{"id":188,"date":"2023-06-21T13:22:42","date_gmt":"2023-06-21T13:22:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/express-and-plot-complex-numbers\/"},"modified":"2023-07-04T03:45:16","modified_gmt":"2023-07-04T03:45:16","slug":"express-and-plot-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/express-and-plot-complex-numbers\/","title":{"raw":"\u25aa   Express and Plot Complex Numbers","rendered":"\u25aa   Express and Plot Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Express square roots of negative numbers as multiples of [latex]i[\/latex]<em>.<\/em><\/li>\r\n \t<li>Plot complex numbers on the complex plane.<\/li>\r\n<\/ul>\r\n<\/div>\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 [latex]i[\/latex]. Consider the square root of \u201325.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{-25}&amp;=\\sqrt{25\\cdot \\left(-1\\right)}\\\\&amp;=\\sqrt{25}\\cdot\\sqrt{-1}\\\\ &amp;=5i\\end{align}[\/latex]<\/p>\r\nWe use [latex]5i[\/latex]<em>\u00a0<\/em>and not [latex]-\\text{5}i[\/latex]\u00a0because the principal root of 25 is the positive root.\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 [latex]a+bi[\/latex]\u00a0where [latex]a[\/latex]\u00a0is the real part and [latex]bi[\/latex]\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\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\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>[latex]a[\/latex] is the real part of the complex number.<\/li>\r\n \t<li>[latex]bi[\/latex] is 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 [latex]b[\/latex] is 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}\\cdot\\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 examples\">\r\n<h3>recall writing square roots in simplest form<\/h3>\r\nRecall that the <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" \/>\r\n\r\nTo simplify a square root, we rewrite it such that there are no perfect squares in the radicand. Use the <strong>product rule for simplifying square roots,\u00a0<\/strong>which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\\sqrt{-75}[\/latex] as [latex]\\sqrt{25}\\cdot \\sqrt{3}\\cdot \\sqrt{-1} = 5\u00a0 \\sqrt{3}i[\/latex].\r\n\r\n&nbsp;\r\n\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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612345\"]\r\n\r\n[latex]\\sqrt{-9}=\\sqrt{9}\\cdot\\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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"745111\"]\r\n\r\n[latex]\\sqrt{-24}=2i\\sqrt{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]61706[\/ohm_question]\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 [latex](a, b)[\/latex], where [latex]a[\/latex] represents the coordinate for the horizontal axis and [latex]b[\/latex] represents 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 [latex]\u20132[\/latex]\u00a0and the imaginary part is [latex]3i[\/latex]. 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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716834\"]\r\n\r\nThe real part of the complex number is [latex]3[\/latex], and the imaginary part is [latex]\u20134i[\/latex]. 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\"]Show 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\r\n[ohm_question]65079[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Express square roots of negative numbers as multiples of [latex]i[\/latex]<em>.<\/em><\/li>\n<li>Plot complex numbers on the complex plane.<\/li>\n<\/ul>\n<\/div>\n<p>We 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 [latex]i[\/latex]. Consider the square root of \u201325.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{-25}&=\\sqrt{25\\cdot \\left(-1\\right)}\\\\&=\\sqrt{25}\\cdot\\sqrt{-1}\\\\ &=5i\\end{align}[\/latex]<\/p>\n<p>We use [latex]5i[\/latex]<em>\u00a0<\/em>and not [latex]-\\text{5}i[\/latex]\u00a0because the principal root of 25 is the positive root.<\/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 [latex]a+bi[\/latex]\u00a0where [latex]a[\/latex]\u00a0is the real part and [latex]bi[\/latex]\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><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>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>[latex]a[\/latex] is the real part of the complex number.<\/li>\n<li>[latex]bi[\/latex] is 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 [latex]b[\/latex] is 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}\\cdot\\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 examples\">\n<h3>recall writing square roots in simplest form<\/h3>\n<p>Recall that the <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" \/><\/p>\n<p>To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. Use the <strong>product rule for simplifying square roots,\u00a0<\/strong>which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\\sqrt{-75}[\/latex] as [latex]\\sqrt{25}\\cdot \\sqrt{3}\\cdot \\sqrt{-1} = 5\u00a0 \\sqrt{3}i[\/latex].<\/p>\n<p>&nbsp;<\/p>\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\">Show Solution<\/span><\/p>\n<div id=\"q612345\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\sqrt{-9}=\\sqrt{9}\\cdot\\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\">Show Solution<\/span><\/p>\n<div id=\"q745111\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\sqrt{-24}=2i\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm61706\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61706&theme=oea&iframe_resize_id=ohm61706&show_question_numbers\" width=\"100%\" height=\"150\"><\/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 [latex](a, b)[\/latex], where [latex]a[\/latex] represents the coordinate for the horizontal axis and [latex]b[\/latex] represents 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 [latex]\u20132[\/latex]\u00a0and the imaginary part is [latex]3i[\/latex]. 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\">Show Solution<\/span><\/p>\n<div id=\"q716834\" class=\"hidden-answer\" style=\"display: none\">\n<p>The real part of the complex number is [latex]3[\/latex], and the imaginary part is [latex]\u20134i[\/latex]. 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\">Show 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=\"ohm65079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65079&theme=oea&iframe_resize_id=ohm65079&show_question_numbers\" width=\"100%\" height=\"150\"><\/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-188\">\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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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>","protected":false},"author":395986,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introduction to Complex Numbers\",\"author\":\"Sousa, James\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/NeTRNpBI17I\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 61706\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question 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