{"id":866,"date":"2014-08-26T06:47:02","date_gmt":"2014-08-25T22:47:02","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=866"},"modified":"2024-02-16T21:11:55","modified_gmt":"2024-02-16T13:11:55","slug":"pythagoras-venn-euclid-and-complex-numbers","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/08\/pythagoras-venn-euclid-and-complex-numbers\/","title":{"rendered":"Pythagoras, Venn, Euclid and Complex Numbers"},"content":{"rendered":"<p><em>[This is a back-issue of one of <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/newsletter\" target=\"_blank\" rel=\"noopener\">this site&#8217;s newsletters<\/a>]<\/em><\/p>\n<p>Last week, I showed you how a puzzle about rectangles gives you a simple formula for pythagorean triplets. This week, I want to show you a bit more about that formula.<\/p>\n<p>But first, have you seen this? It&#8217;s proof that even people in a marketing career need to know a bit of mathematics:<\/p>\n<p><!--more--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/i.imgur.com\/2eXN5IV.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p>\n<p>As someone said: &#8220;Once you&#8217;ve seen this as a Venn Diagram, it&#8217;s hard to un-see&#8221;. Poor Reuters!<\/p>\n<p>Ok, back to pythagorean triangles. The <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/08\/you-wont-believe-this-amazing-formula\/\">formula went like this<\/a>:<\/p>\n<ul>\n<li>Start with a number, T<\/li>\n<li>Work out 1\/T<\/li>\n<li>Then, work out T + 1\/T and T &#8211; 1\/T.<\/li>\n<li>Divide these both by 2, and put them over a common denominator.<\/li>\n<li>Then, if (T+1\/T)\/2 = R\/Q and (T-1\/T)\/2 = P\/Q, then P<sup>2<\/sup> + Q<sup>2<\/sup> = R<sup>2<\/sup>.<\/li>\n<\/ul>\n<p>If you make T equal to M\/N, then this process gives P=M<sup>2<\/sup>-N<sup>2<\/sup>, Q=2MN and R=M<sup>2<\/sup>+N<sup>2<\/sup>. That&#8217;s a much more explicit formula, and is known to us from Euclid&#8217;s &#8220;Elements&#8221;, a famous collection of ancient Greek mathematics.<\/p>\n<p>There&#8217;s a much simpler way to come up with this formula, which uses complex numbers. Yes, complex numbers make some things simple!<\/p>\n<ul>\n<li>Start with a complex number M + iN.<\/li>\n<li>Square it: (M+iN)<sup>2<\/sup> = (M<sup>2<\/sup>-N<sup>2<\/sup>) + (2MN)i.<\/li>\n<li>Take the absolute value of both sides: |M+iN|<sup>2<\/sup> = |(M<sup>2<\/sup>-N<sup>2<\/sup>) + (2MN)i|.<\/li>\n<li>Expand out the squared absolute value: M<sup>2<\/sup>+N<sup>2<\/sup> = |(M<sup>2<\/sup>-N<sup>2<\/sup>) + (2MN)i|.<\/li>\n<li>Let P=M<sup>2<\/sup>-N<sup>2<\/sup>, Q=2MN and R=M<sup>2<\/sup>+N<sup>2<\/sup>: R = |P+iQ|<\/li>\n<li>Square both sides of this: R<sup>2<\/sup> = |P+iQ|<sup>2<\/sup><\/li>\n<li>Expand out the last absolute value: R<sup>2<\/sup> = P<sup>2<\/sup>+Q<sup>2<\/sup>.<\/li>\n<\/ul>\n<p>So there you have it &#8211; Pythagorean triples from complex numbers. Who would have thought!<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[This is a back-issue of one of this site&#8217;s newsletters] Last week, I showed you how a puzzle about rectangles gives you a simple formula for pythagorean triplets. This week, I want to show you a bit more about that formula. But first, have you seen this? It&#8217;s proof that even people in a marketing &hellip; <a href=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/08\/pythagoras-venn-euclid-and-complex-numbers\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Pythagoras, Venn, Euclid and Complex Numbers<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[153],"tags":[],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/866"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=866"}],"version-history":[{"count":3,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/866\/revisions"}],"predecessor-version":[{"id":1284,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/866\/revisions\/1284"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=866"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=866"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=866"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}