{"id":1770,"date":"2017-03-14T00:22:33","date_gmt":"2017-03-14T00:22:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1770"},"modified":"2021-01-05T22:21:52","modified_gmt":"2021-01-05T22:21:52","slug":"introduction-fractals-generated-by-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/chapter\/introduction-fractals-generated-by-complex-numbers\/","title":{"raw":"Fractals Generated by Complex Numbers","rendered":"Fractals Generated by Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the difference between an imaginary number and a complex number<\/li>\r\n \t<li>Identify the real and imaginary parts of a complex number<\/li>\r\n \t<li>Plot a complex number on the complex plane<\/li>\r\n \t<li>Perform arithmetic operations on complex numbers<\/li>\r\n \t<li>Graph physical representations of arithmetic operations on complex numbers as scaling or rotation<\/li>\r\n \t<li>Generate several terms of a recursive relation<\/li>\r\n \t<li>Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set<\/li>\r\n<\/ul>\r\n<\/div>\r\nYou may be familiar with the fractal in the image below. \u00a0The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole. This object is called the Mandelbrot set and is generated by iterating a simple recursive\u00a0rule using complex numbers.\r\n\r\n&nbsp;\r\n\r\n<img class=\"wp-image-1737 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"561\" height=\"381\" \/>\r\n\r\nIn this lesson, you will first learn about the arithmetic of complex numbers so you can understand how a fractal like the Mandelbrot set is generated.\r\n<h2>Arithmetic with Complex Numbers<\/h2>\r\n[footnote]Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA[\/footnote]\r\nThe numbers you are most familiar with are called <strong>real numbers<\/strong>. These include numbers like 4, 275, -200, 10.7, \u00bd, \u03c0, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point:\r\n\r\nTo solve certain problems like [latex]x^{2}=\u20134[\/latex], it became necessary to introduce <strong>imaginary numbers<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Imaginary Number <em>i<\/em><\/h3>\r\nThe imaginary number <em>i<\/em> is defined to be [latex]i=\\sqrt{-1}[\/latex].\r\n\r\nAny real multiple of <em>i<\/em>, like 5<em>i<\/em>, is also an imaginary number.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\sqrt{-9}[\/latex].\r\n\r\nWe can separate [latex]\\sqrt{-9}[\/latex] as [latex]\\sqrt{9}\\sqrt{-1}[\/latex]. We can take the square root of 9, and write the square root of [latex]-1[\/latex] as <em>i<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]<\/p>\r\n\r\n<\/div>\r\nA complex number is the sum of a real number and an imaginary number.\r\n<div class=\"textbox\">\r\n<h3>Complex Number<\/h3>\r\nA <strong>complex number<\/strong> is a number [latex]z=a+bi[\/latex], where\r\n<ul>\r\n \t<li><em>a<\/em> and <em>b<\/em> are real numbers<\/li>\r\n \t<li><em>a <\/em>is the real part of the complex number<\/li>\r\n \t<li><em>b<\/em> is the imaginary part of the complex number<\/li>\r\n<\/ul>\r\n<\/div>\r\nTo plot a complex number like [latex]3-4i[\/latex], we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane.\r\n<div class=\"textbox\">\r\n<h3>Complex Plane<\/h3>\r\nIn the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.\r\n\r\n<img class=\"aligncenter wp-image-1729 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"145\" height=\"101\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nPlot the number [latex]3-4i[\/latex] on the complex plane.\r\n\r\nThe real part of this number is 3, and the imaginary part is [latex]-4[\/latex]. To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.\r\n\r\n<img class=\"aligncenter size-full wp-image-1732\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184650\/Screen-Shot-2017-02-23-at-10.46.27-AM.png\" alt=\"A graph with imaginary y-axis and real x-axis. The point 3, negative 4 is marked.\" width=\"275\" height=\"268\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\nBecause this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number [latex]z=a+bi[\/latex] as if we were plotting the point (a, b) in Cartesian coordinates. Sometimes people write complex numbers as [latex]z=x+yi[\/latex] to highlight this relation.\r\n<h3>Arithmetic on Complex Numbers<\/h3>\r\nBefore we dive into the more complicated uses of complex numbers, let\u2019s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd [latex]3-4i[\/latex] and [latex]2+5i[\/latex].\r\n\r\nAdding [latex](3-4i)+(2+5i)[\/latex], we add the real parts and the imaginary parts.\r\n<p style=\"text-align: center;\">[latex]3+2-4i+5i[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]5+i[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract [latex]2+5i[\/latex] from [latex]3-4i[\/latex].\r\n\r\n[reveal-answer q=\"192798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"192798\"][latex](3-4i)-(2+5i)=1-9i[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=61710&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video, we present more worked examples of arithmetic with complex numbers.\r\n\r\nhttps:\/\/youtu.be\/XJXDcybM84U\r\n\r\nWhen we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nVisualize the addition [latex]3-4i[\/latex] and [latex]-1+5i[\/latex].\r\n\r\nThe initial point is [latex]3-4i[\/latex]. When we add [latex]-1+3i[\/latex], we add [latex]-1[\/latex] to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. This shifts the point [latex]3-4i[\/latex] to [latex]2+1i[\/latex].\r\n\r\n<img class=\"aligncenter size-full wp-image-1733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23185217\/Screen-Shot-2017-02-23-at-10.51.55-AM.png\" alt=\"A graph with an imaginary y-axis and a real x-axis. The point 3, negative 4 is labeled 3 minus 4i. The point 2, 1 is labeled 2 plus 1i. An arrow goes from 3 minus 4i to 2 plus 1i.\" width=\"300\" height=\"291\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131218&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"500\"><\/iframe>\r\n\r\n<\/div>\r\nWe can also multiply complex numbers by a real number, or multiply two complex numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply: [latex]4\\left(2+5i\\right)[\/latex]\r\n\r\nTo multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.\r\n\r\nDistribute and simplify.\r\n<p style=\"text-align: center;\">[latex]4(2+5i)\\\\\\,\\,\\,= 4\\cdot2+4\\cdot5i\\\\\\,\\,\\,=8+20i[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply: [latex](2+5i)(4+i)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2+5i\\right)\\left(4+i\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Expand.}\\\\=8+20i+2i+5i^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Since }i=\\sqrt{-1},i^{2}=-1\\\\=8+20i+2i+5\\left(-1\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify.}\\\\=3+22i\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply [latex]3-4i[\/latex] and [latex]2+3i[\/latex].\r\n[reveal-answer q=\"929203\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929203\"]Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[\/latex][\/hidden-answer]\r\n<iframe id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1418&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nTo understand the effect of multiplication visually, we\u2019ll explore three examples.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nVisualize the product [latex]2(1+2i)[\/latex].\r\n\r\nMultiplying we\u2019d get\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;2\\cdot1+2\\cdot2i\\\\&amp;=2+4i\\\\\\end{align}[\/latex]<\/p>\r\nNotice both the real and imaginary parts have been scaled by 2. Visually, this will stretch the point outwards, away from the origin.\r\n\r\n<img class=\"aligncenter size-full wp-image-1734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23195901\/Screen-Shot-2017-02-23-at-11.58.34-AM.png\" alt=\"Graph with imaginary y-axis and real x-axis. The point 1,2 is marked and labeled 1 plus 2i. The point 2,4 is marked and labeled 2 plus 4i. A red arrow is drawn from the origin and through both points.\" width=\"302\" height=\"279\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nVisualize the product [latex]i\\left(l+2i\\right)[\/latex].\r\n\r\nMultiplying, we\u2019d get\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;i\\cdot1+i\\cdot2i\\\\&amp;=i+2{{i}^{2}}\\\\&amp;=i+2(-1)\\\\&amp;=-2+i\\\\\\end{align}[\/latex]<\/p>\r\nIn this case, the distance from the origin has not changed, but the point has been rotated about the origin, 90\u00b0 counter-clockwise.\r\n\r\n<img class=\"aligncenter size-full wp-image-1735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply [latex]3-4i[\/latex] and [latex]2+3i[\/latex].\r\n[reveal-answer q=\"929203\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929203\"]Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[\/latex][\/hidden-answer]\r\n<iframe id=\"mom30\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1418&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nVisualize the result of multiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex]. Then show the result of multiplying by [latex]1+i[\/latex] again.\r\n\r\nMultiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex],\r\n<p style=\"text-align: center;\">[latex]-4+2i[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;(1+2i)(1+i)\\\\&amp;=1+i+2i+2{{i}^{2}}\\\\&amp;=1+3i+2(-1)\\\\&amp;=-1+3i\\\\\\end{align}[\/latex]<\/p>\r\nMultiplying by [latex]1+i[\/latex] again,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;(-1+3i)(1+i)\\\\&amp;=-1-i+3i+3{{i}^{2}}\\\\&amp;=-1+2i+3(-1)\\\\&amp;=-4+2i\\\\\\end{align}[\/latex]<\/p>\r\nIf we multiplied by [latex]1+i[\/latex] again, we\u2019d get [latex]\u20136\u20132i[\/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by 45\u00b0.\r\n\r\n<img class=\"aligncenter size-full wp-image-1736\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23221746\/Screen-Shot-2017-02-23-at-2.15.58-PM.png\" alt=\"The imaginary-real graph with four points marked, each with a dotted red line extending from the origin to that point. The points are as follows. The point 1,2, represented by 1 plus 2i. The point negative 1, 3, represented by negative 1 plus 3i. The point negative 4, 2, represented by negative 4 plus 2i. The point negative 6, negative 2, represented by negative 6 minus 2i.\" width=\"400\" height=\"346\" \/>\r\n\r\n<\/div>\r\nIn general, multiplication by a complex number can be thought of as a <strong>scaling<\/strong>, changing the distance from the origin, combined with a <strong>rotation<\/strong> about the origin.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom35\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131223&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\nThe following video presents more examples of how to visualize the results of arithmetic on complex numbers.\r\n\r\nIn the following video, we present more worked examples of arithmetic with complex numbers.\r\n\r\nhttps:\/\/youtu.be\/vPZAW7Lhh1E\r\n<h2>Generating Fractals with Complex Numbers<\/h2>\r\n<h3>Complex Recursive Sequences<\/h3>\r\nSome fractals are generated with complex numbers. \u00a0The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.\r\n<div class=\"textbox\">\r\n<h3>Recursive Sequence<\/h3>\r\nA <strong>recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].\r\n\r\n<\/div>\r\nThe sequence of values produced is the recursive sequence.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2,\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.\r\n\r\nWe are given the starting value, [latex]{{z}_{0}}=4[\/latex]. The recursive formula holds for any value of <em>n<\/em>, so if [latex]n = 0[\/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[\/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[\/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[\/latex].\r\n\r\nNotice this defines [latex]{{z}_{1}}[\/latex] in terms of the known [latex]{{z}_{0}}[\/latex], so we can compute the value:\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[\/latex].<\/p>\r\nNow letting [latex]n = 1[\/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[\/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[\/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.\r\n<p style=\"text-align: center;\">[latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[\/latex]<\/p>\r\nContinuing,\r\n<p style=\"text-align: center;\">[latex]{{z}_{3}}={{z}_{2}}+2=8+2=10\\\\{{z}_{4}}={{z}_{3}}+2=10+2=12[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5816&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\\cdot{i}+(1-i),\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.\r\n\r\nWe are given [latex]{{z}_{0}}=4[\/latex]. Using the recursive formula:\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i\\\\{{z}_{2}}={{z}_{1}}\\cdot{i}+(1-i)=(1+3i)\\cdot{i}+(1-i)=i+3{{i}^{2}}+(1-i)=i-3+(1-i)=-2\\\\{{z}_{3}}={{z}_{2}}\\cdot{i}+(1-i)=(-2)\\cdot{i}+(1-i)=-2i+(1-i)=1-3i\\\\{{z}_{4}}={{z}_{3}}\\cdot{i}+(1-i)=(1-3i)\\cdot{i}+(1-i)=i-3{{i}^{2}}+(1-i)=i+3+(1-i)=4\\\\{{z}_{5}}={{z}_{4}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i[\/latex]<\/p>\r\nNotice this sequence is exhibiting an interesting pattern\u2014it began to repeat itself.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131230&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.\r\n\r\nhttps:\/\/youtu.be\/lOyusyTsLTs\r\n<h3>Mandelbrot Set<\/h3>\r\nThe Mandelbrot Set is a set of numbers defined based on recursive sequences.\r\n<div class=\"textbox\">\r\n<h3>Mandelbrot Set<\/h3>\r\nFor any complex number <em>c<\/em>, define the sequence [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]\r\n\r\nIf this sequence always stays close to the origin (within 2 units), then the number <em>c<\/em> is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number <em>c<\/em> is not part of the set.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine if [latex]c=1+i[\/latex] is part of the Mandelbrot set.\r\n\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+1+i=0+1+i=1+i\\\\{{z}_{2}}={{z}_{1}}^{2}+1+i={{(1+i)}^{2}}+1+i=1+3i\\\\{{z}_{3}}={{z}_{2}}^{2}+1+i={{(1+3i)}^{2}}+1+i=-7+7i\\\\{{z}_{4}}={{z}_{3}}^{2}+1+i={{(-7+7i)}^{2}}+1+i=1-97i[\/latex]<\/p>\r\nWe can already see that these values are getting quite large. It does not appear that [latex]c=1+i[\/latex] is part of the Mandelbrot set.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine if [latex]c=0.5i[\/latex] is part of the Mandelbrot set.\r\n\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+0.5i=0+0.5i=0.5i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.5i={{(0.5i)}^{2}}+0.5i=-0.25+0.5i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]<\/p>\r\nWhile not definitive with this few iterations, it does appear that this value is remaining small, suggesting that 0.5<em>i<\/em> is part of the Mandelbrot set.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDetermine if [latex]c=0.4+0.3i[\/latex] is part of the Mandelbrot set.\r\n\r\n[reveal-answer q=\"31678\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"31678\"]\r\n\r\n[latex]{{z}_{1}}={{z}_{0}}^{2}+0.4+0.3i=0+0.4+0.3i=0.4+0.3i[\/latex]\r\n\r\n[latex]{{z}_{2}}={{z}_{1}}^{2}+0.4+0.3i=(0.4+0.3i)^{2}+0.4+0.3i=0.07+.24i+0.4+0.3i=0.47+0.54i[\/latex]\r\n\r\n[latex]{{z}_{3}}={{z}_{2}}^{2}+0.4+0.3i=(0.47+0.54i)^{2}+0.4+0.3i=-0.0707+0.576i+0.4+0.3i=0.3293+0.8076i[\/latex]\r\n\r\n[latex]\\\\{{z}_{4}}={{z}_{3}}^{2}+0.4+0.3i=(0.3293+0.8076i)^{2}+0.4+0.3i\\approx{-0.5438}+0.5319i+0.4+0.3i=-0.1438+0.8319i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-1737 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"187\" height=\"127\" \/>If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.[footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png<\/a>[\/footnote]\r\n\r\nThe boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.\r\n\r\nWatch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.\r\n\r\nhttps:\/\/youtu.be\/ORqk5jAFpWg\r\n\r\nIn addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using [latex]c=1+i [\/latex] above, the sequence was distance 2 from the origin after only two recursions.\r\n\r\nFor some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below[footnote]This series was generated using Scott\u2019s Mandelbrot Set Explorer[\/footnote], light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in\u2014those are used for numbers that grow very slowly.\r\n\r\nThe Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.\r\n<h3>Additional Resources<\/h3>\r\nA much more extensive coverage of fractals can be found on the <a href=\"http:\/\/classes.yale.edu\/fractals\/\">Fractal Geometry site<\/a>. This site includes links to several Java software programs for exploring fractals.\r\n\r\nIf you are impressed with the Mandelbrot set, <a href=\"http:\/\/www.ted.com\/talks\/benoit_mandelbrot_fractals_the_art_of_roughness\">check out this TED talk from 2010<\/a>\u00a0given by Benoit Mandelbrot on fractals and the art of roughness.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the difference between an imaginary number and a complex number<\/li>\n<li>Identify the real and imaginary parts of a complex number<\/li>\n<li>Plot a complex number on the complex plane<\/li>\n<li>Perform arithmetic operations on complex numbers<\/li>\n<li>Graph physical representations of arithmetic operations on complex numbers as scaling or rotation<\/li>\n<li>Generate several terms of a recursive relation<\/li>\n<li>Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set<\/li>\n<\/ul>\n<\/div>\n<p>You may be familiar with the fractal in the image below. \u00a0The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole. This object is called the Mandelbrot set and is generated by iterating a simple recursive\u00a0rule using complex numbers.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1737 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"561\" height=\"381\" \/><\/p>\n<p>In this lesson, you will first learn about the arithmetic of complex numbers so you can understand how a fractal like the Mandelbrot set is generated.<\/p>\n<h2>Arithmetic with Complex Numbers<\/h2>\n<p><a class=\"footnote\" title=\"Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA\" id=\"return-footnote-1770-1\" href=\"#footnote-1770-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><br \/>\nThe numbers you are most familiar with are called <strong>real numbers<\/strong>. These include numbers like 4, 275, -200, 10.7, \u00bd, \u03c0, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point:<\/p>\n<p>To solve certain problems like [latex]x^{2}=\u20134[\/latex], it became necessary to introduce <strong>imaginary numbers<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Imaginary Number <em>i<\/em><\/h3>\n<p>The imaginary number <em>i<\/em> is defined to be [latex]i=\\sqrt{-1}[\/latex].<\/p>\n<p>Any real multiple of <em>i<\/em>, like 5<em>i<\/em>, is also an imaginary number.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\sqrt{-9}[\/latex].<\/p>\n<p>We can separate [latex]\\sqrt{-9}[\/latex] as [latex]\\sqrt{9}\\sqrt{-1}[\/latex]. We can take the square root of 9, and write the square root of [latex]-1[\/latex] as <em>i<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]<\/p>\n<\/div>\n<p>A complex number is the sum of a real number and an imaginary number.<\/p>\n<div class=\"textbox\">\n<h3>Complex Number<\/h3>\n<p>A <strong>complex number<\/strong> is a number [latex]z=a+bi[\/latex], where<\/p>\n<ul>\n<li><em>a<\/em> and <em>b<\/em> are real numbers<\/li>\n<li><em>a <\/em>is the real part of the complex number<\/li>\n<li><em>b<\/em> is the imaginary part of the complex number<\/li>\n<\/ul>\n<\/div>\n<p>To plot a complex number like [latex]3-4i[\/latex], we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane.<\/p>\n<div class=\"textbox\">\n<h3>Complex Plane<\/h3>\n<p>In the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1729 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"145\" height=\"101\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Plot the number [latex]3-4i[\/latex] on the complex plane.<\/p>\n<p>The real part of this number is 3, and the imaginary part is [latex]-4[\/latex]. To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1732\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184650\/Screen-Shot-2017-02-23-at-10.46.27-AM.png\" alt=\"A graph with imaginary y-axis and real x-axis. The point 3, negative 4 is marked.\" width=\"275\" height=\"268\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<p>Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number [latex]z=a+bi[\/latex] as if we were plotting the point (a, b) in Cartesian coordinates. Sometimes people write complex numbers as [latex]z=x+yi[\/latex] to highlight this relation.<\/p>\n<h3>Arithmetic on Complex Numbers<\/h3>\n<p>Before we dive into the more complicated uses of complex numbers, let\u2019s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add [latex]3-4i[\/latex] and [latex]2+5i[\/latex].<\/p>\n<p>Adding [latex](3-4i)+(2+5i)[\/latex], we add the real parts and the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]3+2-4i+5i[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]5+i[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]2+5i[\/latex] from [latex]3-4i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q192798\">Show Solution<\/span><\/p>\n<div id=\"q192798\" class=\"hidden-answer\" style=\"display: none\">[latex](3-4i)-(2+5i)=1-9i[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=61710&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we present more worked examples of arithmetic with complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Complex arithmetic\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XJXDcybM84U?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Visualize the addition [latex]3-4i[\/latex] and [latex]-1+5i[\/latex].<\/p>\n<p>The initial point is [latex]3-4i[\/latex]. When we add [latex]-1+3i[\/latex], we add [latex]-1[\/latex] to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. This shifts the point [latex]3-4i[\/latex] to [latex]2+1i[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23185217\/Screen-Shot-2017-02-23-at-10.51.55-AM.png\" alt=\"A graph with an imaginary y-axis and a real x-axis. The point 3, negative 4 is labeled 3 minus 4i. The point 2, 1 is labeled 2 plus 1i. An arrow goes from 3 minus 4i to 2 plus 1i.\" width=\"300\" height=\"291\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131218&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"500\"><\/iframe><\/p>\n<\/div>\n<p>We can also multiply complex numbers by a real number, or multiply two complex numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply: [latex]4\\left(2+5i\\right)[\/latex]<\/p>\n<p>To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.<\/p>\n<p>Distribute and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]4(2+5i)\\\\\\,\\,\\,= 4\\cdot2+4\\cdot5i\\\\\\,\\,\\,=8+20i[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply: [latex](2+5i)(4+i)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2+5i\\right)\\left(4+i\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Expand.}\\\\=8+20i+2i+5i^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Since }i=\\sqrt{-1},i^{2}=-1\\\\=8+20i+2i+5\\left(-1\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify.}\\\\=3+22i\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]3-4i[\/latex] and [latex]2+3i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q929203\">Show Solution<\/span><\/p>\n<div id=\"q929203\" class=\"hidden-answer\" style=\"display: none\">Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1418&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>To understand the effect of multiplication visually, we\u2019ll explore three examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Visualize the product [latex]2(1+2i)[\/latex].<\/p>\n<p>Multiplying we\u2019d get<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&2\\cdot1+2\\cdot2i\\\\&=2+4i\\\\\\end{align}[\/latex]<\/p>\n<p>Notice both the real and imaginary parts have been scaled by 2. Visually, this will stretch the point outwards, away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23195901\/Screen-Shot-2017-02-23-at-11.58.34-AM.png\" alt=\"Graph with imaginary y-axis and real x-axis. The point 1,2 is marked and labeled 1 plus 2i. The point 2,4 is marked and labeled 2 plus 4i. A red arrow is drawn from the origin and through both points.\" width=\"302\" height=\"279\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Visualize the product [latex]i\\left(l+2i\\right)[\/latex].<\/p>\n<p>Multiplying, we\u2019d get<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&i\\cdot1+i\\cdot2i\\\\&=i+2{{i}^{2}}\\\\&=i+2(-1)\\\\&=-2+i\\\\\\end{align}[\/latex]<\/p>\n<p>In this case, the distance from the origin has not changed, but the point has been rotated about the origin, 90\u00b0 counter-clockwise.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]3-4i[\/latex] and [latex]2+3i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q929203\">Show Solution<\/span><\/p>\n<div id=\"q929203\" class=\"hidden-answer\" style=\"display: none\">Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom30\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1418&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Visualize the result of multiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex]. Then show the result of multiplying by [latex]1+i[\/latex] again.<\/p>\n<p>Multiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]-4+2i[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&(1+2i)(1+i)\\\\&=1+i+2i+2{{i}^{2}}\\\\&=1+3i+2(-1)\\\\&=-1+3i\\\\\\end{align}[\/latex]<\/p>\n<p>Multiplying by [latex]1+i[\/latex] again,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&(-1+3i)(1+i)\\\\&=-1-i+3i+3{{i}^{2}}\\\\&=-1+2i+3(-1)\\\\&=-4+2i\\\\\\end{align}[\/latex]<\/p>\n<p>If we multiplied by [latex]1+i[\/latex] again, we\u2019d get [latex]\u20136\u20132i[\/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by 45\u00b0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1736\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23221746\/Screen-Shot-2017-02-23-at-2.15.58-PM.png\" alt=\"The imaginary-real graph with four points marked, each with a dotted red line extending from the origin to that point. The points are as follows. The point 1,2, represented by 1 plus 2i. The point negative 1, 3, represented by negative 1 plus 3i. The point negative 4, 2, represented by negative 4 plus 2i. The point negative 6, negative 2, represented by negative 6 minus 2i.\" width=\"400\" height=\"346\" \/><\/p>\n<\/div>\n<p>In general, multiplication by a complex number can be thought of as a <strong>scaling<\/strong>, changing the distance from the origin, combined with a <strong>rotation<\/strong> about the origin.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom35\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131223&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<p>The following video presents more examples of how to visualize the results of arithmetic on complex numbers.<\/p>\n<p>In the following video, we present more worked examples of arithmetic with complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Visualizing complex arithmetic\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vPZAW7Lhh1E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Generating Fractals with Complex Numbers<\/h2>\n<h3>Complex Recursive Sequences<\/h3>\n<p>Some fractals are generated with complex numbers. \u00a0The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.<\/p>\n<div class=\"textbox\">\n<h3>Recursive Sequence<\/h3>\n<p>A <strong>recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].<\/p>\n<\/div>\n<p>The sequence of values produced is the recursive sequence.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2,\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.<\/p>\n<p>We are given the starting value, [latex]{{z}_{0}}=4[\/latex]. The recursive formula holds for any value of <em>n<\/em>, so if [latex]n = 0[\/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[\/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[\/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[\/latex].<\/p>\n<p>Notice this defines [latex]{{z}_{1}}[\/latex] in terms of the known [latex]{{z}_{0}}[\/latex], so we can compute the value:<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[\/latex].<\/p>\n<p>Now letting [latex]n = 1[\/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[\/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[\/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[\/latex]<\/p>\n<p>Continuing,<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{3}}={{z}_{2}}+2=8+2=10\\\\{{z}_{4}}={{z}_{3}}+2=10+2=12[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5816&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\\cdot{i}+(1-i),\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.<\/p>\n<p>We are given [latex]{{z}_{0}}=4[\/latex]. Using the recursive formula:<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i\\\\{{z}_{2}}={{z}_{1}}\\cdot{i}+(1-i)=(1+3i)\\cdot{i}+(1-i)=i+3{{i}^{2}}+(1-i)=i-3+(1-i)=-2\\\\{{z}_{3}}={{z}_{2}}\\cdot{i}+(1-i)=(-2)\\cdot{i}+(1-i)=-2i+(1-i)=1-3i\\\\{{z}_{4}}={{z}_{3}}\\cdot{i}+(1-i)=(1-3i)\\cdot{i}+(1-i)=i-3{{i}^{2}}+(1-i)=i+3+(1-i)=4\\\\{{z}_{5}}={{z}_{4}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i[\/latex]<\/p>\n<p>Notice this sequence is exhibiting an interesting pattern\u2014it began to repeat itself.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131230&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Recursive complex sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lOyusyTsLTs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Mandelbrot Set<\/h3>\n<p>The Mandelbrot Set is a set of numbers defined based on recursive sequences.<\/p>\n<div class=\"textbox\">\n<h3>Mandelbrot Set<\/h3>\n<p>For any complex number <em>c<\/em>, define the sequence [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]<\/p>\n<p>If this sequence always stays close to the origin (within 2 units), then the number <em>c<\/em> is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number <em>c<\/em> is not part of the set.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine if [latex]c=1+i[\/latex] is part of the Mandelbrot set.<\/p>\n<p>We start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+1+i=0+1+i=1+i\\\\{{z}_{2}}={{z}_{1}}^{2}+1+i={{(1+i)}^{2}}+1+i=1+3i\\\\{{z}_{3}}={{z}_{2}}^{2}+1+i={{(1+3i)}^{2}}+1+i=-7+7i\\\\{{z}_{4}}={{z}_{3}}^{2}+1+i={{(-7+7i)}^{2}}+1+i=1-97i[\/latex]<\/p>\n<p>We can already see that these values are getting quite large. It does not appear that [latex]c=1+i[\/latex] is part of the Mandelbrot set.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine if [latex]c=0.5i[\/latex] is part of the Mandelbrot set.<\/p>\n<p>We start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+0.5i=0+0.5i=0.5i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.5i={{(0.5i)}^{2}}+0.5i=-0.25+0.5i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]<\/p>\n<p>While not definitive with this few iterations, it does appear that this value is remaining small, suggesting that 0.5<em>i<\/em> is part of the Mandelbrot set.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Determine if [latex]c=0.4+0.3i[\/latex] is part of the Mandelbrot set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q31678\">Show Solution<\/span><\/p>\n<div id=\"q31678\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{{z}_{1}}={{z}_{0}}^{2}+0.4+0.3i=0+0.4+0.3i=0.4+0.3i[\/latex]<\/p>\n<p>[latex]{{z}_{2}}={{z}_{1}}^{2}+0.4+0.3i=(0.4+0.3i)^{2}+0.4+0.3i=0.07+.24i+0.4+0.3i=0.47+0.54i[\/latex]<\/p>\n<p>[latex]{{z}_{3}}={{z}_{2}}^{2}+0.4+0.3i=(0.47+0.54i)^{2}+0.4+0.3i=-0.0707+0.576i+0.4+0.3i=0.3293+0.8076i[\/latex]<\/p>\n<p>[latex]\\\\{{z}_{4}}={{z}_{3}}^{2}+0.4+0.3i=(0.3293+0.8076i)^{2}+0.4+0.3i\\approx{-0.5438}+0.5319i+0.4+0.3i=-0.1438+0.8319i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1737 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"187\" height=\"127\" \/>If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.<a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" id=\"return-footnote-1770-2\" href=\"#footnote-1770-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.<\/p>\n<p>Watch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Mandelbrot sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ORqk5jAFpWg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using [latex]c=1+i[\/latex] above, the sequence was distance 2 from the origin after only two recursions.<\/p>\n<p>For some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below<a class=\"footnote\" title=\"This series was generated using Scott\u2019s Mandelbrot Set Explorer\" id=\"return-footnote-1770-3\" href=\"#footnote-1770-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>, light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in\u2014those are used for numbers that grow very slowly.<\/p>\n<p>The Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.<\/p>\n<h3>Additional Resources<\/h3>\n<p>A much more extensive coverage of fractals can be found on the <a href=\"http:\/\/classes.yale.edu\/fractals\/\">Fractal Geometry site<\/a>. This site includes links to several Java software programs for exploring fractals.<\/p>\n<p>If you are impressed with the Mandelbrot set, <a href=\"http:\/\/www.ted.com\/talks\/benoit_mandelbrot_fractals_the_art_of_roughness\">check out this TED talk from 2010<\/a>\u00a0given by Benoit Mandelbrot on fractals and the art of roughness.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1770\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 131223, 131218. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Complex arithmetic. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/XJXDcybM84U\">https:\/\/youtu.be\/XJXDcybM84U<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Visualizing complex arithmetic. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vPZAW7Lhh1E\">https:\/\/youtu.be\/vPZAW7Lhh1E<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question 1418. <strong>Authored by<\/strong>: WebWork Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Recursive complex sequences. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lOyusyTsLTs\">https:\/\/youtu.be\/lOyusyTsLTs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Mandelbrot sequences. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ORqk5jAFpWg\">https:\/\/youtu.be\/ORqk5jAFpWg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Visualizing mandelbrot sequences and set . <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UJjForMx6-0\">https:\/\/youtu.be\/UJjForMx6-0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1770-1\">Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA <a href=\"#return-footnote-1770-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1770-2\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png<\/a> <a href=\"#return-footnote-1770-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1770-3\">This series was generated using Scott\u2019s Mandelbrot Set Explorer <a href=\"#return-footnote-1770-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Complex arithmetic\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/XJXDcybM84U\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Visualizing complex arithmetic\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vPZAW7Lhh1E\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 131223, 131218\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question 1418\",\"author\":\"WebWork Rochester\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Recursive complex sequences\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/lOyusyTsLTs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Mandelbrot sequences\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ORqk5jAFpWg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Visualizing mandelbrot sequences and set \",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/UJjForMx6-0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"0066457e-475a-435a-bac2-f2ff5f334cd9","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1770","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1770","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1770\/revisions"}],"predecessor-version":[{"id":3201,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1770\/revisions\/3201"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1770\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/wp\/v2\/media?parent=1770"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapter-type?post=1770"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/wp\/v2\/contributor?post=1770"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/wp-json\/wp\/v2\/license?post=1770"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}