{"id":1385,"date":"2016-10-24T21:33:50","date_gmt":"2016-10-24T21:33:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1385"},"modified":"2017-04-14T19:08:04","modified_gmt":"2017-04-14T19:08:04","slug":"express-and-plot-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/express-and-plot-complex-numbers\/","title":{"raw":"Express and Plot Complex Numbers","rendered":"Express and Plot Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Express square roots of negative numbers as multiples of <em>i<\/em><\/li>\r\n \t<li>Plot complex numbers on the complex plane<\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\nWe know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an <strong>imaginary number<\/strong>. The imaginary number [latex]i[\/latex] is defined as the square root of negative 1.\r\n<p style=\"text-align: center;\">[latex]\\sqrt{-1}=i[\/latex]<\/p>\r\nSo, using properties of radicals,\r\n<p style=\"text-align: center;\">[latex]{i}^{2}={\\left(\\sqrt{-1}\\right)}^{2}=-1[\/latex]<\/p>\r\nWe can write the square root of any negative number as a multiple of <em>i<\/em>. Consider the square root of \u201325.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\sqrt{-25}=\\sqrt{25\\cdot \\left(-1\\right)}\\hfill \\\\ \\text{ }=\\sqrt{25}\\sqrt{-1}\\hfill \\\\ \\text{ }=5i\\hfill \\end{array}[\/latex]<\/p>\r\nWe use 5<em>i\u00a0<\/em>and not [latex]-\\text{5}i[\/latex]\u00a0because the principal root of 25 is the positive root.\r\n\r\n<img class=\"aligncenter wp-image-2527 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211956\/CNX_Precalc_Figure_03_01_0012.jpg\" alt=\"Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.\" width=\"487\" height=\"72\" \/>\r\n\r\nA <strong>complex number<\/strong> is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written <em>a\u00a0<\/em>+ <em>bi<\/em>\u00a0where <em>a<\/em>\u00a0is the real part and <em>bi<\/em>\u00a0is the imaginary part. For example, [latex]5+2i[\/latex] is a complex number. So, too, is [latex]3+4\\sqrt{3}i[\/latex].\r\n\r\nImaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative\u00a0real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Imaginary and Complex Numbers<\/h3>\r\nA <strong>complex number<\/strong> is a number of the form [latex]a+bi[\/latex] where\r\n<ul>\r\n \t<li><em>a<\/em>\u00a0is the real part of the complex number.<\/li>\r\n \t<li><em>bi<\/em>\u00a0is the imaginary part of the complex number.<\/li>\r\n<\/ul>\r\nIf [latex]b=0[\/latex], then [latex]a+bi[\/latex] is a real number. If [latex]a=0[\/latex] and <em>b<\/em>\u00a0is not equal to 0, the complex number is called an <strong>imaginary number<\/strong>. An imaginary number is an even root of a negative number.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an imaginary number, express it in standard form.<\/h3>\r\n<ol>\r\n \t<li>Write [latex]\\sqrt{-a}[\/latex] as [latex]\\sqrt{a}\\sqrt{-1}[\/latex].<\/li>\r\n \t<li>Express [latex]\\sqrt{-1}[\/latex] as <em>i<\/em>.<\/li>\r\n \t<li>Write [latex]\\sqrt{a}\\cdot i[\/latex] in simplest form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Expressing an Imaginary Number in Standard Form<\/h3>\r\nExpress [latex]\\sqrt{-9}[\/latex] in standard form.\r\n\r\n[reveal-answer q=\"612345\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"612345\"]\r\n[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]\r\n\r\nIn standard form, this is [latex]0+3i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nExpress [latex]\\sqrt{-24}[\/latex] in standard form.\r\n\r\n[reveal-answer q=\"745111\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"745111\"][latex]\\sqrt{-24}=0+2i\\sqrt{6}\\\\[\/latex][\/hidden-answer]\r\n<iframe id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=61706&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/NeTRNpBI17I\r\n<h2>Plot complex numbers on the complex plane<\/h2>\r\nWe cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the <strong>complex plane<\/strong>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs (<em>a<\/em>, <em>b<\/em>), where <em>a<\/em>\u00a0represents the coordinate for the horizontal axis and <em>b<\/em>\u00a0represents the coordinate for the vertical axis.\r\n\r\n<img class=\"aligncenter wp-image-2530 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211959\/CNX_Precalc_Figure_03_01_0022.jpg\" alt=\"Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/>\r\n\r\nLet\u2019s consider the number [latex]-2+3i[\/latex]. The real part of the complex number is \u20132\u00a0and the imaginary part is 3<em>i<\/em>. We plot the ordered pair [latex]\\left(-2,3\\right)[\/latex] to represent the complex number [latex]-2+3i[\/latex]<strong>.<\/strong>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Complex Plane<\/h3>\r\n<img class=\"aligncenter wp-image-2531 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24212002\/CNX_Precalc_Figure_03_01_003n2.jpg\" alt=\"The complex plane showing that the horizontal axis (in the real plane, the x-axis) is known as the real axis and the vertical axis (in the real plane, the y-axis) is known as the imaginary axis.\" width=\"487\" height=\"350\" \/>\r\n\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<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a complex number, represent its components on the complex plane.<\/h3>\r\n<ol>\r\n \t<li>Determine the real part and the imaginary part of the complex number.<\/li>\r\n \t<li>Move along the horizontal axis to show the real part of the number.<\/li>\r\n \t<li>Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\r\n \t<li>Plot the point.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Plotting a Complex Number on the Complex Plane<\/h3>\r\nPlot the complex number [latex]3 - 4i[\/latex] on the complex plane.\r\n\r\n[reveal-answer q=\"716834\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"716834\"]\r\n\r\nThe real part of the complex number is 3, and the imaginary part is \u20134<em>i<\/em>. We plot the ordered pair [latex]\\left(3,-4\\right)[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-2532 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24212004\/CNX_Precalc_Figure_03_01_0042.jpg\" alt=\"Plot of a complex number, 3 - 4i. Note that the real part (3) is plotted on the x-axis and the imaginary part (-4i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nPlot the complex number [latex]-4-i[\/latex] on the complex plane.\r\n\r\n[reveal-answer q=\"305544\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"305544\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16202748\/CNX_Precalc_Figure_03_01_0052.jpg\"><img class=\"aligncenter size-full wp-image-2837\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16202748\/CNX_Precalc_Figure_03_01_0052.jpg\" alt=\"Graph of the plotted point, -4-i.\" width=\"487\" height=\"443\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Express square roots of negative numbers as multiples of <em>i<\/em><\/li>\n<li>Plot complex numbers on the complex plane<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><br \/>\nWe know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an <strong>imaginary number<\/strong>. The imaginary number [latex]i[\/latex] is defined as the square root of negative 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{-1}=i[\/latex]<\/p>\n<p>So, using properties of radicals,<\/p>\n<p style=\"text-align: center;\">[latex]{i}^{2}={\\left(\\sqrt{-1}\\right)}^{2}=-1[\/latex]<\/p>\n<p>We can write the square root of any negative number as a multiple of <em>i<\/em>. Consider the square root of \u201325.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\sqrt{-25}=\\sqrt{25\\cdot \\left(-1\\right)}\\hfill \\\\ \\text{ }=\\sqrt{25}\\sqrt{-1}\\hfill \\\\ \\text{ }=5i\\hfill \\end{array}[\/latex]<\/p>\n<p>We use 5<em>i\u00a0<\/em>and not [latex]-\\text{5}i[\/latex]\u00a0because the principal root of 25 is the positive root.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2527 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211956\/CNX_Precalc_Figure_03_01_0012.jpg\" alt=\"Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.\" width=\"487\" height=\"72\" \/><\/p>\n<p>A <strong>complex number<\/strong> is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written <em>a\u00a0<\/em>+ <em>bi<\/em>\u00a0where <em>a<\/em>\u00a0is the real part and <em>bi<\/em>\u00a0is the imaginary part. For example, [latex]5+2i[\/latex] is a complex number. So, too, is [latex]3+4\\sqrt{3}i[\/latex].<\/p>\n<p>Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative\u00a0real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Imaginary and Complex Numbers<\/h3>\n<p>A <strong>complex number<\/strong> is a number of the form [latex]a+bi[\/latex] where<\/p>\n<ul>\n<li><em>a<\/em>\u00a0is the real part of the complex number.<\/li>\n<li><em>bi<\/em>\u00a0is the imaginary part of the complex number.<\/li>\n<\/ul>\n<p>If [latex]b=0[\/latex], then [latex]a+bi[\/latex] is a real number. If [latex]a=0[\/latex] and <em>b<\/em>\u00a0is not equal to 0, the complex number is called an <strong>imaginary number<\/strong>. An imaginary number is an even root of a negative number.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an imaginary number, express it in standard form.<\/h3>\n<ol>\n<li>Write [latex]\\sqrt{-a}[\/latex] as [latex]\\sqrt{a}\\sqrt{-1}[\/latex].<\/li>\n<li>Express [latex]\\sqrt{-1}[\/latex] as <em>i<\/em>.<\/li>\n<li>Write [latex]\\sqrt{a}\\cdot i[\/latex] in simplest form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expressing an Imaginary Number in Standard Form<\/h3>\n<p>Express [latex]\\sqrt{-9}[\/latex] in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612345\">Solution<\/span><\/p>\n<div id=\"q612345\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]<\/p>\n<p>In standard form, this is [latex]0+3i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Express [latex]\\sqrt{-24}[\/latex] in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q745111\">Solution<\/span><\/p>\n<div id=\"q745111\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sqrt{-24}=0+2i\\sqrt{6}\\\\[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=61706&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NeTRNpBI17I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Plot complex numbers on the complex plane<\/h2>\n<p>We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the <strong>complex plane<\/strong>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs (<em>a<\/em>, <em>b<\/em>), where <em>a<\/em>\u00a0represents the coordinate for the horizontal axis and <em>b<\/em>\u00a0represents the coordinate for the vertical axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2530 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211959\/CNX_Precalc_Figure_03_01_0022.jpg\" alt=\"Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/><\/p>\n<p>Let\u2019s consider the number [latex]-2+3i[\/latex]. The real part of the complex number is \u20132\u00a0and the imaginary part is 3<em>i<\/em>. We plot the ordered pair [latex]\\left(-2,3\\right)[\/latex] to represent the complex number [latex]-2+3i[\/latex]<strong>.<\/strong><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Complex Plane<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2531 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24212002\/CNX_Precalc_Figure_03_01_003n2.jpg\" alt=\"The complex plane showing that the horizontal axis (in the real plane, the x-axis) is known as the real axis and the vertical axis (in the real plane, the y-axis) is known as the imaginary axis.\" width=\"487\" height=\"350\" \/><\/p>\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<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a complex number, represent its components on the complex plane.<\/h3>\n<ol>\n<li>Determine the real part and the imaginary part of the complex number.<\/li>\n<li>Move along the horizontal axis to show the real part of the number.<\/li>\n<li>Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\n<li>Plot the point.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Plotting a Complex Number on the Complex Plane<\/h3>\n<p>Plot the complex number [latex]3 - 4i[\/latex] on the complex plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716834\">Solution<\/span><\/p>\n<div id=\"q716834\" class=\"hidden-answer\" style=\"display: none\">\n<p>The real part of the complex number is 3, and the imaginary part is \u20134<em>i<\/em>. We plot the ordered pair [latex]\\left(3,-4\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2532 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24212004\/CNX_Precalc_Figure_03_01_0042.jpg\" alt=\"Plot of a complex number, 3 - 4i. Note that the real part (3) is plotted on the x-axis and the imaginary part (-4i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Plot the complex number [latex]-4-i[\/latex] on the complex plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q305544\">Solution<\/span><\/p>\n<div id=\"q305544\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16202748\/CNX_Precalc_Figure_03_01_0052.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2837\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16202748\/CNX_Precalc_Figure_03_01_0052.jpg\" alt=\"Graph of the plotted point, -4-i.\" width=\"487\" height=\"443\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\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-1385\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introduction to Complex Numbers. <strong>Authored by<\/strong>: Sousa, James. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/NeTRNpBI17I\">https:\/\/youtu.be\/NeTRNpBI17I<\/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 ID 61706. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 65709. <strong>Authored by<\/strong>: Kaslik,Pete, mb Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introduction to Complex Numbers\",\"author\":\"Sousa, 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