{"id":385,"date":"2014-12-07T12:00:04","date_gmt":"2014-12-07T20:00:04","guid":{"rendered":"http:\/\/mostlymental.com\/?p=385"},"modified":"2014-12-03T01:59:33","modified_gmt":"2014-12-03T09:59:33","slug":"continued-fractions-the-golden-ratio-and-fibonacci","status":"publish","type":"post","link":"http:\/\/mostlymental.com\/?p=385","title":{"rendered":"Continued Fractions, the Golden Ratio, and Fibonacci"},"content":{"rendered":"<p>Hello everyone!<\/p>\n<p>I was playing around with <a href=\"http:\/\/www.maths.surrey.ac.uk\/hosted-sites\/R.Knott\/Fibonacci\/cfINTRO.html\">continued fractions<\/a> last week, and I stumbled across a nice pattern that I hadn't seen before.\u00a0 I thought it might be interesting to talk about it here.<\/p>\n<p>&nbsp;<\/p>\n<p>Let's start with a motivating question.\u00a0 Consider the number below:<\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_6e6360ef5b24c6018f99f7b9877a469b.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>,<\/p>\n<p style=\"text-align: left;\">extending on infinitely.\u00a0 It's possible to prove that continued fractions always converge (that is, they come out to actual numbers), so <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is well-defined.\u00a0 So what's its value?<\/p>\n<p style=\"text-align: left;\"><!--more--><\/p>\n<p style=\"text-align: left;\">Given what <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> looks like, we might try looking at <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_bd6404ad4b40e99dadaef60ba553a42a.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 When we do, we get<\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_ba0304785a313c63698f494bf1f0bcb6.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p style=\"text-align: left;\">But that expression on the right side is just <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 So we have <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_36f9a6cdd93620b06ad322aab30d3f92.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Rearranging, we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_aaa0252719d2b450bb24d7fcb7886093.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 By the quadratic formula, we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_20f9463a6affcf8089f34ab6ed0deb2a.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9dd4e461268c8034f5c8564e155c67a6.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is positive, so we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_52a5b65b4376de82c2e62c69268ab1e9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p style=\"text-align: left;\">This number might look familiar; it's known as the golden ratio, <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1ed346930917426bc46d41e22cc525ec.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 It shows up everywhere in nature, it's fundamental in music and art, and it has a lot of interesting mathematical properties.\u00a0 For today, let's show that it's intimately related to the Fibonacci sequence.<\/p>\n<p style=\"text-align: left;\">Recall that the Fibonacci numbers, <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_5c394a360d92eee72a487edf79a03ccb.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, are what we get when we start with <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_736552276c229ab57989a00e9e5a9d20.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> and get each term by adding the previous two together.\u00a0 What happens if we divide consecutive terms?<\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_431bd832138f817133616fb26f3d373f.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9a3704850742ff8b70f9558742dfcc67.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7f07238452691de51338db75879d4372.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_2ceb44753b44cdda0c1f10d99e1a93f3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_cbe945a2959fcee06f5bffbd4358f1cd.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_bb43a09b7e703efe204388295508b6a3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: left;\">If we keep going, we see ratios seem to be getting close together.\u00a0 In fact, they seem to be bouncing back and forth across a point somewhere near <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_7429a6d331db6b049d6dc7e5bb8cc630.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p style=\"text-align: left;\">\n<p style=\"text-align: left;\">Given that I talked about the golden ratio earlier, you might guess that the ratios are converging to <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1ed346930917426bc46d41e22cc525ec.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0\u00a0 And in fact, they are.\u00a0 Why?\u00a0 Let's look at the continued fraction we started with, but instead of going out to infinity, stop part way.\u00a0 That is, let's look at the convergents of the continued fraction.<\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_ef9fd148b3a3b57289d5641698a52842.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_0f088f8350c1dd2a418b7530b5b7b3c9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_4c2306ed798198b35629b330b16a676d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_747aa569429e49aa44ae3016b39253a7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_2cbdbb1ba3efefb202bf6a25d885473d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: left;\">The more ones we add on the left side, the closer we get to <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1ed346930917426bc46d41e22cc525ec.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 But the numbers on the right are just the ratios of consecutive Fibonacci numbers.\u00a0 (Do you see why?)\u00a0 So they must also get closer to <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1ed346930917426bc46d41e22cc525ec.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">There's actually another very nice connection between the Fibonacci numbers and <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_1ed346930917426bc46d41e22cc525ec.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Binet's formula relates them directly:<\/p>\n<p style=\"text-align: center;\"><span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_5a24537a87949fe11733cddb55ff2551.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p style=\"text-align: left;\">This is a rather magical fact, so I'll prove it in a later post.<\/p>\n<p style=\"text-align: left;\">That's it for today.\u00a0 I'll see you next week.<\/p>\n<a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-facebook nolightbox\" data-provider=\"facebook\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Facebook\" href=\"https:\/\/www.facebook.com\/sharer.php?u=http%3A%2F%2Fmostlymental.com&amp;t=Continued%20Fractions%2C%20the%20Golden%20Ratio%2C%20and%20Fibonacci&amp;s=100&amp;p[url]=http%3A%2F%2Fmostlymental.com&amp;p[images][0]=&amp;p[title]=Continued%20Fractions%2C%20the%20Golden%20Ratio%2C%20and%20Fibonacci\" style=\"font-size: 0px;width:48px;height:48px;margin:0;margin-bottom:5px;margin-right:5px\"><img loading=\"lazy\" decoding=\"async\" alt=\"Facebook\" title=\"Share on Facebook\" class=\"synved-share-image 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I was playing around with continued fractions last week, and I stumbled across a nice pattern that I hadn't seen before.\u00a0 I thought it might be interesting to talk about it here. &nbsp; Let's start with a motivating question.\u00a0 Consider the number below: , extending on infinitely.\u00a0 It's possible to prove that continued &hellip; <a href=\"http:\/\/mostlymental.com\/?p=385\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Continued Fractions, the Golden Ratio, and Fibonacci<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,2,12],"tags":[],"class_list":["post-385","post","type-post","status-publish","format-standard","hentry","category-combinatorics","category-math","category-number-theory"],"_links":{"self":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=385"}],"version-history":[{"count":10,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/385\/revisions"}],"predecessor-version":[{"id":412,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/385\/revisions\/412"}],"wp:attachment":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=385"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=385"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}