{"id":16446,"date":"2019-10-03T15:48:51","date_gmt":"2019-10-03T15:48:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/introduction-operations-on-complex-numbers\/"},"modified":"2024-05-02T15:48:06","modified_gmt":"2024-05-02T15:48:06","slug":"introduction-operations-on-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/introduction-operations-on-complex-numbers\/","title":{"raw":"Adding and Subtracting Complex Numbers","rendered":"Adding and Subtracting Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add complex numbers<\/li>\r\n \t<li>Subtract complex numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\nAny time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, \u201cHow do you add them?\u201d In this section, you will learn how to add and subtract complex numbers.\r\n\r\nFirst, consider the following expression.\r\n<p style=\"text-align: center;\">[latex](6x+8)+(4x+2)[\/latex]<\/p>\r\nTo simplify this expression, you combine the like terms, [latex]6x[\/latex] and [latex]4x[\/latex].<i>\u00a0<\/i>These are like terms because they have the same variable with the same exponents. Similarly, 8 and 2 are like terms because they are both constants, with no variables.\r\n<p style=\"text-align: center;\">[latex](6x+8)+(4x+2)=10x+10[\/latex]<\/p>\r\nIn the same way, you can simplify expressions with radicals.\r\n<p style=\"text-align: center;\">[latex] (6\\sqrt{3}+8)+(4\\sqrt{3}+2)=10\\sqrt{3}+10[\/latex]<\/p>\r\nYou can add [latex] 6\\sqrt{3}[\/latex] to [latex] 4\\sqrt{3}[\/latex] because the two terms have the same radical, [latex]\\sqrt{3}[\/latex], just as [latex]6x[\/latex] and\u00a0[latex]4x[\/latex] have the same variable and exponent.\r\n\r\nThe number [latex]i[\/latex] looks like a variable, but remember that it is equal to [latex]\\sqrt{-1}[\/latex]. The great thing is you have no new rules to worry about\u2014whether you treat it as a variable or a radical, the exact same rules apply to adding and subtracting <strong>complex numbers<\/strong>. You combine the imaginary parts (the terms with [latex]i[\/latex]),<i> <\/i>and you combine the real parts.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd.\u00a0[latex](\u22123+3i)+(7\u20132i)[\/latex]\r\n\r\n[reveal-answer q=\"929105\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929105\"]\r\n\r\nRearrange the sums to put like terms together.\r\n<p style=\"text-align: center;\">[latex]\u22123+3i+7\u20132i=\u22123+7+3i\u20132i[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]\u22123+7=4[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]3i\u20132i=(3\u20132)i=i[\/latex]<\/p>\r\nThe answer is [latex]4+i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract.\u00a0[latex](\u22123+3i)\u2013(7\u20132i)[\/latex]\r\n\r\n[reveal-answer q=\"203125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"203125\"]\r\n\r\nBe sure to distribute the subtraction sign to all terms in the set of parentheses that follows.\r\n<p style=\"text-align: center;\">[latex](\u22123+3i)\u2013(7\u20132i)=\u22123+3i\u20137+2i[\/latex]<i>\u00a0<\/i><\/p>\r\nRearrange the terms to put like terms together.\r\n<p style=\"text-align: center;\">[latex]\u22123\u20137+3i+2i[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]\u22123\u20137=\u221210[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]3i+2i=(3+2)i=5i[\/latex]<\/p>\r\nThe answer is [latex]-10+5i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3901[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of how to add and subtract complex numbers.\r\n\r\nhttps:\/\/youtu.be\/SGhTjioGqqA","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add complex numbers<\/li>\n<li>Subtract complex numbers<\/li>\n<\/ul>\n<\/div>\n<p>Any time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, \u201cHow do you add them?\u201d In this section, you will learn how to add and subtract complex numbers.<\/p>\n<p>First, consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex](6x+8)+(4x+2)[\/latex]<\/p>\n<p>To simplify this expression, you combine the like terms, [latex]6x[\/latex] and [latex]4x[\/latex].<i>\u00a0<\/i>These are like terms because they have the same variable with the same exponents. Similarly, 8 and 2 are like terms because they are both constants, with no variables.<\/p>\n<p style=\"text-align: center;\">[latex](6x+8)+(4x+2)=10x+10[\/latex]<\/p>\n<p>In the same way, you can simplify expressions with radicals.<\/p>\n<p style=\"text-align: center;\">[latex](6\\sqrt{3}+8)+(4\\sqrt{3}+2)=10\\sqrt{3}+10[\/latex]<\/p>\n<p>You can add [latex]6\\sqrt{3}[\/latex] to [latex]4\\sqrt{3}[\/latex] because the two terms have the same radical, [latex]\\sqrt{3}[\/latex], just as [latex]6x[\/latex] and\u00a0[latex]4x[\/latex] have the same variable and exponent.<\/p>\n<p>The number [latex]i[\/latex] looks like a variable, but remember that it is equal to [latex]\\sqrt{-1}[\/latex]. The great thing is you have no new rules to worry about\u2014whether you treat it as a variable or a radical, the exact same rules apply to adding and subtracting <strong>complex numbers<\/strong>. You combine the imaginary parts (the terms with [latex]i[\/latex]),<i> <\/i>and you combine the real parts.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add.\u00a0[latex](\u22123+3i)+(7\u20132i)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q929105\">Show Solution<\/span><\/p>\n<div id=\"q929105\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rearrange the sums to put like terms together.<\/p>\n<p style=\"text-align: center;\">[latex]\u22123+3i+7\u20132i=\u22123+7+3i\u20132i[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\u22123+7=4[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]3i\u20132i=(3\u20132)i=i[\/latex]<\/p>\n<p>The answer is [latex]4+i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract.\u00a0[latex](\u22123+3i)\u2013(7\u20132i)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q203125\">Show Solution<\/span><\/p>\n<div id=\"q203125\" class=\"hidden-answer\" style=\"display: none\">\n<p>Be sure to distribute the subtraction sign to all terms in the set of parentheses that follows.<\/p>\n<p style=\"text-align: center;\">[latex](\u22123+3i)\u2013(7\u20132i)=\u22123+3i\u20137+2i[\/latex]<i>\u00a0<\/i><\/p>\n<p>Rearrange the terms to put like terms together.<\/p>\n<p style=\"text-align: center;\">[latex]\u22123\u20137+3i+2i[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\u22123\u20137=\u221210[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]3i+2i=(3+2)i=5i[\/latex]<\/p>\n<p>The answer is [latex]-10+5i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3901\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3901&theme=oea&iframe_resize_id=ohm3901&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we show more examples of how to add and subtract complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Adding and Subtracting Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SGhTjioGqqA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-16446\">\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>Ex 1: Adding and Subtracting Complex Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SGhTjioGqqA\">https:\/\/youtu.be\/SGhTjioGqqA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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 fro free at:  http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":169554,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Adding and Subtracting Complex Numbers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/SGhTjioGqqA\",\"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@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download fro free at:  http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"485490bd5efe4c1a964f6209ab347300, 6af6c2c9946842ceaf55cbfd43bef368","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16446","chapter","type-chapter","status-publish","hentry"],"part":16201,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16446","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16446\/revisions"}],"predecessor-version":[{"id":19981,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16446\/revisions\/19981"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16201"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16446\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16446"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16446"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16446"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}