{"id":1472,"date":"2023-04-10T15:05:11","date_gmt":"2023-04-10T15:05:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1472"},"modified":"2025-05-22T20:06:00","modified_gmt":"2025-05-22T20:06:00","slug":"fractals-generated-by-complex-numbers-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-learn-it-3\/","title":{"raw":"Fractals Generated by Complex Numbers: Learn It 3","rendered":"Fractals Generated by Complex Numbers: Learn It 3"},"content":{"raw":"<h2>Arithmetic on Complex Numbers<\/h2>\r\n<p>To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.<\/p>\r\n<section class=\"textbox example\">Add [latex]3-4i[\/latex] and [latex]2+5i[\/latex].<br \/>\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703380\"]<br \/>\r\nAdding [latex](3-4i)+(2+5i)[\/latex], we add the like terms by combining the real parts and the imaginary parts.\r\n\r\n<p style=\"text-align: center;\">[latex]3+2-4i+5i[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]5+i[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6895[\/ohm2_question]<\/section>\r\n<p>When we add complex numbers, we can visualize the addition as a shift, or <strong>translation<\/strong>, of a point in the complex plane.<\/p>\r\n<section class=\"textbox example\">Visualize the addition [latex]3-4i[\/latex] and [latex]-1+5i[\/latex].<br \/>\r\n[reveal-answer q=\"703381\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703381\"]<br \/>\r\nThe initial point is [latex]3-4i[\/latex]. When we add [latex]-1+3i[\/latex], we add [latex]-1[\/latex] to the real part, moving the point [latex]1[\/latex] units to the left, and we add [latex]5[\/latex] to the imaginary part, moving the point [latex]5[\/latex] units vertically. This shifts the point [latex]3-4i[\/latex] to [latex]2+1i[\/latex].<center><img class=\"aligncenter size-full wp-image-1733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23185217\/Screen-Shot-2017-02-23-at-10.51.55-AM.png\" alt=\"A graph with an imaginary y-axis and a real x-axis. The point 3, negative 4 is labeled 3 minus 4i. The point 2, 1 is labeled 2 plus 1i. An arrow goes from 3 minus 4i to 2 plus 1i.\" width=\"300\" height=\"291\" \/><\/center>[\/hidden-answer]<\/section>\r\n<p>We can also multiply complex numbers by a real number, or multiply two complex numbers.<\/p>\r\n<section class=\"textbox example\">Multiply: [latex](2+5i)(4+i)[\/latex]<br \/>\r\n[reveal-answer q=\"703385\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703385\"]\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2+5i\\right)\\left(4+i\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Expand.}\\\\=8+20i+2i+5i^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Since }i=\\sqrt{-1},i^{2}=-1\\\\=8+20i+2i+5\\left(-1\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Group like terms.}\\\\=8-5+20i+2i\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify.}\\\\=3+22i\\end{array}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6896[\/ohm2_question]<\/section>\r\n<p>When we multiply by a complex number, it's like we're doing two things at once: we're changing the size (or <strong>scaling<\/strong>) and turning (or <strong>rotating<\/strong>) the number around the starting point, or origin. To really see what's happening, in the following examples, we'll use the complex plane to helps us visualize these changes.<\/p>\r\n<section class=\"textbox example\">Using the complex plane, visualize the product [latex]2(1+2i)[\/latex].<br \/>\r\n[reveal-answer q=\"703386\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703386\"]<br \/>\r\nMultiplying we\u2019d get:\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;2\\cdot1+2\\cdot2i\\\\&amp;=2+4i\\\\\\end{align}[\/latex]<\/p>\r\n<p>Notice both the real and imaginary parts have been scaled by [latex]2[\/latex]. Visually, this will stretch the point outwards, away from the origin.<\/p>\r\n<center><img class=\"aligncenter size-full wp-image-1734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23195901\/Screen-Shot-2017-02-23-at-11.58.34-AM.png\" alt=\"Graph with imaginary y-axis and real x-axis. The point 1,2 is marked and labeled 1 plus 2i. The point 2,4 is marked and labeled 2 plus 4i. A red arrow is drawn from the origin and through both points.\" width=\"302\" height=\"279\" \/><\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Using the complex plane, visualize the result of multiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex]. Then show the result of multiplying by [latex]1+i[\/latex] again.<br \/>\r\n[reveal-answer q=\"703387\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703387\"]<br \/>\r\nMultiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex],\r\n\r\n<p style=\"text-align: center;\">[latex]-4+2i[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;(1+2i)(1+i)\\\\&amp;=1+i+2i+2{{i}^{2}}\\\\&amp;=1+3i+2(-1)\\\\&amp;=-1+3i\\\\\\end{align}[\/latex]<\/p>\r\n<p>Multiplying by [latex]1+i[\/latex] again,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;(-1+3i)(1+i)\\\\&amp;=-1-i+3i+3{{i}^{2}}\\\\&amp;=-1+2i+3(-1)\\\\&amp;=-4+2i\\\\\\end{align}[\/latex]<\/p>\r\n<p>If we multiplied by [latex]1+i[\/latex] again, we\u2019d get [latex]\u20136\u20132i[\/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by [latex]45\u00b0[\/latex].<\/p>\r\n<center><img class=\"aligncenter size-full wp-image-1736\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23221746\/Screen-Shot-2017-02-23-at-2.15.58-PM.png\" alt=\"The imaginary-real graph with four points marked, each with a dotted red line extending from the origin to that point. The points are as follows. The point 1,2, represented by 1 plus 2i. The point negative 1, 3, represented by negative 1 plus 3i. The point negative 4, 2, represented by negative 4 plus 2i. The point negative 6, negative 2, represented by negative 6 minus 2i.\" width=\"400\" height=\"346\" \/><\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<h2>Arithmetic on Complex Numbers<\/h2>\n<p>To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.<\/p>\n<section class=\"textbox example\">Add [latex]3-4i[\/latex] and [latex]2+5i[\/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\">\nAdding [latex](3-4i)+(2+5i)[\/latex], we add the like terms by combining the real parts and the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]3+2-4i+5i[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]5+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6895\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6895&theme=lumen&iframe_resize_id=ohm6895&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>When we add complex numbers, we can visualize the addition as a shift, or <strong>translation<\/strong>, of a point in the complex plane.<\/p>\n<section class=\"textbox example\">Visualize the addition [latex]3-4i[\/latex] and [latex]-1+5i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703381\">Show Solution<\/button><\/p>\n<div id=\"q703381\" class=\"hidden-answer\" style=\"display: none\">\nThe initial point is [latex]3-4i[\/latex]. When we add [latex]-1+3i[\/latex], we add [latex]-1[\/latex] to the real part, moving the point [latex]1[\/latex] units to the left, and we add [latex]5[\/latex] to the imaginary part, moving the point [latex]5[\/latex] units vertically. This shifts the point [latex]3-4i[\/latex] to [latex]2+1i[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23185217\/Screen-Shot-2017-02-23-at-10.51.55-AM.png\" alt=\"A graph with an imaginary y-axis and a real x-axis. The point 3, negative 4 is labeled 3 minus 4i. The point 2, 1 is labeled 2 plus 1i. An arrow goes from 3 minus 4i to 2 plus 1i.\" width=\"300\" height=\"291\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>We can also multiply complex numbers by a real number, or multiply two complex numbers.<\/p>\n<section class=\"textbox example\">Multiply: [latex](2+5i)(4+i)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703385\">Show Solution<\/button><\/p>\n<div id=\"q703385\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2+5i\\right)\\left(4+i\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Expand.}\\\\=8+20i+2i+5i^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Since }i=\\sqrt{-1},i^{2}=-1\\\\=8+20i+2i+5\\left(-1\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Group like terms.}\\\\=8-5+20i+2i\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify.}\\\\=3+22i\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6896\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6896&theme=lumen&iframe_resize_id=ohm6896&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>When we multiply by a complex number, it&#8217;s like we&#8217;re doing two things at once: we&#8217;re changing the size (or <strong>scaling<\/strong>) and turning (or <strong>rotating<\/strong>) the number around the starting point, or origin. To really see what&#8217;s happening, in the following examples, we&#8217;ll use the complex plane to helps us visualize these changes.<\/p>\n<section class=\"textbox example\">Using the complex plane, visualize the product [latex]2(1+2i)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703386\">Show Solution<\/button><\/p>\n<div id=\"q703386\" class=\"hidden-answer\" style=\"display: none\">\nMultiplying we\u2019d get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&2\\cdot1+2\\cdot2i\\\\&=2+4i\\\\\\end{align}[\/latex]<\/p>\n<p>Notice both the real and imaginary parts have been scaled by [latex]2[\/latex]. Visually, this will stretch the point outwards, away from the origin.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23195901\/Screen-Shot-2017-02-23-at-11.58.34-AM.png\" alt=\"Graph with imaginary y-axis and real x-axis. The point 1,2 is marked and labeled 1 plus 2i. The point 2,4 is marked and labeled 2 plus 4i. A red arrow is drawn from the origin and through both points.\" width=\"302\" height=\"279\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Using the complex plane, visualize the result of multiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex]. Then show the result of multiplying by [latex]1+i[\/latex] again.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703387\">Show Solution<\/button><\/p>\n<div id=\"q703387\" class=\"hidden-answer\" style=\"display: none\">\nMultiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]-4+2i[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&(1+2i)(1+i)\\\\&=1+i+2i+2{{i}^{2}}\\\\&=1+3i+2(-1)\\\\&=-1+3i\\\\\\end{align}[\/latex]<\/p>\n<p>Multiplying by [latex]1+i[\/latex] again,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&(-1+3i)(1+i)\\\\&=-1-i+3i+3{{i}^{2}}\\\\&=-1+2i+3(-1)\\\\&=-4+2i\\\\\\end{align}[\/latex]<\/p>\n<p>If we multiplied by [latex]1+i[\/latex] again, we\u2019d get [latex]\u20136\u20132i[\/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by [latex]45\u00b0[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1736\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23221746\/Screen-Shot-2017-02-23-at-2.15.58-PM.png\" alt=\"The imaginary-real graph with four points marked, each with a dotted red line extending from the origin to that point. The points are as follows. The point 1,2, represented by 1 plus 2i. The point negative 1, 3, represented by negative 1 plus 3i. The point negative 4, 2, represented by negative 4 plus 2i. The point negative 6, negative 2, represented by negative 6 minus 2i.\" width=\"400\" height=\"346\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"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\/1472"}],"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":29,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1472\/revisions"}],"predecessor-version":[{"id":15521,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1472\/revisions\/15521"}],"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\/1472\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1472"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1472"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1472"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1472"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}