{"id":15385,"date":"2021-10-11T23:05:42","date_gmt":"2021-10-11T23:05:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/16-4-1-complex-numbers\/"},"modified":"2021-10-18T03:39:26","modified_gmt":"2021-10-18T03:39:26","slug":"complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/complex-numbers\/","title":{"raw":"Complex Numbers","rendered":"Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Express roots of negative numbers in terms of [latex]i[\/latex]<\/li>\r\n \t<li>Express imaginary numbers as [latex]bi[\/latex] and complex numbers as [latex]a+bi[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\nYou really need only one new number to start working with the square roots of negative numbers. That number is the square root of [latex]\u22121,\\sqrt{-1}[\/latex]. The <i>real numbers<\/i> are those that can be shown on a number line\u2014they seem pretty <i>real<\/i> to us! When something is not real, we often say it is <i>imaginary<\/i>. We call this new number [latex]i[\/latex]\u00a0and it is used<i>\u00a0<\/i>to represent the square root of [latex]\u22121[\/latex].\r\n<p style=\"text-align: center;\">[latex] i=\\sqrt{-1}[\/latex]<\/p>\r\nBecause [latex] \\sqrt{x}\\,\\cdot \\,\\sqrt{x}=x[\/latex], we can also see that [latex] \\sqrt{-1}\\,\\cdot \\,\\sqrt{-1}=-1[\/latex] or [latex] i\\,\\cdot \\,i=-1[\/latex]. We also know that [latex] i\\,\\cdot \\,i={{i}^{2}}[\/latex], so we can conclude that [latex] {{i}^{2}}=-1[\/latex].\r\n<p style=\"text-align: center;\">[latex] {{i}^{2}}=-1[\/latex]<\/p>\r\nThe number [latex]\u22121[\/latex]<i>\u00a0<\/i>allows us to work with roots of all negative numbers, not just [latex] \\sqrt{-1}[\/latex]. There are two important rules to remember: [latex] \\sqrt{-1}=i[\/latex], and [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex]. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times [latex] \\sqrt{-1}[\/latex]. Next you will simplify the square root and rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>. <\/i>Let us try an example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{-4}[\/latex]\r\n\r\n[reveal-answer q=\"793555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"793555\"]\r\n\r\nUse the rule [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex] \\sqrt{-1}[\/latex].\r\n\r\n[latex] \\sqrt{-4}=\\sqrt{4\\cdot -1}=\\sqrt{4}\\sqrt{-1}[\/latex]\r\n\r\nSince\u00a0[latex]4[\/latex] is a perfect square\u00a0[latex](4=2^{2})[\/latex], you can simplify the square root of\u00a0[latex]4[\/latex].\r\n\r\n[latex] \\sqrt{4}\\sqrt{-1}=2\\sqrt{-1}[\/latex]\r\n\r\nUse the definition of [latex]i[\/latex]\u00a0to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i>\r\n\r\n[latex] 2\\sqrt{-1}=2i[\/latex]\r\n\r\nThe answer is [latex]2i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{-18}[\/latex]\r\n\r\n[reveal-answer q=\"760057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"760057\"]\r\n\r\nUse the rule [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex] \\sqrt{-1}[\/latex].\r\n\r\n[latex] \\sqrt{-18}=\\sqrt{18\\cdot -1}=\\sqrt{18}\\sqrt{-1}[\/latex]\r\n\r\nSince\u00a0[latex]18[\/latex] is <i>not<\/i> a perfect square, use the same rule to rewrite it using factors that are perfect squares. In this case,\u00a0[latex]9[\/latex] is the only perfect square factor, and the square root of\u00a0[latex]9[\/latex] is\u00a0[latex]3[\/latex].\r\n\r\n[latex] \\sqrt{18}\\sqrt{-1}=\\sqrt{9}\\sqrt{2}\\sqrt{-1}=3\\sqrt{2}\\sqrt{-1}[\/latex]\r\n\r\nUse the definition of [latex]i[\/latex] to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i>\r\n\r\n[latex] 3\\sqrt{2}\\sqrt{-1}=3\\sqrt{2}i=3i\\sqrt{2}[\/latex]\r\n\r\nRemember to write [latex]i[\/latex] in front of the radical.\r\n\r\nThe answer is [latex]3i\\sqrt[{}]{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] -\\sqrt{-72}[\/latex]\r\n\r\n[reveal-answer q=\"503996\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"503996\"]\r\n\r\nUse the rule [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex] \\sqrt{-1}[\/latex].\r\n\r\n[latex] -\\sqrt{-72}=-\\sqrt{72\\cdot -1}=-\\sqrt{72}\\sqrt{-1}[\/latex]\r\n\r\nSince\u00a0[latex]72[\/latex] is <i>not<\/i> a perfect square, use the same rule to rewrite it using factors that are perfect squares. Notice that\u00a0[latex]72[\/latex] has three perfect squares as factors:\u00a0[latex]4, 9[\/latex], and\u00a0[latex]36[\/latex]. It is easiest to use the largest factor that is a perfect square.\r\n\r\n[latex] -\\sqrt{72}\\sqrt{-1}=-\\sqrt{36}\\sqrt{2}\\sqrt{-1}=-6\\sqrt{2}\\sqrt{-1}[\/latex]\r\n\r\nUse the definition of [latex]i[\/latex] to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i>\r\n\r\n[latex] -6\\sqrt{2}\\sqrt{-1}=-6\\sqrt{2}i=-6i\\sqrt{2}[\/latex]\r\n\r\nRemember to write [latex]i[\/latex] in front of the radical.\r\n\r\nThe answer is [latex]-6i\\sqrt{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have wanted to simplify [latex] -\\sqrt{-72}[\/latex] using different factors. Some may have thought of rewriting this radical as [latex] -\\sqrt{-9}\\sqrt{8}[\/latex], or [latex] -\\sqrt{-4}\\sqrt{18}[\/latex], or [latex] -\\sqrt{-6}\\sqrt{12}[\/latex] for instance. Each of these radicals would have eventually yielded the same answer of [latex] -6i\\sqrt{2}[\/latex].\r\n\r\nIn the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.\r\n\r\nhttps:\/\/youtu.be\/LSp7yNP6Xxc\r\n<div class=\"textbox shaded\">\r\n<h3>Rewriting the Square Root of a Negative Number<\/h3>\r\n<ul>\r\n \t<li>Find perfect squares within the radical.<\/li>\r\n \t<li>Rewrite the radical using the rule [latex] \\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].<\/li>\r\n \t<li>Rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex].<\/li>\r\n<\/ul>\r\nExample: [latex] \\sqrt{-18}=\\sqrt{9}\\sqrt{-2}=\\sqrt{9}\\sqrt{2}\\sqrt{-1}=3i\\sqrt{2}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3907[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Complex Numbers<\/h2>\r\n<img class=\"wp-image-2527 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2016\/06\/22231825\/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<p id=\"fs-id1165135500790\">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] is the real part and [latex]bi[\/latex] is the imaginary part. For example, [latex]5+2i[\/latex] is a complex number. So, too, is [latex]3+4i\\sqrt{3}[\/latex].<span id=\"fs-id1165137832295\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137892327\">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. You can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You will see more of that later.<\/p>\r\n\r\n<table cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr>\r\n<th>Complex Number<\/th>\r\n<th>Real Part<\/th>\r\n<th>Imaginary Part<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]3+7i[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]7i[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]18\u201332i[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<td>[latex]\u221232i[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] -\\frac{3}{5}+i\\sqrt{2}[\/latex]<\/td>\r\n<td>[latex] -\\frac{3}{5}[\/latex]<\/td>\r\n<td>[latex] i\\sqrt{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\frac{\\sqrt{2}}{2}-\\frac{1}{2}i[\/latex]<\/td>\r\n<td>[latex] \\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]-\\frac{1}{2}i[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn a number with a radical as part of <i>b<\/i>, such as [latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex]\u00a0above, the imaginary <i>[latex]i[\/latex]<\/i> should be written in front of the radical. Though writing this number as [latex] -\\frac{3}{5}+\\sqrt{2}i[\/latex] is technically correct, it makes it much more difficult to tell whether <i>[latex]i[\/latex]<\/i> is inside or outside of the radical. Putting it before the radical, as in [latex] -\\frac{3}{5}+i\\sqrt{2}[\/latex], clears up any confusion. Look at these last two examples.\r\n<table cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr>\r\n<th>Number<\/th>\r\n<th>Complex Form:\r\n[latex]a+bi[\/latex]<\/th>\r\n<th>Real Part<\/th>\r\n<th>Imaginary Part<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]17[\/latex]<\/td>\r\n<td>[latex]17+0i[\/latex]<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<td>[latex]0i[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22123i[\/latex]<\/td>\r\n<td>[latex]0\u20133i[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\u22123i[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBy making [latex]b=0[\/latex], any real number can be expressed as a complex number. The real number [latex]<i>a[\/latex]<\/i>\u00a0is written as [latex]a+0i[\/latex] in complex form. Similarly, any imaginary number can be expressed as a complex number. By making [latex]a=0[\/latex], any imaginary number [latex]bi[\/latex] can be written as [latex]0+bi[\/latex] in complex form.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite [latex]83.6[\/latex] as a complex number.\r\n\r\n[reveal-answer q=\"704457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"704457\"]\r\n\r\nRemember that a complex number has the form [latex]a+bi[\/latex]. You need to figure out what <i>a<\/i> and <i>b<\/i> need to be.\r\n\r\n[latex]a+bi[\/latex]\r\n\r\nSince [latex]83.6[\/latex] is a real number, it is the real part (a) of the complex number [latex]a+bi[\/latex].\u00a0<i>\r\n<\/i>\r\n\r\n[latex]83.6+bi[\/latex]\r\n\r\nA real number does not contain any imaginary parts, so the value of [latex]b[\/latex] is\u00a0[latex]0[\/latex].\r\n\r\nThe answer is [latex]83.6+0i[\/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\n[ohm_question]197339[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite [latex]\u22123i[\/latex] as a complex number.\r\n\r\n[reveal-answer q=\"451549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"451549\"]\r\n\r\nRemember that a complex number has the form [latex]a+bi[\/latex].\u00a0You need to figure out what [latex]a[\/latex] and <i>[latex]b[\/latex]<\/i> need to be.\r\n\r\n[latex]a+bi[\/latex]\r\n\r\nSince [latex]\u22123i[\/latex] is an imaginary number, it is the imaginary part [latex]bi[\/latex] of the complex number [latex]a+bi[\/latex].\r\n\r\n[latex]a\u20133i[\/latex]\r\n\r\nThis imaginary number has no real parts, so the value of [latex]a[\/latex] is [latex]0[\/latex].\r\n\r\nThe answer is\u00a0[latex]0\u20133i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, we show more examples of how to write numbers as complex numbers.\r\n\r\nhttps:\/\/youtu.be\/mfoOYdDkuyY\r\n<h2>Summary<\/h2>\r\nSquare roots of negative numbers can be simplified using [latex] \\sqrt{-1}=i[\/latex] and [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex].\u00a0 Complex numbers have the form [latex]a+bi[\/latex], where [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0are real numbers and [latex]i[\/latex]\u00a0is the square root of [latex]\u22121[\/latex]. All real numbers can be written as complex numbers by setting [latex]b=0[\/latex]. Imaginary numbers have the form [latex]bi[\/latex] and can also be written as complex numbers by setting [latex]a=0[\/latex].\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1GcR4XrneYIZ5GbQcxKA42J9SU44nj9Dzst-r0MoZO0Y\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Express roots of negative numbers in terms of [latex]i[\/latex]<\/li>\n<li>Express imaginary numbers as [latex]bi[\/latex] and complex numbers as [latex]a+bi[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>You really need only one new number to start working with the square roots of negative numbers. That number is the square root of [latex]\u22121,\\sqrt{-1}[\/latex]. The <i>real numbers<\/i> are those that can be shown on a number line\u2014they seem pretty <i>real<\/i> to us! When something is not real, we often say it is <i>imaginary<\/i>. We call this new number [latex]i[\/latex]\u00a0and it is used<i>\u00a0<\/i>to represent the square root of [latex]\u22121[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]i=\\sqrt{-1}[\/latex]<\/p>\n<p>Because [latex]\\sqrt{x}\\,\\cdot \\,\\sqrt{x}=x[\/latex], we can also see that [latex]\\sqrt{-1}\\,\\cdot \\,\\sqrt{-1}=-1[\/latex] or [latex]i\\,\\cdot \\,i=-1[\/latex]. We also know that [latex]i\\,\\cdot \\,i={{i}^{2}}[\/latex], so we can conclude that [latex]{{i}^{2}}=-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{{i}^{2}}=-1[\/latex]<\/p>\n<p>The number [latex]\u22121[\/latex]<i>\u00a0<\/i>allows us to work with roots of all negative numbers, not just [latex]\\sqrt{-1}[\/latex]. There are two important rules to remember: [latex]\\sqrt{-1}=i[\/latex], and [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex]. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times [latex]\\sqrt{-1}[\/latex]. Next you will simplify the square root and rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>. <\/i>Let us try an example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{-4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q793555\">Show Solution<\/span><\/p>\n<div id=\"q793555\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex]\\sqrt{-1}[\/latex].<\/p>\n<p>[latex]\\sqrt{-4}=\\sqrt{4\\cdot -1}=\\sqrt{4}\\sqrt{-1}[\/latex]<\/p>\n<p>Since\u00a0[latex]4[\/latex] is a perfect square\u00a0[latex](4=2^{2})[\/latex], you can simplify the square root of\u00a0[latex]4[\/latex].<\/p>\n<p>[latex]\\sqrt{4}\\sqrt{-1}=2\\sqrt{-1}[\/latex]<\/p>\n<p>Use the definition of [latex]i[\/latex]\u00a0to rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i><\/p>\n<p>[latex]2\\sqrt{-1}=2i[\/latex]<\/p>\n<p>The answer is [latex]2i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{-18}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q760057\">Show Solution<\/span><\/p>\n<div id=\"q760057\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex]\\sqrt{-1}[\/latex].<\/p>\n<p>[latex]\\sqrt{-18}=\\sqrt{18\\cdot -1}=\\sqrt{18}\\sqrt{-1}[\/latex]<\/p>\n<p>Since\u00a0[latex]18[\/latex] is <i>not<\/i> a perfect square, use the same rule to rewrite it using factors that are perfect squares. In this case,\u00a0[latex]9[\/latex] is the only perfect square factor, and the square root of\u00a0[latex]9[\/latex] is\u00a0[latex]3[\/latex].<\/p>\n<p>[latex]\\sqrt{18}\\sqrt{-1}=\\sqrt{9}\\sqrt{2}\\sqrt{-1}=3\\sqrt{2}\\sqrt{-1}[\/latex]<\/p>\n<p>Use the definition of [latex]i[\/latex] to rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i><\/p>\n<p>[latex]3\\sqrt{2}\\sqrt{-1}=3\\sqrt{2}i=3i\\sqrt{2}[\/latex]<\/p>\n<p>Remember to write [latex]i[\/latex] in front of the radical.<\/p>\n<p>The answer is [latex]3i\\sqrt[{}]{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]-\\sqrt{-72}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q503996\">Show Solution<\/span><\/p>\n<div id=\"q503996\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] to rewrite this as a product using [latex]\\sqrt{-1}[\/latex].<\/p>\n<p>[latex]-\\sqrt{-72}=-\\sqrt{72\\cdot -1}=-\\sqrt{72}\\sqrt{-1}[\/latex]<\/p>\n<p>Since\u00a0[latex]72[\/latex] is <i>not<\/i> a perfect square, use the same rule to rewrite it using factors that are perfect squares. Notice that\u00a0[latex]72[\/latex] has three perfect squares as factors:\u00a0[latex]4, 9[\/latex], and\u00a0[latex]36[\/latex]. It is easiest to use the largest factor that is a perfect square.<\/p>\n<p>[latex]-\\sqrt{72}\\sqrt{-1}=-\\sqrt{36}\\sqrt{2}\\sqrt{-1}=-6\\sqrt{2}\\sqrt{-1}[\/latex]<\/p>\n<p>Use the definition of [latex]i[\/latex] to rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]<i>.<\/i><\/p>\n<p>[latex]-6\\sqrt{2}\\sqrt{-1}=-6\\sqrt{2}i=-6i\\sqrt{2}[\/latex]<\/p>\n<p>Remember to write [latex]i[\/latex] in front of the radical.<\/p>\n<p>The answer is [latex]-6i\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have wanted to simplify [latex]-\\sqrt{-72}[\/latex] using different factors. Some may have thought of rewriting this radical as [latex]-\\sqrt{-9}\\sqrt{8}[\/latex], or [latex]-\\sqrt{-4}\\sqrt{18}[\/latex], or [latex]-\\sqrt{-6}\\sqrt{12}[\/latex] for instance. Each of these radicals would have eventually yielded the same answer of [latex]-6i\\sqrt{2}[\/latex].<\/p>\n<p>In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots to Imaginary Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LSp7yNP6Xxc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>Rewriting the Square Root of a Negative Number<\/h3>\n<ul>\n<li>Find perfect squares within the radical.<\/li>\n<li>Rewrite the radical using the rule [latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].<\/li>\n<li>Rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex].<\/li>\n<\/ul>\n<p>Example: [latex]\\sqrt{-18}=\\sqrt{9}\\sqrt{-2}=\\sqrt{9}\\sqrt{2}\\sqrt{-1}=3i\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3907\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3907&theme=oea&iframe_resize_id=ohm3907&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Complex Numbers<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2527 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2016\/06\/22231825\/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 id=\"fs-id1165135500790\">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] is the real part and [latex]bi[\/latex] is the imaginary part. For example, [latex]5+2i[\/latex] is a complex number. So, too, is [latex]3+4i\\sqrt{3}[\/latex].<span id=\"fs-id1165137832295\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137892327\">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. You can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You will see more of that later.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr>\n<th>Complex Number<\/th>\n<th>Real Part<\/th>\n<th>Imaginary Part<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]3+7i[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]7i[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]18\u201332i[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<td>[latex]\u221232i[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex]<\/td>\n<td>[latex]-\\frac{3}{5}[\/latex]<\/td>\n<td>[latex]i\\sqrt{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{\\sqrt{2}}{2}-\\frac{1}{2}i[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{1}{2}i[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In a number with a radical as part of <i>b<\/i>, such as [latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex]\u00a0above, the imaginary <i>[latex]i[\/latex]<\/i> should be written in front of the radical. Though writing this number as [latex]-\\frac{3}{5}+\\sqrt{2}i[\/latex] is technically correct, it makes it much more difficult to tell whether <i>[latex]i[\/latex]<\/i> is inside or outside of the radical. Putting it before the radical, as in [latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex], clears up any confusion. Look at these last two examples.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr>\n<th>Number<\/th>\n<th>Complex Form:<br \/>\n[latex]a+bi[\/latex]<\/th>\n<th>Real Part<\/th>\n<th>Imaginary Part<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]17[\/latex]<\/td>\n<td>[latex]17+0i[\/latex]<\/td>\n<td>[latex]17[\/latex]<\/td>\n<td>[latex]0i[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22123i[\/latex]<\/td>\n<td>[latex]0\u20133i[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22123i[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>By making [latex]b=0[\/latex], any real number can be expressed as a complex number. The real number [latex]<i>a[\/latex]<\/i>\u00a0is written as [latex]a+0i[\/latex] in complex form. Similarly, any imaginary number can be expressed as a complex number. By making [latex]a=0[\/latex], any imaginary number [latex]bi[\/latex] can be written as [latex]0+bi[\/latex] in complex form.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]83.6[\/latex] as a complex number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q704457\">Show Solution<\/span><\/p>\n<div id=\"q704457\" class=\"hidden-answer\" style=\"display: none\">\n<p>Remember that a complex number has the form [latex]a+bi[\/latex]. You need to figure out what <i>a<\/i> and <i>b<\/i> need to be.<\/p>\n<p>[latex]a+bi[\/latex]<\/p>\n<p>Since [latex]83.6[\/latex] is a real number, it is the real part (a) of the complex number [latex]a+bi[\/latex].\u00a0<i><br \/>\n<\/i><\/p>\n<p>[latex]83.6+bi[\/latex]<\/p>\n<p>A real number does not contain any imaginary parts, so the value of [latex]b[\/latex] is\u00a0[latex]0[\/latex].<\/p>\n<p>The answer is [latex]83.6+0i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm197339\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=197339&theme=oea&iframe_resize_id=ohm197339&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]\u22123i[\/latex] as a complex number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q451549\">Show Solution<\/span><\/p>\n<div id=\"q451549\" class=\"hidden-answer\" style=\"display: none\">\n<p>Remember that a complex number has the form [latex]a+bi[\/latex].\u00a0You need to figure out what [latex]a[\/latex] and <i>[latex]b[\/latex]<\/i> need to be.<\/p>\n<p>[latex]a+bi[\/latex]<\/p>\n<p>Since [latex]\u22123i[\/latex] is an imaginary number, it is the imaginary part [latex]bi[\/latex] of the complex number [latex]a+bi[\/latex].<\/p>\n<p>[latex]a\u20133i[\/latex]<\/p>\n<p>This imaginary number has no real parts, so the value of [latex]a[\/latex] is [latex]0[\/latex].<\/p>\n<p>The answer is\u00a0[latex]0\u20133i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video, we show more examples of how to write numbers as complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Write Number in the Form of Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/mfoOYdDkuyY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Square roots of negative numbers can be simplified using [latex]\\sqrt{-1}=i[\/latex] and [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex].\u00a0 Complex numbers have the form [latex]a+bi[\/latex], where [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0are real numbers and [latex]i[\/latex]\u00a0is the square root of [latex]\u22121[\/latex]. All real numbers can be written as complex numbers by setting [latex]b=0[\/latex]. Imaginary numbers have the form [latex]bi[\/latex] and can also be written as complex numbers by setting [latex]a=0[\/latex].<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1GcR4XrneYIZ5GbQcxKA42J9SU44nj9Dzst-r0MoZO0Y\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/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-15385\">\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>Write Number in the Form of Complex Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mfoOYdDkuyY\">https:\/\/youtu.be\/mfoOYdDkuyY<\/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>Simplify Square Roots to Imaginary Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LSp7yNP6Xxc\">https:\/\/youtu.be\/LSp7yNP6Xxc<\/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":167848,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Write Number in the Form of Complex Numbers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/mfoOYdDkuyY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Square Roots to Imaginary Numbers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LSp7yNP6Xxc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"485490bd5efe4c1a964f6209ab347300, fea1cb81aaf44427b94614774d30e81d, b178af37b58c45fc9d28e6f5683a982e","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15385","chapter","type-chapter","status-publish","hentry"],"part":10990,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15385\/revisions"}],"predecessor-version":[{"id":15698,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15385\/revisions\/15698"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/10990"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15385\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15385"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15385"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15385"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}