{"id":1772,"date":"2017-03-14T00:23:51","date_gmt":"2017-03-14T00:23:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1772"},"modified":"2019-05-30T16:34:54","modified_gmt":"2019-05-30T16:34:54","slug":"arithmetic-with-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/arithmetic-with-complex-numbers\/","title":{"raw":"Arithmetic With Complex Numbers","rendered":"Arithmetic With 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\n<h2>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<h2>Arithmetic on Complex Numbers<\/h2>\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\r\n&nbsp;","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<h2>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-1772-1\" href=\"#footnote-1772-1\" 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<h2>Arithmetic on Complex Numbers<\/h2>\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<p>&nbsp;<\/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-1772\">\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>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><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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><\/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-1772-1\">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-1772-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"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\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"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-1772","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1772","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1772\/revisions"}],"predecessor-version":[{"id":2974,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1772\/revisions\/2974"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1772\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/media?parent=1772"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapter-type?post=1772"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/contributor?post=1772"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/license?post=1772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}