{"id":267,"date":"2020-04-20T14:46:13","date_gmt":"2020-04-20T14:46:13","guid":{"rendered":"https:\/\/sites.ps.uci.edu\/mathceo\/?page_id=267"},"modified":"2020-12-03T06:35:29","modified_gmt":"2020-12-03T06:35:29","slug":"picturing-sequences","status":"publish","type":"page","link":"https:\/\/sites.ps.uci.edu\/mathceo\/es\/meeting-2\/picturing-sequences","title":{"rendered":"Picturing Sequences"},"content":{"rendered":"<h4 class=\"wp-block-heading\">Graphing Sequences<\/h4>\n\n\n\n<p>Let&#8217;s draw a number sequence.  We can have each number of the sequence represent a point on a two-dimensional graph. <\/p>\n\n\n\n<p>We need two numbers to specify a point on a two-dimensional graph, for example (2, 5).  <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"427\" height=\"424\" src=\"http:\/\/sites.ps.uci.edu\/mathceo\/wp-content\/uploads\/sites\/20\/2020\/04\/twodim_graph.png\" alt=\"\" class=\"wp-image-281\" srcset=\"https:\/\/sites.ps.uci.edu\/mathceo\/wp-content\/uploads\/sites\/20\/2020\/04\/twodim_graph.png 427w, https:\/\/sites.ps.uci.edu\/mathceo\/wp-content\/uploads\/sites\/20\/2020\/04\/twodim_graph-300x298.png 300w, https:\/\/sites.ps.uci.edu\/mathceo\/wp-content\/uploads\/sites\/20\/2020\/04\/twodim_graph-150x150.png 150w\" sizes=\"auto, (max-width: 427px) 100vw, 427px\" \/><\/figure><\/div>\n\n\n\n<p>How do we turn a number sequence into a sequence of pairs of two numbers?<\/p>\n\n\n\n<p><strong>Example:<\/strong> For the Fibonacci sequence, {1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, &#8230;},<\/p>\n\n\n\n<p>an easy way to do so is to let the first number in the pair to be the order number:<\/p>\n\n\n\n<p class=\"has-large-font-size\">{ (1, <span class=\"has-inline-color has-accent-color\">1<\/span>), (2, <span class=\"has-inline-color has-accent-color\">1<\/span>), (3, <span class=\"has-inline-color has-accent-color\">2<\/span>), (4, <span class=\"has-inline-color has-accent-color\">3<\/span>), (5, <span class=\"has-inline-color has-accent-color\">5<\/span>), (6, <span class=\"has-inline-color has-accent-color\">8<\/span>), (7, <span class=\"has-inline-color has-accent-color\">13<\/span>), (8, <span class=\"has-inline-color has-accent-color\">21<\/span>), (9, <span class=\"has-inline-color has-accent-color\">34<\/span>), (10, <span class=\"has-inline-color has-accent-color\">55<\/span>), (11, <span class=\"has-inline-color has-accent-color\">89<\/span>) &#8230;. }<\/p>\n\n\n\n<p>We can plot this sequence of pair of numbers on a <a rel=\"noreferrer noopener\" href=\"https:\/\/www.desmos.com\/calculator\" target=\"_blank\">graphing calculator<\/a>.   <\/p>\n\n\n\n<div style=\"height:66px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Try plotting the other sequences you have created.  You can graph within Google Sheet by clicking on &#8220;Insert&#8221; and &#8220;Chart.&#8221;<\/p>\n\n\n\n<!--nextpage-->\n\n\n\n<h4 class=\"has-text-align-left wp-block-heading\">Sequences in Music<\/h4>\n\n\n\n<p>Music are sounds put together in a pleasing way.  There are many interesting mathematics in music.  For instance, the <a href=\"https:\/\/www.classicfm.com\/discover-music\/fibonacci-sequence-in-music\/\" target=\"_blank\" rel=\"noreferrer noopener\">Fibonacci sequence also appears in many places in music<\/a>.<\/p>\n\n\n\n<hr class=\"wp-block-separator\" \/>\n\n\n\n<p>A song is a sequences of musical notes.  Each musical note is denoted by a letter (A to G). <\/p>\n\n\n\n<p>For example, for <em>Twinkle Twinkle Little Star<\/em>, the tune begins with the  the following sequence (in C Major): <\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"><strong>C C G G A A G F F E E D D C <\/strong>   <\/p>\n\n\n\n<hr class=\"wp-block-separator\" \/>\n\n\n\n<p>We can turn a number sequence into music.  Let&#8217;s do that with the Fibonacci sequence.  There are only 12 different notes within an octave.  So we must turn each number in the Fibonacci sequence into a number within 1-12, (or 0-11).  One way to do this is to divide each number in the sequence by 12, and the remainder is the number we desire:<\/p>\n\n\n\n<p>13 \/ 12   (remainder  = 1)     &gt;&gt;&gt;&gt;   13 mod 12 = 1<\/p>\n\n\n\n<p>21 \/ 12   (remainder = 9)    &gt;&gt;&gt;&gt;   21 mod 12 = 9<\/p>\n\n\n\n<p>34 \/ 12    (remainder = 10)   &gt;&gt;&gt;&gt;    34 mod 12 = 10<\/p>\n\n\n\n<p>55 \/ 12   (remainder =  7)      &gt;&gt;&gt;&gt;    55 mod 12 = 7<\/p>\n\n\n\n<p>89\/12   (remainder = ?)     &gt;&gt;&gt;&gt;.  89 mod 12 = ??<\/p>\n\n\n\n<p>This mathematical shorthand for taking the remainder is &#8220;mod.&#8221;  We would say for example &#8220;13 modulo 12 is 1.&#8221; <\/p>\n\n\n\n<p>The Fibonacci sequence taking the remainder of 12 becomes:<\/p>\n\n\n\n<p class=\"has-background has-text-align-center has-large-font-size has-subtle-background-background-color\">1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, &#8230;.<\/p>\n\n\n\n<p>This can now be set to music.  Listen <a href=\"https:\/\/safeYouTube.net\/w\/dPn7\" target=\"_blank\" rel=\"noreferrer noopener\">here<\/a>.<\/p>","protected":false},"excerpt":{"rendered":"<p>Graphing Sequences Let&#8217;s draw a number sequence. We can have each number of the sequence represent a point on a two-dimensional graph. We need two numbers to specify a point on a two-dimensional graph, for example (2, 5). How do we turn a number sequence into a sequence of pairs of two numbers? Example: For [&hellip;]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":218,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"class_list":["post-267","page","type-page","status-publish","hentry","post"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Picturing Sequences - MATH CEO<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/sites.ps.uci.edu\/mathceo\/es\/meeting-2\/picturing-sequences\" \/>\n<link rel=\"next\" href=\"https:\/\/sites.ps.uci.edu\/mathceo\/es\/meeting-2\/picturing-sequences\/2\" \/>\n<meta property=\"og:locale\" content=\"es_MX\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Picturing Sequences - MATH CEO\" \/>\n<meta property=\"og:description\" content=\"Graphing Sequences Let&#8217;s draw a number sequence. 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