{"id":3582,"date":"2019-12-31T17:58:32","date_gmt":"2019-12-31T17:58:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3582"},"modified":"2021-02-05T23:49:12","modified_gmt":"2021-02-05T23:49:12","slug":"dividing-integers","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/dividing-integers\/","title":{"raw":"Dividing Integers","rendered":"Dividing Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide integers with the same sign<\/li>\r\n \t<li>Divide integers with different signs<\/li>\r\n \t<li>Divide integers by -1<\/li>\r\n<\/ul>\r\n<\/div>\r\nDivision is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex] In words, this expression says that [latex]\\mathbf{\\text{15}}[\/latex] can be divided into [latex]\\mathbf{\\text{3}}[\/latex] groups of [latex]\\mathbf{\\text{5}}[\/latex] each because adding five three times gives [latex]\\mathbf{\\text{15}}[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill &amp; &amp; &amp; &amp; -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill &amp; &amp; &amp; &amp; 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]<\/p>\r\nDivision of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>Division of Signed Numbers<\/h3>\r\nThe sign of the quotient of two numbers depends on their signs.\r\n<table id=\"fs-id2593939\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Same signs<\/th>\r\n<th>Quotient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Two positives\r\n\r\n\u2022Two negatives<\/td>\r\n<td>Positive\r\n\r\nPositive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id2379645\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Different signs<\/th>\r\n<th>Quotient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Positive &amp; negative\r\n\r\n\u2022Negative &amp; positive<\/td>\r\n<td>Negative\r\n\r\nNegative<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRemember, you can always check the answer to a division problem by multiplying.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following:\r\n<ol>\r\n \t<li>[latex]-27\\div 3[\/latex]<\/li>\r\n \t<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15.7812px\">\r\n<td style=\"height: 15.7812px\">1.<\/td>\r\n<td style=\"height: 15.7812px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]-27\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Divide, noting that the signs are different and so the quotient is negative.<\/td>\r\n<td style=\"height: 15px\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide, noting that the signs are the same and so the quotient is positive.<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145323[\/ohm_question]\r\n\r\n<\/div>\r\nJust as we saw with multiplication, when we divide a number by [latex]1[\/latex], the result is the same number. What happens when we divide a number by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill &amp; &amp; &amp; -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill &amp; &amp; &amp; 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill &amp; &amp; &amp; \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]<\/p>\r\nWhen we divide a number by [latex]-1[\/latex] we get its opposite.\r\n<div class=\"textbox shaded\">\r\n<h3>Division by [latex]-1[\/latex]<\/h3>\r\nDividing a number by [latex]-1[\/latex] gives its opposite.\r\n\r\n[latex]a\\div \\left(-1\\right)=-a[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following:\r\n<ol>\r\n \t<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\r\n \t<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"343427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343427\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td>[latex]-16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that the signs were different, so the result was negative.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that the signs were the same, so the result was positive.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145326[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to divide integers with the same and different signs.\r\n\r\nhttps:\/\/youtu.be\/z5ZFiyLi5Y0","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide integers with the same sign<\/li>\n<li>Divide integers with different signs<\/li>\n<li>Divide integers by -1<\/li>\n<\/ul>\n<\/div>\n<p>Division is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex] In words, this expression says that [latex]\\mathbf{\\text{15}}[\/latex] can be divided into [latex]\\mathbf{\\text{3}}[\/latex] groups of [latex]\\mathbf{\\text{5}}[\/latex] each because adding five three times gives [latex]\\mathbf{\\text{15}}[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill & & & & -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill & & & & 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]<\/p>\n<p>Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division of Signed Numbers<\/h3>\n<p>The sign of the quotient of two numbers depends on their signs.<\/p>\n<table id=\"fs-id2593939\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Same signs<\/th>\n<th>Quotient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Two positives<\/p>\n<p>\u2022Two negatives<\/td>\n<td>Positive<\/p>\n<p>Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id2379645\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Different signs<\/th>\n<th>Quotient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Positive &amp; negative<\/p>\n<p>\u2022Negative &amp; positive<\/td>\n<td>Negative<\/p>\n<p>Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Remember, you can always check the answer to a division problem by multiplying.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following:<\/p>\n<ol>\n<li>[latex]-27\\div 3[\/latex]<\/li>\n<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr style=\"height: 15.7812px\">\n<td style=\"height: 15.7812px\">1.<\/td>\n<td style=\"height: 15.7812px\"><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]-27\\div 3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Divide, noting that the signs are different and so the quotient is negative.<\/td>\n<td style=\"height: 15px\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide, noting that the signs are the same and so the quotient is positive.<\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145323\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145323&theme=oea&iframe_resize_id=ohm145323&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Just as we saw with multiplication, when we divide a number by [latex]1[\/latex], the result is the same number. What happens when we divide a number by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill & & & -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill & & & 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill & & & \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]<\/p>\n<p>When we divide a number by [latex]-1[\/latex] we get its opposite.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division by [latex]-1[\/latex]<\/h3>\n<p>Dividing a number by [latex]-1[\/latex] gives its opposite.<\/p>\n<p>[latex]a\\div \\left(-1\\right)=-a[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following:<\/p>\n<ol>\n<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\n<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343427\">Show Solution<\/span><\/p>\n<div id=\"q343427\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that the signs were different, so the result was negative.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that the signs were the same, so the result was positive.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145326&theme=oea&iframe_resize_id=ohm145326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to divide integers with the same and different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Dividing Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z5ZFiyLi5Y0?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-3582\">\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: Dividing Integers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/z5ZFiyLi5Y0\">https:\/\/youtu.be\/z5ZFiyLi5Y0<\/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>Question ID: 145323, 145326. <strong>Authored by<\/strong>: Alyson Day. <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, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":25777,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex: Dividing Integers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/z5ZFiyLi5Y0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 145323, 145326\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\" IMathAS Community License CC-BY +GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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