{"id":624,"date":"2021-08-30T20:40:20","date_gmt":"2021-08-30T20:40:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=624"},"modified":"2023-02-22T14:40:01","modified_gmt":"2023-02-22T14:40:01","slug":"2-3-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/2-3-complex-numbers\/","title":{"raw":"2.3: Complex Numbers","rendered":"2.3: Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define [latex]i[\/latex] as the square root of -1<\/li>\r\n \t<li>Take the square root of a negative number<\/li>\r\n \t<li>Express imaginary numbers as [latex]bi[\/latex] and complex numbers as [latex]a+bi[\/latex]<\/li>\r\n \t<li>Simplify complex numbers to standard form<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key WORDS<\/h3>\r\n<ul>\r\n \t<li><strong>Rational numbers<\/strong>: the set of numbers that can be written as the ratio of two integers<\/li>\r\n \t<li><strong>Irrational numbers<\/strong>:\u00a0the set of numbers that can not be written as the ratio of two integers<\/li>\r\n \t<li><strong>Real numbers<\/strong>: the set of numbers formed by the union of the set of rational numbers with the set of irrational numbers<\/li>\r\n \t<li><strong>Imaginary numbers<\/strong>: the set of numbers formed from the square root of a negative real number<\/li>\r\n \t<li><strong>Complex numbers<\/strong>: the set of numbers formed from a union of the set of real numbers and the set of imaginary numbers<\/li>\r\n \t<li><strong>Standard form of a complex number<\/strong>: [latex]a+bi[\/latex] where [latex]a[\/latex] and\u00a0[latex]b[\/latex] are realnumbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Square Root of a Negative Number<\/h2>\r\nWe have seen that taking the square root of a negative number is not possible in the Real number system.\u00a0<span style=\"font-size: 1em;\">The set of\u00a0<\/span><em><strong>real numbers<\/strong><\/em><span style=\"font-size: 1em;\">\u00a0is the <em><strong>union<\/strong><\/em> of the set of <em><strong>rational numbers<\/strong><\/em> and the set of <em><strong>irrational numbers<\/strong><\/em> and can be shown on a real number line.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">In order to take the square root of a negative number, we have to consider a new number system. But, we really need only one new number to start working with the square roots of negative numbers. That number is the square root of negative one, [latex]\\sqrt{-1}[\/latex] which we define to be the <em><strong>imaginary<\/strong><\/em><\/span><span style=\"font-size: 1rem; text-align: initial;\"><em><strong>\u00a0number<\/strong><\/em> [latex]i[\/latex].<\/span>\r\n\r\nAnother way of saying this is that [latex] {{i}^{2}}=-1[\/latex]. This is because [latex]\\sqrt{i^2}=i[\/latex] and [latex]i=\\sqrt{-1}[\/latex] imply that [latex]\\sqrt{i^2}=\\sqrt{-1}[\/latex] and, consequently, [latex]i^2=-1[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>definition<\/h3>\r\n<p style=\"text-align: center;\">[latex]{i}=\\sqrt{-1}[\/latex]\r\n[latex] {{i}^{2}}=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe number [latex]\u22121[\/latex]<i>\u00a0<\/i>allows us to work with roots of negative Real numbers, not just [latex] \\sqrt{-1}[\/latex]. There are two important square root rules to remember: [latex] \\sqrt{-1}=i[\/latex], and [latex] \\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] for. We will use the latter rule to rewrite the square root of a negative number as the square root of a positive number times [latex] \\sqrt{-1}[\/latex], then rewrite[latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{-4}[\/latex]\r\n<h4>Solution<\/h4>\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<p style=\"text-align: center;\">[latex] \\sqrt{-4}=\\sqrt{4\\cdot -1}=\\sqrt{4}\\sqrt{-1}[\/latex]<\/p>\r\nSince\u00a0[latex]4[\/latex] is a perfect square\u00a0[latex](4=2^{2})[\/latex], we can simplify the square root of\u00a0[latex]4[\/latex]:\r\n<p style=\"text-align: center;\">[latex] \\sqrt{4}\\sqrt{-1}=2\\sqrt{-1}[\/latex]<\/p>\r\nUse the definition of [latex]i[\/latex]\u00a0to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]:\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt{-1}=2i[\/latex]<\/p>\r\nThe answer is:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\sqrt{-4}=2i[\/latex]\r\n\r\n<\/div>\r\nNotice that [latex]\\sqrt{-4}=2i[\/latex] and not [latex]\\pm2i[\/latex]. This is because the radical sign [latex]\\sqrt{}[\/latex] represents the principal square root, which is always \u2265 0.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{-9}[\/latex]\r\n\r\n&nbsp;\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<p style=\"text-align: center;\">[latex] \\sqrt{-9}=\\sqrt{9\\cdot -1}=\\sqrt{9}\\cdot\\sqrt{-1}[\/latex]<\/p>\r\nSince\u00a0[latex]9[\/latex] is a perfect square\u00a0[latex](9=3^{2})[\/latex], we can simplify the square root of\u00a0[latex]9[\/latex]:\r\n<p style=\"text-align: center;\">[latex] \\sqrt{9}\\cdot\\sqrt{-1}=3\\cdot\\sqrt{-1}[\/latex]<\/p>\r\nUse the definition of [latex]i[\/latex] to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]:\r\n<p style=\"text-align: center;\">[latex] 3\\cdot\\sqrt{-1}=3i[\/latex]<\/p>\r\nThe answer is:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\sqrt{-9}=3i[\/latex].\r\n\r\n<\/div>\r\nThe only way to end up with a negative answer is if there is a negative sign in front of the radical.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] -\\sqrt{-64}[\/latex]\r\n\r\n&nbsp;\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<p style=\"text-align: center;\">[latex] -\\sqrt{-64}=-\\sqrt{64\\cdot -1}=-\\sqrt{64}\\sqrt{-1}[\/latex]<\/p>\r\nSince\u00a0[latex]64[\/latex] is a perfect square\u00a0[latex](64=8^{2})[\/latex], we can simplify the square root of\u00a0[latex]64[\/latex]:\r\n<p style=\"text-align: center;\">[latex] -\\sqrt{64}\\cdot\\sqrt{-1}=-8\\cdot\\sqrt{-1}[\/latex]<\/p>\r\nUse the definition of [latex]i[\/latex] to rewrite [latex] \\sqrt{-1}[\/latex] as [latex]i[\/latex]:\r\n<p style=\"text-align: center;\">[latex] -8\\cdot\\sqrt{-1}==-8i[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe answer is\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]-\\sqrt{-64}=-8i[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Rewriting the Square Root of a Negative Number as an imaginary 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\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nRewrite the square roots as imaginary numbers:\r\n<ol>\r\n \t<li>[latex]\\sqrt{-49}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-81}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-100}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm638\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm638\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{-49}=7i[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-81}=9i[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-100}=-10i[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<h2>Imaginary Numbers<\/h2>\r\nThe set of imaginary numbers is defined as the set of all numbers of the form [latex]ai[\/latex], where [latex]a[\/latex] is a real number.\u00a0 [latex]2i,\\; -7i,\\; 9658i,\\; \\frac{3}{4}i,\\; -\\frac{9}{5}i,\\; \\sqrt{3}\\,i[\/latex] are all imaginary numbers. When there is a square root in front of the [latex]i[\/latex], it is best to write the [latex]i[\/latex] in front of the radical: [latex]i\\sqrt{3}[\/latex] so that it is obvious that the [latex]i[\/latex] is not under the radical. In set builder notation, the set of imaginary numbers is defined as [latex]\\{ai\\; \\large | \\normalsize \\;\\;a\\;\\in\\mathbb{R}\\}[\/latex]. The set of imaginary numbers is NOT a subset of the set of real numbers.\r\n<div class=\"textbox shaded\">\r\n<h3>imaginary numbers<\/h3>\r\nThe set of imaginary numbers is defined as the set of all numbers of the form [latex]ai[\/latex], where [latex]a[\/latex] is a real number.\r\n<p style=\"text-align: center;\">Imaginary numbers = [latex]\\mathbb{I}=\\{ai\\; \\large | \\normalsize \\;\\;a\\;\\in\\mathbb{R}\\}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>When we rewrite the square root of a negative number as an imaginary number, we usually have to simplify the radical by looking for perfect square factors under the radical.<\/div>\r\n<div>\r\n\r\nThe following video, shows more examples of how to use the imaginary number [latex]i[\/latex] to simplify a square root with a negative radicand.\r\n\r\nhttps:\/\/youtu.be\/LSp7yNP6Xxc\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nWrite the square roots as imaginary numbers.\r\n<ol>\r\n \t<li>[latex]\\sqrt{-36}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-8}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-16}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-45}[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<ol>\r\n \t<li>[latex]\\sqrt{-36}=\\sqrt{36\\cdot (-1)}=\\sqrt{36}\\cdot\\sqrt{-1}=6i=[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-8}=\\sqrt{-1}\\cdot\\sqrt{8}=i\\cdot\\sqrt{4\\cdot 2}=i\\cdot\\sqrt{4}\\cdot\\sqrt{2}=2i\\sqrt{2}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-16}=-\\sqrt{16}\\cdot\\sqrt{-1}=-4i[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-45}=-\\sqrt{-1}\\cdot\\sqrt{9}\\cdot\\sqrt{5}=-3i\\sqrt{5}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nWrite the square roots as imaginary numbers.\r\n<ol>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-20}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-100}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-63}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"HJM785\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"HJM785\"]\r\n<ol>\r\n \t<li>[latex]3i[\/latex]<\/li>\r\n \t<li>[latex]2i\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]-10i[\/latex]<\/li>\r\n \t<li>[latex]-3i\\sqrt{7}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\"><img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>To rewrite the radicals as imaginary numbers, we used the product rule for square roots:\u00a0[latex] \\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]. This rule works for all real numbers [latex]a[\/latex] and [latex]b[\/latex]. However, be aware that this rule only works backwards\u00a0[latex] \\left (\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}\\right )[\/latex] when [latex]a[\/latex] and [latex]b[\/latex] are greater than or equal to zero. In other words,\u00a0[latex] \\sqrt{a}\\cdot \\sqrt{b}\\neq\\sqrt{ab}[\/latex] if [latex]a[\/latex] or [latex]b[\/latex] are negative.<\/div>\r\nFor example, [latex]\\sqrt{-4}\\cdot\\sqrt{-9}\\neq\\sqrt{-4\\cdot (-9)}[\/latex], because [latex]\\sqrt{-4}\\cdot\\sqrt{-9}=2i\\cdot 3i=6i^2=6\\cdot (-1)=-6[\/latex] and [latex]\\sqrt{-4\\cdot (-9)}=\\sqrt{36}=6[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\sqrt{-4}\\cdot\\sqrt{-25}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-4\\cdot-25}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\sqrt{-4}\\cdot\\sqrt{-25}=2i\\cdot 5i=10i^2=10\\cdot(-1)=-10[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-4\\cdot-25}=\\sqrt{100}=10[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\sqrt{-9}\\cdot\\sqrt{-36}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9\\cdot-36}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{9\\cdot-36}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-9\\cdot 36}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"HJM999\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"HJM999\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{-9}\\cdot\\sqrt{-36}=-18[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9\\cdot-36}=\\sqrt{324}=18[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{9\\cdot-36}=\\sqrt{-324}=18i[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{-9\\cdot 36}=-18i[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Complex Numbers<\/span>\r\n\r\nThe set of real numbers and the set of imaginary numbers have only one number in common: zero. Zero can be written as [latex]0[\/latex] in the set of real numbers and [latex]0i[\/latex] in the set of imaginary numbers. [latex]0i=0[\/latex]. When we union the set of real numbers with the set of imaginary numbers, we get a\u00a0new set of numbers called the\u00a0<em><strong>complex numbers<\/strong><\/em>.\r\n\r\nA complex number is expressed in <em><strong>standard form<\/strong><\/em> 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].\r\n\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\"><span id=\"fs-id1165137832295\">\u00a0<\/span><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>complex numbers<\/h3>\r\nThe set of complex numbers, [latex]\\mathbb{C}[\/latex], is the union of the set of real numbers and the set of imaginary numbers.\r\n\r\nA complex number is in standard form when it is written in the form [latex]a+bi[\/latex] where\u00a0[latex]a[\/latex] and [latex]b[\/latex] are real numbers.\r\n<p style=\"text-align: center;\">[latex]\\mathbb{C}=\\{a+bi\\; \\large | \\normalsize \\;\\;a,b\\;\\in\\mathbb{R}\\}[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165137892327\">Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative\u00a0real number, while\u00a0the square of a real number (either positive or negative) is a positive real number. Complex numbers are a combination of real and imaginary numbers.<\/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\n&nbsp;\r\n\r\nIn a number with a radical as part of <i>b<\/i>, such as [latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex], <i>[latex]i[\/latex]<\/i> should be written in front of the radical. Although 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.\r\n\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<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\nThese examples show that the set of real numbers is a subset of the complex numbers and the set of imaginary numbers is a subset of the complex numbers.\r\n<p style=\"text-align: center;\">[latex]\\mathbb{R}\\subset\\mathbb{C}\\text{ and }\\mathbb{I}\\subset\\mathbb{C}[\/latex]<\/p>\r\n\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&nbsp;\r\n\r\nA complex number has the form [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><\/i>\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<\/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&nbsp;\r\n\r\nA complex number has the form [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\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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the numbers as complex numbers in standard form:\r\n<ol>\r\n \t<li>[latex]17[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{7}{9}[\/latex]<\/li>\r\n \t<li>[latex]-3i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{2}}{5}i[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"647734\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"647734\"]\r\n<ol>\r\n \t<li>[latex]17=17+0i[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{7}{9}=-\\frac{7}{9}+0i[\/latex]<\/li>\r\n \t<li>[latex]-3i=0-3i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{2}}{5}i=0+\\frac{\\sqrt{2}}{5}i[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe video, shows more examples of how to write complex numbers.\r\n\r\nhttps:\/\/youtu.be\/mfoOYdDkuyY\r\n<h2>Standard Form<\/h2>\r\nThe standard form of a complex number is [latex]a+bi[\/latex]. So, [latex]\\frac{5}{2}+\\frac{3}{2}i[\/latex] is a complex number written in standard form. However, if we used the common denominator to add the fractions, [latex]\\frac{5+3i}{2}[\/latex] is NOT in standard form. If a complex number is the numerator of a fraction, we can simplify the fraction to write the answer in standard form by splitting up the fractions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\u00a0[latex]\\frac{5+3i}{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSeparate the fractions into a real part and an imaginary part:\r\n<p style=\"text-align: center;\">[latex]\\frac{5+3i}{2}=\\frac{5}{2}+\\frac{3i}{2}[\/latex]<i>\u00a0<\/i><\/p>\r\nSimplify each fraction, if possible:\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{2}+\\frac{3i}{2}[\/latex]<\/p>\r\nThe answer is [latex]\\frac{5+3i}{2}=\\frac{5}{2}+\\frac{3i}{2}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\u00a0[latex]\\frac{2+3i}{4}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSeparate the fractions into a real part and an imaginary part:\r\n<p style=\"text-align: center;\">[latex]\\frac{2+3i}{4}=\\frac{2}{4}+\\frac{3i}{4}[\/latex]<i>\u00a0<\/i><\/p>\r\nSimplify each fraction:\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{2}+\\frac{3i}{4}[\/latex]<\/p>\r\nThe answer is [latex]\\frac{2+3i}{4}=\\frac{1}{2}+\\frac{3i}{4}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nWrite in standard form:\r\n<ol>\r\n \t<li>[latex]\\frac{4+9i}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{-8+4i}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{7+3i}{9}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"59406\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"59406\"]\r\n<ol>\r\n \t<li>[latex]\\frac{4+9i}{3}=\\frac{4}{3}+3i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{-8+4i}{4}=-2+i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{7+3i}{9}=\\frac{7}{9}+\\frac{1}{3}i[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe same technique can be used when there are radicals involved.\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSimplify to standard form.\r\n<ol>\r\n \t<li>[latex]\\frac{6+i\\sqrt{3}}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{12-4i\\sqrt{5}}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{-24+\\sqrt{-16}}{-8}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{81}-\\sqrt{-27}}{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm265\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm265\"]\r\n<ol>\r\n \t<li>[latex]\\frac{6+i\\sqrt{3}}{4}=\\frac{6}{4}+\\frac{\\sqrt{3}}{4}i=\\frac{3}{2}+\\frac{\\sqrt{3}}{4}i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{12-4i\\sqrt{5}}{2}=\\frac{12}{2}-\\frac{4\\sqrt{5}}{2}i=6-2i\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{-24+\\sqrt{-16}}{-8}=\\frac{-24}{-8}+\\frac{4i}{-8}=3-\\frac{1}{2}i[\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{81}-\\sqrt{-27}}{3}=\\frac{9-3i\\sqrt{3}}{3}=\\frac{9}{3}-\\frac{3i\\sqrt{3}}{3}=3-i\\sqrt{3}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define [latex]i[\/latex] as the square root of -1<\/li>\n<li>Take the square root of a negative number<\/li>\n<li>Express imaginary numbers as [latex]bi[\/latex] and complex numbers as [latex]a+bi[\/latex]<\/li>\n<li>Simplify complex numbers to standard form<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key WORDS<\/h3>\n<ul>\n<li><strong>Rational numbers<\/strong>: the set of numbers that can be written as the ratio of two integers<\/li>\n<li><strong>Irrational numbers<\/strong>:\u00a0the set of numbers that can not be written as the ratio of two integers<\/li>\n<li><strong>Real numbers<\/strong>: the set of numbers formed by the union of the set of rational numbers with the set of irrational numbers<\/li>\n<li><strong>Imaginary numbers<\/strong>: the set of numbers formed from the square root of a negative real number<\/li>\n<li><strong>Complex numbers<\/strong>: the set of numbers formed from a union of the set of real numbers and the set of imaginary numbers<\/li>\n<li><strong>Standard form of a complex number<\/strong>: [latex]a+bi[\/latex] where [latex]a[\/latex] and\u00a0[latex]b[\/latex] are realnumbers<\/li>\n<\/ul>\n<\/div>\n<h2>Square Root of a Negative Number<\/h2>\n<p>We have seen that taking the square root of a negative number is not possible in the Real number system.\u00a0<span style=\"font-size: 1em;\">The set of\u00a0<\/span><em><strong>real numbers<\/strong><\/em><span style=\"font-size: 1em;\">\u00a0is the <em><strong>union<\/strong><\/em> of the set of <em><strong>rational numbers<\/strong><\/em> and the set of <em><strong>irrational numbers<\/strong><\/em> and can be shown on a real number line.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">In order to take the square root of a negative number, we have to consider a new number system. But, we really need only one new number to start working with the square roots of negative numbers. That number is the square root of negative one, [latex]\\sqrt{-1}[\/latex] which we define to be the <em><strong>imaginary<\/strong><\/em><\/span><span style=\"font-size: 1rem; text-align: initial;\"><em><strong>\u00a0number<\/strong><\/em> [latex]i[\/latex].<\/span><\/p>\n<p>Another way of saying this is that [latex]{{i}^{2}}=-1[\/latex]. This is because [latex]\\sqrt{i^2}=i[\/latex] and [latex]i=\\sqrt{-1}[\/latex] imply that [latex]\\sqrt{i^2}=\\sqrt{-1}[\/latex] and, consequently, [latex]i^2=-1[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>definition<\/h3>\n<p style=\"text-align: center;\">[latex]{i}=\\sqrt{-1}[\/latex]<br \/>\n[latex]{{i}^{2}}=-1[\/latex]<\/p>\n<\/div>\n<p>The number [latex]\u22121[\/latex]<i>\u00a0<\/i>allows us to work with roots of negative Real numbers, not just [latex]\\sqrt{-1}[\/latex]. There are two important square root rules to remember: [latex]\\sqrt{-1}=i[\/latex], and [latex]\\sqrt{ab}=\\sqrt{a}\\sqrt{b}[\/latex] for. We will use the latter rule to rewrite the square root of a negative number as the square root of a positive number times [latex]\\sqrt{-1}[\/latex], then rewrite[latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{-4}[\/latex]<\/p>\n<h4>Solution<\/h4>\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 style=\"text-align: center;\">[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], we can simplify the square root of\u00a0[latex]4[\/latex]:<\/p>\n<p style=\"text-align: center;\">[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]:<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt{-1}=2i[\/latex]<\/p>\n<p>The answer is:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\sqrt{-4}=2i[\/latex]<\/p>\n<\/div>\n<p>Notice that [latex]\\sqrt{-4}=2i[\/latex] and not [latex]\\pm2i[\/latex]. This is because the radical sign [latex]\\sqrt{}[\/latex] represents the principal square root, which is always \u2265 0.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{-9}[\/latex]<\/p>\n<p>&nbsp;<\/p>\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 style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9\\cdot -1}=\\sqrt{9}\\cdot\\sqrt{-1}[\/latex]<\/p>\n<p>Since\u00a0[latex]9[\/latex] is a perfect square\u00a0[latex](9=3^{2})[\/latex], we can simplify the square root of\u00a0[latex]9[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{9}\\cdot\\sqrt{-1}=3\\cdot\\sqrt{-1}[\/latex]<\/p>\n<p>Use the definition of [latex]i[\/latex] to rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot\\sqrt{-1}=3i[\/latex]<\/p>\n<p>The answer is:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\sqrt{-9}=3i[\/latex].<\/p>\n<\/div>\n<p>The only way to end up with a negative answer is if there is a negative sign in front of the radical.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]-\\sqrt{-64}[\/latex]<\/p>\n<p>&nbsp;<\/p>\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 style=\"text-align: center;\">[latex]-\\sqrt{-64}=-\\sqrt{64\\cdot -1}=-\\sqrt{64}\\sqrt{-1}[\/latex]<\/p>\n<p>Since\u00a0[latex]64[\/latex] is a perfect square\u00a0[latex](64=8^{2})[\/latex], we can simplify the square root of\u00a0[latex]64[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]-\\sqrt{64}\\cdot\\sqrt{-1}=-8\\cdot\\sqrt{-1}[\/latex]<\/p>\n<p>Use the definition of [latex]i[\/latex] to rewrite [latex]\\sqrt{-1}[\/latex] as [latex]i[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]-8\\cdot\\sqrt{-1}==-8i[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The answer is\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]-\\sqrt{-64}=-8i[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Rewriting the Square Root of a Negative Number as an imaginary 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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Rewrite the square roots as imaginary numbers:<\/p>\n<ol>\n<li>[latex]\\sqrt{-49}[\/latex]<\/li>\n<li>[latex]\\sqrt{-81}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-100}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm638\">Show Answer<\/span><\/p>\n<div id=\"qhjm638\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{-49}=7i[\/latex]<\/li>\n<li>[latex]\\sqrt{-81}=9i[\/latex]<\/li>\n<li>[latex]-\\sqrt{-100}=-10i[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<h2>Imaginary Numbers<\/h2>\n<p>The set of imaginary numbers is defined as the set of all numbers of the form [latex]ai[\/latex], where [latex]a[\/latex] is a real number.\u00a0 [latex]2i,\\; -7i,\\; 9658i,\\; \\frac{3}{4}i,\\; -\\frac{9}{5}i,\\; \\sqrt{3}\\,i[\/latex] are all imaginary numbers. When there is a square root in front of the [latex]i[\/latex], it is best to write the [latex]i[\/latex] in front of the radical: [latex]i\\sqrt{3}[\/latex] so that it is obvious that the [latex]i[\/latex] is not under the radical. In set builder notation, the set of imaginary numbers is defined as [latex]\\{ai\\; \\large | \\normalsize \\;\\;a\\;\\in\\mathbb{R}\\}[\/latex]. The set of imaginary numbers is NOT a subset of the set of real numbers.<\/p>\n<div class=\"textbox shaded\">\n<h3>imaginary numbers<\/h3>\n<p>The set of imaginary numbers is defined as the set of all numbers of the form [latex]ai[\/latex], where [latex]a[\/latex] is a real number.<\/p>\n<p style=\"text-align: center;\">Imaginary numbers = [latex]\\mathbb{I}=\\{ai\\; \\large | \\normalsize \\;\\;a\\;\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>When we rewrite the square root of a negative number as an imaginary number, we usually have to simplify the radical by looking for perfect square factors under the radical.<\/div>\n<div>\n<p>The following video, shows more examples of how to use the imaginary number [latex]i[\/latex] 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>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Write the square roots as imaginary numbers.<\/p>\n<ol>\n<li>[latex]\\sqrt{-36}[\/latex]<\/li>\n<li>[latex]\\sqrt{-8}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-16}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-45}[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<ol>\n<li>[latex]\\sqrt{-36}=\\sqrt{36\\cdot (-1)}=\\sqrt{36}\\cdot\\sqrt{-1}=6i=[\/latex]<\/li>\n<li>[latex]\\sqrt{-8}=\\sqrt{-1}\\cdot\\sqrt{8}=i\\cdot\\sqrt{4\\cdot 2}=i\\cdot\\sqrt{4}\\cdot\\sqrt{2}=2i\\sqrt{2}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-16}=-\\sqrt{16}\\cdot\\sqrt{-1}=-4i[\/latex]<\/li>\n<li>[latex]-\\sqrt{-45}=-\\sqrt{-1}\\cdot\\sqrt{9}\\cdot\\sqrt{5}=-3i\\sqrt{5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Write the square roots as imaginary numbers.<\/p>\n<ol>\n<li>[latex]\\sqrt{-9}[\/latex]<\/li>\n<li>[latex]\\sqrt{-20}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-100}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-63}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qHJM785\">Show Answer<\/span><\/p>\n<div id=\"qHJM785\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]3i[\/latex]<\/li>\n<li>[latex]2i\\sqrt{5}[\/latex]<\/li>\n<li>[latex]-10i[\/latex]<\/li>\n<li>[latex]-3i\\sqrt{7}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>To rewrite the radicals as imaginary numbers, we used the product rule for square roots:\u00a0[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]. This rule works for all real numbers [latex]a[\/latex] and [latex]b[\/latex]. However, be aware that this rule only works backwards\u00a0[latex]\\left (\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}\\right )[\/latex] when [latex]a[\/latex] and [latex]b[\/latex] are greater than or equal to zero. In other words,\u00a0[latex]\\sqrt{a}\\cdot \\sqrt{b}\\neq\\sqrt{ab}[\/latex] if [latex]a[\/latex] or [latex]b[\/latex] are negative.<\/div>\n<p>For example, [latex]\\sqrt{-4}\\cdot\\sqrt{-9}\\neq\\sqrt{-4\\cdot (-9)}[\/latex], because [latex]\\sqrt{-4}\\cdot\\sqrt{-9}=2i\\cdot 3i=6i^2=6\\cdot (-1)=-6[\/latex] and [latex]\\sqrt{-4\\cdot (-9)}=\\sqrt{36}=6[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\sqrt{-4}\\cdot\\sqrt{-25}[\/latex]<\/li>\n<li>[latex]\\sqrt{-4\\cdot-25}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\sqrt{-4}\\cdot\\sqrt{-25}=2i\\cdot 5i=10i^2=10\\cdot(-1)=-10[\/latex]<\/li>\n<li>[latex]\\sqrt{-4\\cdot-25}=\\sqrt{100}=10[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\sqrt{-9}\\cdot\\sqrt{-36}[\/latex]<\/li>\n<li>[latex]\\sqrt{-9\\cdot-36}[\/latex]<\/li>\n<li>[latex]\\sqrt{9\\cdot-36}[\/latex]<\/li>\n<li>[latex]-\\sqrt{-9\\cdot 36}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qHJM999\">Show Answer<\/span><\/p>\n<div id=\"qHJM999\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{-9}\\cdot\\sqrt{-36}=-18[\/latex]<\/li>\n<li>[latex]\\sqrt{-9\\cdot-36}=\\sqrt{324}=18[\/latex]<\/li>\n<li>[latex]\\sqrt{9\\cdot-36}=\\sqrt{-324}=18i[\/latex]<\/li>\n<li>[latex]-\\sqrt{-9\\cdot 36}=-18i[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Complex Numbers<\/span><\/p>\n<p>The set of real numbers and the set of imaginary numbers have only one number in common: zero. Zero can be written as [latex]0[\/latex] in the set of real numbers and [latex]0i[\/latex] in the set of imaginary numbers. [latex]0i=0[\/latex]. When we union the set of real numbers with the set of imaginary numbers, we get a\u00a0new set of numbers called the\u00a0<em><strong>complex numbers<\/strong><\/em>.<\/p>\n<p>A complex number is expressed in <em><strong>standard form<\/strong><\/em> 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].<\/p>\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\"><span id=\"fs-id1165137832295\">\u00a0<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>complex numbers<\/h3>\n<p>The set of complex numbers, [latex]\\mathbb{C}[\/latex], is the union of the set of real numbers and the set of imaginary numbers.<\/p>\n<p>A complex number is in standard form when it is written in the form [latex]a+bi[\/latex] where\u00a0[latex]a[\/latex] and [latex]b[\/latex] are real numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathbb{C}=\\{a+bi\\; \\large | \\normalsize \\;\\;a,b\\;\\in\\mathbb{R}\\}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137892327\">Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative\u00a0real number, while\u00a0the square of a real number (either positive or negative) is a positive real number. Complex numbers are a combination of real and imaginary numbers.<\/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>&nbsp;<\/p>\n<p>In a number with a radical as part of <i>b<\/i>, such as [latex]-\\frac{3}{5}+i\\sqrt{2}[\/latex], <i>[latex]i[\/latex]<\/i> should be written in front of the radical. Although 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.<\/p>\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<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>These examples show that the set of real numbers is a subset of the complex numbers and the set of imaginary numbers is a subset of the complex numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathbb{R}\\subset\\mathbb{C}\\text{ and }\\mathbb{I}\\subset\\mathbb{C}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]83.6[\/latex] as a complex number.<\/p>\n<p>&nbsp;<\/p>\n<p>A complex number has the form [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><\/i><\/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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]\u22123i[\/latex] as a complex number.<\/p>\n<p>&nbsp;<\/p>\n<p>A complex number has the form [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>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the numbers as complex numbers in standard form:<\/p>\n<ol>\n<li>[latex]17[\/latex]<\/li>\n<li>[latex]-\\frac{7}{9}[\/latex]<\/li>\n<li>[latex]-3i[\/latex]<\/li>\n<li>[latex]\\frac{\\sqrt{2}}{5}i[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647734\">Show Answer<\/span><\/p>\n<div id=\"q647734\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]17=17+0i[\/latex]<\/li>\n<li>[latex]-\\frac{7}{9}=-\\frac{7}{9}+0i[\/latex]<\/li>\n<li>[latex]-3i=0-3i[\/latex]<\/li>\n<li>[latex]\\frac{\\sqrt{2}}{5}i=0+\\frac{\\sqrt{2}}{5}i[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The video, shows more examples of how to write 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>Standard Form<\/h2>\n<p>The standard form of a complex number is [latex]a+bi[\/latex]. So, [latex]\\frac{5}{2}+\\frac{3}{2}i[\/latex] is a complex number written in standard form. However, if we used the common denominator to add the fractions, [latex]\\frac{5+3i}{2}[\/latex] is NOT in standard form. If a complex number is the numerator of a fraction, we can simplify the fraction to write the answer in standard form by splitting up the fractions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.\u00a0[latex]\\frac{5+3i}{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Separate the fractions into a real part and an imaginary part:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5+3i}{2}=\\frac{5}{2}+\\frac{3i}{2}[\/latex]<i>\u00a0<\/i><\/p>\n<p>Simplify each fraction, if possible:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{2}+\\frac{3i}{2}[\/latex]<\/p>\n<p>The answer is [latex]\\frac{5+3i}{2}=\\frac{5}{2}+\\frac{3i}{2}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.\u00a0[latex]\\frac{2+3i}{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Separate the fractions into a real part and an imaginary part:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2+3i}{4}=\\frac{2}{4}+\\frac{3i}{4}[\/latex]<i>\u00a0<\/i><\/p>\n<p>Simplify each fraction:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{2}+\\frac{3i}{4}[\/latex]<\/p>\n<p>The answer is [latex]\\frac{2+3i}{4}=\\frac{1}{2}+\\frac{3i}{4}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Write in standard form:<\/p>\n<ol>\n<li>[latex]\\frac{4+9i}{3}[\/latex]<\/li>\n<li>[latex]\\frac{-8+4i}{4}[\/latex]<\/li>\n<li>[latex]\\frac{7+3i}{9}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q59406\">Show Answer<\/span><\/p>\n<div id=\"q59406\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\frac{4+9i}{3}=\\frac{4}{3}+3i[\/latex]<\/li>\n<li>[latex]\\frac{-8+4i}{4}=-2+i[\/latex]<\/li>\n<li>[latex]\\frac{7+3i}{9}=\\frac{7}{9}+\\frac{1}{3}i[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The same technique can be used when there are radicals involved.<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Simplify to standard form.<\/p>\n<ol>\n<li>[latex]\\frac{6+i\\sqrt{3}}{4}[\/latex]<\/li>\n<li>[latex]\\frac{12-4i\\sqrt{5}}{2}[\/latex]<\/li>\n<li>[latex]\\frac{-24+\\sqrt{-16}}{-8}[\/latex]<\/li>\n<li>[latex]\\frac{\\sqrt{81}-\\sqrt{-27}}{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm265\">Show Answer<\/span><\/p>\n<div id=\"qhjm265\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\frac{6+i\\sqrt{3}}{4}=\\frac{6}{4}+\\frac{\\sqrt{3}}{4}i=\\frac{3}{2}+\\frac{\\sqrt{3}}{4}i[\/latex]<\/li>\n<li>[latex]\\frac{12-4i\\sqrt{5}}{2}=\\frac{12}{2}-\\frac{4\\sqrt{5}}{2}i=6-2i\\sqrt{5}[\/latex]<\/li>\n<li>[latex]\\frac{-24+\\sqrt{-16}}{-8}=\\frac{-24}{-8}+\\frac{4i}{-8}=3-\\frac{1}{2}i[\/latex]<\/li>\n<li>[latex]\\frac{\\sqrt{81}-\\sqrt{-27}}{3}=\\frac{9-3i\\sqrt{3}}{3}=\\frac{9}{3}-\\frac{3i\\sqrt{3}}{3}=3-i\\sqrt{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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-624\">\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>Provided by<\/strong>: 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). <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><li>Write Number in the form of a complex number. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <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>2.3: Complex Numbers. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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":422605,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Write Number in the Form of Complex Numbers.\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning.\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/youtu.be\/mfoOYdDkuyY.\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Square Roots to Imaginary 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