{"id":21630,"date":"2026-02-25T17:42:20","date_gmt":"2026-02-25T12:12:20","guid":{"rendered":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/?p=21630"},"modified":"2026-05-25T09:18:04","modified_gmt":"2026-05-25T03:48:04","slug":"%e0%a4%95%e0%a4%be%e0%a4%89%e0%a4%82%e0%a4%9f%e0%a4%b0%e0%a5%8d%e0%a4%b8-%e0%a4%95%e0%a5%80-%e0%a4%b8%e0%a4%ae%e0%a5%80%e0%a4%95%e0%a5%8d%e0%a4%b7%e0%a4%be","status":"publish","type":"post","link":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/%e0%a4%95%e0%a4%be%e0%a4%89%e0%a4%82%e0%a4%9f%e0%a4%b0%e0%a5%8d%e0%a4%b8-%e0%a4%95%e0%a5%80-%e0%a4%b8%e0%a4%ae%e0%a5%80%e0%a4%95%e0%a5%8d%e0%a4%b7%e0%a4%be\/","title":{"rendered":"\u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u0938\u0930\u0932 \u0936\u092c\u094d\u0926\u094b\u0902 \u092e\u0947\u0902, \u091c\u093f\u0938 \u092d\u0940 \u091a\u0940\u091c\u093c \u0915\u094b \u0917\u093f\u0928\u093e \u091c\u093e \u0938\u0915\u0947, \u0909\u0938\u0947 \u0915\u093e\u0909\u0902\u091f\u0930 (counter) \u092e\u093e\u0928\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092a\u0924\u094d\u0925\u0930, \u091a\u093e\u0915 \u0915\u0947 \u091f\u0941\u0915\u0921\u093c\u0947, \u092c\u094b\u0924\u0932\u094b\u0902 \u0915\u0947 \u0922\u0915\u094d\u0915\u0928, \u092c\u0940\u091c, \u092c\u091f\u0928 \u2013 \u092f\u0947 \u0938\u092d\u0940 \u092a\u0930\u093f\u091a\u093f\u0924 \u0935\u0938\u094d\u0924\u0941\u090f\u0901 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0915\u093e\u0909\u0902\u091f\u0930 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0907\u0938\u094d\u0924\u0947\u092e\u093e\u0932 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0915\u093e\u0930\u094d\u0921\u092c\u094b\u0930\u094d\u0921 \u0915\u0947 \u092a\u0930 2 \u0938\u0947\u092e\u0940 \u092d\u0941\u091c\u093e \u0935\u093e\u0932\u093e \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0917\u094d\u0930\u093f\u0921 \u092c\u0928\u093e\u0915\u0930 \u0914\u0930 \u092b\u093f\u0930 \u0909\u0938\u0947 \u0915\u093e\u091f\u0915\u0930 \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0915\u093e\u0909\u0902\u091f\u0930 \u092c\u0928\u093e\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0915\u093e\u0930\u094d\u0921\u092c\u094b\u0930\u094d\u0921 \u0915\u094b \u0915\u093e\u091f\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u0948\u0902\u091a\u0940 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 \u0938\u094d\u200d\u091f\u0940\u0932 \u0915\u091f\u0930 \u092c\u0939\u0941\u0924 \u0924\u0947\u091c\u093c\u0940 \u0938\u0947 \u0915\u093e\u092e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0915\u093e\u0930\u094d\u0921\u092c\u094b\u0930\u094d\u0921 \u0938\u0947 \u0935\u0943\u0924\u094d\u0924\u093e\u0915\u093e\u0930 \u0915\u093e\u0909\u0902\u091f\u0930 \u092c\u0928\u093e\u0928\u093e \u0925\u094b\u0921\u093c\u093e \u0925\u0915\u093e\u090a \u0939\u0948 \u0932\u0947\u0915\u093f\u0928 \u0905\u0938\u092e\u094d\u200d\u092d\u0935 \u0928\u0939\u0940\u0902\u0964 \u0915\u0948\u0930\u092e \u0915\u0940 \u0917\u094b\u091f\u093f\u092f\u093e\u0901, \u092c\u094b\u0924\u0932\u094b\u0902 \u0915\u0947 \u0922\u0915\u094d\u0915\u0928 (\u0906\u0926\u0930\u094d\u0936 \u0930\u0942\u092a \u0938\u0947 \u090f\u0915 \u0939\u0940 \u0906\u0915\u093e\u0930 \u0915\u0947), \u091f\u0942\u0925\u092a\u0947\u0938\u094d\u091f \u0915\u0947 \u0922\u0915\u094d\u0915\u0928 \u0917\u094b\u0932 \u0915\u093e\u0909\u0902\u091f\u0930 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092c\u0947\u0939\u0924\u0930\u0940\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u091b\u094b\u091f\u0947 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u094b \u091a\u092e\u0915\u0940\u0932\u0947 \u0930\u0902\u0917\u094b\u0902 \u092e\u0947\u0902 \u0930\u0902\u0917\u0928\u093e \u0909\u091a\u093f\u0924 \u0930\u0939\u0924\u093e \u0939\u0948\u0964      <\/p>\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"380\" height=\"720\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-figure.jpg\" alt=\"counters-figure\" class=\"wp-image-21726\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-figure.jpg 380w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-figure-190x360.jpg 190w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-figure-107x202.jpg 107w\" sizes=\"auto, (max-width: 380px) 100vw, 380px\" \/><\/figure>\n\n<p class=\"wp-block-paragraph\">\u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u092c\u091a\u094d\u091a\u0947 \u0915\u094b \u092e\u093e\u0924\u094d\u0930\u093e (\u0917\u093f\u0928\u0924\u0940) \u0914\u0930 \u0909\u0938 \u092e\u093e\u0924\u094d\u0930\u093e \u0915\u093e \u092a\u094d\u0930\u0924\u093f\u0928\u093f\u0927\u093f\u0924\u094d\u0935 \u0915\u0930\u0928\u0947 \u0935\u093e\u0932\u0947 \u0905\u0902\u0915 \u092f\u093e \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0928\u093e\u092e \u0915\u0947 \u092c\u0940\u091a \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0938\u0947 \u0938\u092e\u094d\u200d\u092c\u0928\u094d\u200d\u0927 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 (\u091a\u093f\u0924\u094d\u0930-1)\u0964 \u092f\u0939 \u0935\u0939 \u092a\u0939\u0932\u0940 \u092e\u0948\u0928\u093f\u092a\u0941\u0932\u0947\u091f\u093f\u0935\u094d\u0938 \u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0939\u0948 \u091c\u093f\u0938\u0938\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u092c\u091a\u094d\u091a\u0947 \u0915\u093e \u092a\u0930\u093f\u091a\u092f \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0915\u094b \u092e\u093e\u0924\u094d\u0930\u093e \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0915\u0947\u0928\u094d\u0926\u094d\u0930\u093f\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0928 \u0915\u093f \u0907\u0938 \u092c\u093e\u0924 \u092a\u0930 \u0915\u093f \u0935\u0947 \u0915\u093f\u0924\u0928\u0940 \u091c\u0917\u0939 \u092e\u0947\u0902 \u092b\u0948\u0932\u0947 \u0939\u0941\u090f \u0939\u0948\u0902 (\u091a\u093f\u0924\u094d\u0930-2)\u0964 \u092e\u0948\u0928\u093f\u092a\u0941\u0932\u0947\u091f\u093f\u0935\u094d\u0938 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902, \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u094b \u0939\u093e\u0925\u094b\u0902 \u0938\u0947 \u0909\u0920\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 (\u0914\u0930 \u090f\u0915 \u0915\u091f\u094b\u0930\u0947 \u092f\u093e \u091f\u094b\u0915\u0930\u0940 \u092e\u0947\u0902 \u0905\u0932\u0917 \u0930\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948) \u2013 \u0914\u0930 \u091c\u094b \u0917\u093f\u0928\u0947 \u091c\u093e \u091a\u0941\u0915\u0947 \u0939\u0948\u0902 \u0909\u0928\u094d\u200d\u0939\u0947\u0902 \u0905\u0928\u0917\u093f\u0928\u094b\u0902 \u0938\u0947 \u0905\u0932\u0917 \u0915\u0930\u0928\u093e \u0906\u0938\u093e\u0928 \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u092f\u0939 \u0915\u0939\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f \u0917\u093f\u0928\u0924\u0940 \u0915\u0947 \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0905\u092d\u094d\u092f\u093e\u0938 \u0915\u0947 \u092c\u093f\u0928\u093e, \u092c\u091a\u094d\u091a\u093e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u092e\u091c\u092c\u0942\u0924 \u0928\u0940\u0902\u0935 \u0928\u0939\u0940\u0902 \u092c\u0928\u093e \u092a\u093e\u090f\u0917\u093e\u0964 \u0907\u0938\u0932\u093f\u090f, \u0936\u0941\u0930\u0941\u0906\u0924 \u092e\u0947\u0902 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u092c\u0939\u0941\u0924 \u092e\u0939\u0924\u094d\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0902\u0964     <\/p>\n\n<div class=\"wp-block-group is-vertical is-content-justification-center is-layout-flex wp-container-core-group-is-layout-97ed9c64 wp-block-group-is-layout-flex\">\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"606\" height=\"661\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-1-1-e1772021243682.jpg\" alt=\"counters-fig-1\" class=\"wp-image-21659\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-1-1-e1772021243682.jpg 606w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-1-1-e1772021243682-330x360.jpg 330w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-1-1-e1772021243682-185x202.jpg 185w\" sizes=\"auto, (max-width: 606px) 100vw, 606px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-1<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"748\" height=\"655\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-2-e1772021278404.jpg\" alt=\"counters-fig-2\" class=\"wp-image-21662\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-2-e1772021278404.jpg 748w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-2-e1772021278404-411x360.jpg 411w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-2-e1772021278404-231x202.jpg 231w\" sizes=\"auto, (max-width: 748px) 100vw, 748px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-2<\/figcaption><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">\u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0947 \u092b\u093c\u093e\u092f\u0926\u0947 \u0915\u0947\u0935\u0932 \u0917\u093f\u0928\u0924\u0940 \u0924\u0915 \u0939\u0940 \u0938\u0940\u092e\u093f\u0924 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902\u0964 \u0935\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u0948\u091f\u0930\u094d\u0928 \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u092e\u0926\u0926 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0932\u093f\u090f, \u0935\u0947 \u092c\u0940\u091c\u0940\u092f \u0938\u094b\u091a (algebraic thinking) \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0936\u093e\u0928\u0926\u093e\u0930 \u0936\u0941\u0930\u0941\u0906\u0924\u0940 \u092c\u093f\u0928\u094d\u200d\u0926\u0941 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u092c\u0922\u093c\u0924\u0947 \u0939\u0941\u090f (\u091a\u093f\u0924\u094d\u0930-3) \u0914\u0930 \u0926\u094b\u0939\u0930\u093e\u090f \u091c\u093e\u0928\u0947 \u0935\u093e\u0932\u0947 (\u091a\u093f\u0924\u094d\u0930-4) \u092a\u0948\u091f\u0930\u094d\u0928 \u092c\u0940\u091c\u0940\u092f \u0935\u094d\u092f\u0902\u091c\u0915\u094b\u0902 \u0914\u0930 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u0947 \u0906\u0938-\u092a\u093e\u0938 \u091a\u0930\u094d\u091a\u093e \u0936\u0941\u0930\u0942 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964     <\/p>\n\n<div class=\"wp-block-group is-vertical is-content-justification-center is-layout-flex wp-container-core-group-is-layout-524f8de7 wp-block-group-is-layout-flex\">\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"243\" height=\"228\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-3-1-e1772021315655.jpg\" alt=\"counters-fig-3\" class=\"wp-image-21665\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-3-1-e1772021315655.jpg 243w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-3-1-e1772021315655-215x202.jpg 215w\" sizes=\"auto, (max-width: 243px) 100vw, 243px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-3<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"616\" height=\"129\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-4-1-e1772021345861.jpg\" alt=\"counters-fig-4\" class=\"wp-image-21668\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-4-1-e1772021345861.jpg 616w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-4-1-e1772021345861-360x75.jpg 360w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-4<\/figcaption><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u0935\u0947 \u0905\u0938\u0924\u0924 \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u0947\u091f\u094b\u0902 (discrete number sets), \u091c\u0948\u0938\u0947 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 (whole numbers) \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 (integers) \u0915\u0947 \u0932\u093f\u090f \u091c\u094b\u0921\u093c \u0914\u0930 \u0917\u0941\u0923\u093e \u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0916\u0942\u092c\u0938\u0942\u0930\u0924\u0940 \u0938\u0947 \u0926\u0930\u094d\u0936\u093e\u0924\u0947 \u0939\u0948\u0902\u0964 \u092f\u0939\u093e\u0901 \u0915\u0941\u091b \u0909\u0926\u093e\u0939\u0930\u0923 \u0926\u093f\u090f \u0917\u090f \u0939\u0948\u0902 : <\/p>\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1046\" height=\"720\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-5.jpg\" alt=\"counters-fig-5\" class=\"wp-image-21705\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-5.jpg 1046w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-5-523x360.jpg 523w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-5-293x202.jpg 293w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-5-768x529.jpg 768w\" sizes=\"auto, (max-width: 1046px) 100vw, 1046px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-5<\/figcaption><\/figure>\n\n<p class=\"wp-block-paragraph\">\u091a\u093f\u0924\u094d\u0930-5 \u0926\u094b \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923\u094b\u0902 \u0938\u0947 \u090f\u0915 \u0939\u0940 \u092f\u094b\u0917 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948, \u091c\u092c\u0915\u093f \u091a\u093f\u0924\u094d\u0930-6 \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948 \u0915\u093f \u091a\u093e\u0939\u0947 \u0915\u093f\u0928\u094d\u0939\u0940\u0902 \u092d\u0940 \u0926\u094b \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u092a\u0939\u0932\u0947 \u091c\u094b\u0921\u093c\u093e \u091c\u093e\u090f, \u092f\u094b\u0917 \u0938\u092e\u093e\u0928 \u0930\u0939\u0924\u093e \u0939\u0948 (\u0905\u0927\u093f\u0915 \u0935\u093f\u0935\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f \u0938\u0928\u094d\u200d\u0926\u0930\u094d\u092d [2] \u0926\u0947\u0916\u0947\u0902)\u0964 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f, \u091a\u0942\u0901\u0915\u093f \u0926\u094b \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902 \u2013 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0914\u0930 \u090b\u0923\u093e\u0924\u094d\u092e\u0915, \u0939\u092e\u0947\u0902 \u0926\u094b \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u092e\u0942\u0932 \u0930\u0942\u092a \u0938\u0947, \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0936\u0942\u0928\u094d\u092f \u0938\u0947 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0915\u0940 \u0926\u0942\u0930\u0940 (\u092f\u093e\u0928\u0940 \u0928\u093f\u0930\u092a\u0947\u0915\u094d\u0937 \u092e\u093e\u0928\/absolute value) \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u0940 \u0939\u0948 \u0914\u0930 \u0930\u0902\u0917 (\u092f\u093e \u092a\u094d\u0930\u0915\u093e\u0930) \u092f\u0939 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0935\u0939 \u0936\u0942\u0928\u094d\u092f \u0915\u0947 \u0915\u093f\u0938 \u0913\u0930 \u0939\u0948 (\u092f\u093e\u0928\u0940 \u091a\u093f\u0939\u094d\u0928)\u0964 \u091a\u093f\u0924\u094d\u0930-7 \u0914\u0930 \u091a\u093f\u0924\u094d\u0930-8 \u0915\u094d\u0930\u092e\u0936\u0903 3 \u0914\u0930 -2 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938\u0947 \u0938\u094d\u092a\u0937\u094d\u091f \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u091a\u093f\u0924\u094d\u0930-5 \u0914\u0930 \u091a\u093f\u0924\u094d\u0930-6 \u0915\u094b \u0910\u0938\u0947 \u0930\u0902\u0917\u0940\u0928 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0947 \u0938\u093e\u0925 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u092d\u0940 \u0907\u0938\u0940 \u0924\u0930\u0939 \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 (\u0935\u093f\u0935\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f \u0938\u0928\u094d\u200d\u0926\u0930\u094d\u092d [3] \u0926\u0947\u0916\u0947\u0902)\u0964    <\/p>\n\n<div class=\"wp-block-group is-vertical is-content-justification-center is-layout-flex wp-container-core-group-is-layout-524f8de7 wp-block-group-is-layout-flex\">\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1120\" height=\"720\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig6.jpg\" alt=\"counters-fig-6\" class=\"wp-image-21701\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig6.jpg 1120w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig6-560x360.jpg 560w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig6-314x202.jpg 314w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig6-768x494.jpg 768w\" sizes=\"auto, (max-width: 1120px) 100vw, 1120px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-6<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1365\" height=\"220\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945.jpg\" alt=\"counters-fig-7\" class=\"wp-image-21674\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945.jpg 1365w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945-640x103.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945-1280x206.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945-360x58.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-7-1-e1772021379945-768x124.jpg 768w\" sizes=\"auto, (max-width: 1365px) 100vw, 1365px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-7<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1365\" height=\"207\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149.jpg\" alt=\"counters-fig-8\" class=\"wp-image-21677\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149.jpg 1365w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149-640x97.jpg 640w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149-1280x194.jpg 1280w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149-360x55.jpg 360w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-8-1-e1772021400149-768x116.jpg 768w\" sizes=\"auto, (max-width: 1365px) 100vw, 1365px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-8<\/figcaption><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">\u091a\u093f\u0924\u094d\u0930-9 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u0917\u0941\u0923\u093e \u0915\u0940 \u0915\u094d\u0930\u092e\u0935\u093f\u0928\u093f\u092e\u0947\u092f\u0924\u093e (commutativity) \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u090f\u0915 \u0939\u0940 \u0935\u094d\u092f\u0942\u0939 (array) \u0905\u092a\u0928\u0940 \u0926\u093f\u0936\u093e \u092c\u0926\u0932\u0928\u0947 \u092a\u0930 \u0926\u094b \u0917\u0941\u0923\u0928\u092b\u0932\u094b\u0902 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0926\u0942\u0938\u0930\u0940 \u0913\u0930, \u091a\u093f\u0924\u094d\u0930-10 \u0914\u0930 \u091a\u093f\u0924\u094d\u0930-11 \u0915\u094d\u0930\u092e\u0936\u0903 \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0935\u093f\u0924\u0930\u0923 \u0917\u0941\u0923 (distributive property) \u0915\u094b \u0926\u093f\u0916\u093e\u0924\u0947 \u0939\u0948\u0902 (\u0935\u093f\u0935\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f \u0938\u0928\u094d\u200d\u0926\u0930\u094d\u092d [2] \u0914\u0930 [3] \u0926\u0947\u0916\u0947\u0902)\u0964  <\/p>\n\n<div class=\"wp-block-group is-vertical is-content-justification-center is-layout-flex wp-container-core-group-is-layout-524f8de7 wp-block-group-is-layout-flex\">\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"794\" height=\"515\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-9-e1772020730854.jpg\" alt=\"counters-fig-9\" class=\"wp-image-21680\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-9-e1772020730854.jpg 794w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-9-e1772020730854-555x360.jpg 555w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-9-e1772020730854-311x202.jpg 311w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-9-e1772020730854-768x498.jpg 768w\" sizes=\"auto, (max-width: 794px) 100vw, 794px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-9<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"622\" height=\"492\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-10-e1772020768599.jpg\" alt=\"counters-fig-10\" class=\"wp-image-21683\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-10-e1772020768599.jpg 622w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-10-e1772020768599-455x360.jpg 455w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-10-e1772020768599-255x202.jpg 255w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-10<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"735\" height=\"652\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-11-e1772020812546.jpg\" alt=\"counters-fig-11\" class=\"wp-image-21686\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-11-e1772020812546.jpg 735w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-11-e1772020812546-406x360.jpg 406w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-11-e1772020812546-228x202.jpg 228w\" sizes=\"auto, (max-width: 735px) 100vw, 735px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-11<\/figcaption><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">\u0935\u0947 \u0917\u0941\u0923\u0928\u0916\u0923\u094d\u200d\u0921 (factors) \u0916\u094b\u091c\u0928\u0947, \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 (primes) \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 (\u091c\u093f\u0938\u092e\u0947\u0902 \u092f\u0939 \u092d\u0940 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948 \u0915\u093f 1 \u0905\u092d\u093e\u091c\u094d\u092f \u0915\u094d\u092f\u094b\u0902 \u0928\u0939\u0940\u0902 \u0939\u0948), \u0914\u0930 \u0935\u0930\u094d\u0917 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u092d\u0940 \u092e\u0926\u0926 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902! \u091a\u093f\u0924\u094d\u0930-12, 12 \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u0947 \u0938\u093e\u0925 \u0938\u092d\u0940 \u0938\u092e\u094d\u200d\u092d\u0935 \u0935\u094d\u092f\u0942\u0939\u094b\u0902 (arrays) \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0935\u094d\u092f\u0942\u0939 \u092e\u0947\u0902 \u092a\u0902\u0915\u094d\u0924\u093f\u092f\u094b\u0902 (\u092f\u093e \u0938\u094d\u0924\u092e\u094d\u200d\u092d\u094b\u0902) \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 12 \u0915\u0947 \u090f\u0915 \u0917\u0941\u0923\u0928\u0916\u0923\u094d\u200d\u0921 \u0938\u0947 \u092e\u0947\u0932 \u0916\u093e\u0924\u0940 \u0939\u0948 \u0914\u0930 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u092e\u094d\u200d\u092d\u093e\u0935\u093f\u0924 \u0935\u094d\u092f\u0942\u0939\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0917\u0941\u0923\u0928\u0916\u0923\u094d\u200d\u0921\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964  <\/p>\n\n<figure class=\"wp-block-image size-full article-web-image\"><img loading=\"lazy\" decoding=\"async\" width=\"522\" height=\"446\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-12-e1772020872903.jpg\" alt=\"counters-fig-12\" class=\"wp-image-21689\" srcset=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-12-e1772020872903.jpg 522w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-12-e1772020872903-421x360.jpg 421w, https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-12-e1772020872903-236x202.jpg 236w\" sizes=\"auto, (max-width: 522px) 100vw, 522px\" \/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-12<\/figcaption><\/figure>\n\n<p class=\"has-text-color has-link-color wp-elements-5667e07ae69c3a6aad3f1ba0acd80515 wp-block-paragraph\" style=\"color:#e68d31\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u0915\u0948\u0938\u0947 \u0935\u0943\u0924\u094d\u0924 \u0915\u0940 \u0938\u0930\u0923\u093f\u092f\u093e\u0901 (arrays) \u0916\u0921\u093c\u0947-\u0906\u0921\u093c\u0947 \u0915\u0947 \u091c\u094b\u0921\u093c\u0947 \u092e\u0947\u0902 \u0939\u0948\u0902\u0964 \u0905\u0917\u0930 \u0935\u0943\u0924\u094d\u0924 \u0915\u0940 \u091c\u0917\u0939 \u0935\u0930\u094d\u0917 \u0939\u094b\u0924\u0947 \u0924\u094b \u0915\u0948\u0938\u0947 \u0939\u094b\u0924\u0947? <\/p>\n\n<figure class=\"wp-block-table\">\r\n<table class=\"has-fixed-layout\" style=\"width:100%; text-align:center; border-collapse:collapse;\">\r\n<tbody>\r\n<tr style=\"height:50px;\">\r\n<th style=\"width:15%;\">\u0928\u092e\u094d\u200d\u092c\u0930<\/th>\r\n<th style=\"width:15%;\">\u0938\u0930\u0923\u093f\u092f\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u200d\u092f\u093e <\/th>\r\n<th style=\"width:15%;\">\u0928\u092e\u094d\u200d\u092c\u0930<\/th>\r\n<th style=\"width:15%;\">\u0938\u0930\u0923\u093f\u092f\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u200d\u092f\u093e <\/th>\r\n<th style=\"width:40%;\">\u0905\u0935\u0932\u094b\u0915\u0928<\/th>\r\n<\/tr>\r\n\r\n<tr style=\"height:50px;\">\r\n<td>1<\/td>\r\n<td>1<\/td>\r\n<td>9<\/td>\r\n<td>3<\/td>\r\n<td rowspan=\"8\" style=\"text-align:left; vertical-align:middle;\">\r\n<ul>\r\n<li>\u0915\u093f\u0928 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0915\u0947\u0935\u0932 2 \u0939\u0940 \u0938\u0902\u092d\u0935 \u0938\u0930\u0923\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902?<\/li>\r\n<li>\u0915\u093f\u0928 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 2 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0938\u0930\u0923\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902?<\/li>\r\n<li>\u0924\u094b, \u0915\u094d\u092f\u093e 1 \u090f\u0915 \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e (prime) \u0939\u0948? \u0915\u094d\u092f\u093e \u092f\u0939 \u090f\u0915 \u0938\u0902\u092f\u0941\u0915\u094d\u0924 \u0938\u0902\u0916\u094d\u092f\u093e(composite) \u0939\u0948? <\/li>\r\n<li>\u0915\u093f\u0928 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u0930\u0923\u0940 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0935\u093f\u0937\u092e (odd) \u0939\u094b\u0924\u0940 \u0939\u0948?<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n\r\n<tr style=\"height:50px;\"><td>2<\/td><td>2<\/td><td>10<\/td><td>4<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>3<\/td><td>2<\/td><td>11<\/td><td>2<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>4<\/td><td>3<\/td><td>12<\/td><td>6<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>5<\/td><td>2<\/td><td>13<\/td><td>2<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>6<\/td><td>4<\/td><td>14<\/td><td>4<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>7<\/td><td>2<\/td><td>15<\/td><td>4<\/td><\/tr>\r\n<tr style=\"height:50px;\"><td>8<\/td><td>4<\/td><td>16<\/td><td>5<\/td><\/tr>\r\n\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\n<p class=\"wp-block-paragraph\">&#8216;\u0935\u0930\u094d\u0917 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0917\u0941\u0923\u0928\u0916\u0923\u094d\u200d\u0921\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0935\u093f\u0937\u092e (odd) \u0915\u094d\u092f\u094b\u0902 \u0939\u094b\u0924\u0940 \u0939\u0948&#8217; \u0915\u093e \u0926\u0942\u0938\u0930\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u0905\u0902\u0915\u0917\u0923\u093f\u0924 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u092a\u094d\u0930\u092e\u0947\u092f (Fundamental Theorem of Arithmetic) \u0914\u0930 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u093e \u0939\u0948 (\u0914\u0930 \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u092c\u0939\u0941\u0924 \u0905\u0927\u093f\u0915 \u0915\u0920\u093f\u0928 \u0939\u0948)!<\/p>\n\n<p class=\"wp-block-paragraph\">\u0905\u0935\u0932\u094b\u0915\u0928 \u0938\u094d\u0924\u092e\u094d\u200d\u092d \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0947 \u0905\u092e\u0942\u0930\u094d\u0924\u093f\u0915\u0930\u0923 (abstraction) \u0915\u0947 \u092e\u093e\u0930\u094d\u0917 \u0915\u094b \u0938\u0941\u0917\u092e \u092c\u0928\u093e\u0924\u093e \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0935\u093f\u0926\u094d\u092f\u093e\u0930\u094d\u0925\u0940 \u0905\u092a\u0928\u0940 \u0938\u094b\u091a \u0915\u094b \u091a\u093f\u0924\u094d\u0930\u092e\u092f \u092a\u094d\u0930\u0938\u094d\u200d\u0924\u0941\u0924\u093f\u0915\u0930\u0923 \u0914\u0930 \u092b\u093f\u0930 \u092e\u093e\u0928\u0938\u093f\u0915 \u091b\u0935\u093f\u092f\u094b\u0902 \u0924\u0915 \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u093e \u0936\u0941\u0930\u0942 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n\n<p class=\"wp-block-paragraph\">\u0907\u0938\u0915\u0947 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924, \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f &#8216;\u092c\u093f\u0928\u093e \u0936\u092c\u094d\u0926\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u092e\u093e\u0923&#8217; [Proofs Without Words (PWW)] \u0924\u0948\u092f\u093e\u0930 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092c\u0947\u0939\u0924\u0930\u0940\u0928 \u0939\u0948\u0902\u0964 \u090f\u0915 \u0938\u0902\u0938\u093e\u0927\u0928 \u0938\u092e\u0942\u0939 \u0928\u0947 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0917\u0949\u0938 (Gauss) \u0915\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u0940 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u092c\u091a\u092a\u0928 \u0915\u0940 \u0916\u094b\u091c \u0915\u093e \u090f\u0915 \u092e\u0949\u0921\u0932 \u092c\u0928\u093e\u092f\u093e \u0939\u0948! \u091a\u093f\u0924\u094d\u0930-13 \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u094b \u090f\u0915 \u0935\u0930\u094d\u0917 \u092e\u0947\u0902 \u091c\u0941\u0921\u093c\u0924\u0947 \u0939\u0941\u090f \u0926\u093f\u0916\u093e\u0924\u093e \u0939\u0948\u0964 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u200d\u092f\u093e\u0913\u0902 \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u0939\u0930 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u0947 \u0932\u093f\u090f PWW \u092c\u0928\u093e\u0928\u093e \u0938\u092e\u094d\u200d\u092d\u0935 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964   <\/p>\n\n<figure class=\"wp-block-image size-full text-center\"><img loading=\"lazy\" decoding=\"async\" width=\"155\" height=\"149\" src=\"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-13-e1772020933329.jpeg\" alt=\"counters-fig-13\" class=\"wp-image-21656\"\/><figcaption class=\"wp-element-caption\">\u091a\u093f\u0924\u094d\u0930-13<\/figcaption><\/figure>\n\n<p class=\"wp-block-paragraph\">\u091c\u092c \u0939\u092e \u0917\u0923\u093f\u0924 \u0915\u0947 \u0910\u0938\u0947 \u0909\u091a\u094d\u091a \u0914\u0930 \u0930\u094b\u092e\u093e\u0902\u091a\u0915 \u0938\u094d\u0924\u0930\u094b\u0902 \u092a\u0930 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u092f\u0939 \u0924\u092c \u092e\u0926\u0926\u0917\u093e\u0930 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u0915\u093e\u0909\u0902\u091f\u0930\u094d\u0938 \u090f\u0915 \u091c\u0948\u0938\u0947 \u0939\u094b\u0902\u0964 \u0926\u094b \u0932\u094b\u0915\u092a\u094d\u0930\u093f\u092f \u0906\u0915\u093e\u0930 \u0939\u0948\u0902 \u2013 \u0935\u0943\u0924\u094d\u0924\u093e\u0915\u093e\u0930 \u0914\u0930 \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u2013 \u091c\u093f\u0928\u092e\u0947\u0902 \u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u0947 \u0905\u092a\u0928\u0947 \u092b\u093c\u093e\u092f\u0926\u0947 \u0914\u0930 \u0928\u0941\u0915\u093c\u0938\u093e\u0928 \u0939\u0948\u0902\u0964 <\/p>\n\n<figure class=\"wp-block-table\">\r\n<table class=\"has-fixed-layout\" style=\"width:100%; border-collapse:collapse;\">\r\n<tbody>\r\n\r\n<tr>\r\n<th style=\"width:20%;\"><\/th>\r\n<th style=\"width:40%; text-align:center;\">\u0935\u0943\u0924\u094d\u0924<\/th>\r\n<th style=\"width:40%; text-align:center;\">\u0935\u0930\u094d\u0917<\/th>\r\n<\/tr>\r\n\r\n<tr>\r\n<th style=\"text-align:center;\">\u092b\u093c\u093e\u092f\u0926\u093e<\/th>\r\n<td>\r\n\u0938\u092c\u0938\u0947 \u0938\u092e\u092e\u093f\u0924 (symmetric) \u0906\u0915\u093e\u0930 <br\/>\u2234 \u0905\u092d\u093f\u0935\u093f\u0928\u094d\u092f\u093e\u0938 (orientation) \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 <br\/>\u21d2 \u092c\u0939\u0941\u0924 \u091b\u094b\u091f\u0947 \u092c\u091a\u094d\u091a\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u093e <br\/>\u21d2 \u092a\u0948\u091f\u0930\u094d\u0928 \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u093e \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u0905\u0927\u093f\u0915\u0924\u092e \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0926\u0947\u0924\u093e \u0939\u0948\u0964 <br\/> <br\/>\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u091c\u092c \u0907\u0928\u094d\u0939\u0947\u0902 \u090f\u0915 \u0915\u0924\u093e\u0930 \u092e\u0947\u0902 \u0932\u0917\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 \u0907\u0928\u094d\u0939\u0947\u0902 \u0917\u093f\u0928\u0928\u093e \u0906\u0938\u093e\u0928 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0915\u093e\u0909\u0902\u091f\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0916\u093e\u0932\u0940 \u091c\u0917\u0939 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964\r\n<\/td>\r\n<td>\r\n\u090f\u0915 \u0928\u093f\u092f\u092e\u093f\u0924 \u0906\u0915\u093e\u0930 \u091c\u094b \u091f\u093e\u0907\u0932 \u0915\u0940 \u0924\u0930\u0939 \u091c\u0941\u095c \u0938\u0915\u0924\u093e \u0939\u0948 <br\/>\u2234 \u0935\u0930\u094d\u0917 \u0917\u094d\u0930\u093f\u0921 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0915\u0941\u091b \u092d\u0940 \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092c\u0947\u0939\u0924\u0930\u0940\u0928, \u091c\u0948\u0938\u0947 : \u092a\u0949\u0932\u093f\u092f\u094b\u092e\u093f\u0928\u094b (polyominoes) \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0915\u0940 \u0907\u0915\u093e\u0908 (Unit of area) <br\/>\u2234 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0914\u0930 \u092a\u0930\u093f\u092e\u093e\u092a (perimeter) \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092c\u0947\u0939\u0924\u0930\u0940\u0928 <br\/>\u21d2 \u091c\u092c \u0907\u0928\u094d\u0939\u0947\u0902 \u090f\u0915 \u0915\u0924\u093e\u0930 \u092e\u0947\u0902 \u0932\u0917\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u092f\u0947 \u090f\u0915 \u0906\u092f\u0924 (rectangle) \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0909\u0938\u0915\u0940 \u0932\u092e\u094d\u200d\u092c\u093e\u0908 \u0915\u0947 \u0938\u092e\u093e\u0928\u0941\u092a\u093e\u0924\u0940 (proportionate) \u0939\u094b\u0924\u093e \u0939\u0948 <br\/>\u2234 \u0907\u0938\u0947 \u0932\u092e\u094d\u200d\u092c\u093e\u0908 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 (generalized) \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\r\n<\/td>\r\n<\/tr>\r\n\r\n<tr>\r\n<th style=\"text-align:center;\">\u0928\u0941\u0915\u093c\u0938\u093e\u0928<\/th>\r\n<td>\r\n\u091f\u093e\u0907\u0932 \u0915\u0940 \u0924\u0930\u0939 \u0928\u0939\u0940\u0902 \u091c\u0941\u095c\u0924\u093e\u0964 \r\n<\/td>\r\n\r\n<td>\u091c\u092c \u0907\u0928\u094d\u0939\u0947\u0902 \u090f\u0915 \u0930\u0947\u0916\u093e \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0915\u0947 \u0906\u092f\u0924 \u092c\u0928\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0907\u0928\u094d\u0939\u0947\u0902 \u0917\u093f\u0928\u0928\u093e \u0915\u0920\u093f\u0928 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948 \u091c\u092c \u0924\u0915 \u0915\u093f \u0907\u0928\u0915\u0940 \u0938\u0940\u092e\u093e\u090f\u0901 (borders) \u0938\u094d\u092a\u0937\u094d\u091f \u0928 \u0939\u094b\u0902\u0964\r\n\u091c\u092c \u0924\u0915 \u0907\u0928\u094d\u0939\u0947\u0902 \u0920\u0940\u0915 \u0938\u0947 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0928 \u0915\u093f\u092f\u093e \u091c\u093e\u090f, \u092f\u0947 \u0938\u0941\u0928\u094d\u200d\u0926\u0930 \u092f\u093e \u0938\u093e\u092b\u093c-\u0938\u0941\u0925\u0930\u0947 \u0928\u0939\u0940\u0902 \u0926\u093f\u0916 \u0938\u0915\u0924\u0947\u0964 \u0935\u0943\u0924\u094d\u0924\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 \u0915\u092e \u0938\u092e\u092e\u093f\u0924\u093f (symmetry)\u0964<br\/>\u2234 \u092f\u0926\u093f \u0935\u0943\u0924\u094d\u0924\u093e\u0915\u093e\u0930 \u091f\u093e\u0907\u0932\u094b\u0902 \u0915\u0947 \u092c\u091c\u093e\u092f \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u091f\u093e\u0907\u0932\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0915\u0941\u091b \u0935\u093f\u0936\u0947\u0937 \u0938\u092e\u092e\u093f\u0924\u093f \u0938\u094d\u092a\u0937\u094d\u091f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 <\/td>\r\n<\/tr>\r\n\r\n<tr>\r\n<th style=\"text-align:center;\">\u091a\u093f\u0924\u094d\u0930<\/th>\r\n<td style=\"text-align:center;\">\r\n<img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/02\/Circle-counters.jpg\" alt=\"Circle counters\"\/>\r\n<\/td>\r\n<td style=\"text-align:center;\">\r\n<img decoding=\"async\" src=\"\/wp-content\/uploads\/2026\/02\/Square-counters.jpg\" alt=\"Square counters\"\/>\r\n<\/td>\r\n<\/tr>\r\n\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n","protected":false},"excerpt":{"rendered":"<p>\u0938\u0930\u0932 \u0936\u092c\u094d\u0926\u094b\u0902 \u092e\u0947\u0902, \u091c\u093f\u0938 \u092d\u0940 \u091a\u0940\u091c\u093c \u0915\u094b \u0917\u093f\u0928\u093e \u091c\u093e \u0938\u0915\u0947, \u0909\u0938\u0947 \u0915\u093e\u0909\u0902\u091f\u0930 (counter) \u092e\u093e\u0928\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092a\u0924\u094d\u0925\u0930, \u091a\u093e\u0915 \u0915\u0947 \u091f\u0941\u0915\u0921\u093c\u0947, \u092c\u094b\u0924\u0932\u094b\u0902 \u0915\u0947 \u0922\u0915\u094d\u0915\u0928, \u092c\u0940\u091c, \u092c\u091f\u0928 \u2013 \u092f\u0947 \u0938\u092d\u0940 \u092a\u0930\u093f\u091a\u093f\u0924 \u0935\u0938\u094d\u0924\u0941\u090f\u0901 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0915\u093e\u0909\u0902\u091f\u0930 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0907\u0938\u094d\u0924\u0947\u092e\u093e\u0932 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0915\u093e\u0930\u094d\u0921\u092c\u094b\u0930\u094d\u0921 \u0915\u0947 \u092a\u0930 2 \u0938\u0947\u092e\u0940 \u092d\u0941\u091c\u093e \u0935\u093e\u0932\u093e \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0917\u094d\u0930\u093f\u0921 \u092c\u0928\u093e\u0915\u0930 \u0914\u0930 \u092b\u093f\u0930 \u0909\u0938\u0947 [&hellip;]<\/p>\n","protected":false},"author":17,"featured_media":21657,"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":[241],"tags":[],"stage-class":[1009,1006],"magazine_category":[354],"class_list":["post-21630","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category--hi","stage-class-preparatory-class-3-5-hi","stage-class-foundational-pre-primary-class-1-2-hi","magazine_category-issue-29-november-2021-hi"],"acf":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/wp-content\/uploads\/2026\/02\/counters-fig-13-e1772020933329.jpeg","_links":{"self":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/21630","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=21630"}],"version-history":[{"count":2,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/21630\/revisions"}],"predecessor-version":[{"id":24355,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/posts\/21630\/revisions\/24355"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/media\/21657"}],"wp:attachment":[{"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/media?parent=21630"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/categories?post=21630"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/tags?post=21630"},{"taxonomy":"stage-class","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/stage-class?post=21630"},{"taxonomy":"magazine_category","embeddable":true,"href":"https:\/\/atrightangles.azimpremjiuniversity.edu.in\/hi\/wp-json\/wp\/v2\/magazine_category?post=21630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}