{"id":4533,"date":"2023-06-16T01:43:28","date_gmt":"2023-06-16T01:43:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4533"},"modified":"2025-08-26T23:35:53","modified_gmt":"2025-08-26T23:35:53","slug":"fractals-generated-by-complex-numbers-apply-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-apply-it-1\/","title":{"raw":"Fractals Generated by Complex Numbers: Apply It 1","rendered":"Fractals Generated by Complex Numbers: Apply 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>Navigating Complex Waters<\/h2>\r\n<center>\r\n[caption id=\"attachment_6656\" align=\"aligncenter\" width=\"750\"]<img class=\"wp-image-6656\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-300x133.jpg\" alt=\"A ship at sea\" width=\"750\" height=\"331\" \/> Figure 1. Use your knowledge of fractals and complex numbers to navigate the sea[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Imagine you are a navigator in the age of exploration, charting new territories and plotting courses through unfamiliar waters. The seas you're sailing are not composed of salt and water, but the realm of numbers - complex numbers. This is no ordinary expedition, as you're exploring the abstract and fascinating world of imaginary and complex numbers, mapping them onto planes, observing their behavior under different operations, tracing recursive patterns, and even spotting the elusive Mandelbrot Set. Buckle up, and let's start the journey!<\/p>\r\n<h3>The Imaginary and the Complex<\/h3>\r\n<p>Imagine, pun intended, you encountered an unknown sea creature, described by the number [latex]4 + 5i[\/latex]. Can you identify if this is a real number or a complex number?<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]9549[\/ohm2_question]<\/section>\r\n<h3>Plotting the Course<\/h3>\r\n<p>As a navigator, you need to plot your coordinates on a map. If your current position is represented by the complex number [latex]-3 + 4i[\/latex], can you plot this position on the complex plane?<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]9550[\/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>Navigating Complex Waters<\/h2>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_6656\" aria-describedby=\"caption-attachment-6656\" style=\"width: 750px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6656\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-300x133.jpg\" alt=\"A ship at sea\" width=\"750\" height=\"331\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-300x133.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-1024x452.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-768x339.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-1200x530.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-65x29.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-225x99.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1-350x155.jpg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/06\/12180809\/vidar-nordli-mathisen-zqgOkt7sQlY-unsplash_50-1.jpg 1440w\" sizes=\"(max-width: 750px) 100vw, 750px\" \/><figcaption id=\"caption-attachment-6656\" class=\"wp-caption-text\">Figure 1. Use your knowledge of fractals and complex numbers to navigate the sea<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Imagine you are a navigator in the age of exploration, charting new territories and plotting courses through unfamiliar waters. The seas you&#8217;re sailing are not composed of salt and water, but the realm of numbers &#8211; complex numbers. This is no ordinary expedition, as you&#8217;re exploring the abstract and fascinating world of imaginary and complex numbers, mapping them onto planes, observing their behavior under different operations, tracing recursive patterns, and even spotting the elusive Mandelbrot Set. Buckle up, and let&#8217;s start the journey!<\/p>\n<h3>The Imaginary and the Complex<\/h3>\n<p>Imagine, pun intended, you encountered an unknown sea creature, described by the number [latex]4 + 5i[\/latex]. Can you identify if this is a real number or a complex number?<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm9549\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=9549&theme=lumen&iframe_resize_id=ohm9549&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Plotting the Course<\/h3>\n<p>As a navigator, you need to plot your coordinates on a map. If your current position is represented by the complex number [latex]-3 + 4i[\/latex], can you plot this position on the complex plane?<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm9550\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=9550&theme=lumen&iframe_resize_id=ohm9550&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":23,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ship on sea\",\"author\":\"Vidar Nordli-Mathisen\",\"organization\":\"Unsplash\",\"url\":\"https:\/\/unsplash.com\/photos\/zqgOkt7sQlY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"apply_it","content_attributions":[{"type":"copyrighted_video","description":"Ship on sea","author":"Vidar Nordli-Mathisen","organization":"Unsplash","url":"https:\/\/unsplash.com\/photos\/zqgOkt7sQlY","project":"","license":"arr","license_terms":""}],"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\/4533"}],"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\/23"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4533\/revisions"}],"predecessor-version":[{"id":15699,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4533\/revisions\/15699"}],"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\/4533\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4533"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4533"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4533"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}