{"id":4926,"date":"2023-06-23T03:30:11","date_gmt":"2023-06-23T03:30:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4926"},"modified":"2024-10-30T14:21:04","modified_gmt":"2024-10-30T14:21:04","slug":"fractals-get-stronger","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-get-stronger\/","title":{"raw":"Fractals: Get Stronger","rendered":"Fractals: Get Stronger"},"content":{"raw":"<h2>Iterated Fractals<\/h2>\r\n<p>Using the initiator and generator shown, draw the next two stages of the iterated fractal.<\/p>\r\n<blockquote>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>1.<\/p>\r\n<p><img class=\"aligncenter size-full wp-image-1741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1.png\" sizes=\"(max-width: 190px) 100vw, 190px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1.png 190w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1-65x23.png 65w\" alt=\"Initiator is a horizontal line. Generator is a horizontal line that then goes up at a right angle, right at a right angle, down at a right angle, and then continues horizontally.\" width=\"190\" height=\"68\" \/><\/p>\r\n<\/td>\r\n<td>\r\n<p>2.<\/p>\r\n<p><img class=\"aligncenter size-full wp-image-1742\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2.png\" sizes=\"(max-width: 203px) 100vw, 203px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2.png 203w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2-65x22.png 65w\" alt=\"Initiator is a horizontal line. Generator is a zig-zag.\" width=\"203\" height=\"68\" \/><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>3.<\/p>\r\n<p><img class=\"aligncenter size-full wp-image-1743\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3.png\" sizes=\"(max-width: 189px) 100vw, 189px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3.png 189w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3-65x23.png 65w\" alt=\"Initiator is an upward-sloping line. Generator is that line with smaller lines branching off of it.\" width=\"189\" height=\"67\" \/><\/p>\r\n<\/td>\r\n<td>\r\n<p>4.<\/p>\r\n<p><img class=\"aligncenter size-full wp-image-1744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4.png\" sizes=\"(max-width: 202px) 100vw, 202px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4.png 202w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4-65x16.png 65w\" alt=\"Initiator is a horizontal line. Generator is two short horizontal lines side-by-side.\" width=\"202\" height=\"51\" \/><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>5.<\/p>\r\n<p><img class=\"aligncenter size-full wp-image-1745\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5.png\" sizes=\"(max-width: 187px) 100vw, 187px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5.png 187w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5-65x35.png 65w\" alt=\"Initiator is a square. Generator is eight more squares arranged to form the border of a large square.\" width=\"187\" height=\"100\" \/><\/p>\r\n<\/td>\r\n<td>\r\n<p>6.<\/p>\r\n<p><img class=\"aligncenter wp-image-1746 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6.png\" sizes=\"(max-width: 198px) 100vw, 198px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6.png 198w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6-65x33.png 65w\" alt=\"Initiator is an equilateral triangle. Generator is three equilateral triangles that touch each other at an angle.\" width=\"198\" height=\"100\" \/><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"7\">\r\n\t<li>Create your own version of Sierpinski gasket with added randomness.<\/li>\r\n\t<li>Create a version of the branching tree fractal from example #[latex]3[\/latex] with added randomness.<\/li>\r\n<\/ol>\r\n<\/blockquote>\r\n<h2>Fractal Dimension<\/h2>\r\n<ol start=\"9\">\r\n\t<li>Determine the fractal dimension of the Koch curve.<\/li>\r\n\t<li>Determine the fractal dimension of the curve generated in exercise #[latex]1[\/latex]<\/li>\r\n\t<li>Determine the fractal dimension of the Sierpinski carpet generated in exercise #[latex]5[\/latex]<\/li>\r\n\t<li>Determine the fractal dimension of the Cantor set generated in exercise #[latex]4[\/latex]<\/li>\r\n<\/ol>\r\n<h2>Complex Numbers<\/h2>\r\n<ol start=\"13\">\r\n\t<li>Plot each number in the complex plane:\r\n\r\n<ol>\r\n\t<li>[latex]4[\/latex]<\/li>\r\n\t<li>[latex]\u20133i[\/latex]<\/li>\r\n\t<li>[latex]\u20132+3i[\/latex]<\/li>\r\n\t<li>[latex]2 + i[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Plot each number in the complex plane:\r\n\r\n<ol>\r\n\t<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]-2[\/latex]<\/span><\/li>\r\n\t<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]4i[\/latex]<\/span><\/li>\r\n\t<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]1+2i[\/latex]<\/span><\/li>\r\n\t<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]-1-i[\/latex]<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Compute:\r\n\r\n<ol>\r\n\t<li>[latex](2+3i) + (3-4i)[\/latex]<\/li>\r\n\t<li>[latex](3-5i) - (-2-i)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Compute:\r\n\r\n<ol>\r\n\t<li>[latex](1-i) + (2+4i)[\/latex]<\/li>\r\n\t<li>[latex](\u00a0-2-3i) - (4-2i)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Multiply:\r\n\r\n<ol>\r\n\t<li>[latex]3(2+4i)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex](2i)(-1-5i)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex](2-4i)(1+3i)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Multiply:\r\n\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">[latex]2(-1+3i)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex](3i)(2-6i)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex](1-i)(2+5i)[\/latex]<br \/>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo><\/mo><\/mrow><mrow><mo><\/mo><\/mrow><\/math>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]1-i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?<\/li>\r\n\t<li>Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]0.75+0.5i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction.<\/li>\r\n<\/ol>\r\n<h2>Recursive Sequences<\/h2>\r\n<ol start=\"21\">\r\n\t<li>Given the recursive relationship z<sub>n+1<\/sub> =\u00a0 iz<sub>n<\/sub>+i, z<sub>0<\/sub> = 2, generate the next 3 terms of the recursive sequence.<\/li>\r\n\t<li>Given the recursive relationship z<sub>n+1<\/sub> = 2z<sub>n<\/sub>+i, z<sub>0<\/sub> = 3-2i, generate the next 3 terms of the recursive sequence.<\/li>\r\n\t<li>Using c = [latex]-0.25[\/latex], calculate the first [latex]4[\/latex] terms of the Mandelbrot sequence.<\/li>\r\n\t<li>Using c = [latex]1[\/latex]-i, calculate the first [latex]4[\/latex] terms of the Mandelbrot sequence., calculate the first 4 terms of the Mandelbrot sequence.<\/li>\r\n<\/ol>\r\n<p>For a given value of\u00a0<em>c<\/em>, the Mandelbrot sequence can be described as\u00a0<em>escaping<\/em>\u00a0(growing large), a\u00a0<em>attracted<\/em>\u00a0(it approaches a fixed value), or\u00a0<em>periodic<\/em>\u00a0(it jumps between several fixed values). A periodic cycle is typically described the number if values it jumps between; a [latex]2[\/latex]-cycle jumps between [latex]2[\/latex] values, and a [latex]4[\/latex]-cycle jumps between [latex]4[\/latex] values.<\/p>\r\n<p>For questions [latex]25 \u2013 30[\/latex], you\u2019ll want to use a calculator that can compute with complex numbers, or use\u00a0an online calculator\u00a0which can compute a Mandelbrot sequence. For each value of\u00a0<em>c<\/em>, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?<\/p>\r\n<ol start=\"25\">\r\n\t<li>[latex]c = -0.5+0.25i.[\/latex]<\/li>\r\n\t<li>[latex]c = 0.25+-0.25i.[\/latex]<\/li>\r\n\t<li>[latex]c = -1.2.[\/latex]<\/li>\r\n\t<li>[latex]c = i.[\/latex]<\/li>\r\n\t<li>[latex]c = 0.5+0.25i.[\/latex]<\/li>\r\n\t<li>[latex]c = -0.5+0.5i.[\/latex]<\/li>\r\n\t<li>[latex]c = -0.12+0.75i.[\/latex]<\/li>\r\n\t<li>[latex]c = -0.5+0.5i.[\/latex]<\/li>\r\n<\/ol>","rendered":"<h2>Iterated Fractals<\/h2>\n<p>Using the initiator and generator shown, draw the next two stages of the iterated fractal.<\/p>\n<blockquote>\n<table>\n<tbody>\n<tr>\n<td>\n<p>1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1.png\" sizes=\"(max-width: 190px) 100vw, 190px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1.png 190w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23234524\/exercise1-65x23.png 65w\" alt=\"Initiator is a horizontal line. Generator is a horizontal line that then goes up at a right angle, right at a right angle, down at a right angle, and then continues horizontally.\" width=\"190\" height=\"68\" \/><\/p>\n<\/td>\n<td>\n<p>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1742\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2.png\" sizes=\"(max-width: 203px) 100vw, 203px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2.png 203w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235224\/exercise2-65x22.png 65w\" alt=\"Initiator is a horizontal line. Generator is a zig-zag.\" width=\"203\" height=\"68\" \/><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1743\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3.png\" sizes=\"(max-width: 189px) 100vw, 189px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3.png 189w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235324\/exercise3-65x23.png 65w\" alt=\"Initiator is an upward-sloping line. Generator is that line with smaller lines branching off of it.\" width=\"189\" height=\"67\" \/><\/p>\n<\/td>\n<td>\n<p>4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4.png\" sizes=\"(max-width: 202px) 100vw, 202px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4.png 202w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235407\/exercise4-65x16.png 65w\" alt=\"Initiator is a horizontal line. Generator is two short horizontal lines side-by-side.\" width=\"202\" height=\"51\" \/><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>5.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1745\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5.png\" sizes=\"(max-width: 187px) 100vw, 187px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5.png 187w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235611\/exercise5-65x35.png 65w\" alt=\"Initiator is a square. Generator is eight more squares arranged to form the border of a large square.\" width=\"187\" height=\"100\" \/><\/p>\n<\/td>\n<td>\n<p>6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1746 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6.png\" sizes=\"(max-width: 198px) 100vw, 198px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6.png 198w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23235831\/exercise6-65x33.png 65w\" alt=\"Initiator is an equilateral triangle. Generator is three equilateral triangles that touch each other at an angle.\" width=\"198\" height=\"100\" \/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"7\">\n<li>Create your own version of Sierpinski gasket with added randomness.<\/li>\n<li>Create a version of the branching tree fractal from example #[latex]3[\/latex] with added randomness.<\/li>\n<\/ol>\n<\/blockquote>\n<h2>Fractal Dimension<\/h2>\n<ol start=\"9\">\n<li>Determine the fractal dimension of the Koch curve.<\/li>\n<li>Determine the fractal dimension of the curve generated in exercise #[latex]1[\/latex]<\/li>\n<li>Determine the fractal dimension of the Sierpinski carpet generated in exercise #[latex]5[\/latex]<\/li>\n<li>Determine the fractal dimension of the Cantor set generated in exercise #[latex]4[\/latex]<\/li>\n<\/ol>\n<h2>Complex Numbers<\/h2>\n<ol start=\"13\">\n<li>Plot each number in the complex plane:\n<ol>\n<li>[latex]4[\/latex]<\/li>\n<li>[latex]\u20133i[\/latex]<\/li>\n<li>[latex]\u20132+3i[\/latex]<\/li>\n<li>[latex]2 + i[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Plot each number in the complex plane:\n<ol>\n<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]-2[\/latex]<\/span><\/li>\n<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]4i[\/latex]<\/span><\/li>\n<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]1+2i[\/latex]<\/span><\/li>\n<li><span style=\"font-size: 17.44px; text-wrap: nowrap;\">[latex]-1-i[\/latex]<\/span><\/li>\n<\/ol>\n<\/li>\n<li>Compute:\n<ol>\n<li>[latex](2+3i) + (3-4i)[\/latex]<\/li>\n<li>[latex](3-5i) - (-2-i)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Compute:\n<ol>\n<li>[latex](1-i) + (2+4i)[\/latex]<\/li>\n<li>[latex](\u00a0-2-3i) - (4-2i)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Multiply:\n<ol>\n<li>[latex]3(2+4i)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex](2i)(-1-5i)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex](2-4i)(1+3i)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Multiply:\n<ol>\n<li class=\"whitespace-normal break-words\">[latex]2(-1+3i)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex](3i)(2-6i)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex](1-i)(2+5i)[\/latex]<br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo><\/mo><\/mrow><mrow><mo><\/mo><\/mrow><\/math>\n<\/li>\n<\/ol>\n<\/li>\n<li>Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]1-i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?<\/li>\n<li>Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]0.75+0.5i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction.<\/li>\n<\/ol>\n<h2>Recursive Sequences<\/h2>\n<ol start=\"21\">\n<li>Given the recursive relationship z<sub>n+1<\/sub> =\u00a0 iz<sub>n<\/sub>+i, z<sub>0<\/sub> = 2, generate the next 3 terms of the recursive sequence.<\/li>\n<li>Given the recursive relationship z<sub>n+1<\/sub> = 2z<sub>n<\/sub>+i, z<sub>0<\/sub> = 3-2i, generate the next 3 terms of the recursive sequence.<\/li>\n<li>Using c = [latex]-0.25[\/latex], calculate the first [latex]4[\/latex] terms of the Mandelbrot sequence.<\/li>\n<li>Using c = [latex]1[\/latex]-i, calculate the first [latex]4[\/latex] terms of the Mandelbrot sequence., calculate the first 4 terms of the Mandelbrot sequence.<\/li>\n<\/ol>\n<p>For a given value of\u00a0<em>c<\/em>, the Mandelbrot sequence can be described as\u00a0<em>escaping<\/em>\u00a0(growing large), a\u00a0<em>attracted<\/em>\u00a0(it approaches a fixed value), or\u00a0<em>periodic<\/em>\u00a0(it jumps between several fixed values). A periodic cycle is typically described the number if values it jumps between; a [latex]2[\/latex]-cycle jumps between [latex]2[\/latex] values, and a [latex]4[\/latex]-cycle jumps between [latex]4[\/latex] values.<\/p>\n<p>For questions [latex]25 \u2013 30[\/latex], you\u2019ll want to use a calculator that can compute with complex numbers, or use\u00a0an online calculator\u00a0which can compute a Mandelbrot sequence. For each value of\u00a0<em>c<\/em>, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?<\/p>\n<ol start=\"25\">\n<li>[latex]c = -0.5+0.25i.[\/latex]<\/li>\n<li>[latex]c = 0.25+-0.25i.[\/latex]<\/li>\n<li>[latex]c = -1.2.[\/latex]<\/li>\n<li>[latex]c = i.[\/latex]<\/li>\n<li>[latex]c = 0.5+0.25i.[\/latex]<\/li>\n<li>[latex]c = -0.5+0.5i.[\/latex]<\/li>\n<li>[latex]c = -0.12+0.75i.[\/latex]<\/li>\n<li>[latex]c = -0.5+0.5i.[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":23,"menu_order":18,"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":"practice","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\/4926"}],"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":46,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4926\/revisions"}],"predecessor-version":[{"id":15446,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4926\/revisions\/15446"}],"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\/4926\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4926"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4926"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4926"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4926"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}