{"id":1425,"date":"2023-04-06T15:16:04","date_gmt":"2023-04-06T15:16:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1425"},"modified":"2025-05-22T19:53:28","modified_gmt":"2025-05-22T19:53:28","slug":"fractals-generated-by-complex-numbers-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-learn-it-1\/","title":{"raw":"Fractals Generated by Complex Numbers: Learn It 1","rendered":"Fractals Generated by Complex Numbers: 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>Do math operations with complex numbers, and learn how these operations can be shown as scaling or rotation<\/li>\r\n\t<li>Create a series of numbers that are made by repeating a rule, and learn how to find specific terms in the series<\/li>\r\n\t<li>Determine whether a complex number is part of a special set of numbers called the Mandelbrot Set<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Complex Numbers<\/h2>\r\n<p>The numbers you are most familiar with are called <strong>real numbers<\/strong>. These include numbers like [latex]4, 275, -200, 10.7, \\frac{1}{2}, \u03c0[\/latex], and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number [latex]3[\/latex], we plot a point on the number line equidistant between [latex]2[\/latex] and [latex]4[\/latex]. To solve certain problems like [latex]x^{2}=\u20134[\/latex], it became necessary to introduce <strong>imaginary numbers<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>imaginary number<\/h3>\r\n<p>The <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Any real multiple of [latex]i[\/latex], like 5[latex]i[\/latex], is also an imaginary number.<\/p>\r\n<\/div>\r\n<\/section>\r\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]?<br \/>\r\n<p style=\"text-align: center;\">Example: [latex]\\sqrt{9}=3[\/latex] because [latex]3 \\ast 3 = 9[\/latex].<\/p>\r\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>\r\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>\r\n<\/section>\r\n<section class=\"textbox example\">Simplify [latex]\\sqrt{-9}[\/latex]<br \/>\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703380\"]<br \/>\r\nWe can separate [latex]\\sqrt{-9}[\/latex] as [latex]\\sqrt{9\\cdot-1}[\/latex] as [latex]\\sqrt{9}\\cdot\\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\r\n<p style=\"text-align: center;\">[latex]\\sqrt{-9}=\\sqrt{9}\\cdot\\sqrt{-1}=3i[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>A <strong>complex number<\/strong> is the sum of a real number and an imaginary number.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>complex number<\/h3>\r\n<p>A <strong>complex number<\/strong> is a number [latex]z = a + b i[\/latex], where<\/p>\r\n<p>&nbsp;<\/p>\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>\r\n<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>Do math operations with complex numbers, and learn how these operations can be shown as scaling or rotation<\/li>\n<li>Create a series of numbers that are made by repeating a rule, and learn how to find specific terms in the series<\/li>\n<li>Determine whether a complex number is part of a special set of numbers called the Mandelbrot Set<\/li>\n<\/ul>\n<\/section>\n<h2>Complex Numbers<\/h2>\n<p>The numbers you are most familiar with are called <strong>real numbers<\/strong>. These include numbers like [latex]4, 275, -200, 10.7, \\frac{1}{2}, \u03c0[\/latex], and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number [latex]3[\/latex], we plot a point on the number line equidistant between [latex]2[\/latex] and [latex]4[\/latex]. To solve certain problems like [latex]x^{2}=\u20134[\/latex], it became necessary to introduce <strong>imaginary numbers<\/strong>.<\/p>\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\\cdot-1}[\/latex] as [latex]\\sqrt{9}\\cdot\\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}\\cdot\\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.<\/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 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":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1425"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":33,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1425\/revisions"}],"predecessor-version":[{"id":15514,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1425\/revisions\/15514"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1425\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1425"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1425"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1425"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}