{"id":282,"date":"2008-08-27T11:26:36","date_gmt":"2008-08-27T16:26:36","guid":{"rendered":"http:\/\/punyamishra.com\/2008\/08\/27\/visualizing-mathematics\/"},"modified":"2008-11-10T22:36:18","modified_gmt":"2008-11-11T03:36:18","slug":"visualizing-mathematics","status":"publish","type":"post","link":"https:\/\/punyamishra.com\/2008\/08\/27\/visualizing-mathematics\/","title":{"rendered":"Visualizing mathematics"},"content":{"rendered":"<p>I love visual proofs of mathematical theorems. One visual proof I use quite often in my design courses (CEP817 or CEP917) is a visual proof of the fact that the sum of consecutive odd numbers is a square number.<br \/>\n<!--more--><br \/>\nIn other words:<\/p>\n<p>1 + 3 = 4 = 2<sup>2<\/sup><\/p>\n<p>1 + 3 + 5 = 9 = 3<sup>2<\/sup><\/p>\n<p>1 + 3 + 5 + 7 = 16 = 4<sup>2<\/sup><\/p>\n<p>&#8230; and so on.<\/p>\n<p>Mathematically speaking that translates to<br \/>\n<strong><br \/>\n<center>1 + 3 + &#8230; + (2N-1) = N<sup>2<\/sup><\/center><\/strong><\/p>\n<p>Now one can go about proving this fact this through algebra OR one could just do it visually. Check out this image below, <\/p>\n<p><img decoding=\"async\" src=\"http:\/\/punyamishra.com\/images\/oddsquares.jpg\"><\/p>\n<p>&#8230; or better still go to this <a target=\"_blank\" href=\"http:\/\/www.cut-the-knot.org\/Curriculum\/Algebra\/SumOfOddNumbers.shtml\">applet<\/a><\/p>\n<p>Pretty cool huh?<\/p>\n<p>Looking for the above applet led me to this other cool visual proof for<br \/>\n<strong><br \/>\n<center>(a + 1)<sup>2<\/sup> = a<sup>2<\/sup> + 2a + 1<\/center><\/strong><\/p>\n<p><a target=\"_blank\" href=\"http:\/\/digitalroam.typepad.com\/digital_roam\/2006\/03\/open_water_.html\">Check it out<\/a>. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>I love visual proofs of mathematical theorems. One visual proof I use quite often in my design courses (CEP817 or CEP917) is a visual proof of the fact that the sum of consecutive odd numbers is a square number.<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[10,11,5,37,9,29,27,8,1,49],"tags":[],"_links":{"self":[{"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/posts\/282"}],"collection":[{"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/comments?post=282"}],"version-history":[{"count":0,"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/posts\/282\/revisions"}],"wp:attachment":[{"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/media?parent=282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/categories?post=282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/punyamishra.com\/wp-json\/wp\/v2\/tags?post=282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}