{"id":871,"date":"2014-09-01T22:48:32","date_gmt":"2014-09-01T14:48:32","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=871"},"modified":"2024-02-16T21:11:55","modified_gmt":"2024-02-16T13:11:55","slug":"60-degrees","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/09\/60-degrees\/","title":{"rendered":"60 Degrees?"},"content":{"rendered":"<p>[This is a back-issue of <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/newsletter\" target=\"_blank\" rel=\"noopener\">this site&#8217;s newsletter<\/a>]<\/p>\n<p>Saw these online, thought I&#8217;d share them with you all &#8211;<\/p>\n<ul>\n<li>Parallel lines have so much in common. It&#8217;s a shame they&#8217;ll never meet.<\/li>\n<li>I took the shell off my racing snail, hoping it would make him faster. If anything, though, he just become more sluggish.<\/li>\n<li>How do you think the unthinkable? Why, with an itheberg, of course!<\/li>\n<li>I tried to catch some fog yesterday. Mist.<\/li>\n<\/ul>\n<div>Badum-tsss! And it&#8217;s time for Monday Morning Math!<\/div>\n<p><!--more--><\/p>\n<div>A few weeks ago, I promised some math movies. They&#8217;re ready! They&#8217;re ready! I&#8217;ll show them to you <em>very soon<\/em>, I promise. (Or, check out my <a href=\"http:\/\/www.youtube.com\/mike40033\">youtube channel<\/a> for a preview)<\/div>\n<div>Last week, we were chatting s complex numbers and pythagorean triangles.<\/div>\n<div>With a pythagorean triangle, if the two sides near the right angle are a and b, the other side is the square root of a<sup>2<\/sup>+b<sup>2<\/sup>. On the other hand, the absolute value of a complex number a+bi is the same, it&#8217;s the square root of a<sup>2<\/sup>+b<sup>2<\/sup>. Last week I showed that this &#8220;coincidence&#8221; can be used to find a formula for pythagorean triples &#8211; a triangle with sides P=a<sup>2<\/sup>-b<sup>2<\/sup>, Q=2ab and R=a<sup>2<\/sup>+b<sup>2<\/sup> is a right-angled triangle.<\/div>\n<div>What about a triangles with other angles, say 60<sup>o<\/sup>? Can we pull off a similar trick?<\/div>\n<div>A bit of trigonometry tells us that if a triangle has sides a and b, and in between them is a 60<sup>o<\/sup> angle, the length of the third side is the square root of a<sup>2<\/sup>-ab+b<sup>2<\/sup>. It would be nice if there was some kind of number system, something like the complex numbers, where the absolute value of a+bi was a<sup>2<\/sup>-ab+b<sup>2<\/sup> instead of a<sup>2<\/sup>+b<sup>2<\/sup>.<\/div>\n<div>Well, as it happens, there is!<\/div>\n<div>When you learn about complex numbers, you meet a mystery number i, whose square is -1. That is, it&#8217;s a root of the polynomial x<sup>2<\/sup>+1, which has no real roots. There&#8217;s no particular reason why that polynomial is special, though &#8211; we could have used any polynomial with no real roots.<\/div>\n<div>Suppose we used x<sup>2<\/sup>+x+1. It&#8217;s got no real roots, but let&#8217;s invent a root for it, and call it w. We&#8217;ll end up with a system of numbers very much like the complex numbers &#8211; the elements are a+bw instead of a+bi. Adding them works the way you&#8217;d expect, but the rules for multiplying them are slightly different now: simply because w<sup>2<\/sup> is -w-1, not -1.<\/div>\n<div>If we square a+bw, we get a<sup>2<\/sup> + 2abw + b<sup>2<\/sup>w<sup>2<\/sup>, which is a<sup>2<\/sup> + 2abw &#8211; b<sup>2<\/sup>w &#8211; b<sup>2<\/sup>, so that (a+bw)<sup>2<\/sup> = (a<sup>2<\/sup>-b<sup>2<\/sup>) + (2ab-b<sup>2<\/sup>)w.<\/div>\n<div>Despite the different multiplication rule, we could use these numbers just like we use complex numbers &#8211; we could figure out what the &#8220;conjugate&#8221; of a+bw is (it&#8217;s not a-bw, it turns out). And we could work out the absolute value of a+bw. We could do everything with these numbers that we can do with complex numbers. The reason is simply that these new numbers a+bw are just the complex numbers in disguise.<\/div>\n<div>Now, when we do work out absolute values, it turns out that |a+bw| is not a<sup>2<\/sup>+b<sup>2<\/sup>, but a<sup>2<\/sup>-ab+b<sup>2<\/sup>. And that&#8217;s just what we need to solev the triangles puzzle!<\/div>\n<div>Because |(a+bw)<sup>2<\/sup>| = |a+bw|<sup>2<\/sup>, and (a+bw)<sup>2<\/sup> = (a<sup>2<\/sup>-b<sup>2<\/sup>) + (2ab-b<sup>2<\/sup>)w, this means |(a<sup>2<\/sup>-b<sup>2<\/sup>) + (2ab-b<sup>2<\/sup>)w| = (a<sup>2<\/sup>-ab+b<sup>2<\/sup>). Let&#8217;s simplify things by making P equal a<sup>2<\/sup>-b<sup>2<\/sup>, Q equal 2ab-b<sup>2<\/sup> and R will be a<sup>2<\/sup>-ab+b<sup>2<\/sup>. Then, we&#8217;ve got |P+Qw|=R, that is, P<sup>2<\/sup>-PQ-Q<sup>2<\/sup> = R<sup>2<\/sup>, which is exactly right for P, Q and R are to be the sides of a triangle with and angle of 60<sup>o<\/sup> between P and Q.<\/div>\n<div>Some examples:<\/div>\n<ul>\n<li>Choose a=2 and b=1, and you get P =\u00a0a<sup>2<\/sup>-b<sup>2<\/sup>\u00a0= 3, Q = 2ab-b<sup>2<\/sup>\u00a0= 3, and\u00a0R = a<sup>2<\/sup>-ab+b<sup>2<\/sup>\u00a0= 3. An equilateral triangle!<\/li>\n<li>Choosing other values of a and b give other triangles with a sixty degree angle. If a=3 and b=2, you get P=5, Q=8 and R=7.<\/li>\n<li>Pick any values of a and b you like, and you can generate as many 60 degree triangles as you like!<\/li>\n<\/ul>\n<div>There&#8217;s a similar trick for generating 120 degree triangles as well. It involves numbers a+bv for which v2=1-v, and |a+bv|=a<sup>2<\/sup>+ab+b<sup>2<\/sup>.<\/div>\n<p>If you found this email fascinating, here&#8217;s a challenge for you. Can you figure out why there&#8217;s no formula for generating triangles with whole number edges, and a 45 degree angle?<\/p>\n<p>That&#8217;s all for this week! Happy mathing!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[This is a back-issue of this site&#8217;s newsletter] Saw these online, thought I&#8217;d share them with you all &#8211; Parallel lines have so much in common. It&#8217;s a shame they&#8217;ll never meet. I took the shell off my racing snail, hoping it would make him faster. If anything, though, he just become more sluggish. How &hellip; <a href=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/09\/60-degrees\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">60 Degrees?<\/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\/871"}],"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=871"}],"version-history":[{"count":2,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/871\/revisions"}],"predecessor-version":[{"id":1283,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/871\/revisions\/1283"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=871"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=871"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=871"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}