u093f\u092f\u093e\u0901 \u0906\u092a\u0915\u094b \u0938\u094d\u0925\u093f\u0924\u093f-\u0926\u0930-\u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u092f\u093e\u0928\u0940 \u0915\u093f \u0906\u0917\u092e\u0928 \u0935\u093f\u0927\u093f \u0938\u0947 \u092e\u0941\u0915\u094d\u0924 \u0915\u0930\u0924\u0940 \u0939\u0948\u0902 \u0914\u0930 \u0928\u093f\u0917\u092e\u0928\u093f\u0915 (deductive) \u0935\u093f\u0927\u093f \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u0947\u0930\u093f\u0924 \u0915\u0930\u0924\u0940 \u0939\u0948\u0902\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u0915\u0940 \u092f\u094b\u0917\u094d\u092f\u0924\u093e \u0907\u0928 \u0926\u0943\u0936\u094d\u092f \u092a\u0926\u094d\u0927\u0924\u093f\u092f\u094b\u0902 \u0915\u0947 \u0928\u093f\u0917\u092e\u0928\u093f\u0915 \u092a\u0939\u0932\u0942 \u0915\u0947 \u0915\u0947\u0928\u094d\u0926\u094d\u0930 \u092e\u0947\u0902 \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0924\u094b, \u0939\u092e \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092f\u094b\u0917 \u0915\u0947 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f \u0914\u0930 \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0907\u0938 \u092a\u0926\u094d\u0927\u0924\u093f \u0915\u0940 \u092a\u0921\u093c\u0924\u093e\u0932 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0925\u0947\u0964 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u0939\u092e \u0905\u0928\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0905\u0930\u094d\u0925\u093e\u0924 (i) \u092d\u093f\u0928\u094d\u0928 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 (\u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0939\u093f\u0924) \u092f\u093e\u0928\u0940 \u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902, (ii) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902, (iii) \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 (iv) \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u2014 \u0938\u0902\u0915\u094d\u0937\u0947\u092a \u092e\u0947\u0902 \u0915\u0939\u0947\u0902 \u0924\u094b, \u0935\u0947 \u0938\u092d\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u091c\u093f\u0928\u0938\u0947 \u092e\u093e\u0927\u094d\u092f\u092e\u093f\u0915 \u0938\u094d\u0924\u0930 \u0924\u0915 \u0915\u0947 \u092c\u091a\u094d\u091a\u0947 \u0930\u0942\u092c\u0930\u0942 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u2014 \u0915\u0947 \u0938\u093e\u0925 \u092f\u094b\u0917 \u090f\u0935\u0902 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0932\u093f\u090f \u0907\u0928 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0940 \u092a\u0921\u093c\u0924\u093e\u0932 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0925\u0947\u0964 \u0907\u0928\u092e\u0947\u0902 \u0938\u0947 \u0915\u0941\u091b \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0939\u092e\u0947\u0902 \u0915\u0941\u091b \u092e\u0941\u0926\u094d\u0926\u094b\u0902 \u0915\u093e \u0938\u093e\u092e\u0928\u093e \u0915\u0930\u0928\u093e \u092a\u0921\u093c\u093e\u0964 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u0947 \u0905\u0928\u094d\u0924 \u092e\u0947\u0902 \u0939\u092e \u0909\u0928 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0938\u093e\u0925 \u0939\u0940 \u0915\u0941\u091b \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0939\u092e \u2018<em>\u0932\u0917\u092d\u0917<\/em> \u092a\u094d\u0930\u092e\u093e\u0923\u2019 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 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class=\"wp-block-paragraph\">\u092f\u0939 \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u093e \u092e\u0939\u0924\u094d\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948 \u0915\u093f \u0935\u093f\u0936\u093f\u0937\u094d\u091f \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u093e \u0935\u093f\u091c\u093c\u0941\u0905\u0932\u093e\u0907\u091c\u093c\u0947\u0936\u0928 \u0909\u0928 \u0926\u0943\u0936\u094d\u092f\u094b\u0902 \u0938\u0947 \u092d\u093f\u0928\u094d\u0928 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0935\u093f\u091a\u093e\u0930\u093e\u0927\u0940\u0928 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0924\u0915 \u092c\u0922\u093c\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092c\u093e\u0926 \u0935\u093e\u0932\u093e \u092e\u0942\u0932\u0924: \u0936\u092c\u094d\u0926\u0930\u0939\u093f\u0924 \u092a\u094d\u0930\u092e\u093e\u0923 (Proof Without Words &#8211; PWW) \u0939\u0948\u0964 \u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u0915\u0908 \u0932\u094b\u0917 PWW \u0915\u094b \u0928\u093f\u0917\u092e\u0928\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923 \u0938\u0947 \u0905\u0932\u0917 \u092e\u093e\u0928\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u0939\u092e\u0928\u0947 \u0905\u092c \u0924\u0915 \u0915\u094d\u092f\u093e \u0915\u093f\u092f\u093e \u0939\u0948?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0939\u092e\u0928\u0947 \u0905\u092c \u0924\u0915 \u091c\u0941\u0932\u093e\u0908 2018 \u0914\u0930 \u092e\u093e\u0930\u094d\u091a 2019 \u092e\u0947\u0902 \u0915\u094d\u0930\u092e\u0936: \u090f\u0915 \u0932\u0947\u0916 \u0914\u0930 \u090f\u0915 \u092a\u094b\u0938\u094d\u091f\u0930 \u092e\u0947\u0902 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0935\u093f\u0937\u092f \u092a\u094d\u0930\u0915\u093e\u0936\u093f\u0924 \u0915\u093f\u090f \u0939\u0948\u0902 :<\/p>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a\">\n<li>\u092a\u0939\u0932\u0947 \u0935\u093f\u091c\u093c\u0941\u0905\u0932\u093e\u0907\u091c\u093c\u0947\u0936\u0928 \u0914\u0930 \u092b\u093f\u0930 \u090f\u0915 \u0917\u0924\u093f\u0935\u093f\u0927\u093f \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u092f\u094b\u0917 \u0915\u0947 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f \u0914\u0930 \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u092f\u0939 \u0926\u094b\u0928\u094b\u0902 \u0939\u0940 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0932\u093f\u090f \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0915\u093f\u090f \u091c\u093e 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\u0939\u0948\u0902\u0964<\/li>\n\n\n\n<li>\u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e\u0913\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0938\u094d\u0915\u0947\u0932\u093f\u0902\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0917\u0941\u0923\u0928 \u0915\u093e \u090f\u0915 \u0935\u0948\u0915\u0932\u094d\u092a\u093f\u0915 \u0926\u0943\u0936\u094d\u092f \u092e\u0949\u0921\u0932\u0964 \u0905\u0938\u0924\u0924 (non-discrete) \u0914\u0930 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092f\u093e\u0928\u0940 \u0915\u093f \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0938\u092e\u093e\u0928 \u0917\u0941\u0923\u0928 \u0924\u0925\u094d\u092f\u094b\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964<\/li>\n\n\n\n<li>\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0928 \u0926\u094b \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u092d\u0940 \u092a\u093e\u0901\u091a \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u0964 \u0907\u0938\u092e\u0947\u0902 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0930\u0902\u0917\u0940\u0928 \u0932\u092e\u094d\u092c\u093e\u0908 \u0914\u0930 \u0930\u0902\u0917\u0940\u0928 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0915\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0930\u0923 \u0936\u093e\u092e\u093f\u0932 \u0939\u094b\u0917\u093e\u0964 \u0930\u0902\u0917\u0940\u0928 \u0906\u092f\u0924\u0928 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0928\u0939\u0940\u0902 \u0939\u094b\u0917\u0940, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0940 \u0939\u0940 \u0924\u0930\u0939 \u0907\u0928\u0915\u0947 \u0932\u093f\u090f \u092d\u0940 \u0928\u093f\u0917\u092e\u0928\u093f\u0915 \u0924\u0930\u094d\u0915 \u0938\u0947 \u0915\u093e\u092e \u091a\u0932 \u091c\u093e\u090f\u0917\u093e\u0964<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u0905\u092c \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u092e\u0947\u0902 \u090b\u0923\u0947\u0924\u094d\u0924\u0930\u0924\u093e (non-negativity) \u0915\u093e \u0932\u093e\u092d \u0928\u0939\u0940\u0902 \u0939\u0948\u0964 \u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u090f\u0915 \u0905\u0938\u0924\u0924 \u0938\u092e\u0941\u091a\u094d\u091a\u092f (discrete set) \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0906\u092e\u0924\u094c\u0930 \u092a\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0938\u0947 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u093e \u092a\u0930\u093f\u091a\u092f \u0909\u091a\u094d\u091a \u092a\u094d\u0930\u093e\u0925\u092e\u093f\u0915 \u0915\u0915\u094d\u0937\u093e\u0913\u0902 \u092e\u0947\u0902 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u092c \u0935\u0947 \u092a\u0948\u091f\u0930\u094d\u0928 \u0914\u0930 \u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0924\u0930\u094d\u0915\u094b\u0902 (logical arguments) \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0905\u0927\u093f\u0915 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0939\u093e\u0932\u093e\u0901\u0915\u093f \u092d\u0932\u0947 \u0939\u0940 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e, \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0914\u0930 \u0935\u093f\u0924\u0930\u0923 \u0936\u092c\u094d\u0926\u094b\u0902 \u0938\u0947 \u0909\u0928\u0915\u093e \u0938\u093e\u092e\u0928\u093e \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0906\u092e\u0924\u094c\u0930 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924\u093f-\u0926\u0930-\u0938\u094d\u0925\u093f\u0924\u093f \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0905\u0932\u093e\u0935\u093e \u0907\u0928\u094d\u0939\u0947\u0902 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0914\u0930 \u0924\u0930\u0939 \u0938\u0947 \u0928\u0939\u0940\u0902 \u0938\u092e\u091d\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0905\u0927\u093f\u0915\u093e\u0902\u0936 \u092a\u093e\u0920\u094d\u092f\u092a\u0941\u0938\u094d\u0924\u0915\u094b\u0902 \u092e\u0947\u0902 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0917\u0941\u0923\u0928 \u0915\u0940 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0915\u094b, \u0906\u0938\u093e\u0928\u0940-\u0938\u0947 \u092e\u093e\u0928 \u0932\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0916\u093c\u093e\u0938\u0924\u094c\u0930 \u092a\u0930 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0917\u0941\u0923\u093f\u0924 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0917\u0941\u0923\u093f\u0924 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0930\u0942\u092a \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u0947 \u0938\u092e\u092f\u0964 \u0939\u092e\u0947\u0902 \u0932\u0917\u0924\u093e \u0939\u0948 \u0915\u093f \u0917\u0941\u0923\u0947 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094b \u0905\u0932\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u0914\u0930 \u0907\u0938\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0938\u0947 \u092a\u0939\u0932\u0947 \u092c\u0924\u093e\u092f\u093e \u091c\u093e\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"504\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-1.jpg\" alt=\"Exploring Properties fig-1\" class=\"wp-image-27315\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-1.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-1-640x252.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-1-360x142.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-1-768x302.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><figcaption class=\"wp-element-caption\"><strong>\u091a\u093f\u0924\u094d\u0930-1<\/strong><\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u0941\u0932\u0906\u0909\u091f \u0915\u093e \u0905\u0928\u0941\u0938\u0930\u0923 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0939\u092e\u0928\u0947 \u0939\u0930\u0947 \u0914\u0930 \u0932\u093e\u0932 \u0930\u0902\u0917 \u0915\u0947 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u094d\u0930\u092e\u0936: \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0907\u0915\u093e\u0907\u092f\u094b\u0902 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u0939\u0948\u0964 \u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902, \u0917\u094b\u0932 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0940 \u092c\u091c\u093e\u092f \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u093e \u092c\u0947\u0939\u0924\u0930 \u0939\u0948\u0964 \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0915\u093e\u0909\u0902\u091f\u0930 \u0915\u0947 \u092b\u093c\u093e\u092f\u0926\u0947 \u0939\u092e \u092c\u093e\u0926 \u092e\u0947\u0902 \u092c\u0924\u093e\u090f\u0901\u0917\u0947\u0964 \u0924\u094b, \u091b\u0939 \u0915\u094b \u091b\u0939 \u0939\u0930\u0947 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902, \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u091a\u093e\u0930 (-4) \u0915\u094b \u091a\u093e\u0930 \u0932\u093e\u0932 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0914\u0930 \u0936\u0942\u0928\u094d\u092f \u0915\u094b \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0940 \u0905\u0928\u0941\u092a\u0938\u094d\u0925\u093f\u0924\u093f \u092f\u093e \u092c\u0930\u093e\u092c\u0930 \u0938\u0902\u0916\u094d\u092f\u093e \u092e\u0947\u0902 \u0932\u093e\u0932 \u0914\u0930 \u0939\u0930\u0947 \u0930\u0902\u0917 \u0915\u0947 \u0915\u093e\u0909\u0902\u091f\u0930 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 (<b>\u091a\u093f\u0924\u094d\u0930-1<\/b>)\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u092f\u0939\u093e\u0901 \u092f\u094b\u0917 \u0915\u0947 \u0932\u093f\u090f \u0935\u0948\u0938\u093e \u0939\u0940 \u092e\u0949\u0921\u0932 \u0924\u0948\u092f\u093e\u0930 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u091c\u0948\u0938\u093e \u0915\u093f \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0925\u093e\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0915\u094b \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0930\u0902\u0917 \u0915\u0947 \u0909\u0924\u0928\u0947 \u0939\u0940 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u0907\u0928 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u0926\u093f\u090f \u0917\u090f \u092f\u094b\u0917 \u0935\u094d\u092f\u0902\u091c\u0915 \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930 \u092c\u093e\u090f\u0901 \u0938\u0947 \u0926\u093e\u090f\u0901 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0938\u092d\u0940 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u0938\u0902\u092f\u0941\u0915\u094d\u0924 \u0938\u092e\u0942\u0939 \u092f\u094b\u0917\u092b\u0932 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u092e\u0947\u0902 \u0936\u0942\u0928\u094d\u092f \u091c\u094b\u095c\u094b\u0902 (zero pairs) \u0915\u094b \u0939\u091f\u093e\u0928\u0947 \u0915\u093e \u090f\u0915 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924 \u091a\u0930\u0923 \u092d\u0940 \u0939\u0948\u0964 \u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u092f\u0939 \u091a\u0930\u0923 \u0939\u092e\u093e\u0930\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0915\u0947 \u0932\u093f\u090f \u092e\u0939\u0924\u094d\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u092f\u094b\u0917 \u0915\u093e \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u0915\u0947 \u092a\u0940\u091b\u0947 \u092e\u0942\u0932 \u0935\u093f\u091a\u093e\u0930 \u0915\u093f\u0938\u0940 \u092f\u094b\u0917\u092b\u0932 \u0915\u094b \u0926\u094b \u0905\u0928\u0941\u0915\u0942\u0932 \u092a\u094d\u0930\u0947\u0915\u094d\u0937\u0923 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 (vantage points) \u0938\u0947 \u0926\u0947\u0916\u0928\u093e \u0939\u0948, \u091c\u0948\u0938\u093e \u0915\u093f \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u092d\u093f\u0928\u094d\u0928\u094b\u0902 \u0915\u0947 \u092e\u093e\u092e\u0932\u0947 \u092e\u0947\u0902 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"has-custom-css wp-custom-css-ddf7d860 wp-block-paragraph\">\u091c\u092c \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0940 \u092c\u093e\u0924 \u0906\u0924\u0940 \u0939\u0948, \u0924\u094b \u091a\u093e\u0930 \u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u092c\u0928\u0924\u0940 \u0939\u0948\u0902 :<\/p>\n\n\n\n<ol class=\"wp-block-list mb-0\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915 (\u0914\u0930 \u2234 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915)<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a nested-list mb-0\">\n<li>\u092f\u094b\u0917\u092b\u0932 &gt; 0<\/li>\n\n\n\n<li>\u092f\u094b\u0917\u092b\u0932 &lt; 0<\/li>\n<\/ul>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u0938\u094d\u0925\u093f\u0924\u093f 1 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0964 \u0938\u094d\u0925\u093f\u0924\u093f 2b \u0915\u0947 \u0932\u093f\u090f \u0939\u092e\u0928\u0947 \u091a\u093f\u0924\u094d\u0930-2 \u0914\u0930 3 \u092e\u0947\u0902 \u090f\u0915 \u0909\u0926\u093e\u0939\u0930\u0923 \u0926\u093f\u0916\u093e\u092f\u093e \u0939\u0948\u0964 \u092a\u093e\u0920\u0915 \u0938\u092e\u093e\u0928 \u0924\u0930\u0940\u0915\u093c\u0947 \u0938\u0947 2a \u0914\u0930 3 \u0915\u0940 \u092a\u095c\u0924\u093e\u0932 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 (\u0914\u0930 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u093f\u090f)\u0964<\/p>\n\n\n\n<div class=\"wp-block-group is-vertical is-layout-flex wp-container-core-group-is-layout-4fc3f8e1 wp-block-group-is-layout-flex\">\n<figure class=\"wp-block-image size-full mb-0\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"528\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-2.jpg\" alt=\"Exploring Properties fig-2\" class=\"wp-image-27312\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-2.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-2-640x264.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-2-360x149.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-2-768x317.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><figcaption class=\"wp-element-caption\"><strong>\u091a\u093f\u0924\u094d\u0930-2 :<\/strong> (\u201210 + 6)<\/figcaption><\/figure>\n\n\n\n<hr style=\"width: 100%;border: 0;border-top: 2.5px solid #1b1515;margin: 8px 0px;\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"538\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-3.jpg\" alt=\"Exploring Properties fig-3\" class=\"wp-image-27309\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-3.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-3-640x269.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-3-360x151.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-3-768x323.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><figcaption class=\"wp-element-caption\"><strong>\u091a\u093f\u0924\u094d\u0930-3 :<\/strong> 6 + (\u201210)<\/figcaption><\/figure>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u091a\u093f\u0924\u094d\u0930 2, B \u0915\u0947 \u092a\u0930\u093f\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u094d\u092f \u0938\u0947 \u0939\u0948 \u0914\u0930 \u092f\u0939 (\u221210) + 6 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948; \u091c\u092c\u0915\u093f \u091a\u093f\u0924\u094d\u0930-3, \u0909\u0938\u0940 \u092f\u094b\u0917\u092b\u0932 \u0915\u094b A \u0915\u0947 \u092a\u0930\u093f\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u094d\u092f \u0938\u0947 \u0926\u093f\u0916\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u092f\u0939 6 + (\u221210) \u0939\u0948\u0964 \u091a\u0942\u0901\u0915\u093f \u0915\u0947\u0935\u0932 \u092a\u0930\u093f\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u094d\u092f \u092e\u0947\u0902 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0939\u094b\u0924\u093e \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u092f\u094b\u0917\u092b\u0932 \u0905\u092a\u0930\u093f\u0935\u0930\u094d\u0924\u093f\u0924 \u0930\u0939\u0924\u093e \u0939\u0948; \u092f\u093e\u0928\u0940 (\u221210) + 6 = 6 + (\u221210) \u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092f\u0939 \u092c\u093e\u0924 \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0926\u094b \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u0932\u093e\u0917\u0942 \u0939\u094b\u0924\u0940 \u0939\u0948 \u0914\u0930 \u0907\u0938\u0938\u0947 \u0915\u094b\u0908 \u092b\u093c\u0930\u094d\u0915\u093c \u0928\u0939\u0940\u0902 \u092a\u0921\u093c\u0924\u093e \u0915\u093f \u0935\u0947 \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u0915\u093f\u0924\u0928\u0940 \u0926\u0942\u0930 \u0939\u0948\u0902, \u092f\u093e\u0928\u0940 \u0915\u093f \u0909\u0928\u094d\u0939\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u0924\u0928\u0947 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u094b \u0938\u0915\u0924\u0940 \u0939\u0948\u0964<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u092f\u094b\u0917 \u0915\u093e \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0928\u093f\u092f\u092e<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"528\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-4.jpg\" alt=\"Exploring Properties fig-4\" class=\"wp-image-27306\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-4.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-4-640x264.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-4-360x149.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/08\/Exploring-Properties-fig-4-768x317.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><figcaption class=\"wp-element-caption\"><strong>\u091a\u093f\u0924\u094d\u0930-4<\/strong><\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u092e\u0942\u0932 \u0935\u093f\u091a\u093e\u0930 \u092f\u0939 \u0939\u0948 \u0915\u093f \u0905\u0917\u0930 \u0939\u092e\u0947\u0902 \\(x\\) + \\(y\\) + \\(z\\) \u091c\u094b\u0921\u093c\u0928\u093e \u0939\u0948, \u0924\u094b \u0907\u0938\u0938\u0947 \u0915\u094b\u0908 \u092b\u093c\u0930\u094d\u0915\u093c \u0928\u0939\u0940\u0902 \u092a\u0921\u093c\u0924\u093e \u0915\u093f \u0939\u092e \\(x\\) \u0914\u0930 \\(y\\) \u0915\u094b \u092a\u0939\u0932\u0947 \u091c\u094b\u0921\u093c\u0924\u0947 \u0939\u0948\u0902 \u092f\u093e \\(y\\) \u0914\u0930 \\(z\\) \u0915\u094b\u0964 \u092f\u0939 \u0935\u0948\u0938\u093e \u0939\u0940 \u0939\u0948, \u091c\u0948\u0938\u093e \u0915\u093f \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0938\u0947 \u090f\u0915 \u0917\u0924\u093f\u0935\u093f\u0927\u093f \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u0940 \u0924\u0930\u0939 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0915\u094b\u0908 \u092d\u0940 \u0924\u0940\u0928 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u091a\u0941\u0928\u0947\u0902, \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f 7, \u22124 \u0914\u0930 \u221210\u0964 \u0909\u0928\u094d\u0939\u0947\u0902 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u0938\u092e\u0942\u0939 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0947\u0902, \u092f\u093e\u0928\u0940 \u0915\u093f 7 \u0939\u0930\u0947 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u092a\u0939\u0932\u093e \u0938\u092e\u0942\u0939, 4 \u0932\u093e\u0932 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u0926\u0942\u0938\u0930\u093e \u0938\u092e\u0942\u0939 \u0906\u0926\u093f (<strong>\u091a\u093f\u0924\u094d\u0930-4<\/strong>)\u0964 \u0905\u092c 7 + (- 4) + (- 10) \u0915\u093e \u092f\u094b\u0917\u092b\u0932 \u0924\u0940\u0928\u094b\u0902 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u090f\u0915 \u0938\u093e\u0925 \u092e\u093f\u0932\u093e\u0928\u0947 \u092a\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e\u0964 \u092f\u0926\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u0926\u094b \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0924\u094b \u091a\u0930\u0923 1 \u092e\u0947\u0902 \u092a\u0939\u0932\u0947 \u0914\u0930 \u0926\u0942\u0938\u0930\u0947 \u0938\u092e\u0942\u0939 &#8211; \u0915\u094b \u092e\u093f\u0932\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u092e\u0924\u0932\u092c \u0939\u0948 \u0915\u093f 7+ (-4) \u0915\u094b \u091c\u094b\u0921\u093c\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u091a\u0930\u0923 2 \u092e\u0947\u0902 \u0924\u0940\u0938\u0930\u0947 \u0938\u092e\u0942\u0939 \u0915\u094b \u0907\u0938\u092e\u0947\u0902 \u092e\u093f\u0932\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u092f\u093e\u0928\u0940 \u0915\u093f -10 \u0915\u094b 7+ (-4) \u0915\u0947 \u092f\u094b\u0917\u092b\u0932 \u092e\u0947\u0902 \u091c\u094b\u0921\u093c\u0928\u093e \u092f\u093e\u0928\u0940 [7 + (-4)] +(-10)| \u0935\u0939\u0940\u0902 \u0926\u0942\u0938\u0930\u0940 \u0913\u0930, \u091a\u0930\u0923 1 \u092e\u0947\u0902 \u0906\u092a \u0926\u0942\u0938\u0930\u0947 \u0914\u0930 \u0924\u0940\u0938\u0930\u0947 \u0938\u092e\u0942\u0939 \u0915\u094b \u092e\u093f\u0932\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u092f\u093e\u0928\u0940 \u0915\u093f (-4) + (-10) \u0915\u0930\u0928\u093e \u0914\u0930 \u091a\u0930\u0923 2 \u092e\u0947\u0902 \u092a\u0939\u0932\u0947 \u0938\u092e\u0942\u0939 \u0915\u094b \u0907\u0938\u092e\u0947\u0902 \u092e\u093f\u0932\u093e\u0928\u093e, \u092f\u093e\u0928\u0940 \u0915\u093f 7 \u0915\u094b \u091c\u094b\u0921\u093c\u0928\u093e\u0964 \u0926\u094b\u0928\u094b\u0902 \u0939\u0940 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0924\u0940\u0928\u094b\u0902 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u0915\u0930 \u090f\u0915 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0928 \u0924\u094b \u0915\u094b\u0908 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924 \u0915\u093e\u0909\u0902\u091f\u0930 \u092e\u093f\u0932\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0928 \u0939\u0940 \u0915\u093f\u0938\u0940 \u0915\u093e\u0909\u0902\u091f\u0930 \u0915\u094b \u0939\u091f\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0905\u0924\u0903 \u0926\u094b\u0928\u094b\u0902 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u092f\u094b\u0917\u092b\u0932 \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f, \u092f\u093e\u0928\u0940 \u0915\u093f [7 + (-4)] + (-10) =7 + [(-4) + (-10)]\u0964 \u0936\u0942\u0928\u094d\u092f \u091c\u094b\u0921\u093c\u094b\u0902 \u092f\u093e\u0928\u0940 \u0939\u0930\u0947-\u0932\u093e\u0932 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u091c\u094b\u0921\u093c\u094b\u0902 \u0915\u094b \u0924\u0940\u0928\u094b\u0902 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u0939\u091f\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u0907\u0928 \u0924\u0940\u0928 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u094b \u0938\u094d\u200d\u0935\u0947\u091a\u094d\u200d\u091b\u093e (arbitary) \u0938\u0947 \u091a\u0941\u0928\u093e \u0917\u092f\u093e \u0925\u093e \u0914\u0930 \u0907\u0928\u094d\u200d\u0939\u0947\u0902 \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0924\u0940\u0928 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0924\u0915 \u092c\u0922\u093c\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 (\u0915\u092e-\u0938\u0947-\u0915\u092e \u090f\u0915 \u0935\u0948\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0947 \u0924\u094c\u0930 \u092a\u0930), \u092b\u093f\u0930 \u091a\u093e\u0939\u0947 \u0935\u0947 \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u0915\u093f\u0924\u0928\u0940 \u092d\u0940 \u0926\u0942\u0930\u0940 \u092a\u0930 \u0915\u094d\u092f\u094b\u0902 \u0928 \u0939\u094b\u0902\u0964<\/p>\n\n\n\n<p class=\"has-custom-css wp-custom-css-0f808b57 wp-block-paragraph\">\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 14 \u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u092c\u0928\u0924\u0940 \u0939\u0948\u0902 :<\/p>\n\n\n\n<ol class=\"wp-block-list mb-0 has-custom-css wp-custom-css-09515b30\">\n<li class=\"has-text-color has-link-color wp-elements-1\" style=\"color:#006991\">\u0938\u092d\u0940 3 \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li class=\"has-text-color has-link-color wp-elements-2\" style=\"color:#006991\">2 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 1 \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ol>\n\n\n\n<div class=\"wp-block-group list-box-section position-relative has-custom-css wp-custom-css-2b2ff5f8\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<ul class=\"wp-block-list list-type-alpha start-a nested-list mb-1\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915  + \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ul>\n\n\n\n<style data-wp-block-html=\"css\">\n.possibility-box {\n    position: absolute;\n    right: -150px;\n    top: 32px;\n    width: 132px;\n    padding: 4px;\n    border: 2.5px solid #898686;\n}\n.possibility-box p,.possibility-box li{\nfont-size:20px !important;\nmargin:0;\n}\n\n@media (max-width: 768px) {\n    .possibility-box {\n    position: unset !important;\n    width: 100%;\n    padding: 4px;\n    border: 2.5px solid #898686;\nmargin-left: 26px;\n}\n}\n<\/style>\n\n<div class=\"possibility-box\">\n  <p class=\"m-0\">2 \u0935 3 \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915\n\u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u0926\u094b\n\u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u0939\u0948\u0902<\/p>\n  <ol class=\"m-0\" style=\"list-style-type: lower-roman;\">\n    <li class=\"m-0\">\u092f\u094b\u0917\u092b\u0932 &gt; 0<\/li>\n    <li class=\"m-0\">\u092f\u094b\u0917\u092b\u0932 &lt; 0<\/li>\n  <\/ol>\n<\/div>\n<\/div><\/div>\n\n\n\n<ol start=\"3\" class=\"wp-block-list mb-0 has-custom-css wp-custom-css-add91e9d\">\n<li class=\"has-text-color has-link-color wp-elements-3\" style=\"color:#006991\">1 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 2 \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a nested-list mb-1\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915  + \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n\n\n\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915  + \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ul>\n\n\n\n<ol start=\"4\" class=\"wp-block-list has-custom-css wp-custom-css-ce80f767\">\n<li class=\"has-text-color has-link-color wp-elements-4\" style=\"color:#006991\">\u0938\u092d\u0940 3 \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u090a\u092a\u0930 \u0926\u093f\u0916\u093e\u092f\u093e \u0917\u092f\u093e \u091a\u093f\u0924\u094d\u0930 \u0938\u094d\u0925\u093f\u0924\u093f 3a \u0915\u093e \u090f\u0915 \u0909\u0926\u093e\u0939\u0930\u0923 \u0939\u0948, \u091c\u0939\u093e\u0901 \u092f\u094b\u0917\u092b\u0932 &lt; 0 \u0939\u0948\u0964 \u0936\u0947\u0937 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u093e\u0901 \u0939\u092e \u092a\u093e\u0920\u0915 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u095c\u0924\u093e\u0932 \u0915\u0930\u0928\u0947  \u0915\u0947 \u0932\u093f\u090f \u091b\u094b\u0921\u093c \u0930\u0939\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f <strong>\u0917\u0941\u0923\u0947<\/strong> \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u093e \u0925\u094b\u0921\u093c\u093e \u092e\u0941\u0936\u094d\u0915\u093f\u0932 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0915\u094b\u0908 \u092d\u0940 \u092e\u0949\u0921\u0932 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u092a\u0942\u0930\u0940 \u0924\u0930\u0939 \u0938\u0947 \u0915\u093e\u092e \u0928\u0939\u0940\u0902 \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u0917\u0941\u0923\u0947 \u0938\u0947 \u091c\u0941\u0921\u093c\u0940 \u0924\u0940\u0928 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u093e\u0901 \u0939\u0948\u0902- (a) \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 \u0927\u0928\u093e\u0924\u094d\u092e\u0915, (b) \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 (c) \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0964 \u090f\u0928\u0938\u0940\u0908\u0906\u0930\u091f\u0940 \u092a\u093e\u0920\u094d\u092f\u092a\u0941\u0938\u094d\u0924\u0915 \u0928\u0947 \u092a\u0939\u093e\u0921\u093c\u094b\u0902 \u0915\u094b \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0926\u0947\u0915\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0905\u091a\u094d\u091b\u093e \u0915\u093e\u092e \u0915\u093f\u092f\u093e \u0939\u0948\u0964 \u0915\u0915\u094d\u0937\u093e 7 \u0915\u0940 \u092a\u093e\u0920\u094d\u092f\u092a\u0941\u0938\u094d\u0924\u0915 \u0915\u0947 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0905\u0927\u094d\u092f\u093e\u092f \u092e\u0947\u0902 \u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u0924\u0940\u0928 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u0905\u0932\u0917-\u0905\u0932\u0917 \u0914\u0930 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0915\u094b \u092e\u093e\u0928\u0947 \u092c\u093f\u0928\u093e \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964 \u091c\u092c\u0915\u093f \u0915\u0908 \u0905\u0928\u094d\u092f \u092a\u093e\u0920\u094d\u092f\u092a\u0941\u0938\u094d\u0924\u0915\u0947\u0902 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0915\u094b \u092e\u093e\u0928\u0924\u0947 \u0939\u0941\u090f (a) \u0914\u0930 (b) \u0915\u094b \u090f\u0915 \u0938\u092e\u093e\u0928 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948\u0902\u0964 \u0939\u092e \u092a\u0939\u0932\u0947 \u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0917\u0941\u0923\u0928\u092b\u0932\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u092f\u093e \u0938\u092e\u091d\u0928\u0947 \u0915\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u092b\u093f\u0930 \u0907\u0928 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0906\u0907\u090f \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092e\u0941\u0916\u094d\u092f \u0924\u0925\u094d\u092f\u094b\u0902 \u0915\u094b \u090f\u0915-\u090f\u0915 \u0915\u0930\u0915\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 (\u092e\u093e\u0928\u093e \u0915\u093f \\(m\\) \u0914\u0930 \\(n\\)  \u0915\u094b\u0908 \u0926\u094b \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u0948\u0902) :<\/p>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a\">\n<li><strong>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 = \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/strong><br>\u0907\u0938\u0947 \u092c\u093e\u0930\u092e\u094d\u092c\u093e\u0930 \u092f\u094b\u0917 (repeated addition) \u0915\u0940 \u0927\u093e\u0930\u0923\u093e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u092f\u093e\u0928\u0940 \u0915\u093f (-4) \u00d7 3 = (-4) + (-4) + (-4) = -12 | \u0905\u092c \u0915\u093e\u0909\u0902\u091f\u0930 (\u091c\u094b \u0915\u093f \u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0939\u094b\u0902\u0917\u0947) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u092c\u093e\u0930\u092e\u094d\u092c\u093e\u0930 \u092f\u094b\u0917 \u0915\u0947 \u0932\u093f\u090f \u092e\u0949\u0921\u0932 \u0924\u0948\u092f\u093e\u0930 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u0924\u0930\u094d\u0915 \u0926\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0939 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0940 \u0939\u0948 \u092f\u093e\u0928\u0940 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u0930\u0902\u0917 \u0915\u094b \u091b\u094b\u0921\u093c \u0926\u0947\u0902 \u0924\u094b \u092f\u0939 4 \u00d7 3 \u091c\u0948\u0938\u093e \u0939\u0940 \u0939\u0948\u0964 \u092f\u093e \u0926\u0942\u0938\u0930\u0947 \u0936\u092c\u094d\u0926\u094b\u0902 \u092e\u0947\u0902, \u0930\u0902\u0917 \u0915\u094b \u0905\u0928\u0926\u0947\u0916\u093e \u0915\u0930\u0947\u0902 \u0924\u094b \u092f\u0939 \u092d\u0940 4 \u00d7 3 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u093e \u0938\u092e\u093e\u0928 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 (array) \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u092c\u0938 \u0907\u0938 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u092e\u0947\u0902 4 x 3 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0915\u093e\u0909\u0902\u091f\u0930 \u0939\u0948\u0902 \u0914\u0930 \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 (4 \u00d7 3) \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f (-4) x 3 = -(4 \u00d7 3)\u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f 4 \u0914\u0930 3 \u0915\u094b \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u092c\u0926\u0932\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0914\u0930 \u0938\u092d\u0940 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u092f\u0939\u0940 \u0924\u0930\u094d\u0915 \u0932\u093e\u0917\u0942 \u0939\u094b\u0917\u093e\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u092e\u0928\u0947 \u090f\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u0939\u0948, \u091c\u094b \u0939\u0948 (-\\(m\\)) \u00d7 \\(n\\) = \u2212(\\(m\\) \u00d7 \\(n\\))&#8230; (\u0938\u092e\u0940\u0915\u0930\u0923 1)\u0964<\/li>\n\n\n\n<li><strong>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 = \u090b\u0923\u093e\u0924\u094d\u092e\u0915<\/strong><br>\u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0940 \u092a\u0921\u093c\u0924\u093e\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u0939\u0932\u0947 \\(m\\) \u0915\u0947 \u092a\u0939\u093e\u0921\u093c\u0947 \u0915\u094b \\(n\\) = 1 \u0938\u0947 \\(n\\) = 0 \u0915\u0940 \u0924\u0930\u092b\u093c \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u092b\u093f\u0930 \u092a\u0939\u093e\u0921\u093c\u0947 \u092e\u0947\u0902 \u0926\u093f\u0916 \u0930\u0939\u0947 \u092a\u0948\u091f\u0930\u094d\u0928 \u0915\u094b \u092c\u0928\u093e\u090f \u0930\u0916\u0924\u0947 \u0939\u0941\u090f \u0907\u0938\u0947 \\(n\\) &lt; 0 \u0915\u0947 \u0932\u093f\u090f \u0906\u0917\u0947 \u092c\u0922\u093c\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u0947-\u091c\u0948\u0938\u0947 \u0939\u092e \\(m\\) \u0915\u0947 \u092a\u0939\u093e\u0921\u093c\u0947 \u0915\u094b \\(n\\) = 4, 3, 2-\u0906\u0926\u093f \u0938\u0947 \u0906\u0917\u0947 \u092c\u0922\u093c\u093e\u0924\u0947 \u0939\u0948\u0902, \u0939\u092e \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0917\u0941\u0923\u0928\u092b\u0932 \u092e\u0947\u0902 \u0938\u0947 \\(n\\) \u0918\u091f \u0930\u0939\u093e \u0939\u0948\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u092f\u093e \u0905\u0928\u094d\u092f \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u092d\u0940 \u0939\u092e\u0947\u0902 \\(m\\) \u00d7 0 = 0 \u092e\u093f\u0932\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u0905\u0917\u0932\u093e \u091a\u0930\u0923 \\(m\\) \u00d7 (-1) = 0 &#8211; \\(m\\) = &#8211;\\(m\\) \u0939\u094b\u0917\u093e\u0964 \u0907\u0938 \u092a\u0926\u094d\u0927\u0924\u093f \u0915\u094b \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 \u0938\u094d\u0915\u093f\u092a \u0915\u093e\u0909\u0902\u091f\u093f\u0902\u0917 \u0915\u0947 \u0938\u093e\u0925 \u091c\u094b\u0921\u093c\u0915\u0930 \u0926\u0947\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0924\u093e\u0915\u093f \u092f\u0939 \u0938\u092e\u091d \u0906 \u0938\u0915\u0947 \u0915\u093f \\(m\\) \u00d7 (-\\(n\\)) \u0905\u0938\u0932 \u092e\u0947\u0902 \\(m\\) \u0932\u092e\u094d\u092c\u093e\u0908 \u0915\u094b \\(n\\) \u092c\u093e\u0930 \u0938\u094d\u0915\u093f\u092a \u0915\u0930\u0928\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u094d\u0915\u093f\u092a \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u0936\u0941\u0930\u0942 \u0939\u094b\u0915\u0930 \u092c\u093e\u0908\u0902 \u0913\u0930 \u092f\u093e \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u092a\u0915\u094d\u0937 \u092e\u0947\u0902 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 &#8211;\\(n\\) \u0915\u093e \\(n\\) \u092c\u093e\u0930 \u092c\u093e\u0930\u092e\u094d\u092c\u093e\u0930 \u092f\u094b\u0917 \u0939\u0948, \u092f\u093e\u0928\u0940 \u0915\u093f (-\\(m\\)) \u00d7 \\(n\\) \u0964 \u0905\u0924\u0903 \\(m\\) \u00d7 (-\\(n\\)) = (-\\(m\\)) x \\(n\\) \u0914\u0930 \u0907\u0938\u0932\u093f\u090f \\(m\\) \u00d7 (-\\(n\\)) = &#8211; (\\(m\\) x \\(n\\))&#8230; (\u0938\u092e\u0940\u0915\u0930\u0923 2)\u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092f\u0939 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/li>\n\n\n\n<li><strong>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 = \u0927\u0928\u093e\u0924\u094d\u092e\u0915<\/strong><br>\u092f\u0939 \u092a\u093f\u091b\u0932\u0940 \u0926\u094b\u0928\u094b\u0902 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u091c\u094b\u0921\u093c\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0940 \u0936\u0941\u0930\u0941\u0906\u0924 (-\\(m\\)) \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092a\u0939\u093e\u0921\u093c\u093e \u092c\u0928\u093e\u0928\u0947 \u0938\u0947 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u092b\u093f\u0930 \u0907\u0938 \u092a\u0939\u093e\u0921\u093c\u0947 \u0915\u094b \\(n\\) &lt; 0 \u0915\u0940 \u0913\u0930 \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u0939\u092e \u091c\u0948\u0938\u0947-\u091c\u0948\u0938\u0947 \\(n\\) = 3 ,2,1 \u0906\u0926\u093f \u0938\u0947 \u0906\u0917\u0947 \u092c\u0922\u093c\u0924\u0947 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902, \u0917\u0941\u0923\u0928\u092b\u0932 -3\\(m\\), -2\\(m\\), &#8211;\\(m\\) \u0939\u094b\u0924\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0917\u0941\u0923\u0928\u092b\u0932 \u092e\u0947\u0902 \\(m\\) \u091c\u0941\u0921\u093c\u0924\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0924\u094b, (\u2212\\(m\\)) \u00d7 0 = 0 \u0939\u094b\u0917\u093e\u0964 \u0907\u0938\u0947 \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 \u0938\u094d\u0915\u093f\u092a \u0915\u093e\u0909\u091f\u093f\u0902\u0917 \u0915\u0947 \u0938\u093e\u0925 \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930 (-\\(m\\))(-1) = 0 + \\(m\\) = \\(m\\) \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 (-\\(m\\)) \u00d7 (-\\(n\\)) \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u0926\u093e\u0908\u0902 \u0913\u0930 \u092f\u093e \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u092a\u0915\u094d\u0937 \u0915\u0940 \u0924\u0930\u092b\u093c \\(m\\) \u091a\u0930\u0923\u094b\u0902 \u0915\u093e \\(n\\) \u092c\u093e\u0930 \u0938\u094d\u0915\u093f\u092a \u0939\u0948 \u091c\u094b \\(m\\) \u00d7 \\(n\\) \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0940 \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f (-\\(m\\)) \u00d7 (-\\(n\\)) = \\(m\\) \u00d7 \\(n\\) (\u0938\u092e\u0940\u0915\u0930\u0923 3)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0907\u0938 \u0924\u0930\u0939 \u0915\u0940 \u0938\u092e\u091d \u092c\u0928\u093e\u0928\u0947 \u0935\u093e\u0932\u093e \u0910\u0938\u093e \u0915\u094b\u0908 \u092e\u0949\u0921\u0932 \u0909\u092a\u0932\u092c\u094d\u0927 \u0928\u0939\u0940\u0902 \u0939\u0948\u2014 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f (\u22123\/5) \u00d7 (\u22127\/4) \u0915\u094b \u0915\u0948\u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0928\u093e \u091a\u093e\u0939\u093f\u090f? \u0910\u0938\u0947 \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u0915\u0940 \u0938\u092e\u091d \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u093e\u0925\u092e\u093f\u0915 \u0938\u094d\u0924\u0930 \u092a\u0930 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091c\u0940\u0935\u0928 \u0938\u0947 \u091c\u0941\u095c\u0924\u093e \u0939\u0941\u0906 \u0915\u094b\u0908 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0909\u0926\u093e\u0939\u0930\u0923 \u0928\u0939\u0940\u0902 \u092e\u093f\u0932\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u0939\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e\u0913\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0938\u094d\u0915\u0947\u0932\u093f\u0902\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0926\u0943\u0936\u094d\u092f \u092e\u0949\u0921\u0932 \u0915\u094b \u0905\u0917\u0932\u0947 \u0932\u0947\u0916 \u092e\u0947\u0902 \u0926\u0947\u0915\u0930 \u0907\u0938 \u0930\u093f\u0915\u094d\u0924\u0924\u093e \u0915\u094b \u092d\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0905\u092c \u0939\u092e <strong>\u0917\u0941\u0923\u0928 \u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902<\/strong> \u092a\u0930 \u0906\u0924\u0947 \u0939\u0948\u0902\u0964 \u092f\u0939\u093e\u0901 \u0939\u092e \u092a\u0926\u094d\u0927\u0924\u093f \u092a\u0930 \u0915\u092e \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902\u0917\u0947 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0939\u092e \u0915\u0941\u091b \u092e\u0939\u0924\u094d\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092a\u0930\u093f\u0923\u093e\u092e \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u091a\u0941\u0915\u0947 \u0939\u0948\u0902\u0964 \u092f\u0947 \u092d\u0940 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u094b \u092f\u0939 \u0926\u093f\u0916\u093e\u0928\u0947 \u0915\u093e \u090f\u0915 \u0938\u0930\u0932 \u0924\u0930\u0940\u0915\u093c\u093e \u0939\u0948 \u0915\u093f \u0917\u0923\u093f\u0924\u0940\u092f \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0948\u0938\u0947 \u0926\u093f\u090f \u0915\u093f\u090f \u091c\u093e\u0924\u0947 \u0939\u0948\u0902, \u092f\u093e\u0928\u0940 \u0915\u093f \u0915\u0941\u091b \u0928\u092f\u093e \u0928\u093f\u0917\u092e\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u093f\u0926\u094d\u0927 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u094b \u091a\u0930\u0923\u092c\u0926\u094d\u0927 \u0924\u0930\u0940\u0915\u093c\u0947 \u0938\u0947 \u0915\u0948\u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0906\u0907\u090f \u0939\u092e \u0907\u0928 \u0924\u0940\u0928 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u094b \u092b\u093f\u0930 \u0938\u0947 \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902 :<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0926\u094b \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \\(n\\) \u0914\u0930 \\(m\\) \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\\(m\\) \u00d7 (\u2212\\(n\\)) = \u2212(\\(m\\) \u00d7 \\(n\\))<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 \\(n\\) = \u2212(\\(m\\) \u00d7 \\(n\\))<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 (\u2212\\(n\\)) = \\(m\\) \u00d7 \\(n\\)<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u0917\u0941\u0923\u0928 \u0915\u093e \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0938\u0947 \u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u0924\u0940\u0928 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938\u094d\u0924\u0947\u092e\u093e\u0932 \u0915\u0940 \u091c\u093e\u0928\u0947 \u0935\u093e\u0932\u0940 \u092a\u0926\u094d\u0927\u0924\u093f \u0915\u0947 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0947\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u092a\u094d\u0930\u092e\u093e\u0923 \u0909\u091a\u094d\u091a \u092a\u094d\u0930\u093e\u0925\u092e\u093f\u0915 \u0938\u094d\u0924\u0930 \u0915\u0947 \u0932\u093f\u090f \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0938\u0930\u0932 \u0939\u0948\u0902 \u0914\u0930 \u092f\u0939 \u091c\u093e\u0928\u0928\u0947 \u092e\u0947\u0902 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u0940 \u092e\u0926\u0926 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0917\u0923\u093f\u0924 \u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0915\u0948\u0938\u0947 \u0935\u093f\u0915\u0938\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u092f\u094b\u0917 \u0915\u0947 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0915\u0940 \u0939\u0940 \u0924\u0930\u0939 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u092d\u0940 \u0924\u0940\u0928 \u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u0939\u094b \u0938\u0915\u0924\u0940 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0939\u092e \u092a\u0939\u0932\u0947 \u0939\u0940 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u091a\u0941\u0915\u0947 \u0939\u0948\u0902<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li>\\(m\\) \u00d7 \\(n\\) = \\(n\\) \u00d7 \\(m\\)&#8230; (\u0938\u092e\u0940\u0915\u0930\u0923 4)<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u092f\u093e\u0928\u0940 \u0915\u093f \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f, \u091c\u094b \u0917\u0941\u0923\u0928 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u0941\u0932\u0906\u0909\u091f \u092e\u0947\u0902 \u0926\u0940 \u0917\u0908 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0924\u094b \u0905\u092c \u0939\u092e \u0936\u0947\u0937 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902 \u0927\u0928\u093e\u0924\u094d\u092e\u0915-\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0930\u094d\u0925\u093e\u0924 \\(m\\) \u00d7 (\u2212\\(n\\)) = (\u2212\\(n\\)) \u00d7 \\(m\\) \u0914\u0930 \u090b\u0923\u093e\u0924\u094d\u092e\u0915-\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0930\u094d\u0925\u093e\u0924 (\u2212\\(m\\)) \u00d7 (\u2212\\(n\\)) = (\u2212\\(n\\)) \u00d7 (\u2212\\(m\\)) \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \\(m\\) \u0914\u0930 \\(n\\) \u0915\u0947 \u0932\u093f\u090f\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u092e\u093e\u092e\u0932\u094b\u0902 \u0915\u0940 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e \u0915\u093e \u0935\u093f\u0936\u0947\u0937 \u092e\u0939\u0924\u094d\u0924\u094d\u0935 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/p>\n\n\n\n<div class=\"wp-block-columns has-custom-css is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex wp-custom-css-7b7aa4ab\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:16%\">\n<p class=\"has-text-align-left wp-block-paragraph\">\\(m\\) \u00d7 (\u2212\\(n\\))<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:16%\">\n<p class=\"mb-1 wp-block-paragraph\">= \u2212(\\(m\\) \u00d7 \\(n\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \u2212(\\(n\\) \u00d7 \\(m\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= (\u2212\\(n\\)) \u00d7 \\(m\\)<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (4) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (2) \u0938\u0947<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:16%\">\n<p class=\"wp-block-paragraph\">(\u2212\\(m\\)) \u00d7 (\u2212\\(n\\))<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">= \\(m\\) \u00d7 \\(n\\)<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \\(n\\) \u00d7 \\(m\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">= (\u2212\\(n\\)) \u00d7 (-\\(m\\))<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (4) \u0938\u0947<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\">\u0917\u0941\u0923\u0928 \u0915\u093e \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u092f\u094b\u0917 \u0915\u0947 \u0938\u093e\u0939\u091a\u0930\u094d\u092f \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u0915\u0940 \u0924\u0930\u0939 \u0939\u0940 \u0907\u0938\u0915\u0940 \u092d\u0940 8 \u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u0939\u0948\u0902 :<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\\(m\\) \u00d7 \\(n\\) \u00d7 \\(p\\)<\/li>\n\n\n\n<li>\\(m\\) \u00d7 \\(n\\) \u00d7 (\u2212\\(p\\))<\/li>\n\n\n\n<li>\\(m\\) \u00d7 (\u2212\\(n\\)) \u00d7 \\(p\\)<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 \\(n\\) \u00d7 \\(p\\)<\/li>\n\n\n\n<li>\\(m\\) \u00d7 (\u2212\\(n\\)) \u00d7 (\u2212\\(p\\))<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 \\(n\\) \u00d7 (\u2212\\(p\\))<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 (\u2212\\(n\\)) \u00d7 \\(p\\)<\/li>\n\n\n\n<li>(\u2212\\(m\\)) \u00d7 (\u2212\\(n\\)) \u00d7 (\u2212\\(p\\))<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092f\u0939\u093e\u0901 \u0926\u0940 \u0917\u0908 \u0938\u094d\u0925\u093f\u0924\u093f 1 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0939\u0948 \u091c\u093f\u0938\u0947 \u0917\u0941\u0923\u0928 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u0941\u0932\u0906\u0909\u091f \u092e\u0947\u0902 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0925\u093e, \u092f\u093e\u0928\u0940 \u0915\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 (\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\) = \\(m\\) \u00d7 (\\(n\\) \u00d7 \\(p\\)) \u2026(\u0938\u092e\u0940\u0915\u0930\u0923 5) \u092a\u0939\u0932\u0947 \u0938\u0947 \u0939\u0948\u0964<\/p>\n\n\n\n<ol start=\"5\" class=\"wp-block-list\">\n<li>(\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\) = \\(m\\) \u00d7 (\\(n\\) \u00d7 \\(p\\))<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u0939\u092e\u0928\u0947 \u092f\u0939\u093e\u0901 \u0938\u094d\u0925\u093f\u0924\u093f 3, 5 \u0914\u0930 8 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u0939\u0948 \u0914\u0930 \u092c\u093e\u0915\u093c\u0940 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u092a\u093e\u0920\u0915 \u0926\u094d\u0935\u093e\u0930\u093e \u0915\u0930\u0915\u0947 \u0926\u0947\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u091b\u094b\u095c\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:24%\">\n<ol start=\"3\" class=\"wp-block-list\">\n<li>[\\(m\\) \u00d7 (\u2212\\(n\\))] \u00d7 \\(p\\)<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:22%\">\n<p class=\"mb-1 wp-block-paragraph\">= \u2212(\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\)<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \u2212[(\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\)]<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \u2212[\\(m\\) \u00d7 (\\(n\\) \u00d7 \\(p\\))]<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \\(m\\) \u00d7 [\u2212(\\(n\\) \u00d7 \\(p\\))]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">= \\(m\\) \u00d7 [(\u2212\\(n\\)) \u00d7 \\(p\\)]<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (2) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (5) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0938\u0947<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (2) \u0938\u0947<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u0938\u094d\u0925\u093f\u0924\u093f 2 \u0914\u0930 4 \u0915\u094b \u092d\u0940 \u0907\u0938\u0940 \u0924\u0930\u0939 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<div class=\"wp-block-columns mb-1 is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:28%\">\n<ol start=\"5\" class=\"wp-block-list\">\n<li>[\\(m\\) \u00d7 (\u2212\\(n\\))] \u00d7 (\u2212\\(p\\))<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:27%\">\n<p class=\"mb-1 wp-block-paragraph\">= \u2212(\\(m\\) \u00d7 \\(n\\)) \u00d7 (\u2212\\(p\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= (\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\)<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \\(m\\) \u00d7 (\\(n\\) \u00d7 \\(p\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \\(m\\) \u00d7 [(\u2212\\(n\\)) \u00d7 (\u2212\\(p\\))]<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (5) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u0938\u094d\u0925\u093f\u0924\u093f 6 \u0914\u0930 7 \u0915\u094b \u092d\u0940 \u0907\u0938\u0940 \u0924\u0930\u0939 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:32%\">\n<ol start=\"8\" class=\"wp-block-list\">\n<li>[(\u2212\\(m\\)) \u00d7 (\u2212\\(n\\))] \u00d7 (\u2212\\(p\\))<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:28%\">\n<p class=\"mb-1 wp-block-paragraph\">= (\\(m\\) \u00d7 \\(n\\)) \u00d7 (\u2212\\(p\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \u2212[(\\(m\\) \u00d7 \\(n\\)) \u00d7 \\(p\\)]<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= \u2212[\\(m\\) \u00d7 (\\(n\\) \u00d7 \\(p\\))]<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= (\u2212\\(m\\)) \u00d7 (\\(n\\) \u00d7 \\(p\\))<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">= (\u2212\\(m\\)) \u00d7 [(\u2212\\(n\\)) \u00d7 (\u2212\\(p\\))<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:18%\">\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (5) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (5) \u0938\u0947<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0938\u092e\u0940\u0915\u0930\u0923 (3) \u0938\u0947<\/p>\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\">\u0917\u0941\u0923\u0928 \u0915\u093e \u0935\u093f\u0924\u0930\u0923 \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u0926\u094b\u0928\u094b\u0902 \u0938\u0947 \u0905\u0932\u0917 \u0935\u093f\u0924\u0930\u0923 \u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u092e\u0947\u0902 \u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0926\u094b\u0928\u094b\u0902 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0907\u0938\u0932\u093f\u090f \u0939\u092e \u0907\u0938\u0915\u093e \u0915\u094b\u0908 \u092a\u094d\u0930\u092e\u093e\u0923 \u0928\u0939\u0940\u0902 \u0926\u0947 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0932\u0947\u0915\u093f\u0928 \u092f\u0939\u093e\u0901 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0938\u0947 \u0915\u093e\u092e \u092c\u0928 \u091c\u093e\u090f\u0917\u093e\u0964 \u092f\u0939\u093e\u0901 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092e\u0940\u0915\u0930\u0923 (1)-(3) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0917\u093e \u092f\u093e \u092b\u093f\u0930 \u0926\u093f\u090f \u0917\u090f \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0917\u093e\u0964<\/p>\n\n\n\n<p class=\"mb-1 wp-block-paragraph\">\u0907\u0938\u0915\u0940 8 \u0938\u092e\u094d\u092d\u093e\u0935\u0928\u093e\u090f\u0901 \u0939\u094b \u0938\u0915\u0924\u0940 \u0939\u0948\u0902 :<\/p>\n\n\n\n<ol class=\"wp-block-list mb-1\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 \u092f\u094b\u0917\u092b\u0932<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a nested-list mb-1\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915)<\/li>\n\n\n\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915)<\/li>\n<\/ul>\n\n\n\n<ol style=\"list-style-type:lower-roman\" class=\"wp-block-list nested-list mb-1\">\n<li>\u092f\u094b\u0917\u092b\u0932 &gt; 0<\/li>\n\n\n\n<li>\u092f\u094b\u0917\u092b\u0932 &lt; 0<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-c nested-list mb-1\">\n<li>\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915)<\/li>\n<\/ul>\n\n\n\n<ol start=\"2\" class=\"wp-block-list mb-1\">\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 \u092f\u094b\u0917\u092b\u0932<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-a nested-list mb-1\">\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915)<\/li>\n\n\n\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915)<\/li>\n<\/ul>\n\n\n\n<ol style=\"list-style-type:lower-roman\" class=\"wp-block-list nested-list mb-1\">\n<li>\u092f\u094b\u0917\u092b\u0932 &gt; 0<\/li>\n\n\n\n<li>\u092f\u094b\u0917\u092b\u0932 &lt; 0<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list list-type-alpha start-c nested-list\">\n<li>\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u00d7 (\u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u0938\u094d\u0925\u093f\u0924\u093f 1a \u0917\u0941\u0923\u0928 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u0941\u0932\u0906\u0909\u091f \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u0908 \u0917\u0908 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u092f\u094b\u0917 \u0915\u0940 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e (\u091c\u093f\u0938\u0947 \u0939\u092e \u092a\u0939\u0932\u0947 \u0939\u0940 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u091a\u0941\u0915\u0947 \u0939\u0948\u0902) \u0938\u0947 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 + \u0927\u0928\u093e\u0924\u094d\u092e\u0915 = \u0927\u0928\u093e\u0924\u094d\u092e\u0915 + \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0939\u092e\u0928\u0947 \u092f\u0939\u093e\u0901 \u0938\u094d\u0925\u093f\u0924\u093f 1b(ii) \u0914\u0930 2a \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u0939\u0948\u0964 \u0936\u0947\u0937 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u0915\u094b \u0915\u0930\u0915\u0947 \u0926\u0947\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0939\u092e \u092a\u093e\u0920\u0915 \u0915\u094b \u092a\u094d\u0930\u094b\u0924\u094d\u0938\u093e\u0939\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<p class=\"has-text-align-left has-custom-css wp-custom-css-bf4d0348 wp-block-paragraph\">\u092c\u0949\u0930\u094d\u0921\u0930 \u0915\u0947 \u0915\u093e\u0909\u0902\u091f\u0930 \u0917\u0941\u0923\u0928\u0916\u0923\u094d\u0921\u094b\u0902 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0928\u0938\u0947 \u0905\u0932\u0917 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0928\u0947 \u0935\u093e\u0932\u0947 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0915\u0947 \u0915\u093e\u0909\u0902\u091f\u0930 \u0939\u0932\u094d\u0915\u0947 \u0930\u0902\u0917 \u0938\u0947 \u0926\u093f\u0916\u093e\u090f \u0917\u090f \u0939\u0948\u0902\u0964<\/p>\n\n\n\n<figure class=\"wp-block-table\" style=\"flex:80%;\">\r\n<table width=\"100%\" border=\"1\" cellspacing=\"0\" cellpadding=\"4\">\r\n<tbody>\r\n\r\n<tr>\r\n    <td width=\"8%\" align=\"center\" valign=\"middle\">\r\n        <span style=\"display:inline-block; transform:rotate(-90deg); white-space:nowrap; margin-left:2px;\">\u091a\u0930\u0923 1<\/span>\r\n    <\/td>\r\n\r\n    <td width=\"18%\" valign=\"top\">\r\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S1-1.jpg\" alt=\"Exploring Properties Figure-S1-1\" width=\"100%\"\/>\r\n        <br\/>\r\n    <span style=\"padding-top:8px; display:inline-block;\">4 \u00d7 2<\/span>\r\n    <\/td>\r\n\r\n    <td width=\"28%\" valign=\"top\">\r\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S1-2-1.jpg\" alt=\"Exploring Properties Figure-S1-2\" width=\"100%\"\/>\r\n       <br\/>\r\n    <span style=\"padding-top:8px; display:inline-block;\">4 \u00d7 (-5)<\/span>\r\n    <\/td>\r\n\r\n    <td width=\"46%\" valign=\"top\">\r\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S1-3.jpg\" alt=\"Exploring Properties Figure-S1-3\" width=\"100%\"\/>\r\n        <br\/>\r\n    <span style=\"padding-top:8px; display:inline-block;\">4 \u00d7 [2 + (-5)]<\/span>\r\n    <\/td>\r\n<\/tr>\r\n\r\n<tr>\r\n    <td width=\"8%\" align=\"center\" valign=\"middle\">\r\n        <span style=\"display:inline-block; transform:rotate(-90deg); white-space:nowrap; margin-left:2px;\">\u091a\u0930\u0923 2<\/span>\r\n    <\/td>\r\n\r\n    <td width=\"46%\" colspan=\"2\" valign=\"top\">\r\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S1-2.jpg\" alt=\"Exploring Properties Figure-S1-2\" width=\"100%\"\/>\r\n       <br\/>\r\n    <span style=\"padding-top:8px; display:inline-block;\">4 \u00d7 2 + 4 \u00d7 (-5)<\/span>\r\n    <\/td>\r\n\r\n    <td width=\"46%\" valign=\"bottom\">\r\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S2-2.jpg\" alt=\"Exploring Properties Figure-S2-2\" width=\"100%\"\/>\r\n      <br\/>\r\n    <span style=\"padding-top:8px; display:inline-block;\">\u0936\u0942\u0928\u094d\u092f \u091c\u094b\u0921\u093c\u094b\u0902 \u0915\u094b \u0939\u091f\u093e\u0915\u0930 \u092d\u0930\u0928\u093e<\/span>\r\n<\/td><\/tr>\r\n\r\n<tr>\r\n    <td width=\"8%\" align=\"center\" valign=\"middle\" style=\"position:relative;\">\r\n    <span style=\"position:absolute; top:50%; left:50%; transform:translate(-50%, -50%) rotate(-90deg); white-space:nowrap;\">\u091a\u0930\u0923 3<\/span>\r\n<\/td>\r\n\r\n    <td width=\"46%\" colspan=\"2\" valign=\"bottom\" style=\"position:relative;\">\r\n    <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S3.jpg\" alt=\"Exploring-Properties-Figure-S3\" width=\"45%\" align=\"right\"\/>\r\n    <span style=\"position:absolute; left:4px; bottom:4px; width:50%;\">\u0936\u0942\u0928\u094d\u092f \u091c\u094b\u0921\u093c\u094b\u0902 \u0935 \u092c\u0949\u0930\u094d\u0921\u0930 \u0915\u094b \u0939\u091f\u093e\u0928\u093e<\/span>\r\n<\/td>\r\n\r\n<td width=\"46%\" valign=\"bottom\" style=\"position:relative;\">\r\n    <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-Figure-S3.jpg\" alt=\"Exploring-Properties-Figure-S3\" width=\"45%\" align=\"right\"\/>\r\n    <span style=\"position:absolute; left:4px; bottom:4px; width:50%;\">\u092c\u0949\u0930\u094d\u0921\u0930 \u0915\u094b \u0939\u091f\u093e\u0928\u093e<\/span>\r\n<\/td>\r\n<\/tr>\r\n\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\n\n\n\n<figure class=\"wp-block-table\">\n<table width=\"100%\" border=\"1\" cellspacing=\"0\" cellpadding=\"10\">\n<tbody>\n<tr>\n\n    <td width=\"20%\" valign=\"top\">\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-fig-5-1.jpg\" alt=\"Exploring Properties fig-5-1\" width=\"100%\">\n        <p style:=\"\">(-4) \u00d7 2<\/p>\n    <\/td>\n\n    <td width=\"34%\" valign=\"top\">\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-fig-5-2.jpg\" alt=\"Exploring Properties fig-5-2\" width=\"100%\">\n        <p style:=\"\">(-4) \u00d7 5<\/p>\n    <\/td>\n\n    <td width=\"44%\" valign=\"top\">\n        <img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/09\/Exploring-Properties-fig-5-3.jpg\" alt=\"Exploring Properties fig-5-3\" width=\"100%\">\n        <p style:=\"\">(-4) \u00d7 [2 + 5]<\/p>\n    <\/td>\n\n<\/tr>\n<\/tbody>\n<\/table>\n<figcaption class=\"wp-element-caption\"><b>\u091a\u093f\u0924\u094d\u0930 5<\/b> <\/figcaption>\n<\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u091a\u0942\u0901\u0915\u093f \u0905\u0928\u094d\u0924\u093f\u092e \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0926\u094b\u0928\u094b\u0902 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u0939\u0948\u0902, 4 \u00d7 2 + 4 \u00d7 \u2212 5 = 4 \u00d7 [ 2 + ( \u22125 ) ] \u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f 4 \u00d7 ( \u22125 ) \u092e\u0947\u0902 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0915\u0947 \u0932\u093f\u090f \u0930\u0902\u0917 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0938\u092e\u0940\u0915\u0930\u0923 (2) \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2a \u0915\u0947 \u0909\u0926\u093e\u0939\u0930\u0923 \u092e\u0947\u0902 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u0947 \u0930\u0902\u0917 \u0915\u094b \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092e\u0940\u0915\u0930\u0923 (1) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u091a\u093f\u0924\u094d\u0930-5 \u0915\u0940 \u0924\u0940\u0938\u0930\u0940 \u0924\u0938\u094d\u0935\u0940\u0930 \u0915\u094b \u092a\u0939\u0932\u0940 \u0914\u0930 \u0926\u0942\u0938\u0930\u0940 \u0924\u0938\u094d\u0935\u0940\u0930 \u092f\u093e\u0928\u0940 \u0915\u093f (\u22124) \u00d7 [2 + 5] = (\u22124) \u00d7 2 + (\u22124) \u00d7 5 \u0915\u0947 \u092f\u094b\u0917\u092b\u0932 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u0926\u0947\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092a\u093f\u091b\u0932\u0947 \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u0915\u0940 \u0924\u0930\u0939 \u092f\u0939\u093e\u0901 \u092d\u0940 \u091a\u0941\u0928\u0947 \u0917\u090f \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0940 \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u0926\u0942\u0930\u0940 \u0938\u094d\u200d\u0935\u0947\u091a\u094d\u200d\u091b\u093e \u0938\u0947 \u0924\u092f \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0924\u0940\u0928 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u092e\u093e\u0928\u094d\u092f \u0939\u094b\u0917\u0940\u0964 \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947, \u092f\u0926\u093f \u092a\u0939\u0932\u093e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0936\u0942\u0928\u094d\u092f \u0939\u0948 \u0924\u094b \u0938\u092d\u0940 \u0917\u0941\u0923\u0928\u092b\u0932 \u0936\u0942\u0928\u094d\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u092f\u0939 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e \u092c\u093f\u0928\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0947 \u0938\u0939\u0940 \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0940 \u0924\u0930\u0939, \u092f\u0926\u093f \u0926\u0942\u0938\u0930\u093e \u092f\u093e \u0924\u0940\u0938\u0930\u093e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0936\u0942\u0928\u094d\u092f \u0939\u0948 \u0924\u094b \u090f\u0915 \u0917\u0941\u0923\u0928\u092b\u0932 \u0936\u0942\u0928\u094d\u092f \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948; \u092f\u093e\u0928\u0940 \u0915\u093f \u090f\u0915 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u0917\u093c\u093e\u092f\u092c \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e \u092b\u093f\u0930 \u0938\u0947 \u092c\u093f\u0928\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0947 \u0938\u0924\u094d\u092f \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948\u0964<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u090f\u0915 \u0932\u093e\u0907\u0928 \u092e\u0947\u0902 \u091c\u092e\u093e\u090f \u0917\u090f \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0930\u0902\u0917\u0940\u0928 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u0930\u0902\u0917 \u0915\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938 \u0924\u0930\u0939 \u092f\u0939 \u0938\u093e\u0902\u0915\u0947\u0924\u093f\u0915 \u0932\u092e\u094d\u092c\u093e\u0908 \u0939\u094b\u0917\u0940, \u091c\u093f\u0938\u0947 \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u0915\u0947 \u0938\u093e\u0925 \u091c\u094b\u0921\u093c\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092f\u0926\u093f \u0907\u0928 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u094b \u090f\u0915 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u094d\u092f\u093e\u0938 \u092e\u0947\u0902 \u091c\u092e\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0909\u0928\u0915\u0947 \u0930\u0902\u0917 \u0915\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948; \u092f\u093e\u0928\u0940 \u0915\u093f \u0907\u0938\u0938\u0947 \u0938\u093e\u0902\u0915\u0947\u0924\u093f\u0915 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932  \u0928\u093f\u0915\u093e\u0932\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0939\u092e\u0947\u0902 \u0938\u093e\u0902\u0915\u0947\u0924\u093f\u0915 \u0906\u092f\u0924\u0928 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0928\u0939\u0940\u0902 \u092a\u0921\u093c\u0940\u0964 \u092f\u0939 \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 2-3 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0915\u093e\u092e \u0915\u0930\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092f\u093e \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0928\u0939\u0940\u0902\u0964<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0914\u0930 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u0907\u0938 \u0924\u093e\u0930\u0924\u092e\u094d\u092f \u092e\u0947\u0902 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0938\u092c\u0938\u0947 \u0906\u0938\u093e\u0928 \u0925\u0940\u0902 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0935\u0947 \u0905\u0938\u0924\u0924 \u0914\u0930 \u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u0939\u0948\u0902 \u0914\u0930 \u0907\u0938\u0932\u093f\u090f \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0909\u0928\u0915\u0947 \u0932\u093f\u090f \u092e\u0949\u0921\u0932 \u0924\u0948\u092f\u093e\u0930 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092d\u093f\u0928\u094d\u0928\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0905\u0938\u0924\u0924\u0924\u093e \u0915\u093e \u0932\u093e\u092d \u0928\u0939\u0940\u0902 \u092e\u093f\u0932\u0924\u093e \u0939\u0948\u0964 \u0932\u0947\u0915\u093f\u0928 \u0905\u091a\u094d\u091b\u0940 \u092c\u093e\u0924 \u092f\u0939 \u0939\u0948 \u0915\u093f \u0909\u0928\u0915\u0947 \u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u0939\u094b\u0928\u0947 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 (\u0915\u092d\u0940-\u0915\u092d\u0940 \u0932\u092e\u094d\u092c\u093e\u0908 \u0915\u0947 \u0905\u0928\u0941\u092a\u093e\u0924 \u092e\u0947\u0902) \u0914\u0930 \u0906\u092f\u0924\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0928\u0915\u0947 \u0932\u093f\u090f \u092e\u0949\u0921\u0932 \u0924\u0948\u092f\u093e\u0930 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092e\u0947\u0902 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0936\u093e\u092e\u093f\u0932 \u0925\u0940\u0902, \u092a\u0930 \u0905\u092a\u0928\u0940 \u0905\u0938\u0924\u0924\u093e \u0915\u0947 \u0915\u093e\u0930\u0923 \u0930\u0902\u0917\u0940\u0928 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0928\u0915\u0947 \u0932\u093f\u090f \u092e\u0949\u0921\u0932 \u0924\u0948\u092f\u093e\u0930 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u093e\u0964 \u0932\u0947\u0915\u093f\u0928 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0914\u0930 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0928 \u0924\u094b \u0905\u0938\u0924\u0924 \u0939\u0948\u0902, \u0928\u093e \u0939\u0940 \u090b\u0923\u0947\u0924\u094d\u0924\u0930\u0964 \u0907\u0938\u0932\u093f\u090f \u0935\u0947 \u0938\u092c\u0938\u0947 \u091a\u0941\u0928\u094c\u0924\u0940\u092a\u0942\u0930\u094d\u0923 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0938\u093e\u092c\u093f\u0924 \u0939\u0941\u090f\u0964 \u0909\u0928\u0938\u0947 \u0939\u092e \u0905\u0917\u0932\u0947 \u0926\u094b \u0932\u0947\u0916\u094b\u0902 \u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0902\u0917\u0947\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092f\u094b\u0917 \u0914\u0930 \u0917\u0941\u0923\u0928 \u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e\u094b\u0902 \u0915\u0940 \u092a\u095c\u0924\u093e\u0932 \u0915\u0940 \u0907\u0938 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u093e \u092f\u0939 \u0924\u0940\u0938\u0930\u093e \u0932\u0947\u0916 \u0939\u0948\u0964 (i) \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 (ii) \u090b\u0923\u0947\u0924\u094d\u0924\u0930 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u09021 (Non-negative rational numbers) \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u0907\u0938 \u0932\u0947\u0916 \u092e\u0947\u0902 \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u0907\u0938 \u0936\u0943\u0902\u0916\u0932\u093e, \u0907\u0938\u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0914\u0930 \u0907\u0938\u0938\u0947 \u0939\u092e\u093e\u0930\u0940 \u091c\u094b \u0906\u0915\u093e\u0902\u0915\u094d\u0937\u093e\u090f\u0901 [&hellip;]<\/p>\n","protected":false},"author":17,"featured_media":28636,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_wpcom_ai_launchpad_first_post":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[239],"tags":[],"stage-class":[1008],"magazine_category":[340],"class_list":["post-27912","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category---hi","stage-class-middle-class-6-8-hi","magazine_category-issue-22-july-2019-hi"],"acf":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/09\/Exploring-Properties-of-Addition-thumbnail-1.png","_links":{"self":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/27912","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/comments?post=27912"}],"version-history":[{"count":13,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/27912\/revisions"}],"predecessor-version":[{"id":28128,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/27912\/revisions\/28128"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/media\/28636"}],"wp:attachment":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/media?parent=27912"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/categories?post=27912"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/tags?post=27912"},{"taxonomy":"stage-class","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/stage-class?post=27912"},{"taxonomy":"magazine_category","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/magazine_category?post=27912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}