{"id":1829,"date":"2024-06-10T21:49:04","date_gmt":"2024-06-10T21:49:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1829"},"modified":"2025-08-14T01:05:22","modified_gmt":"2025-08-14T01:05:22","slug":"complex-numbers-and-operations-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/complex-numbers-and-operations-learn-it-1\/","title":{"raw":"Complex Numbers and Operations: Learn It 1","rendered":"Complex Numbers and Operations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Understand the difference between imaginary numbers and complex numbers<\/li>\r\n \t<li>Learn how to plot a complex number on a special graph called the complex plane<\/li>\r\n \t<li>Perform calculations with complex numbers and visualize how these operations change their position and size when graphed<\/li>\r\n \t<li><span data-sheets-root=\"1\">Find the points where a quadratic equation crosses the x-axis, including both real and complex solutions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>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].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Consider the square root of [latex]\u201325[\/latex].\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] because the principal root of [latex]25[\/latex] is the positive root.\r\n\r\n<\/section><section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>imaginary number<\/h3>\r\nThe <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex].\r\n\r\n&nbsp;\r\n\r\nAny real multiple of [latex]i[\/latex], like 5[latex]i[\/latex], is also an imaginary number.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox recall\">Recall that the square root of a number [latex]\\sqrt{n}[\/latex] is another way of asking the question\u00a0<em>what number when multiplied by itself results in the number\u00a0<\/em>[latex]n[\/latex]?\r\n<p style=\"text-align: center;\">Example: [latex]\\sqrt{9}=3[\/latex] because [latex]3 \\ast 3 = 9[\/latex].<\/p>\r\nIt is also true that [latex](-3)\\ast (-3) = 9[\/latex] although we agreed that using the radical symbol requests only the principle root, the positive one. But, there is no number that when multiplied by itself results in a negative number.\r\n\r\nThe property of integer multiplication states that both a negative number squared and a positive number squared result in a positive number. Indeed, you saw in the review section to this module that the square root of a negative number does not exist in the set of real numbers. Mathematicians realized the helpfulness of being to do calculations with such numbers though, so they assigned a value to [latex]\\sqrt{-1}[\/latex], calling it the <em>imaginary unit<\/em> [latex]i[\/latex].\r\n\r\n<\/section><section class=\"textbox example\">Simplify [latex]\\sqrt{-9}[\/latex].\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703380\"]\r\nWe can separate [latex]\\sqrt{-9}[\/latex] as [latex]\\sqrt{9}\\sqrt{-1}[\/latex]. We can take the square root of [latex]9[\/latex], and write the square root of [latex]-1[\/latex] as [latex]i[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>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.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>complex number<\/h3>\r\nA <strong>complex number<\/strong> is a number [latex]z = a + b i[\/latex], where\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li>[latex]a[\/latex]\u00a0and [latex]b[\/latex] are real numbers<\/li>\r\n \t<li>[latex]a[\/latex] is the real part of the complex number<\/li>\r\n \t<li>[latex]b[\/latex] is the imaginary part of the complex number<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">For example, [latex]5+2i[\/latex] is a complex number.\r\n\r\n[caption id=\"attachment_2527\" align=\"aligncenter\" width=\"487\"]<img class=\"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\" \/> Example with real and imaginary parts labeled[\/caption]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6894[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the difference between imaginary numbers and complex numbers<\/li>\n<li>Learn how to plot a complex number on a special graph called the complex plane<\/li>\n<li>Perform calculations with complex numbers and visualize how these operations change their position and size when graphed<\/li>\n<li><span data-sheets-root=\"1\">Find the points where a quadratic equation crosses the x-axis, including both real and complex solutions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>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].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the square root of [latex]\u201325[\/latex].<\/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] because the principal root of [latex]25[\/latex] is the positive root.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>imaginary number<\/h3>\n<p>The <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Any real multiple of [latex]i[\/latex], like 5[latex]i[\/latex], is also an imaginary number.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox recall\">Recall that the square root of a number [latex]\\sqrt{n}[\/latex] is another way of asking the question\u00a0<em>what number when multiplied by itself results in the number\u00a0<\/em>[latex]n[\/latex]?<\/p>\n<p style=\"text-align: center;\">Example: [latex]\\sqrt{9}=3[\/latex] because [latex]3 \\ast 3 = 9[\/latex].<\/p>\n<p>It is also true that [latex](-3)\\ast (-3) = 9[\/latex] although we agreed that using the radical symbol requests only the principle root, the positive one. But, there is no number that when multiplied by itself results in a negative number.<\/p>\n<p>The property of integer multiplication states that both a negative number squared and a positive number squared result in a positive number. Indeed, you saw in the review section to this module that the square root of a negative number does not exist in the set of real numbers. Mathematicians realized the helpfulness of being to do calculations with such numbers though, so they assigned a value to [latex]\\sqrt{-1}[\/latex], calling it the <em>imaginary unit<\/em> [latex]i[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">Simplify [latex]\\sqrt{-9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703380\">Show Solution<\/button><\/p>\n<div id=\"q703380\" class=\"hidden-answer\" style=\"display: none\">\nWe can separate [latex]\\sqrt{-9}[\/latex] as [latex]\\sqrt{9}\\sqrt{-1}[\/latex]. We can take the square root of [latex]9[\/latex], and write the square root of [latex]-1[\/latex] as [latex]i[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9}\\sqrt{-1}=3i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\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.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>complex number<\/h3>\n<p>A <strong>complex number<\/strong> is a number [latex]z = a + b i[\/latex], where<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>[latex]a[\/latex]\u00a0and [latex]b[\/latex] are real numbers<\/li>\n<li>[latex]a[\/latex] is the real part of the complex number<\/li>\n<li>[latex]b[\/latex] is the imaginary part of the complex number<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">For example, [latex]5+2i[\/latex] is a complex number.<\/p>\n<figure id=\"attachment_2527\" aria-describedby=\"caption-attachment-2527\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"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\" \/><figcaption id=\"caption-attachment-2527\" class=\"wp-caption-text\">Example with real and imaginary parts labeled<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6894\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6894&theme=lumen&iframe_resize_id=ohm6894&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1829"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1829\/revisions"}],"predecessor-version":[{"id":7724,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1829\/revisions\/7724"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1829\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1829"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1829"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1829"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}