{"id":2775,"date":"2025-08-13T18:38:43","date_gmt":"2025-08-13T18:38:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2775"},"modified":"2025-10-21T16:36:58","modified_gmt":"2025-10-21T16:36:58","slug":"polar-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-functions-background-youll-need-2\/","title":{"raw":"Polar Functions: Background You'll Need 2","rendered":"Polar Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Compute operations with complex numbers<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">The imaginary number i s[latex]i=\\sqrt{-1}[\/latex]. A complex number is written in the form: [latex]a+bi[\/latex] where [latex]a[\/latex] is the real part and [latex]b[\/latex] represents the coefficient of the imaginary part.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<ol>\r\n \t<li>[latex](3 + 2i) + (5 - 7i) = 8 - 5i[\/latex]<\/li>\r\n \t<li>[latex](2 + 3i)(4 + i)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3 + 2i}{1 - i}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"998377\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"998377\"]\r\n<ol>\r\n \t<li data-start=\"714\" data-end=\"806\">Combine like terms:<br data-start=\"760\" data-end=\"763\" \/>[latex](3 + 2i) + (5 - 7i) = 8 - 5i[\/latex]<\/li>\r\n \t<li data-start=\"714\" data-end=\"806\">Use the distributive property and the fact that [latex]i^2 = -1[\/latex]:\r\n[latex]\r\n\\begin{align}\r\n(2 + 3i)(4 + i) &amp;= 8 + 2i + 12i + 3i^2 \\\\\r\n&amp;= 8 + 14i + 3(-1) \\\\\r\n&amp;= 5 + 14i\r\n\\end{align}\r\n[\/latex]<\/li>\r\n \t<li data-start=\"714\" data-end=\"806\">Multiply by the <strong data-start=\"1055\" data-end=\"1068\">conjugate<\/strong> of the denominator to remove [latex]i[\/latex]:\r\n[latex]\r\n\\begin{align}\r\n\\dfrac{3 + 2i}{1 - i} &amp;= \\dfrac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} \\\\\r\n&amp;= \\dfrac{3 + 3i + 2i + 2i^2}{1 - i^2} \\\\\r\n&amp;= \\dfrac{3 + 5i - 2}{1 + 1} \\\\\r\n&amp;= \\dfrac{1 + 5i}{2} \\\\\r\n&amp;= \\dfrac{1}{2} + \\dfrac{5}{2}i\r\n\\end{align}\r\n[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313881[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313882[\/ohm_question]<\/section>\r\n<h3 data-start=\"1271\" data-end=\"1312\">Squaring a Complex Number<\/h3>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1314\" data-end=\"1429\">To find the square of a complex number [latex](a + bi)^2[\/latex], apply the distributive property:<\/p>\r\n<p data-start=\"1431\" data-end=\"1589\">[latex]\r\n\\begin{align*}\r\n(a + bi)^2 &amp;= (a + bi)(a + bi) \\\\\r\n&amp;= a^2 + abi + abi + b^2i^2 \\\\\r\n&amp;= a^2 + 2abi + b^2(-1) \\\\\r\n&amp;= (a^2 - b^2) + 2abi\r\n\\end{align*}\r\n[\/latex]<\/p>\r\n<p data-start=\"1591\" data-end=\"1695\">So, the square of [latex](a + bi)[\/latex] is another complex number<br data-start=\"1658\" data-end=\"1661\" \/>[latex](a^2 - b^2) + 2abi[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313883[\/ohm_question]<\/section><section class=\"textbox interact\" aria-label=\"Interact\">The TI-84+ series can perform operations with complex numbers.\r\n<ol data-start=\"2393\" data-end=\"2559\">\r\n \t<li data-start=\"2393\" data-end=\"2438\">\r\n<p data-start=\"2396\" data-end=\"2438\">Press <code data-start=\"2402\" data-end=\"2408\">MODE<\/code> \u2192 choose a + bi format.<\/p>\r\n<\/li>\r\n \t<li data-start=\"2439\" data-end=\"2481\">\r\n<p data-start=\"2442\" data-end=\"2481\">Enter your expression: <code data-start=\"2465\" data-end=\"2479\">(3 + 4i) ^ 2<\/code><\/p>\r\n<\/li>\r\n \t<li data-start=\"2482\" data-end=\"2559\">\r\n<p data-start=\"2485\" data-end=\"2559\">Press <code data-start=\"2491\" data-end=\"2498\">ENTER<\/code> to calculate.<br data-start=\"2512\" data-end=\"2515\" \/>Your calculator should display \u20137 + 24i.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Compute operations with complex numbers<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">The imaginary number i s[latex]i=\\sqrt{-1}[\/latex]. A complex number is written in the form: [latex]a+bi[\/latex] where [latex]a[\/latex] is the real part and [latex]b[\/latex] represents the coefficient of the imaginary part.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ol>\n<li>[latex](3 + 2i) + (5 - 7i) = 8 - 5i[\/latex]<\/li>\n<li>[latex](2 + 3i)(4 + i)[\/latex]<\/li>\n<li>[latex]\\dfrac{3 + 2i}{1 - i}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q998377\">Show Solution<\/button><\/p>\n<div id=\"q998377\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li data-start=\"714\" data-end=\"806\">Combine like terms:<br data-start=\"760\" data-end=\"763\" \/>[latex](3 + 2i) + (5 - 7i) = 8 - 5i[\/latex]<\/li>\n<li data-start=\"714\" data-end=\"806\">Use the distributive property and the fact that [latex]i^2 = -1[\/latex]:<br \/>\n[latex]\\begin{align}  (2 + 3i)(4 + i) &= 8 + 2i + 12i + 3i^2 \\\\  &= 8 + 14i + 3(-1) \\\\  &= 5 + 14i  \\end{align}[\/latex]<\/li>\n<li data-start=\"714\" data-end=\"806\">Multiply by the <strong data-start=\"1055\" data-end=\"1068\">conjugate<\/strong> of the denominator to remove [latex]i[\/latex]:<br \/>\n[latex]\\begin{align}  \\dfrac{3 + 2i}{1 - i} &= \\dfrac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} \\\\  &= \\dfrac{3 + 3i + 2i + 2i^2}{1 - i^2} \\\\  &= \\dfrac{3 + 5i - 2}{1 + 1} \\\\  &= \\dfrac{1 + 5i}{2} \\\\  &= \\dfrac{1}{2} + \\dfrac{5}{2}i  \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313881\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313881&theme=lumen&iframe_resize_id=ohm313881&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313882\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313882&theme=lumen&iframe_resize_id=ohm313882&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 data-start=\"1271\" data-end=\"1312\">Squaring a Complex Number<\/h3>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1314\" data-end=\"1429\">To find the square of a complex number [latex](a + bi)^2[\/latex], apply the distributive property:<\/p>\n<p data-start=\"1431\" data-end=\"1589\">[latex]\\begin{align*}  (a + bi)^2 &= (a + bi)(a + bi) \\\\  &= a^2 + abi + abi + b^2i^2 \\\\  &= a^2 + 2abi + b^2(-1) \\\\  &= (a^2 - b^2) + 2abi  \\end{align*}[\/latex]<\/p>\n<p data-start=\"1591\" data-end=\"1695\">So, the square of [latex](a + bi)[\/latex] is another complex number<br data-start=\"1658\" data-end=\"1661\" \/>[latex](a^2 - b^2) + 2abi[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313883\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313883&theme=lumen&iframe_resize_id=ohm313883&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">The TI-84+ series can perform operations with complex numbers.<\/p>\n<ol data-start=\"2393\" data-end=\"2559\">\n<li data-start=\"2393\" data-end=\"2438\">\n<p data-start=\"2396\" data-end=\"2438\">Press <code data-start=\"2402\" data-end=\"2408\">MODE<\/code> \u2192 choose a + bi format.<\/p>\n<\/li>\n<li data-start=\"2439\" data-end=\"2481\">\n<p data-start=\"2442\" data-end=\"2481\">Enter your expression: <code data-start=\"2465\" data-end=\"2479\">(3 + 4i) ^ 2<\/code><\/p>\n<\/li>\n<li data-start=\"2482\" data-end=\"2559\">\n<p data-start=\"2485\" data-end=\"2559\">Press <code data-start=\"2491\" data-end=\"2498\">ENTER<\/code> to calculate.<br data-start=\"2512\" data-end=\"2515\" \/>Your calculator should display \u20137 + 24i.<\/p>\n<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"background_you_need","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\/2775"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2775\/revisions"}],"predecessor-version":[{"id":4780,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2775\/revisions\/4780"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/247"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2775\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2775"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2775"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2775"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2775"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}