{"id":3134,"date":"2025-08-15T22:11:37","date_gmt":"2025-08-15T22:11:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3134"},"modified":"2026-01-13T19:41:52","modified_gmt":"2026-01-13T19:41:52","slug":"complex-numbers-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/complex-numbers-apply-it\/","title":{"raw":"Complex Numbers: Apply It","rendered":"Complex Numbers: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Express square roots of negative numbers as multiples of i.<\/li>\r\n \t<li>Plot complex numbers on the complex plane.<\/li>\r\n \t<li>Add, subtract, and multiply complex numbers.<\/li>\r\n \t<li>Rationalize complex denominators<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Complex Numbers in Medical Imaging<\/h2>\r\n<p class=\"whitespace-normal break-words\">Medical imaging technologies like MRI and CT scanners rely on complex numbers to transform raw signal data into the detailed images doctors use for diagnosis. These systems collect frequency information that must be mathematically processed using complex number operations to reconstruct clear, accurate medical images.<\/p>\r\n\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">MRI Signal Processing<\/h3>\r\n<p class=\"whitespace-normal break-words\">Magnetic Resonance Imaging (MRI) machines detect radio frequency signals emitted by hydrogen atoms in the body. These signals are naturally complex, containing both amplitude and phase information that complex numbers can represent mathematically.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">An MRI scanner receives signals from a patient's brain tissue. The raw signal from one location is [latex]S = 150 + 200i[\/latex], where the real part represents the signal strength in phase with the reference, and the imaginary part represents the signal strength [latex]90^\\circ[\/latex] out of phase.\r\n<p class=\"whitespace-pre-wrap break-words\">To determine the total signal intensity (what appears as brightness in the final image), radiologists calculate the magnitude: [latex]|S| = \\sqrt{150^2 + 200^2} = \\sqrt{22,500 + 40,000} = \\sqrt{62,500} = 250[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Different tissues produce different phase signatures: gray matter might produce [latex]S_1 = 150 + 200i[\/latex], while white matter produces [latex]S_2 = 180 + 120i[\/latex], allowing doctors to differentiate brain structures.<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>magnitude<\/h3>\r\nIn medical imaging, the magnitude [latex]|z| = \\sqrt{a^2 + b^2}[\/latex] determines pixel brightness, while the phase angle helps identify tissue composition and detect abnormalities.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311090[\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Fourier Transform Reconstruction<\/h3>\r\n<p class=\"whitespace-normal break-words\">Medical scanners collect data in the frequency domain and use complex-valued Fourier transforms glossary: mathematical technique converting between time\/space and frequency domains to reconstruct spatial images. This process requires extensive complex number arithmetic.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How to: Processing Medical Image Data<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Collect complex frequency data from scanner detectors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply complex multiplication to adjust signal phases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add multiple complex signals from different detector positions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate magnitudes for final pixel intensities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use phase information to enhance tissue contrast<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine real and imaginary components to reconstruct spatial information<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A CT scanner collects projection data from multiple angles around a patient's chest. Three key frequency components are:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]F_1 = 80 + 60i[\/latex] (low frequency component)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]F_2 = 40 - 30i[\/latex] (medium frequency component)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]F_3 = 20 + 15i[\/latex] (high frequency component)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">To reconstruct one pixel, the system combines these frequencies:\\begin{aligned}\r\nP &amp;= F_1 + F_2 + F_3 \\\\\r\n&amp;= (80 + 60i) + (40 - 30i) + (20 + 15i) \\\\\r\n&amp;= (80 + 40 + 20) + (60 - 30 + 15)i \\\\\r\n&amp;= 140 + 45i\r\n\\end{aligned}<\/p>\r\n<p class=\"whitespace-normal break-words\">The pixel intensity is [latex]|P| = \\sqrt{140^2 + 45^2} = \\sqrt{19,600 + 2,025} = \\sqrt{21,625} = 147[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This intensity value gets converted to a grayscale level in the final medical image.<\/p>\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">When adding complex numbers, combine like terms: add all real parts together and add all imaginary parts together.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311089[\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Contrast Enhancement and Filtering<\/h3>\r\n<p class=\"whitespace-normal break-words\">Medical imaging systems use complex number multiplication to enhance contrast between healthy and diseased tissues, making diagnoses more accurate.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A mammography system applies a complex filter [latex]H = 1.2 + 0.8i[\/latex] to enhance tumor detection. When this filter processes a tissue signal [latex]T = 100 + 50i[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{align} T_{enhanced} &amp;= H \\cdot T \\\\ &amp;= (1.2 + 0.8i)(100 + 50i) \\\\ &amp;= 1.2 \\cdot 100 + 1.2 \\cdot 50i + 0.8i \\cdot 100 + 0.8i \\cdot 50i \\\\ &amp;= 120 + 60i + 80i + 40i^2 \\\\ &amp;= 120 + 140i + 40(-1) \\\\ &amp;= 120 + 140i - 40 \\\\ &amp;= 80 + 140i \\end{align}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Original signal magnitude: [latex]|T| = \\sqrt{100^2 + 50^2} = \\sqrt{12,500} = 111.8[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Enhanced signal magnitude: [latex]|T_{enhanced}| = \\sqrt{80^2 + 140^2} = \\sqrt{26,000} = 161.2[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The filter increased signal strength by [latex]\\frac{161.2}{111.8}=0.44 = 44[\/latex]%, making potential abnormalities more visible to radiologists.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]311091[\/ohm_question]<\/p>\r\n\r\n<\/section>\r\n<h3>Phase-Sensitive Imaging Techniques<\/h3>\r\n<p class=\"whitespace-normal break-words\">Advanced medical imaging uses phase information from complex numbers to detect subtle tissue changes that magnitude alone cannot reveal.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">An experimental brain imaging technique compares signals from two adjacent regions:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Region A (healthy): [latex]S_A = 180 + 240i[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region B (potentially damaged): [latex]S_B = 220 + 140i[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">The phase difference analysis involves dividing one signal by the other: [latex]\\frac{S_B}{S_A} = \\frac{220 + 140i}{180 + 240i}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Rationalizing the denominator: [latex]\\frac{S_B}{S_A} = \\frac{220 + 140i}{180 + 240i} \\cdot \\frac{180 - 240i}{180 - 240i} = \\frac{(220 + 140i)(180 - 240i)}{180^2 + 240^2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]= \\frac{39,600 - 52,800i + 25,200i - 33,600i^2}{32,400 + 57,600} = \\frac{39,600 - 27,600i + 33,600}{90,000}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]= \\frac{73,200 - 27,600i}{90,000} = 0.813 - 0.307i[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The magnitude [latex]|0.813 - 0.307i| = 0.869[\/latex]. This indicates Region B has [latex]1.000-0.869=-.13=13[\/latex]% lower signal intensity, potentially indicating tissue damage.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311092[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Express square roots of negative numbers as multiples of i.<\/li>\n<li>Plot complex numbers on the complex plane.<\/li>\n<li>Add, subtract, and multiply complex numbers.<\/li>\n<li>Rationalize complex denominators<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Complex Numbers in Medical Imaging<\/h2>\n<p class=\"whitespace-normal break-words\">Medical imaging technologies like MRI and CT scanners rely on complex numbers to transform raw signal data into the detailed images doctors use for diagnosis. These systems collect frequency information that must be mathematically processed using complex number operations to reconstruct clear, accurate medical images.<\/p>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">MRI Signal Processing<\/h3>\n<p class=\"whitespace-normal break-words\">Magnetic Resonance Imaging (MRI) machines detect radio frequency signals emitted by hydrogen atoms in the body. These signals are naturally complex, containing both amplitude and phase information that complex numbers can represent mathematically.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">An MRI scanner receives signals from a patient&#8217;s brain tissue. The raw signal from one location is [latex]S = 150 + 200i[\/latex], where the real part represents the signal strength in phase with the reference, and the imaginary part represents the signal strength [latex]90^\\circ[\/latex] out of phase.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">To determine the total signal intensity (what appears as brightness in the final image), radiologists calculate the magnitude: [latex]|S| = \\sqrt{150^2 + 200^2} = \\sqrt{22,500 + 40,000} = \\sqrt{62,500} = 250[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Different tissues produce different phase signatures: gray matter might produce [latex]S_1 = 150 + 200i[\/latex], while white matter produces [latex]S_2 = 180 + 120i[\/latex], allowing doctors to differentiate brain structures.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>magnitude<\/h3>\n<p>In medical imaging, the magnitude [latex]|z| = \\sqrt{a^2 + b^2}[\/latex] determines pixel brightness, while the phase angle helps identify tissue composition and detect abnormalities.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311090&theme=lumen&iframe_resize_id=ohm311090&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Fourier Transform Reconstruction<\/h3>\n<p class=\"whitespace-normal break-words\">Medical scanners collect data in the frequency domain and use complex-valued Fourier transforms glossary: mathematical technique converting between time\/space and frequency domains to reconstruct spatial images. This process requires extensive complex number arithmetic.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How to: Processing Medical Image Data<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Collect complex frequency data from scanner detectors<\/li>\n<li class=\"whitespace-normal break-words\">Apply complex multiplication to adjust signal phases<\/li>\n<li class=\"whitespace-normal break-words\">Add multiple complex signals from different detector positions<\/li>\n<li class=\"whitespace-normal break-words\">Calculate magnitudes for final pixel intensities<\/li>\n<li class=\"whitespace-normal break-words\">Use phase information to enhance tissue contrast<\/li>\n<li class=\"whitespace-normal break-words\">Combine real and imaginary components to reconstruct spatial information<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A CT scanner collects projection data from multiple angles around a patient&#8217;s chest. Three key frequency components are:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]F_1 = 80 + 60i[\/latex] (low frequency component)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]F_2 = 40 - 30i[\/latex] (medium frequency component)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]F_3 = 20 + 15i[\/latex] (high frequency component)<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">To reconstruct one pixel, the system combines these frequencies:\\begin{aligned}<br \/>\nP &amp;= F_1 + F_2 + F_3 \\\\<br \/>\n&amp;= (80 + 60i) + (40 &#8211; 30i) + (20 + 15i) \\\\<br \/>\n&amp;= (80 + 40 + 20) + (60 &#8211; 30 + 15)i \\\\<br \/>\n&amp;= 140 + 45i<br \/>\n\\end{aligned}<\/p>\n<p class=\"whitespace-normal break-words\">The pixel intensity is [latex]|P| = \\sqrt{140^2 + 45^2} = \\sqrt{19,600 + 2,025} = \\sqrt{21,625} = 147[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This intensity value gets converted to a grayscale level in the final medical image.<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">When adding complex numbers, combine like terms: add all real parts together and add all imaginary parts together.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311089\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311089&theme=lumen&iframe_resize_id=ohm311089&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Contrast Enhancement and Filtering<\/h3>\n<p class=\"whitespace-normal break-words\">Medical imaging systems use complex number multiplication to enhance contrast between healthy and diseased tissues, making diagnoses more accurate.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A mammography system applies a complex filter [latex]H = 1.2 + 0.8i[\/latex] to enhance tumor detection. When this filter processes a tissue signal [latex]T = 100 + 50i[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{align} T_{enhanced} &= H \\cdot T \\\\ &= (1.2 + 0.8i)(100 + 50i) \\\\ &= 1.2 \\cdot 100 + 1.2 \\cdot 50i + 0.8i \\cdot 100 + 0.8i \\cdot 50i \\\\ &= 120 + 60i + 80i + 40i^2 \\\\ &= 120 + 140i + 40(-1) \\\\ &= 120 + 140i - 40 \\\\ &= 80 + 140i \\end{align}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Original signal magnitude: [latex]|T| = \\sqrt{100^2 + 50^2} = \\sqrt{12,500} = 111.8[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Enhanced signal magnitude: [latex]|T_{enhanced}| = \\sqrt{80^2 + 140^2} = \\sqrt{26,000} = 161.2[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The filter increased signal strength by [latex]\\frac{161.2}{111.8}=0.44 = 44[\/latex]%, making potential abnormalities more visible to radiologists.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm311091\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311091&theme=lumen&iframe_resize_id=ohm311091&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3>Phase-Sensitive Imaging Techniques<\/h3>\n<p class=\"whitespace-normal break-words\">Advanced medical imaging uses phase information from complex numbers to detect subtle tissue changes that magnitude alone cannot reveal.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">An experimental brain imaging technique compares signals from two adjacent regions:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Region A (healthy): [latex]S_A = 180 + 240i[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Region B (potentially damaged): [latex]S_B = 220 + 140i[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">The phase difference analysis involves dividing one signal by the other: [latex]\\frac{S_B}{S_A} = \\frac{220 + 140i}{180 + 240i}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Rationalizing the denominator: [latex]\\frac{S_B}{S_A} = \\frac{220 + 140i}{180 + 240i} \\cdot \\frac{180 - 240i}{180 - 240i} = \\frac{(220 + 140i)(180 - 240i)}{180^2 + 240^2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]= \\frac{39,600 - 52,800i + 25,200i - 33,600i^2}{32,400 + 57,600} = \\frac{39,600 - 27,600i + 33,600}{90,000}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]= \\frac{73,200 - 27,600i}{90,000} = 0.813 - 0.307i[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The magnitude [latex]|0.813 - 0.307i| = 0.869[\/latex]. This indicates Region B has [latex]1.000-0.869=-.13=13[\/latex]% lower signal intensity, potentially indicating tissue damage.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311092\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311092&theme=lumen&iframe_resize_id=ohm311092&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":506,"module-header":"apply_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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3134"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3134\/revisions"}],"predecessor-version":[{"id":5332,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3134\/revisions\/5332"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3134\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3134"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3134"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3134"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}