{"id":11,"date":"2017-08-23T06:42:32","date_gmt":"2017-08-23T06:42:32","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=11"},"modified":"2019-08-02T10:47:26","modified_gmt":"2019-08-02T10:47:26","slug":"complex-numbers","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/complex-numbers\/","title":{"rendered":"Complex Numbers"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebraic-expressions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/number-sense\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/operations-with-real-numbers\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Complex Numbers<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson you will study how to graph complex numbers in the plane and perform operations with complex numbers.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <strong><em>real numbers<\/em><\/strong> are the field of all rational and irrational numbers.<\/li>\n<\/ul>\n<section>\n<h3>What are complex numbers?<\/h3>\n<p><strong>What are complex numbers?<\/strong><\/p>\n<p>An <em><strong>imaginary number<\/strong><\/em> is the square\u00a0root of a negative number. A <abbr title=\"is the field of numbers in the form a + bi where a and b are real numbers and i is an imaginary number equal to the square root of a negative number.\">complex\u00a0number<\/abbr> is the field of numbers in the form\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image002.gif\" width=\"34\" height=\"11\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0, which is called the standard form of the complex number, where <em>a\u00a0<\/em>and <em>b<\/em> are real numbers and <em>i <\/em>is an imaginary\u00a0number equal to\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image004.gif\" width=\"27\" height=\"18\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In the standard form of the complex number, <em>a <\/em>is the real\u00a0part of the complex number and <em>bi<\/em> is the imaginary part.<\/p>\n<p>A <strong><em>conjugate<\/em><\/strong> is created by changing\u00a0the sign of the imaginary part of a complex number.<\/p>\n<p><strong>How do I perform operations with complex numbers?<\/strong><\/p>\n<p>We will study four operations with complex numbers:<\/p>\n<p>(1) addition,<br \/>\n(2) subtraction,<br \/>\n(3) multiplication, and<br \/>\n(4) division.<\/p>\n<p><strong>First Operation\u2014Addition<\/strong><\/p>\n<p>To add complex numbers, add the real parts and then add the\u00a0imaginary parts. For example,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image006.gif\" width=\"154\" height=\"18\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image008.gif\" width=\"105\" height=\"18\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image010.gif\" width=\"46\" height=\"11\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image012.gif\" width=\"87\" height=\"13\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image014.gif\" width=\"32\" height=\"11\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image016.gif\" width=\"38\" height=\"11\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. To simplify, add the real parts, add\u00a0the imaginary parts and then combine both parts.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image018.gif\" width=\"162\" height=\"35\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p><strong>Second Operation\u2014Subtraction<\/strong><\/p>\n<p>Subtraction works in much the same way as addition. To\u00a0simplify, subtract the real parts, and then subtract the imaginary\u00a0parts. For example,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image020.gif\" width=\"151\" height=\"18\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct result of subtracting\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image022.gif\" width=\"102\" height=\"18\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<p><!--Q--><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image024.gif\" width=\"23\" height=\"11\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image026.gif\" width=\"48\" height=\"11\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image028.gif\" width=\"48\" height=\"11\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image030.gif\" width=\"38\" height=\"11\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<p><!--A--><\/p>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. To simplify, subtract the real parts,\u00a0subtract the imaginary parts, and then combine the two.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p2_clip_image032.gif\" width=\"199\" height=\"39\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>Third Operation\u2014Multiplication<\/h3>\n<p>Multiplication can be done using the FOIL method discussed\u00a0earlier. For example,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image002.gif\" width=\"295\" height=\"40\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Mathematically\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image004.gif\" width=\"42\" height=\"13\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image006.gif\" width=\"128\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Combine like terms for a final answer of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image008.gif\" width=\"40\" height=\"11\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>It\u2019s helpful to memorize the following:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image010.gif\" width=\"42\" height=\"85\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image012.gif\" width=\"92\" height=\"18\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image014.gif\" width=\"31\" height=\"11\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image016.gif\" width=\"47\" height=\"11\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image018.gif\" width=\"33\" height=\"11\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image020.gif\" width=\"38\" height=\"11\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B. To simplify, use the FOIL method.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image022.gif\" width=\"206\" height=\"136\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p><strong>Fourth Operation\u2014Division <\/strong><\/p>\n<p>To divide complex numbers, multiply by the conjugate of the\u00a0denominator. For example, divide\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image024.gif\" width=\"35\" height=\"29\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The conjugate of the denominator is\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image026.gif\" width=\"32\" height=\"11\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Multiply the numerator and denominator by the conjugate, which\u00a0will guarantee that you are multiplying by a form of one and not\u00a0changing the value of the expression, to get\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image028.gif\" width=\"85\" height=\"32\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Use the FOIL method to multiply the numerators.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image030.gif\" width=\"164\" height=\"112\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Then use the FOIL method to multiply the denominators.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image032.gif\" width=\"205\" height=\"112\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Finally, write the numerator over the denominator.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image034.gif\" width=\"51\" height=\"29\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image036.gif\" width=\"34\" height=\"29\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" style=\"padding: 5px;\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image038.gif\" width=\"49\" height=\"29\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" style=\"padding: 5px;\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image040.gif\" width=\"49\" height=\"29\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" style=\"padding: 5px;\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image042.gif\" width=\"33\" height=\"29\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" style=\"padding: 5px;\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image044.gif\" width=\"17\" height=\"29\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. To simplify, multiply the numerator\u00a0and denominator by the conjugate,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image046.gif\" width=\"25\" height=\"11\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image048.gif\" width=\"105\" height=\"88\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image050.gif\" width=\"91\" height=\"88\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Finally, place the numerator over the denominator.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image051.gif\" width=\"49\" height=\"29\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h4>How do I graph complex numbers?<\/h4>\n<p>Each point (<em>a<\/em>, <em>b<\/em>) represents the complex number\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image053.gif\" width=\"34\" height=\"11\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in the complex plane. The real axis is the horizontal axis and the\u00a0imaginary axis is the vertical axis. For example, to plot the\u00a0point\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image055.gif\" width=\"32\" height=\"11\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0move 3 right and 2 up from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-997 size-medium\" src=\"http:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-298x300.png\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-298x300.png 298w, https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-150x150.png 150w, https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph.png 484w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which point is shown on the graph?<br \/><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-997 size-medium\" src=\"http:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-298x300.png\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-298x300.png 298w, https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph-150x150.png 150w, https:\/\/americanboard.org\/Subjects\/mathematics\/wp-content\/uploads\/sites\/5\/2017\/08\/graph.png 484w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p><a name=\"three\"><\/a><\/p>\n<p><!--Q--><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image057.gif\" width=\"33\" height=\"11\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image059.gif\" width=\"32\" height=\"11\" name=\"graphics34\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image061.gif\" width=\"32\" height=\"11\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image063.gif\" width=\"31\" height=\"11\" name=\"graphics36\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. The real number axis is the\u00a0horizontal axis and the imaginary axis is the vertical axis. The\u00a0point shown is\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s2_p3_clip_image065.gif\" width=\"33\" height=\"11\" name=\"graphics37\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li><em><strong>Like terms<\/strong><\/em> have the same variable raised to the same exponent.<\/li>\n<li>Monomials can be added if they\u00a0are like terms.<\/li>\n<li>Monomials can be multiplied\u00a0regardless of their variables and exponents.<\/li>\n<li>Binomials can be added or\u00a0subtracted by combining like terms.<\/li>\n<li>One way to multiply binomials\u00a0is to use the FOIL method.<\/li>\n<li>To multiply polynomials,\u00a0multiply each term in the first polynomial by each term in the\u00a0second polynomial.<\/li>\n<li>A complex number is the field\u00a0of numbers in the form\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s3_p1_clip_image002.gif\" width=\"34\" height=\"11\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0where <em>a <\/em>and <em>b<\/em> are real numbers and <em><em>i<\/em> <\/em>is an imaginary\u00a0number equal to<em> <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s3_p1_clip_image004.gif\" width=\"27\" height=\"18\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>. <\/em><\/li>\n<li>To add complex numbers, add\u00a0the real parts and then add the imaginary parts.<\/li>\n<li>To subtract complex numbers,\u00a0subtract the real parts and then subtract the imaginary parts.<\/li>\n<li>To multiply complex numbers,\u00a0use the FOIL method.<\/li>\n<li>To divide complex numbers,\u00a0multiply by the conjugate of the denominator.<\/li>\n<li>To graph complex numbers in the plane, the real part is\u00a0graphed on the horizontal axis and the imaginary part is graphed\u00a0on the vertical axis.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebraic-expressions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/number-sense\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/operations-with-real-numbers\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Complex Numbers Objective In this lesson you will study how to graph complex numbers in the plane and perform operations with complex numbers. Previously Covered: The real numbers are the field of all rational and irrational numbers. What are complex numbers? What are complex numbers? An imaginary number is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-11","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/11","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=11"}],"version-history":[{"count":14,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/11\/revisions"}],"predecessor-version":[{"id":1000,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/11\/revisions\/1000"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=11"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}