{"id":9488,"date":"2017-05-02T23:15:59","date_gmt":"2017-05-02T23:15:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9488"},"modified":"2020-09-03T10:47:41","modified_gmt":"2020-09-03T10:47:41","slug":"finding-the-reciprocal-of-a-number","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/finding-the-reciprocal-of-a-number\/","title":{"raw":"2.2.c - Finding the Reciprocal of a Number","rendered":"2.2.c &#8211; Finding the Reciprocal of a Number"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the reciprocal of a fraction<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe fractions [latex]\\Large\\frac{2}{3}[\/latex] and [latex]\\Large\\frac{3}{2}[\/latex] are related to each other in a special way. So are [latex]\\Large-\\frac{10}{7}[\/latex] and [latex]\\Large-\\frac{7}{10}[\/latex]. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\cdot \\frac{3}{2}\\normalsize=1\\text{ and }-\\Large\\frac{10}{7}\\left(-\\frac{7}{10}\\right)\\normalsize=1[\/latex]<\/p>\r\n<p style=\"text-align: left\">Such pairs of numbers are called reciprocals.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Reciprocal<\/h3>\r\nThe reciprocal of the fraction [latex]\\Large\\frac{a}{b}[\/latex] is [latex]\\Large\\frac{b}{a}[\/latex], where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex],\r\nA number and its reciprocal have a product of [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot \\frac{b}{a}\\normalsize=1[\/latex]<\/p>\r\n\r\n<\/div>\r\nHere are some examples of reciprocals:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Original number<\/th>\r\n<th>Reciprocal<\/th>\r\n<th>Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] 3=\\dfrac{3}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\dfrac{1}{3}=\\dfrac{7}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}=\\normalsize 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.\u00a0 You can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex].\r\n\r\nMake sure that if it's a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.\u00a0 To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220923\/CNX_BMath_Figure_04_02_035_img.png\" alt=\"\" \/>\r\nTo find the reciprocal, keep the same sign and invert the fraction.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the reciprocal of each number. Then check that the product of each number and its reciprocal is [latex]1[\/latex].\r\n<ol id=\"eip-id1168469776775\" class=\"circled\">\r\n \t<li>[latex]\\Large\\frac{4}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\Large-\\frac{1}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\Large-\\frac{14}{5}[\/latex]<\/li>\r\n \t<li>[latex]7[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\nTo find the reciprocals, we keep the sign and invert the fractions.\r\n<table id=\"eip-374\" class=\"unnumbered 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>Find the reciprocal of [latex]\\Large\\frac{4}{9}[\/latex]<\/td>\r\n<td>The reciprocal of [latex]\\Large\\frac{4}{9}[\/latex] is [latex]\\Large\\frac{9}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the number and its reciprocal.<\/td>\r\n<td>[latex]\\Large\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply numerators and denominators.<\/td>\r\n<td>[latex]\\Large\\frac{36}{36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-37774\" class=\"unnumbered 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>Find the reciprocal of [latex]\\Large-\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\Large-\\frac{6}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]\\Large-\\frac{1}{6}\\normalsize\\cdot \\left(-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1170196522976\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]\\Large-\\frac{14}{5}[\/latex]<\/td>\r\n<td>[latex]\\Large-\\frac{5}{14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]\\Large-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{70}{70}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1170195508737\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]7[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\Large\\frac{7}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the reciprocal of [latex]\\Large\\frac{7}{1}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{1}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]7\\cdot\\Large\\left(\\frac{1}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"400\"]141842[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.\r\n\r\nhttps:\/\/youtu.be\/IM991IqCi44\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\r\n<h2>Division by Zero<\/h2>\r\nYou know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction\r\n<p style=\"text-align: center\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\r\nWe can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?\r\n<p style=\"text-align: center\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\r\n<p style=\"text-align: left\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\r\nNow let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have \"what times [latex]0[\/latex] equals [latex]8[\/latex]?\"\r\n<p style=\"text-align: center\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\r\nThis doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better defined, all division by zero is undefined.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the reciprocal of a fraction<\/li>\n<\/ul>\n<\/div>\n<p>The fractions [latex]\\Large\\frac{2}{3}[\/latex] and [latex]\\Large\\frac{3}{2}[\/latex] are related to each other in a special way. So are [latex]\\Large-\\frac{10}{7}[\/latex] and [latex]\\Large-\\frac{7}{10}[\/latex]. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\cdot \\frac{3}{2}\\normalsize=1\\text{ and }-\\Large\\frac{10}{7}\\left(-\\frac{7}{10}\\right)\\normalsize=1[\/latex]<\/p>\n<p style=\"text-align: left\">Such pairs of numbers are called reciprocals.<\/p>\n<div class=\"textbox shaded\">\n<h3>Reciprocal<\/h3>\n<p>The reciprocal of the fraction [latex]\\Large\\frac{a}{b}[\/latex] is [latex]\\Large\\frac{b}{a}[\/latex], where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex],<br \/>\nA number and its reciprocal have a product of [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot \\frac{b}{a}\\normalsize=1[\/latex]<\/p>\n<\/div>\n<p>Here are some examples of reciprocals:<\/p>\n<table>\n<thead>\n<tr>\n<th>Original number<\/th>\n<th>Reciprocal<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3=\\dfrac{3}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\dfrac{1}{3}=\\dfrac{7}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}=\\normalsize 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.\u00a0 You can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex].<\/p>\n<p>Make sure that if it&#8217;s a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.\u00a0 To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220923\/CNX_BMath_Figure_04_02_035_img.png\" alt=\"\" \/><br \/>\nTo find the reciprocal, keep the same sign and invert the fraction.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the reciprocal of each number. Then check that the product of each number and its reciprocal is [latex]1[\/latex].<\/p>\n<ol id=\"eip-id1168469776775\" class=\"circled\">\n<li>[latex]\\Large\\frac{4}{9}[\/latex]<\/li>\n<li>[latex]\\Large-\\frac{1}{6}[\/latex]<\/li>\n<li>[latex]\\Large-\\frac{14}{5}[\/latex]<\/li>\n<li>[latex]7[\/latex]<\/li>\n<\/ol>\n<p>Solution:<br \/>\nTo find the reciprocals, we keep the sign and invert the fractions.<\/p>\n<table id=\"eip-374\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]\\Large\\frac{4}{9}[\/latex]<\/td>\n<td>The reciprocal of [latex]\\Large\\frac{4}{9}[\/latex] is [latex]\\Large\\frac{9}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply the number and its reciprocal.<\/td>\n<td>[latex]\\Large\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply numerators and denominators.<\/td>\n<td>[latex]\\Large\\frac{36}{36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-37774\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]\\Large-\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\Large-\\frac{6}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]\\Large-\\frac{1}{6}\\normalsize\\cdot \\left(-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170196522976\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]\\Large-\\frac{14}{5}[\/latex]<\/td>\n<td>[latex]\\Large-\\frac{5}{14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]\\Large-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{70}{70}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170195508737\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]7[\/latex] as a fraction.<\/td>\n<td>[latex]\\Large\\frac{7}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the reciprocal of [latex]\\Large\\frac{7}{1}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{1}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]7\\cdot\\Large\\left(\\frac{1}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141842\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141842&theme=oea&iframe_resize_id=ohm141842&show_question_numbers\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine the Reciprocal of Integers, Fractions, and Mixed Numbers\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/IM991IqCi44?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\n<h2>Division by Zero<\/h2>\n<p>You know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\n<p>We can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?<\/p>\n<p style=\"text-align: center\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\n<p style=\"text-align: left\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\n<p>Now let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have &#8220;what times [latex]0[\/latex] equals [latex]8[\/latex]?&#8221;<\/p>\n<p style=\"text-align: center\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\n<p>This doesn&#8217;t make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better defined, all division by zero is undefined.<\/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-9488\">\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>Ex: Determine the Reciprocal of Integers, Fractions, and Mixed Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IM991IqCi44\">https:\/\/youtu.be\/IM991IqCi44<\/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: 141842, 146026. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2: Determine the Absolute Value of an Integer. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lY5ksjix5Kg\">https:\/\/youtu.be\/lY5ksjix5Kg<\/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 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":17533,"menu_order":11,"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\":\"original\",\"description\":\"Ex: Determine the Reciprocal of Integers, Fractions, and Mixed Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/IM991IqCi44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Determine the Absolute Value of an Integer\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/lY5ksjix5Kg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 141842, 146026\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"0678e17bee3a4b39a1b7badfae5b722b, 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