{"id":226,"date":"2025-02-13T22:45:17","date_gmt":"2025-02-13T22:45:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers\/"},"modified":"2025-08-15T13:54:59","modified_gmt":"2025-08-15T13:54:59","slug":"polar-form-of-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers\/","title":{"raw":"Polar Form of Complex Numbers: Learn It 1","rendered":"Polar Form of Complex Numbers: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Plot complex numbers in the complex plane.<\/li>\r\n \t<li>Write complex numbers in polar form.<\/li>\r\n \t<li>Convert a complex number from polar to rectangular form.<\/li>\r\n \t<li>Find products and quotients of complex numbers in polar form.<\/li>\r\n \t<li>Find powers and roots of complex numbers in polar form.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nComplex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.\r\n<h2>Plotting Complex Numbers in the Complex Plane<\/h2>\r\nPlotting a <strong>complex number<\/strong> [latex]a+bi[\/latex] is similar to plotting a real number, except that the horizontal axis represents the real part of the number, [latex]a[\/latex], and the vertical axis represents the imaginary part of the number, [latex]bi[\/latex].\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a complex number [latex]a+bi[\/latex], plot it in the complex plane.<\/strong>\r\n<ol>\r\n \t<li>Label the horizontal axis as the <em>real<\/em> axis and the vertical axis as the <em>imaginary axis.<\/em><\/li>\r\n \t<li>Plot the point in the complex plane by moving [latex]a[\/latex] units in the horizontal direction and [latex]b[\/latex] units in the vertical direction.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Plot the complex number [latex]2 - 3i[\/latex] in the <strong>complex plane<\/strong>.[reveal-answer q=\"821515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"821515\"]From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180842\/CNX_Precalc_Figure_08_05_0012.jpg\" alt=\"Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).\" width=\"487\" height=\"331\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Plot the point [latex]1+5i[\/latex] in the complex plane.[reveal-answer q=\"522113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"522113\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180857\/CNX_Precalc_Figure_08_05_0022.jpg\" alt=\"Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).\" \/>[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Finding the Absolute Value of a Complex Number<\/h2>\r\nThe first step toward working with a complex number in <strong>polar form<\/strong> is to find the absolute value. The absolute value of a complex number is the same as its <strong>magnitude<\/strong>, or [latex]|z|[\/latex]. It measures the distance from the origin to a point in the plane.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180845\/CNX_Precalc_Figure_08_05_0032.jpg\" alt=\"Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.\" width=\"487\" height=\"368\" \/>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>absolute value of a complex number<\/h3>\r\nGiven [latex]z=x+yi[\/latex], a complex number, the absolute value of [latex]z[\/latex] is defined as\r\n<p style=\"text-align: center;\">[latex]|z|=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/p>\r\nIt is the distance from the origin to the point [latex]\\left(x,y\\right)[\/latex].\r\n\r\nNotice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin, [latex]\\left(0,\\text{ }0\\right)[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the absolute value of [latex]z=\\sqrt{5}-i[\/latex].[reveal-answer q=\"219333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"219333\"]Using the formula, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;|z|=\\sqrt{{x}^{2}+{y}^{2}}\\\\ &amp;|z|=\\sqrt{{\\left(\\sqrt{5}\\right)}^{2}+{\\left(-1\\right)}^{2}} \\\\ &amp;|z|=\\sqrt{5+1} \\\\ &amp;|z|=\\sqrt{6} \\end{align}[\/latex]<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180847\/CNX_Precalc_Figure_08_05_0042.jpg\" alt=\"Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.\" width=\"487\" height=\"331\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nFind the absolute value of the complex number [latex]z=12 - 5i[\/latex].\r\n\r\n[reveal-answer q=\"514025\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"514025\"]\r\n\r\n13\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given [latex]z=3 - 4i[\/latex], find [latex]|z|[\/latex].[reveal-answer q=\"748063\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"748063\"]Using the formula, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;|z|=\\sqrt{{x}^{2}+{y}^{2}} \\\\ &amp;|z|=\\sqrt{{\\left(3\\right)}^{2}+{\\left(-4\\right)}^{2}} \\\\ &amp;|z|=\\sqrt{9+16} \\\\ &amp;|z|=\\sqrt{25}\\\\ &amp;|z|=5 \\end{align}[\/latex]<\/p>\r\nThe absolute value [latex]z[\/latex] is 5.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180849\/CNX_Precalc_Figure_08_05_0052-1.jpg\" alt=\"Plot of (3-4i) in the complex plane and its magnitude |z| =5.\" width=\"487\" height=\"331\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nGiven [latex]z=1 - 7i[\/latex], find [latex]|z|[\/latex].\r\n\r\n[reveal-answer q=\"473571\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473571\"]\r\n\r\n[latex]|z|=\\sqrt{50}=5\\sqrt{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]173801[\/ohm_question]<\/section>\r\n<dl id=\"fs-id1165133162992\" class=\"definition\">\r\n \t<dd id=\"fs-id1165133162998\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Plot complex numbers in the complex plane.<\/li>\n<li>Write complex numbers in polar form.<\/li>\n<li>Convert a complex number from polar to rectangular form.<\/li>\n<li>Find products and quotients of complex numbers in polar form.<\/li>\n<li>Find powers and roots of complex numbers in polar form.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.<\/p>\n<h2>Plotting Complex Numbers in the Complex Plane<\/h2>\n<p>Plotting a <strong>complex number<\/strong> [latex]a+bi[\/latex] is similar to plotting a real number, except that the horizontal axis represents the real part of the number, [latex]a[\/latex], and the vertical axis represents the imaginary part of the number, [latex]bi[\/latex].<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a complex number [latex]a+bi[\/latex], plot it in the complex plane.<\/strong><\/p>\n<ol>\n<li>Label the horizontal axis as the <em>real<\/em> axis and the vertical axis as the <em>imaginary axis.<\/em><\/li>\n<li>Plot the point in the complex plane by moving [latex]a[\/latex] units in the horizontal direction and [latex]b[\/latex] units in the vertical direction.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Plot the complex number [latex]2 - 3i[\/latex] in the <strong>complex plane<\/strong>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q821515\">Show Solution<\/button><\/p>\n<div id=\"q821515\" class=\"hidden-answer\" style=\"display: none\">From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180842\/CNX_Precalc_Figure_08_05_0012.jpg\" alt=\"Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).\" width=\"487\" height=\"331\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Plot the point [latex]1+5i[\/latex] in the complex plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q522113\">Show Solution<\/button><\/p>\n<div id=\"q522113\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180857\/CNX_Precalc_Figure_08_05_0022.jpg\" alt=\"Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).\" \/><\/div>\n<\/div>\n<\/section>\n<h2>Finding the Absolute Value of a Complex Number<\/h2>\n<p>The first step toward working with a complex number in <strong>polar form<\/strong> is to find the absolute value. The absolute value of a complex number is the same as its <strong>magnitude<\/strong>, or [latex]|z|[\/latex]. It measures the distance from the origin to a point in the plane.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180845\/CNX_Precalc_Figure_08_05_0032.jpg\" alt=\"Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.\" width=\"487\" height=\"368\" \/><\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>absolute value of a complex number<\/h3>\n<p>Given [latex]z=x+yi[\/latex], a complex number, the absolute value of [latex]z[\/latex] is defined as<\/p>\n<p style=\"text-align: center;\">[latex]|z|=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/p>\n<p>It is the distance from the origin to the point [latex]\\left(x,y\\right)[\/latex].<\/p>\n<p>Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin, [latex]\\left(0,\\text{ }0\\right)[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the absolute value of [latex]z=\\sqrt{5}-i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q219333\">Show Solution<\/button><\/p>\n<div id=\"q219333\" class=\"hidden-answer\" style=\"display: none\">Using the formula, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&|z|=\\sqrt{{x}^{2}+{y}^{2}}\\\\ &|z|=\\sqrt{{\\left(\\sqrt{5}\\right)}^{2}+{\\left(-1\\right)}^{2}} \\\\ &|z|=\\sqrt{5+1} \\\\ &|z|=\\sqrt{6} \\end{align}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180847\/CNX_Precalc_Figure_08_05_0042.jpg\" alt=\"Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.\" width=\"487\" height=\"331\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>Find the absolute value of the complex number [latex]z=12 - 5i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q514025\">Show Solution<\/button><\/p>\n<div id=\"q514025\" class=\"hidden-answer\" style=\"display: none\">\n<p>13<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]z=3 - 4i[\/latex], find [latex]|z|[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q748063\">Show Solution<\/button><\/p>\n<div id=\"q748063\" class=\"hidden-answer\" style=\"display: none\">Using the formula, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&|z|=\\sqrt{{x}^{2}+{y}^{2}} \\\\ &|z|=\\sqrt{{\\left(3\\right)}^{2}+{\\left(-4\\right)}^{2}} \\\\ &|z|=\\sqrt{9+16} \\\\ &|z|=\\sqrt{25}\\\\ &|z|=5 \\end{align}[\/latex]<\/p>\n<p>The absolute value [latex]z[\/latex] is 5.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180849\/CNX_Precalc_Figure_08_05_0052-1.jpg\" alt=\"Plot of (3-4i) in the complex plane and its magnitude |z| =5.\" width=\"487\" height=\"331\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>Given [latex]z=1 - 7i[\/latex], find [latex]|z|[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q473571\">Show Solution<\/button><\/p>\n<div id=\"q473571\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]|z|=\\sqrt{50}=5\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm173801\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173801&theme=lumen&iframe_resize_id=ohm173801&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<dl id=\"fs-id1165133162992\" class=\"definition\">\n<dd id=\"fs-id1165133162998\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/226"}],"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\/6"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/226\/revisions"}],"predecessor-version":[{"id":2961,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/226\/revisions\/2961"}],"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\/226\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=226"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=226"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=226"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}