{"id":191,"date":"2023-06-21T13:22:42","date_gmt":"2023-06-21T13:22:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-complex-numbers\/"},"modified":"2023-07-03T20:16:24","modified_gmt":"2023-07-03T20:16:24","slug":"summary-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-complex-numbers\/","title":{"raw":"Summary: Complex Numbers","rendered":"Summary: Complex Numbers"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135261454\">\n \t<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\n \t<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\n \t<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\n \t<li>Complex numbers can be multiplied and divided.<\/li>\n \t<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\n \t<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\n \t<li>The powers of [latex]i[\/latex]&nbsp;are cyclic, repeating every fourth one.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135320095\" class=\"definition\">\n \t<dt><strong>complex conjugate<\/strong><\/dt>\n \t<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135320107\" class=\"definition\">\n \t<dt><strong>complex number<\/strong><\/dt>\n \t<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex]&nbsp;is the real part, and [latex]bi[\/latex]&nbsp;is the imaginary part<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260439\" class=\"definition\">\n \t<dt><strong>complex plane<\/strong><\/dt>\n \t<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260450\" class=\"definition\">\n \t<dt><strong>imaginary number<\/strong><\/dt>\n \t<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]&nbsp;where [latex]i=\\sqrt{-1}\\\\[\/latex]<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135261454\">\n<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\n<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\n<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\n<li>Complex numbers can be multiplied and divided.<\/li>\n<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\n<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\n<li>The powers of [latex]i[\/latex]&nbsp;are cyclic, repeating every fourth one.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135320095\" class=\"definition\">\n<dt><strong>complex conjugate<\/strong><\/dt>\n<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135320107\" class=\"definition\">\n<dt><strong>complex number<\/strong><\/dt>\n<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex]&nbsp;is the real part, and [latex]bi[\/latex]&nbsp;is the imaginary part<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260439\" class=\"definition\">\n<dt><strong>complex plane<\/strong><\/dt>\n<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260450\" class=\"definition\">\n<dt><strong>imaginary number<\/strong><\/dt>\n<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]&nbsp;where [latex]i=\\sqrt{-1}\\\\[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-191\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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