{"id":515,"date":"2021-08-02T19:21:29","date_gmt":"2021-08-02T19:21:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=515"},"modified":"2022-01-18T20:04:35","modified_gmt":"2022-01-18T20:04:35","slug":"multiplying-and-dividing-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/multiplying-and-dividing-real-numbers\/","title":{"raw":"Multiplying and Dividing Real Numbers","rendered":"Multiplying and Dividing Real Numbers"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply and divide real numbers\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply two or more real numbers<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Divide real numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiply Real Numbers<\/h2>\r\nWith whole numbers, you can think of multiplication as repeated addition. The <strong>product <\/strong>(or result of multiplication)\u00a0[latex]3 \\cdot 5[\/latex] can be interpreted as\r\n<p style=\"text-align: center;\">[latex]3 \\cdot 5=5+5+5=15[\/latex]<\/p>\r\nSo, to multiply [latex]3(-5)[\/latex], can be found as follows.\r\n<p style=\"text-align: center;\">[latex]3(-5)=(-5)+(-5)+(-5)= -15[\/latex]<\/p>\r\nWe can multiply two numbers in any order and the result is the same, so\r\n<p style=\"text-align: center;\">[latex](-5)(3)=15[\/latex]<\/p>\r\nThe product of a positive number and a negative number (or a negative and a positive) is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\r\nTo multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their <strong>absolute values<\/strong>, or their distance from zero. The product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]-2(4)[\/latex]\r\n\r\n[reveal-answer q=\"564281\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"564281\"]\r\n\r\nSince one number is positive and the other is negative, we multiply their absolute values and the product is negative.\r\n\r\n[latex]-2(4)=-(2 \\cdot 4)=-8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of how to multiply decimal numbers with different <strong>signs<\/strong>.\r\n\r\nhttps:\/\/youtu.be\/7gY0S3LUUyQ\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\r\nTo multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\nTo multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex](-12)(-3)[\/latex]\r\n\r\n[reveal-answer q=\"682812\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"682812\"]\r\n\r\nSince both numbers are negative, we multiply their absolute values and the product is positive.\r\n\r\n[latex](-12)(-3)=12 \\cdot 3 = 36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\r\nIf there are an <strong>even<\/strong> number ([latex]0, 2, 4[\/latex], ...) of negative factors to multiply, the product is <strong>positive<\/strong>.\r\nIf there are an <strong>odd<\/strong> number ([latex]1, 3, 5[\/latex], ...) of negative factors, the product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex](-4)(-2)(-5)[\/latex].\r\n\r\n[reveal-answer q=\"462362\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"462362\"]\r\n\r\nSince there are 3 negative factors, and 3 is odd, the product is negative.\r\n\r\n[latex](-4)(-2)(-5) = -(4 \\cdot 2 \\cdot 5) = -40[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of multiplying more than two signed <strong>integers, <\/strong>or counting numbers (1, 2, 3, etc.).\r\n\r\nhttps:\/\/youtu.be\/rx8F9SPd0HE\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]228233[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Divide Real Numbers<\/h2>\r\nYou may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. Reciprocal is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).\u00a0A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Rules of Division<\/h3>\r\nWhen one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]24 \\div (-3)[\/latex]\r\n\r\n[reveal-answer q=\"863625\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"863625\"]\r\n\r\nSince one number is positive and the other is negative, the quotient is negative.\r\n\r\n[latex]24 \\div (-3) = -(24 \\div 3) = -8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex](-36) \\div (-9)[\/latex]\r\n\r\n[reveal-answer q=\"374455\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"374455\"]\r\n\r\nSince both numbers are negative, the quotient is positive.\r\n\r\n[latex](-36) \\div (-9) = 36 \\div 9 = 4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply and divide real numbers\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply two or more real numbers<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Divide real numbers<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Multiply Real Numbers<\/h2>\n<p>With whole numbers, you can think of multiplication as repeated addition. The <strong>product <\/strong>(or result of multiplication)\u00a0[latex]3 \\cdot 5[\/latex] can be interpreted as<\/p>\n<p style=\"text-align: center;\">[latex]3 \\cdot 5=5+5+5=15[\/latex]<\/p>\n<p>So, to multiply [latex]3(-5)[\/latex], can be found as follows.<\/p>\n<p style=\"text-align: center;\">[latex]3(-5)=(-5)+(-5)+(-5)= -15[\/latex]<\/p>\n<p>We can multiply two numbers in any order and the result is the same, so<\/p>\n<p style=\"text-align: center;\">[latex](-5)(3)=15[\/latex]<\/p>\n<p>The product of a positive number and a negative number (or a negative and a positive) is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\n<p>To multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their <strong>absolute values<\/strong>, or their distance from zero. The product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]-2(4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q564281\">Show Answer<\/span><\/p>\n<div id=\"q564281\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since one number is positive and the other is negative, we multiply their absolute values and the product is negative.<\/p>\n<p>[latex]-2(4)=-(2 \\cdot 4)=-8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of how to multiply decimal numbers with different <strong>signs<\/strong>.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Multiplying Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7gY0S3LUUyQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\n<p>To multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<p>To multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex](-12)(-3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q682812\">Show Answer<\/span><\/p>\n<div id=\"q682812\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since both numbers are negative, we multiply their absolute values and the product is positive.<\/p>\n<p>[latex](-12)(-3)=12 \\cdot 3 = 36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\n<p>If there are an <strong>even<\/strong> number ([latex]0, 2, 4[\/latex], &#8230;) of negative factors to multiply, the product is <strong>positive<\/strong>.<br \/>\nIf there are an <strong>odd<\/strong> number ([latex]1, 3, 5[\/latex], &#8230;) of negative factors, the product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex](-4)(-2)(-5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q462362\">Show Answer<\/span><\/p>\n<div id=\"q462362\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since there are 3 negative factors, and 3 is odd, the product is negative.<\/p>\n<p>[latex](-4)(-2)(-5) = -(4 \\cdot 2 \\cdot 5) = -40[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of multiplying more than two signed <strong>integers, <\/strong>or counting numbers (1, 2, 3, etc.).<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Multiplying Three or More Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rx8F9SPd0HE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm228233\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=228233&theme=oea&iframe_resize_id=ohm228233&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Divide Real Numbers<\/h2>\n<p>You may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. Reciprocal is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).\u00a0A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rules of Division<\/h3>\n<p>When one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]24 \\div (-3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q863625\">Show Answer<\/span><\/p>\n<div id=\"q863625\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since one number is positive and the other is negative, the quotient is negative.<\/p>\n<p>[latex]24 \\div (-3) = -(24 \\div 3) = -8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex](-36) \\div (-9)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q374455\">Show Answer<\/span><\/p>\n<div id=\"q374455\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since both numbers are negative, the quotient is positive.<\/p>\n<p>[latex](-36) \\div (-9) = 36 \\div 9 = 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-515\">\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><li>QID 228233. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Multiplying Signed Decimals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/7gY0S3LUUyQ\">https:\/\/youtu.be\/7gY0S3LUUyQ<\/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>Ex: Multiplying Three or More Integers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rx8F9SPd0HE\">https:\/\/youtu.be\/rx8F9SPd0HE<\/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>Unit 9: Real Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Multiplying Signed Decimals\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/7gY0S3LUUyQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Multiplying Three or More Integers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/rx8F9SPd0HE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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