{"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":"2020-05-16T22:20:21","modified_gmt":"2020-05-16T22:20:21","slug":"picturing-sequences","status":"publish","type":"page","link":"https:\/\/sites.ps.uci.edu\/mathceo-old\/meeting-2\/picturing-sequences\/","title":{"rendered":"Picturing Sequences"},"content":{"rendered":"\n

Graphing Sequences<\/h4>\n\n\n\n

Let’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

We need two numbers to specify a point on a two-dimensional graph, for example (2, 5). <\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

How do we turn a number sequence into a sequence of pairs of two numbers?<\/p>\n\n\n\n

Example:<\/strong> For the Fibonacci sequence, {1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …},<\/p>\n\n\n\n

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

{ (1, 1<\/span>), (2, 1<\/span>), (3, 2<\/span>), (4, 3<\/span>), (5, 5<\/span>), (6, 8<\/span>), (7, 13<\/span>), (8, 21<\/span>), (9, 34<\/span>), (10, 55<\/span>), (11, 89<\/span>) …. }<\/p>\n\n\n\n

We can plot this sequence of pair of numbers on a graphing calculator<\/a>. <\/p>\n\n\n\n

<\/div>\n\n\n\n

Try plotting the other sequences you have created. You can graph within Google Sheet by clicking on “Insert” and “Chart.”<\/p>\n\n\n\n\n\n\n\n

Sequences in Music<\/h4>\n\n\n\n

Music are sounds put together in a pleasing way. There are many interesting mathematics in music. For instance, the Fibonacci sequence also appears in many places in music<\/a>.<\/p>\n\n\n\n

\n\n\n\n

A song is a sequences of musical notes. Each musical note is denoted by a letter (A to G). <\/p>\n\n\n\n

For example, for Twinkle Twinkle Little Star<\/em>, the tune begins with the the following sequence (in C Major): <\/p>\n\n\n\n

C C G G A A G F F E E D D C <\/strong> <\/p>\n\n\n\n

\n\n\n\n

We can turn a number sequence into music. Let’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

13 \/ 12 (remainder = 1) >>>> 13 mod 12 = 1<\/p>\n\n\n\n

21 \/ 12 (remainder = 9) >>>> 21 mod 12 = 9<\/p>\n\n\n\n

34 \/ 12 (remainder = 10) >>>> 34 mod 12 = 10<\/p>\n\n\n\n

55 \/ 12 (remainder = 7) >>>> 55 mod 12 = 7<\/p>\n\n\n\n

89\/12 (remainder = ?) >>>>. 89 mod 12 = ??<\/p>\n\n\n\n

This mathematical shorthand for taking the remainder is “mod.” We would say for example “13 modulo 12 is 1.” <\/p>\n\n\n\n

The Fibonacci sequence taking the remainder of 12 becomes:<\/p>\n\n\n\n

1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, ….<\/p>\n\n\n\n

This can now be set to music. Listen here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Graphing Sequences Let’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 […]<\/p>\n","protected":false},"author":16,"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"],"_links":{"self":[{"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/pages\/267","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/users\/16"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/comments?post=267"}],"version-history":[{"count":8,"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/pages\/267\/revisions"}],"predecessor-version":[{"id":282,"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/pages\/267\/revisions\/282"}],"up":[{"embeddable":true,"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/pages\/218"}],"wp:attachment":[{"href":"https:\/\/sites.ps.uci.edu\/mathceo-old\/wp-json\/wp\/v2\/media?parent=267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}