{"id":4507,"date":"2017-06-07T18:50:08","date_gmt":"2017-06-07T18:50:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiplying-and-dividing-real-numbers\/"},"modified":"2017-08-15T12:04:20","modified_gmt":"2017-08-15T12:04:20","slug":"read-multiplying-and-dividing-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-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 Objectives<\/h3>\r\n<ul>\r\n \t<li>Multiply and divide real numbers\r\n<ul>\r\n \t<li>Multiply two or more real numbers.<\/li>\r\n \t<li>Divide real numbers<\/li>\r\n \t<li>Simplify expressions with both multiplication and division<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nMultiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations \"undo\" each other.\r\n<h2>Multiply Real Numbers<\/h2>\r\nMultiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product.\r\n\r\nWith whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185004\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/>\r\n\r\nSo to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185006\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/>\r\n\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 absolute values. 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]\u22123.8(0.6)[\/latex].\r\n[reveal-answer q=\"456211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"456211\"]Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\r\nThe product of a negative and a positive is negative.\r\n<h4>Answer<\/h4>\r\n[latex]\u22123.8(0.6)=\u22122.28[\/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 signs.\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] ~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]\r\n\r\n[reveal-answer q=\"322816\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322816\"]Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product's numerator. Then, multiply the denominators together to get the product's denominator. Rewrite in lowest terms, if needed.\r\n<p style=\"text-align: center\">[latex] \\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\r\nThe product of two negative numbers is positive.\r\n<h4>Answer<\/h4>\r\n[latex] \\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of multiplying two signed fractions, including simplification of the answer.\r\n\r\nhttps:\/\/youtu.be\/yUdJ46pTblo\r\n\r\nTo summarize:\r\n<ul>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n<\/ul>\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 (0, 2, 4, ...) of negative factors to multiply, the product is <strong>positive<\/strong>.\r\nIf there are an <strong>odd<\/strong> number (1, 3, 5, ...) 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]3(\u22126)(2)(\u22123)(\u22121)[\/latex].\r\n\r\n[reveal-answer q=\"149062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149062\"]Multiply the absolute values of the numbers.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\r\nCount the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\r\nSince there are an odd number of negative factors, the product is negative.\r\n<h4>Answer<\/h4>\r\n[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of multiplying more than two signed integers.\r\n\r\nhttps:\/\/youtu.be\/rx8F9SPd0HE\r\n<h2>Divide Real Numbers<\/h2>\r\nYou may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).\r\n\r\nAn easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:\r\n<ul>\r\n \t<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex] \\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\r\n<\/ul>\r\nWhen you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.\r\n\r\nThink about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.\r\n<p style=\"text-align: center\">[latex] 26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\r\nNotice that 2 and [latex] \\frac{1}{2}[\/latex] are reciprocals.\r\n\r\nTry again, dividing a bag of 36 marbles into smaller bags.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number of bags<\/th>\r\n<th>Dividing by number of bags<\/th>\r\n<th>Multiplying by reciprocal<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<thead><\/thead>\r\n<tbody>\r\n<tr>\r\n<td>3<\/td>\r\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\r\n<td>[latex] 36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex] 28\\div \\frac{4}{3}[\/latex]\r\n\r\n[reveal-answer q=\"210216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"210216\"]Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex] \\frac{4}{3}[\/latex] is [latex]\\frac{3}{4} [\/latex].\r\n<p style=\"text-align: center\">[latex] 28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]28\\div\\frac{4}{3}=21[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's see what this means when one or more of the numbers is negative. A 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 dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.\r\n\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\\left(-\\frac{5}{6}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"716581\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716581\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center\">[latex] 24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\r\nMultiply. Since one number is positive and one is negative, the product is negative.\r\n<p style=\"text-align: center\">[latex] \\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] 24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/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] 4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]\r\n\r\n[reveal-answer q=\"557653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557653\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center\">[latex] \\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\r\nMultiply. There is an even number of negative numbers, so the product is positive.\r\n<p style=\"text-align: center\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\r\nWrite the fraction in lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex] 4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video explains how to divide signed fractions.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/>\r\n\r\nRemember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].\r\n\r\nIn each case, the overall fraction is negative because there's only one negative in the division.\r\n\r\n<\/div>\r\nThe following video explains how to divide signed fractions.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Multiply and divide real numbers\n<ul>\n<li>Multiply two or more real numbers.<\/li>\n<li>Divide real numbers<\/li>\n<li>Simplify expressions with both multiplication and division<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>Multiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations &#8220;undo&#8221; each other.<\/p>\n<h2>Multiply Real Numbers<\/h2>\n<p>Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven&#8217;t learned what effect a negative sign has on the product.<\/p>\n<p>With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185004\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/><\/p>\n<p>So to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185006\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/><\/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 absolute values. The product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]\u22123.8(0.6)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q456211\">Show Solution<\/span><\/p>\n<div id=\"q456211\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\n<p>The product of a negative and a positive is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\u22123.8(0.6)=\u22122.28[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of how to multiply decimal numbers with different signs.<\/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]~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q322816\">Show Solution<\/span><\/p>\n<div id=\"q322816\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product&#8217;s numerator. Then, multiply the denominators together to get the product&#8217;s denominator. Rewrite in lowest terms, if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\n<p>The product of two negative numbers is positive.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of multiplying two signed fractions, including simplification of the answer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Multiplying Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yUdJ46pTblo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>To summarize:<\/p>\n<ul>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<\/ul>\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 (0, 2, 4, &#8230;) of negative factors to multiply, the product is <strong>positive<\/strong>.<br \/>\nIf there are an <strong>odd<\/strong> number (1, 3, 5, &#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]3(\u22126)(2)(\u22123)(\u22121)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149062\">Show Solution<\/span><\/p>\n<div id=\"q149062\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\n<p>Count the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\n<p>Since there are an odd number of negative factors, the product is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of multiplying more than two signed integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<h2>Divide Real Numbers<\/h2>\n<p>You may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).<\/p>\n<p>An easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:<\/p>\n<ul>\n<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex]\\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\n<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\n<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\n<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\n<\/ul>\n<p>When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.<\/p>\n<p>Think about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.<\/p>\n<p style=\"text-align: center\">[latex]26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\n<p>Notice that 2 and [latex]\\frac{1}{2}[\/latex] are reciprocals.<\/p>\n<p>Try again, dividing a bag of 36 marbles into smaller bags.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Number of bags<\/th>\n<th>Dividing by number of bags<\/th>\n<th>Multiplying by reciprocal<\/th>\n<\/tr>\n<\/tbody>\n<thead><\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\n<td>[latex]36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]28\\div \\frac{4}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q210216\">Show Solution<\/span><\/p>\n<div id=\"q210216\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex]\\frac{4}{3}[\/latex] is [latex]\\frac{3}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]28\\div\\frac{4}{3}=21[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s see what this means when one or more of the numbers is negative. A 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 dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.<\/p>\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\\left(-\\frac{5}{6}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716581\">Show Solution<\/span><\/p>\n<div id=\"q716581\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center\">[latex]24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\n<p>Multiply. Since one number is positive and one is negative, the product is negative.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557653\">Show Solution<\/span><\/p>\n<div id=\"q557653\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\n<p>Multiply. There is an even number of negative numbers, so the product is positive.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\n<p>Write the fraction in lowest terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video explains how to divide signed fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/><\/p>\n<p>Remember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].<\/p>\n<p>In each case, the overall fraction is negative because there&#8217;s only one negative in the division.<\/p>\n<\/div>\n<p>The following video explains how to divide signed fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?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-4507\">\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: 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><strong>Authored by<\/strong>: Ex: Multiplying Signed Fractions Mathispower4u . <strong>Provided by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yUdJ46pTblo\">https:\/\/youtu.be\/yUdJ46pTblo<\/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 1: Dividing Signed Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OPHdadhDJoI\">https:\/\/youtu.be\/OPHdadhDJoI<\/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\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Multiplying Signed Decimals\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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