{"id":1762,"date":"2023-10-12T00:32:06","date_gmt":"2023-10-12T00:32:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-complex-numbers\/"},"modified":"2025-10-24T15:48:34","modified_gmt":"2025-10-24T15:48:34","slug":"introduction-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-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 class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Express square roots of negative numbers as multiples of\u2009[latex]i[\/latex].<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Plot complex numbers on the complex plane.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Add and subtract complex numbers.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Multiply and divide complex numbers.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nThe study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. The set of real numbers has voids as well. For example, we still have no solution to equations such as\r\n<p style=\"text-align: center;\">[latex]{x}^{2}+4=0[\/latex]<\/p>\r\nOur best guesses might be +2 or \u20132. But if we test +2 in this equation, it does not work. If we test \u20132, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately there is another system of numbers that provides solutions to problems such as these. In this section we will explore this number system and how to work within it.\r\n<h2>Express and Plot Complex Numbers<\/h2>\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\r\n&nbsp;\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>\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 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<iframe id=\"mom200\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61706&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/NeTRNpBI17I\r\n<h2>Plot complex numbers on the complex plane<\/h2>\r\nWe cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the <strong>complex plane<\/strong>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs [latex](a, b)[\/latex], where [latex]a[\/latex]\u00a0represents the coordinate for the horizontal axis and [latex]b[\/latex]\u00a0represents the coordinate for the vertical axis.\r\n\r\n<img class=\"aligncenter wp-image-2530 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211959\/CNX_Precalc_Figure_03_01_0022.jpg\" alt=\"Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/>\r\n\r\nLet\u2019s consider the number [latex]-2+3i[\/latex]. The real part of the complex number is [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]-4i[\/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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Add, Subtract, and Multiply Complex Numbers<\/h2>\r\nJust as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Addition and Subtraction of Complex Numbers<\/h3>\r\nAdding complex numbers:\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\r\nSubtracting complex numbers:\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two complex numbers, find the sum or difference.<\/h3>\r\n<ol>\r\n \t<li>Identify the real and imaginary parts of each number.<\/li>\r\n \t<li>Add or subtract the real parts.<\/li>\r\n \t<li>Add or subtract the imaginary parts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding Complex Numbers<\/h3>\r\nAdd [latex]3 - 4i[\/latex] and [latex]2+5i[\/latex].\r\n\r\n[reveal-answer q=\"5937\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5937\"]\r\n\r\nWe add the real parts and add the imaginary parts.\r\n<p style=\"text-align: center;\">[latex]\\left(3 - 4i\\right)+\\left(2+5i\\right)=\\left(3+2\\right)+\\left(-4+5\\right)i=5+i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract [latex]2+5i[\/latex] from [latex]3 - 4i[\/latex].\r\n\r\n[reveal-answer q=\"732700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"732700\"]\r\n\r\n[latex]\\left(3 - 4i\\right)-\\left(2+5i\\right)=1 - 9i[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61710&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/SGhTjioGqqA\r\n<h2>Multiplying Complex Numbers<\/h2>\r\nMultiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.\r\n<h2>Multiplying a Complex Number by a Real Number<\/h2>\r\nLet\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}3(6+2i)&amp;=(3\\cdot6)+(3\\cdot2i)&amp;&amp;\\text{Distribute.}\\\\&amp;=18+6i&amp;&amp;\\text{Simplify.}\\end{align}[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given a complex number and a real number, multiply to find the product.<\/h3>\r\n<ol>\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying a Complex Number by a Real Number<\/h3>\r\nFind the product [latex]4\\left(2+5i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"928099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"928099\"]\r\n<p style=\"text-align: center;\">[latex]4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)=8+20i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the product [latex]-4\\left(2+6i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"568092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568092\"]\r\n\r\n[latex]-8 - 24i[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40462&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Multiplying Complex Numbers Together<\/h2>\r\nNow, let\u2019s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/p>\r\nBecause [latex]{i}^{2}=-1[\/latex], we have\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/p>\r\nTo simplify, we combine the real parts, and we combine the imaginary parts.\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given two complex numbers, multiply to find the product.<\/h3>\r\n<ol>\r\n \t<li>Use the distributive property or the FOIL method.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying a Complex Number by a Complex Number<\/h3>\r\nMultiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"388605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388605\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(4+3i\\right)\\left(2 - 5i\\right)&amp;=4\\cdot 2 + 4\\cdot \\left(-5i\\right)+3i\\cdot2+3i\\cdot \\left(-5i\\right)\\\\ &amp;=8-20i+6i-15i^2\\\\&amp;=8+15-20i+6i\\\\ &amp;=23 - 14i\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply [latex]\\left(3 - 4i\\right)\\left(2+3i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"576399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"576399\"]\r\n\r\n[latex]18+i[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3903&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/O9xQaIi0NX0\r\n<h2>Divide Complex Numbers<\/h2>\r\nDivision of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the <strong>complex conjugate<\/strong> of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex].\r\n\r\nNote that complex conjugates have a reciprocal relationship: The complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex], and the complex conjugate of [latex]a-bi[\/latex] is [latex]a+bi[\/latex]. Importantly, complex conjugate pairs have a special property. Their product is always real.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}(a+bi)(a-bi)&amp;=a^2-abi+abi-b^2i^2\\\\[2mm]&amp;=a^2-b^2(-1)\\\\[2mm]&amp;=a^2+b^2\\end{align}[\/latex]<\/p>\r\nSuppose we want to divide [latex]c+di[\/latex] by [latex]a+bi[\/latex], where neither [latex]a[\/latex] nor [latex]b[\/latex] equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{c+di}{a+bi}[\/latex] where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex].<\/p>\r\nMultiply the numerator and denominator by the complex conjugate of the denominator.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(c+di\\right)}{\\left(a+bi\\right)}\\cdot \\dfrac{\\left(a-bi\\right)}{\\left(a-bi\\right)}=\\dfrac{\\left(c+di\\right)\\left(a-bi\\right)}{\\left(a+bi\\right)\\left(a-bi\\right)}[\/latex]<\/p>\r\nApply the distributive property.\r\n<p style=\"text-align: center;\">[latex]=\\dfrac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}[\/latex]<\/p>\r\nSimplify, remembering that [latex]{i}^{2}=-1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;=\\dfrac{ca-cbi+adi-bd\\left(-1\\right)}{{a}^{2}-abi+abi-{b}^{2}\\left(-1\\right)} \\\\[2mm] &amp;=\\dfrac{\\left(ca+bd\\right)+\\left(ad-cb\\right)i}{{a}^{2}+{b}^{2}}\\end{align}[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Complex Conjugate<\/h3>\r\nThe <strong>complex conjugate<\/strong> of a complex number [latex]a+bi[\/latex] is [latex]a-bi[\/latex]. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.\r\n<ul>\r\n \t<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\r\n \t<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Complex Conjugates<\/h3>\r\nFind the complex conjugate of each number.\r\n<ol>\r\n \t<li>[latex]2+i\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{1}{2}i[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"349660\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"349660\"]\r\n<ol>\r\n \t<li>The number is already in the form [latex]a+bi[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]2-i\\sqrt{5}[\/latex].<\/li>\r\n \t<li>We can rewrite this number in the form [latex]a+bi[\/latex] as [latex]0-\\frac{1}{2}i[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]0+\\frac{1}{2}i[\/latex]. This can be written simply as [latex]\\frac{1}{2}i[\/latex].<\/li>\r\n<\/ol>\r\n<h4>Analysis of the Solution<\/h4>\r\nAlthough we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by [latex]i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two complex numbers, divide one by the other.<\/h3>\r\n<ol>\r\n \t<li>Write the division problem as a fraction.<\/li>\r\n \t<li>Determine the complex conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Dividing Complex Numbers<\/h3>\r\nDivide [latex]\\left(2+5i\\right)[\/latex] by [latex]\\left(4-i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"932961\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"932961\"]\r\n\r\nWe begin by writing the problem as a fraction.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}[\/latex]<\/p>\r\nThen we multiply the numerator and denominator by the complex conjugate of the denominator.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\dfrac{\\left(4+i\\right)}{\\left(4+i\\right)}[\/latex]<\/p>\r\nTo multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\dfrac{\\left(4+i\\right)}{\\left(4+i\\right)}&amp;=\\dfrac{8+2i+20i+5{i}^{2}}{16+4i - 4i-{i}^{2}}\\\\[2mm] &amp;=\\dfrac{8+2i+20i+5\\left(-1\\right)}{16+4i - 4i-\\left(-1\\right)} &amp;&amp; \\text{Because } {i}^{2}=-1 \\\\[2mm] &amp;=\\frac{3+22i}{17} \\\\[2mm] &amp;=\\dfrac{3}{17}+\\frac{22}{17}i &amp;&amp; \\text{Separate real and imaginary parts}.\\end{align}[\/latex]<\/p>\r\nNote that this expresses the quotient in standard form.\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<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61715&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Substituting a Complex Number into a Polynomial Function<\/h3>\r\nLet [latex]f\\left(x\\right)={x}^{2}-5x+2[\/latex]. Evaluate [latex]f\\left(3+i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"323196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"323196\"]\r\n\r\nSubstitute [latex]x=3+i[\/latex] into the function [latex]f\\left(x\\right)={x}^{2}-5x+2[\/latex] and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f(3+i)&amp;=(3+i)^2-5(3+i)+2&amp;&amp;\\text{Substitute } 3+i \\text{ for }x\\\\[2mm]&amp;=(3+6i+i^2)-(15+5i)+2&amp;&amp;\\text{Multiply}\\\\[2mm]&amp;=9+6i+(-1)-15-5i+2&amp;&amp;\\text{Substitute }-1\\text{ for }i^2 \\\\[2mm]&amp;=-5+i&amp;&amp;\\text{Combine like terms}\\end{align}[\/latex]<\/p>\r\n\r\n<h4>\u00a0Analysis of the Solution<\/h4>\r\nWe write [latex]f\\left(3+i\\right)=-5+i[\/latex]. Notice that the input is [latex]3+i[\/latex] and the output is [latex]-5+i[\/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\nLet [latex]f\\left(x\\right)=2{x}^{2}-3x[\/latex]. Evaluate [latex]f\\left(8-i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"324665\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324665\"]\r\n\r\n[latex]102 - 29i[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=120193&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Substituting an Imaginary Number in a Rational Function<\/h3>\r\nLet [latex]f\\left(x\\right)=\\dfrac{2+x}{x+3}[\/latex]. Evaluate [latex]f\\left(10i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"462657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"462657\"]\r\n\r\nSubstitute [latex]x=10i[\/latex] and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\dfrac{2+10i}{10i+3} &amp;&amp; \\text{Substitute }10i\\text{ for }x\\\\[2mm] &amp;\\dfrac{2+10i}{3+10i} &amp;&amp; \\text{Rewrite the denominator in standard form}\\\\[2mm] &amp;\\dfrac{2+10i}{3+10i}\\cdot \\dfrac{3 - 10i}{3 - 10i} &amp;&amp; \\text{Multiply the numerator and denominator by the complex conjugate of the denominator}\\\\[2mm] &amp;\\dfrac{6 - 20i+30i - 100{i}^{2}}{9 - 30i+30i - 100{i}^{2}} &amp;&amp; \\text{Multiply using the distributive property or the FOIL method} \\\\[2mm] &amp;\\dfrac{6 - 20i+30i - 100\\left(-1\\right)}{9 - 30i+30i - 100\\left(-1\\right)} &amp;&amp; \\text{Substitute }-1\\text{ for } {i}^{2} \\\\[2mm] &amp;\\dfrac{106+10i}{109} &amp;&amp; \\text{Simplify} \\\\[2mm] &amp;\\dfrac{106}{109}+\\dfrac{10}{109}i &amp;&amp; \\text{Separate the real and imaginary parts} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nLet [latex]f\\left(x\\right)=\\dfrac{x+1}{x - 4}[\/latex]. Evaluate [latex]f\\left(-i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"453508\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453508\"]\r\n\r\n[latex]-\\dfrac{3}{17}+\\dfrac{5}{17}i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/XBJjbJAwM1c\r\n<h2>Simplifying Powers of [latex]i[\/latex]<\/h2>\r\nThe powers of [latex]i[\/latex]\u00a0are cyclic. Let\u2019s look at what happens when we raise [latex]i[\/latex]\u00a0to increasing powers.\r\n\r\n[latex]{i}^{1}=i[\/latex]\r\n[latex]{i}^{2}=-1[\/latex]\r\n[latex]{i}^{3}={i}^{2}\\cdot i=-1\\cdot i=-i[\/latex]\r\n[latex]{i}^{4}={i}^{3}\\cdot i=-i\\cdot i=-{i}^{2}=-\\left(-1\\right)=1[\/latex]\r\n[latex]{i}^{5}={i}^{4}\\cdot i=1\\cdot i=i[\/latex]\r\n\r\nWe can see that when we get to the fifth power of [latex]i[\/latex], it is equal to the first power. As we continue to multiply [latex]i[\/latex]\u00a0by itself for increasing powers, we will see a cycle of 4. Let\u2019s examine the next 4 powers of [latex]i[\/latex].\r\n\r\n[latex]{i}^{6}={i}^{5}\\cdot i=i\\cdot i={i}^{2}=-1[\/latex]\r\n[latex]{i}^{7}={i}^{6}\\cdot i={i}^{2}\\cdot i={i}^{3}=-i[\/latex]\r\n[latex]{i}^{8}={i}^{7}\\cdot i={i}^{3}\\cdot i={i}^{4}=1[\/latex]\r\n[latex]{i}^{9}={i}^{8}\\cdot i={i}^{4}\\cdot i={i}^{5}=i[\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Powers of [latex]i[\/latex]<\/h3>\r\nEvaluate [latex]{i}^{35}[\/latex].\r\n\r\n[reveal-answer q=\"879646\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"879646\"]\r\n\r\nSince [latex]{i}^{4}=1[\/latex], we can simplify the problem by factoring out as many factors of [latex]{i}^{4}[\/latex] as possible. To do so, first determine how many times [latex]4[\/latex] goes into [latex]35[\/latex]: [latex]35=4\\cdot 8+3[\/latex].\r\n\r\n[latex]{i}^{35}={i}^{4\\cdot 8+3}={i}^{4\\cdot 8}\\cdot {i}^{3}={\\left({i}^{4}\\right)}^{8}\\cdot {i}^{3}={1}^{8}\\cdot {i}^{3}={i}^{3}=-i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can we write [latex]{i}^{35}[\/latex] in other helpful ways?<\/strong>\r\n\r\n<em>As we saw in Example: Simplifying Powers of <\/em>[latex]i[\/latex]<em>, we reduced [latex]{i}^{35}[\/latex] to [latex]{i}^{3}[\/latex] by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of [latex]{i}^{35}[\/latex] may be more useful. The table below\u00a0shows some other possible factorizations.<\/em>\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Factorization of [latex]{i}^{35}[\/latex]<\/strong><\/td>\r\n<td>[latex]{i}^{34}\\cdot i[\/latex]<\/td>\r\n<td>[latex]{i}^{33}\\cdot {i}^{2}[\/latex]<\/td>\r\n<td>[latex]{i}^{31}\\cdot {i}^{4}[\/latex]<\/td>\r\n<td>[latex]{i}^{19}\\cdot {i}^{16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Reduced form<\/strong><\/td>\r\n<td>[latex]{\\left({i}^{2}\\right)}^{17}\\cdot i[\/latex]<\/td>\r\n<td>[latex]{i}^{33}\\cdot \\left(-1\\right)[\/latex]<\/td>\r\n<td>[latex]{i}^{31}\\cdot 1[\/latex]<\/td>\r\n<td>[latex]{i}^{19}\\cdot {\\left({i}^{4}\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Simplified form<\/strong><\/td>\r\n<td>[latex]{\\left(-1\\right)}^{17}\\cdot i[\/latex]<\/td>\r\n<td>[latex]-{i}^{33}[\/latex]<\/td>\r\n<td>[latex]{i}^{31}[\/latex]<\/td>\r\n<td>[latex]{i}^{19}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<em>Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em>\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135261454\">\r\n \t<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\r\n \t<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\r\n \t<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\r\n \t<li>Complex numbers can be multiplied and divided.<\/li>\r\n \t<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\r\n \t<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\r\n \t<li>The powers of [latex]i[\/latex]\u00a0are cyclic, repeating every fourth one.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135320095\" class=\"definition\">\r\n \t<dt><strong>complex conjugate<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135320107\" class=\"definition\">\r\n \t<dt><strong>complex number<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex]\u00a0is the real part, and [latex]bi[\/latex]\u00a0is the imaginary part<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133260439\" class=\"definition\">\r\n \t<dt><strong>complex plane<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133260450\" class=\"definition\">\r\n \t<dt><strong>imaginary number<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]\u00a0where [latex]i=\\sqrt{-1}\\\\[\/latex]<\/dd>\r\n<\/dl>\r\n<img class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/>\r\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 2.1 - Complex Numbers<\/span><\/h3>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Express square roots of negative numbers as multiples of\u2009[latex]i[\/latex].<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Plot complex numbers on the complex plane.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Add and subtract complex numbers.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Multiply and divide complex numbers.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. The set of real numbers has voids as well. For example, we still have no solution to equations such as<\/p>\n<p style=\"text-align: center;\">[latex]{x}^{2}+4=0[\/latex]<\/p>\n<p>Our best guesses might be +2 or \u20132. But if we test +2 in this equation, it does not work. If we test \u20132, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately there is another system of numbers that provides solutions to problems such as these. In this section we will explore this number system and how to work within it.<\/p>\n<h2>Express and Plot Complex Numbers<\/h2>\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<p>&nbsp;<\/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>\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 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=\"mom200\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61706&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NeTRNpBI17I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Plot complex numbers on the complex plane<\/h2>\n<p>We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the <strong>complex plane<\/strong>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs [latex](a, b)[\/latex], where [latex]a[\/latex]\u00a0represents the coordinate for the horizontal axis and [latex]b[\/latex]\u00a0represents the coordinate for the vertical axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2530 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24211959\/CNX_Precalc_Figure_03_01_0022.jpg\" alt=\"Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.\" width=\"487\" height=\"443\" \/><\/p>\n<p>Let\u2019s consider the number [latex]-2+3i[\/latex]. The real part of the complex number is [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]-4i[\/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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65079&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h2>Add, Subtract, and Multiply Complex Numbers<\/h2>\n<p>Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Addition and Subtraction of Complex Numbers<\/h3>\n<p>Adding complex numbers:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\n<p>Subtracting complex numbers:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two complex numbers, find the sum or difference.<\/h3>\n<ol>\n<li>Identify the real and imaginary parts of each number.<\/li>\n<li>Add or subtract the real parts.<\/li>\n<li>Add or subtract the imaginary parts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Complex Numbers<\/h3>\n<p>Add [latex]3 - 4i[\/latex] and [latex]2+5i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5937\">Show Solution<\/span><\/p>\n<div id=\"q5937\" class=\"hidden-answer\" style=\"display: none\">\n<p>We add the real parts and add the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3 - 4i\\right)+\\left(2+5i\\right)=\\left(3+2\\right)+\\left(-4+5\\right)i=5+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]2+5i[\/latex] from [latex]3 - 4i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q732700\">Show Solution<\/span><\/p>\n<div id=\"q732700\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(3 - 4i\\right)-\\left(2+5i\\right)=1 - 9i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61710&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Adding and Subtracting Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SGhTjioGqqA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Complex Numbers<\/h2>\n<p>Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\n<h2>Multiplying a Complex Number by a Real Number<\/h2>\n<p>Let\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}3(6+2i)&=(3\\cdot6)+(3\\cdot2i)&&\\text{Distribute.}\\\\&=18+6i&&\\text{Simplify.}\\end{align}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a complex number and a real number, multiply to find the product.<\/h3>\n<ol>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying a Complex Number by a Real Number<\/h3>\n<p>Find the product [latex]4\\left(2+5i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q928099\">Show Solution<\/span><\/p>\n<div id=\"q928099\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)=8+20i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the product [latex]-4\\left(2+6i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568092\">Show Solution<\/span><\/p>\n<div id=\"q568092\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-8 - 24i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40462&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Multiplying Complex Numbers Together<\/h2>\n<p>Now, let\u2019s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/p>\n<p>Because [latex]{i}^{2}=-1[\/latex], we have<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/p>\n<p>To simplify, we combine the real parts, and we combine the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two complex numbers, multiply to find the product.<\/h3>\n<ol>\n<li>Use the distributive property or the FOIL method.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying a Complex Number by a Complex Number<\/h3>\n<p>Multiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388605\">Show Solution<\/span><\/p>\n<div id=\"q388605\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(4+3i\\right)\\left(2 - 5i\\right)&=4\\cdot 2 + 4\\cdot \\left(-5i\\right)+3i\\cdot2+3i\\cdot \\left(-5i\\right)\\\\ &=8-20i+6i-15i^2\\\\&=8+15-20i+6i\\\\ &=23 - 14i\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]\\left(3 - 4i\\right)\\left(2+3i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q576399\">Show Solution<\/span><\/p>\n<div id=\"q576399\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]18+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3903&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2:  Multiply Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/O9xQaIi0NX0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Divide Complex Numbers<\/h2>\n<p>Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the <strong>complex conjugate<\/strong> of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex].<\/p>\n<p>Note that complex conjugates have a reciprocal relationship: The complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex], and the complex conjugate of [latex]a-bi[\/latex] is [latex]a+bi[\/latex]. Importantly, complex conjugate pairs have a special property. Their product is always real.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}(a+bi)(a-bi)&=a^2-abi+abi-b^2i^2\\\\[2mm]&=a^2-b^2(-1)\\\\[2mm]&=a^2+b^2\\end{align}[\/latex]<\/p>\n<p>Suppose we want to divide [latex]c+di[\/latex] by [latex]a+bi[\/latex], where neither [latex]a[\/latex] nor [latex]b[\/latex] equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{c+di}{a+bi}[\/latex] where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex].<\/p>\n<p>Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(c+di\\right)}{\\left(a+bi\\right)}\\cdot \\dfrac{\\left(a-bi\\right)}{\\left(a-bi\\right)}=\\dfrac{\\left(c+di\\right)\\left(a-bi\\right)}{\\left(a+bi\\right)\\left(a-bi\\right)}[\/latex]<\/p>\n<p>Apply the distributive property.<\/p>\n<p style=\"text-align: center;\">[latex]=\\dfrac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}[\/latex]<\/p>\n<p>Simplify, remembering that [latex]{i}^{2}=-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=\\dfrac{ca-cbi+adi-bd\\left(-1\\right)}{{a}^{2}-abi+abi-{b}^{2}\\left(-1\\right)} \\\\[2mm] &=\\dfrac{\\left(ca+bd\\right)+\\left(ad-cb\\right)i}{{a}^{2}+{b}^{2}}\\end{align}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Complex Conjugate<\/h3>\n<p>The <strong>complex conjugate<\/strong> of a complex number [latex]a+bi[\/latex] is [latex]a-bi[\/latex]. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.<\/p>\n<ul>\n<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\n<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Complex Conjugates<\/h3>\n<p>Find the complex conjugate of each number.<\/p>\n<ol>\n<li>[latex]2+i\\sqrt{5}[\/latex]<\/li>\n<li>[latex]-\\frac{1}{2}i[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q349660\">Show Solution<\/span><\/p>\n<div id=\"q349660\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The number is already in the form [latex]a+bi[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]2-i\\sqrt{5}[\/latex].<\/li>\n<li>We can rewrite this number in the form [latex]a+bi[\/latex] as [latex]0-\\frac{1}{2}i[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]0+\\frac{1}{2}i[\/latex]. This can be written simply as [latex]\\frac{1}{2}i[\/latex].<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by [latex]i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two complex numbers, divide one by the other.<\/h3>\n<ol>\n<li>Write the division problem as a fraction.<\/li>\n<li>Determine the complex conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Dividing Complex Numbers<\/h3>\n<p>Divide [latex]\\left(2+5i\\right)[\/latex] by [latex]\\left(4-i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q932961\">Show Solution<\/span><\/p>\n<div id=\"q932961\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by writing the problem as a fraction.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}[\/latex]<\/p>\n<p>Then we multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\dfrac{\\left(4+i\\right)}{\\left(4+i\\right)}[\/latex]<\/p>\n<p>To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\dfrac{\\left(4+i\\right)}{\\left(4+i\\right)}&=\\dfrac{8+2i+20i+5{i}^{2}}{16+4i - 4i-{i}^{2}}\\\\[2mm] &=\\dfrac{8+2i+20i+5\\left(-1\\right)}{16+4i - 4i-\\left(-1\\right)} && \\text{Because } {i}^{2}=-1 \\\\[2mm] &=\\frac{3+22i}{17} \\\\[2mm] &=\\dfrac{3}{17}+\\frac{22}{17}i && \\text{Separate real and imaginary parts}.\\end{align}[\/latex]<\/p>\n<p>Note that this expresses the quotient in standard form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61715&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Substituting a Complex Number into a Polynomial Function<\/h3>\n<p>Let [latex]f\\left(x\\right)={x}^{2}-5x+2[\/latex]. Evaluate [latex]f\\left(3+i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q323196\">Show Solution<\/span><\/p>\n<div id=\"q323196\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]x=3+i[\/latex] into the function [latex]f\\left(x\\right)={x}^{2}-5x+2[\/latex] and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(3+i)&=(3+i)^2-5(3+i)+2&&\\text{Substitute } 3+i \\text{ for }x\\\\[2mm]&=(3+6i+i^2)-(15+5i)+2&&\\text{Multiply}\\\\[2mm]&=9+6i+(-1)-15-5i+2&&\\text{Substitute }-1\\text{ for }i^2 \\\\[2mm]&=-5+i&&\\text{Combine like terms}\\end{align}[\/latex]<\/p>\n<h4>\u00a0Analysis of the Solution<\/h4>\n<p>We write [latex]f\\left(3+i\\right)=-5+i[\/latex]. Notice that the input is [latex]3+i[\/latex] and the output is [latex]-5+i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Let [latex]f\\left(x\\right)=2{x}^{2}-3x[\/latex]. Evaluate [latex]f\\left(8-i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324665\">Show Solution<\/span><\/p>\n<div id=\"q324665\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]102 - 29i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=120193&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Substituting an Imaginary Number in a Rational Function<\/h3>\n<p>Let [latex]f\\left(x\\right)=\\dfrac{2+x}{x+3}[\/latex]. Evaluate [latex]f\\left(10i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q462657\">Show Solution<\/span><\/p>\n<div id=\"q462657\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]x=10i[\/latex] and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\dfrac{2+10i}{10i+3} && \\text{Substitute }10i\\text{ for }x\\\\[2mm] &\\dfrac{2+10i}{3+10i} && \\text{Rewrite the denominator in standard form}\\\\[2mm] &\\dfrac{2+10i}{3+10i}\\cdot \\dfrac{3 - 10i}{3 - 10i} && \\text{Multiply the numerator and denominator by the complex conjugate of the denominator}\\\\[2mm] &\\dfrac{6 - 20i+30i - 100{i}^{2}}{9 - 30i+30i - 100{i}^{2}} && \\text{Multiply using the distributive property or the FOIL method} \\\\[2mm] &\\dfrac{6 - 20i+30i - 100\\left(-1\\right)}{9 - 30i+30i - 100\\left(-1\\right)} && \\text{Substitute }-1\\text{ for } {i}^{2} \\\\[2mm] &\\dfrac{106+10i}{109} && \\text{Simplify} \\\\[2mm] &\\dfrac{106}{109}+\\dfrac{10}{109}i && \\text{Separate the real and imaginary parts} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Let [latex]f\\left(x\\right)=\\dfrac{x+1}{x - 4}[\/latex]. Evaluate [latex]f\\left(-i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453508\">Show Solution<\/span><\/p>\n<div id=\"q453508\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\dfrac{3}{17}+\\dfrac{5}{17}i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Dividing Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XBJjbJAwM1c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplifying Powers of [latex]i[\/latex]<\/h2>\n<p>The powers of [latex]i[\/latex]\u00a0are cyclic. Let\u2019s look at what happens when we raise [latex]i[\/latex]\u00a0to increasing powers.<\/p>\n<p>[latex]{i}^{1}=i[\/latex]<br \/>\n[latex]{i}^{2}=-1[\/latex]<br \/>\n[latex]{i}^{3}={i}^{2}\\cdot i=-1\\cdot i=-i[\/latex]<br \/>\n[latex]{i}^{4}={i}^{3}\\cdot i=-i\\cdot i=-{i}^{2}=-\\left(-1\\right)=1[\/latex]<br \/>\n[latex]{i}^{5}={i}^{4}\\cdot i=1\\cdot i=i[\/latex]<\/p>\n<p>We can see that when we get to the fifth power of [latex]i[\/latex], it is equal to the first power. As we continue to multiply [latex]i[\/latex]\u00a0by itself for increasing powers, we will see a cycle of 4. Let\u2019s examine the next 4 powers of [latex]i[\/latex].<\/p>\n<p>[latex]{i}^{6}={i}^{5}\\cdot i=i\\cdot i={i}^{2}=-1[\/latex]<br \/>\n[latex]{i}^{7}={i}^{6}\\cdot i={i}^{2}\\cdot i={i}^{3}=-i[\/latex]<br \/>\n[latex]{i}^{8}={i}^{7}\\cdot i={i}^{3}\\cdot i={i}^{4}=1[\/latex]<br \/>\n[latex]{i}^{9}={i}^{8}\\cdot i={i}^{4}\\cdot i={i}^{5}=i[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Powers of [latex]i[\/latex]<\/h3>\n<p>Evaluate [latex]{i}^{35}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q879646\">Show Solution<\/span><\/p>\n<div id=\"q879646\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]{i}^{4}=1[\/latex], we can simplify the problem by factoring out as many factors of [latex]{i}^{4}[\/latex] as possible. To do so, first determine how many times [latex]4[\/latex] goes into [latex]35[\/latex]: [latex]35=4\\cdot 8+3[\/latex].<\/p>\n<p>[latex]{i}^{35}={i}^{4\\cdot 8+3}={i}^{4\\cdot 8}\\cdot {i}^{3}={\\left({i}^{4}\\right)}^{8}\\cdot {i}^{3}={1}^{8}\\cdot {i}^{3}={i}^{3}=-i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can we write [latex]{i}^{35}[\/latex] in other helpful ways?<\/strong><\/p>\n<p><em>As we saw in Example: Simplifying Powers of <\/em>[latex]i[\/latex]<em>, we reduced [latex]{i}^{35}[\/latex] to [latex]{i}^{3}[\/latex] by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of [latex]{i}^{35}[\/latex] may be more useful. The table below\u00a0shows some other possible factorizations.<\/em><\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Factorization of [latex]{i}^{35}[\/latex]<\/strong><\/td>\n<td>[latex]{i}^{34}\\cdot i[\/latex]<\/td>\n<td>[latex]{i}^{33}\\cdot {i}^{2}[\/latex]<\/td>\n<td>[latex]{i}^{31}\\cdot {i}^{4}[\/latex]<\/td>\n<td>[latex]{i}^{19}\\cdot {i}^{16}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Reduced form<\/strong><\/td>\n<td>[latex]{\\left({i}^{2}\\right)}^{17}\\cdot i[\/latex]<\/td>\n<td>[latex]{i}^{33}\\cdot \\left(-1\\right)[\/latex]<\/td>\n<td>[latex]{i}^{31}\\cdot 1[\/latex]<\/td>\n<td>[latex]{i}^{19}\\cdot {\\left({i}^{4}\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Simplified form<\/strong><\/td>\n<td>[latex]{\\left(-1\\right)}^{17}\\cdot i[\/latex]<\/td>\n<td>[latex]-{i}^{33}[\/latex]<\/td>\n<td>[latex]{i}^{31}[\/latex]<\/td>\n<td>[latex]{i}^{19}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><em>Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135261454\">\n<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\n<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\n<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\n<li>Complex numbers can be multiplied and divided.<\/li>\n<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\n<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\n<li>The powers of [latex]i[\/latex]\u00a0are cyclic, repeating every fourth one.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135320095\" class=\"definition\">\n<dt><strong>complex conjugate<\/strong><\/dt>\n<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135320107\" class=\"definition\">\n<dt><strong>complex number<\/strong><\/dt>\n<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex]\u00a0is the real part, and [latex]bi[\/latex]\u00a0is the imaginary part<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260439\" class=\"definition\">\n<dt><strong>complex plane<\/strong><\/dt>\n<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260450\" class=\"definition\">\n<dt><strong>imaginary number<\/strong><\/dt>\n<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]\u00a0where [latex]i=\\sqrt{-1}\\\\[\/latex]<\/dd>\n<\/dl>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/><\/p>\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 2.1 &#8211; Complex Numbers<\/span><\/h3>\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-1762\">\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><li>Question ID 120193. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><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>Ex: Dividing Complex Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/XBJjbJAwM1c\">https:\/\/youtu.be\/XBJjbJAwM1c<\/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>Ex 1: Adding and Subtracting Complex Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SGhTjioGqqA\">https:\/\/youtu.be\/SGhTjioGqqA<\/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 61710, 61715. <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 40462. <strong>Authored by<\/strong>: Jenck, Michael. <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 3903. <strong>Authored by<\/strong>: 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>Ex 2: Multiply Complex Numbers. <strong>Authored by<\/strong>: Sousa, James  (Mathispower4u). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/O9xQaIi0NX0\">https:\/\/youtu.be\/O9xQaIi0NX0<\/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>Ex: Dividing Complex Numbers . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/XBJjbJAwM1c\">https:\/\/youtu.be\/XBJjbJAwM1c<\/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 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":708740,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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