{"id":3672,"date":"2020-01-29T02:25:13","date_gmt":"2020-01-29T02:25:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3672"},"modified":"2020-01-29T03:24:55","modified_gmt":"2020-01-29T03:24:55","slug":"identifying-rational-and-irrational-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/identifying-rational-and-irrational-numbers\/","title":{"raw":"Identifying Rational and Irrational Numbers","rendered":"Identifying Rational and Irrational Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify rational numbers from a list of numbers<\/li>\r\n \t<li>Identify irrational numbers from a list of numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\nNow we'll take another look at the kinds of numbers we have worked with in all previous lessons. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.\r\n\r\nWe have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?\r\n<table id=\"fs-id1166497471716\" class=\"unnumbered\" style=\"width: 85%\" summary=\"The table has three rows and two columns. The first column contains names of different types of numbers. The second column lists examples of each of these types of numbers. Row one is labeled \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>counting numbers<\/td>\r\n<td>[latex]1,2,3,4\\dots [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>whole numbers<\/td>\r\n<td>[latex]0,1,2,3,4\\dots[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>integers<\/td>\r\n<td>[latex]\\dots -3,-2,-1,0,1,2,3,4\\dots [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Rational Numbers<\/h3>\r\nWhat type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.\r\n<div class=\"textbox shaded\">\r\n<h3>Rational Numbers<\/h3>\r\nA rational number is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne o[\/latex].\r\n\r\n<\/div>\r\nAll fractions, both positive and negative, are rational numbers. A few examples are\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{4}{5}\\normalsize ,-\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\text{and}-\\Large\\frac{20}{3}[\/latex]<\/p>\r\nEach numerator and each denominator is an integer.\r\n\r\nWe need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.\r\n\r\nAre integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.\r\n<p style=\"text-align: center\">[latex]3=\\Large\\frac{3}{1}\\normalsize ,\\space-8=\\Large\\frac{-8}{1}\\normalsize ,\\space0=\\Large\\frac{0}{1}[\/latex]<\/p>\r\nSince any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.\r\n\r\nWhat about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer [latex]-8[\/latex] could be written as the decimal [latex]-8.0[\/latex]. So, clearly, some decimals are rational.\r\n\r\nThink about the decimal [latex]7.3[\/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[\/latex] means [latex]7\\Large\\frac{3}{10}[\/latex], we can write it as an improper fraction, [latex]\\Large\\frac{73}{10}[\/latex]. So [latex]7.3[\/latex] is the ratio of the integers [latex]73[\/latex] and [latex]10[\/latex]. It is a rational number.\r\n\r\nIn general, any decimal that ends after a number of digits such as [latex]7.3[\/latex] or [latex]-1.2684[\/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite each as the ratio of two integers:\r\n\r\n1. [latex]-15[\/latex]\r\n\r\n2. [latex]6.81[\/latex]\r\n\r\n3. [latex]-3\\Large\\frac{6}{7}[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469454543\" 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><\/td>\r\n<td>[latex]-15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the integer as a fraction with denominator 1.<\/td>\r\n<td>[latex]\\Large\\frac{-15}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467276182\" 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><\/td>\r\n<td>[latex]6.81[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the decimal as a mixed number.<\/td>\r\n<td>[latex]6\\Large\\frac{81}{100}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then convert it to an improper fraction.<\/td>\r\n<td>[latex]\\Large\\frac{681}{100}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467114800\" 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><\/td>\r\n<td>[latex]-3\\Large\\frac{6}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert the mixed number to an improper fraction.<\/td>\r\n<td>[latex]-\\Large\\frac{27}{7}[\/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]145911[\/ohm_question]\r\n\r\n<\/div>\r\nLet's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since [latex]a=\\Large\\frac{a}{1}[\/latex] for any integer, [latex]a[\/latex]. We can also change any integer to a decimal by adding a decimal point and a zero.\r\n<p style=\"padding-left: 60px\">Integer\u00a0[latex]-2,-1,0,1,2,3[\/latex]<\/p>\r\n<p style=\"padding-left: 60px\">Decimal [latex]-2.0,-1.0,0.0,1.0,2.0,3.0[\/latex]<\/p>\r\nThese decimal numbers stop.\r\n\r\nWe have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.\r\n<p style=\"padding-left: 60px\">Ratio of Integers [latex]\\Large\\frac{4}{5}\\normalsize ,\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{20}{3}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px\">Decimal Forms [latex]0.8,-0.875,3.25,-6.666\\ldots,-6.\\overline{66}[\/latex]<\/p>\r\nThese decimals either stop or repeat.\r\n\r\nWhat do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.\r\n<table id=\"fs-id1458671\" class=\"unnumbered\" style=\"width: 85%\" summary=\"The table is labeled \">\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\">Rational Numbers<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><\/td>\r\n<td><strong>Fractions<\/strong><\/td>\r\n<td><strong>Integers<\/strong><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Number<\/td>\r\n<td>[latex]\\Large\\frac{4}{5}\\normalsize ,-\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{-20}{3}[\/latex]<\/td>\r\n<td>[latex]-2,-1,0,1,2,3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Ratio of Integer<\/td>\r\n<td>[latex]\\Large\\frac{4}{5}\\normalsize ,\\Large\\frac{-7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{-20}{3}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{-2}{1}\\normalsize ,\\Large\\frac{-1}{1}\\normalsize ,\\Large\\frac{0}{1}\\normalsize ,\\Large\\frac{1}{1}\\normalsize ,\\Large\\frac{2}{1}\\normalsize ,\\Large\\frac{3}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Decimal number<\/td>\r\n<td>[latex]0.8,-0.875,3.25,-6.\\overline{6}[\/latex]<\/td>\r\n<td>[latex]-2.0,-1.0,0.0,1.0,2.0,3.0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Irrational Numbers<\/h3>\r\nAre there any decimals that do not stop or repeat? Yes. The number [latex]\\pi [\/latex] (the Greek letter pi, pronounced \u2018pie\u2019), which is very important in describing circles, has a decimal form that does not stop or repeat.\r\n<p style=\"padding-left: 60px\">[latex]\\pi =\\text{3.141592654.......}[\/latex]<\/p>\r\nSimilarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,\r\n<p style=\"padding-left: 60px\">[latex]\\sqrt{5}=\\text{2.236067978.....}[\/latex]<\/p>\r\nA decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.\r\n<div class=\"textbox shaded\">\r\n<h3>Irrational Number<\/h3>\r\nAn irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.\r\n\r\n<\/div>\r\nLet's summarize a method we can use to determine whether a number is rational or irrational.\r\nIf the decimal form of a number\r\n<ul id=\"fs-id1460638\">\r\n \t<li>stops or repeats, the number is rational.<\/li>\r\n \t<li>does not stop and does not repeat, the number is irrational.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify each of the following as rational or irrational:\r\n1. [latex]0.58\\overline{3}[\/latex]\r\n2. [latex]0.475[\/latex]\r\n3. [latex]3.605551275\\dots [\/latex]\r\n[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214538\"]\r\n\r\nSolution:\r\n1. [latex]0.58\\overline{3}[\/latex]\r\nThe bar above the [latex]3[\/latex] indicates that it repeats. Therefore, [latex]0.58\\overline{3}[\/latex] is a repeating decimal, and is therefore a rational number.\r\n\r\n2. [latex]0.475[\/latex]\r\nThis decimal stops after the [latex]5[\/latex], so it is a rational number.\r\n\r\n3. [latex]3.605551275\\dots[\/latex]\r\nThe ellipsis [latex](\\dots)[\/latex] means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.\r\n\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]145910[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nLet's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify each of the following as rational or irrational:\r\n1. [latex]\\sqrt{36}[\/latex]\r\n2. [latex]\\sqrt{44}[\/latex]\r\n[reveal-answer q=\"237122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"237122\"]\r\n\r\nSolution:\r\n1. The number [latex]36[\/latex] is a perfect square, since [latex]{6}^{2}=36[\/latex]. So [latex]\\sqrt{36}=6[\/latex]. Therefore [latex]\\sqrt{36}[\/latex] is rational.\r\n2. Remember that [latex]{6}^{2}=36[\/latex] and [latex]{7}^{2}=49[\/latex], so [latex]44[\/latex] is not a perfect square.\r\nThis means [latex]\\sqrt{44}[\/latex] is irrational.\r\n\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]145915[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;background-color: #ffffff\">In the following video we show more examples of how to determine whether a number is irrational or rational.<\/span>\r\n\r\nhttps:\/\/youtu.be\/5lYbSxSBu0Y","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify rational numbers from a list of numbers<\/li>\n<li>Identify irrational numbers from a list of numbers<\/li>\n<\/ul>\n<\/div>\n<p>Now we&#8217;ll take another look at the kinds of numbers we have worked with in all previous lessons. We&#8217;ll work with properties of numbers that will help you improve your number sense. And we&#8217;ll practice using them in ways that we&#8217;ll use when we solve equations and complete other procedures in algebra.<\/p>\n<p>We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?<\/p>\n<table id=\"fs-id1166497471716\" class=\"unnumbered\" style=\"width: 85%\" summary=\"The table has three rows and two columns. The first column contains names of different types of numbers. The second column lists examples of each of these types of numbers. Row one is labeled\">\n<tbody>\n<tr valign=\"top\">\n<td>counting numbers<\/td>\n<td>[latex]1,2,3,4\\dots[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>whole numbers<\/td>\n<td>[latex]0,1,2,3,4\\dots[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>integers<\/td>\n<td>[latex]\\dots -3,-2,-1,0,1,2,3,4\\dots[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Rational Numbers<\/h3>\n<p>What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rational Numbers<\/h3>\n<p>A rational number is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne o[\/latex].<\/p>\n<\/div>\n<p>All fractions, both positive and negative, are rational numbers. A few examples are<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{4}{5}\\normalsize ,-\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\text{and}-\\Large\\frac{20}{3}[\/latex]<\/p>\n<p>Each numerator and each denominator is an integer.<\/p>\n<p>We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.<\/p>\n<p>Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.<\/p>\n<p style=\"text-align: center\">[latex]3=\\Large\\frac{3}{1}\\normalsize ,\\space-8=\\Large\\frac{-8}{1}\\normalsize ,\\space0=\\Large\\frac{0}{1}[\/latex]<\/p>\n<p>Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.<\/p>\n<p>What about decimals? Are they rational? Let&#8217;s look at a few to see if we can write each of them as the ratio of two integers. We&#8217;ve already seen that integers are rational numbers. The integer [latex]-8[\/latex] could be written as the decimal [latex]-8.0[\/latex]. So, clearly, some decimals are rational.<\/p>\n<p>Think about the decimal [latex]7.3[\/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[\/latex] means [latex]7\\Large\\frac{3}{10}[\/latex], we can write it as an improper fraction, [latex]\\Large\\frac{73}{10}[\/latex]. So [latex]7.3[\/latex] is the ratio of the integers [latex]73[\/latex] and [latex]10[\/latex]. It is a rational number.<\/p>\n<p>In general, any decimal that ends after a number of digits such as [latex]7.3[\/latex] or [latex]-1.2684[\/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write each as the ratio of two integers:<\/p>\n<p>1. [latex]-15[\/latex]<\/p>\n<p>2. [latex]6.81[\/latex]<\/p>\n<p>3. [latex]-3\\Large\\frac{6}{7}[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469454543\" class=\"unnumbered 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]-15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the integer as a fraction with denominator 1.<\/td>\n<td>[latex]\\Large\\frac{-15}{1}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467276182\" class=\"unnumbered 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]6.81[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the decimal as a mixed number.<\/td>\n<td>[latex]6\\Large\\frac{81}{100}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Then convert it to an improper fraction.<\/td>\n<td>[latex]\\Large\\frac{681}{100}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467114800\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3\\Large\\frac{6}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert the mixed number to an improper fraction.<\/td>\n<td>[latex]-\\Large\\frac{27}{7}[\/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=\"ohm145911\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145911&theme=oea&iframe_resize_id=ohm145911&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Let&#8217;s look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since [latex]a=\\Large\\frac{a}{1}[\/latex] for any integer, [latex]a[\/latex]. We can also change any integer to a decimal by adding a decimal point and a zero.<\/p>\n<p style=\"padding-left: 60px\">Integer\u00a0[latex]-2,-1,0,1,2,3[\/latex]<\/p>\n<p style=\"padding-left: 60px\">Decimal [latex]-2.0,-1.0,0.0,1.0,2.0,3.0[\/latex]<\/p>\n<p>These decimal numbers stop.<\/p>\n<p>We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.<\/p>\n<p style=\"padding-left: 60px\">Ratio of Integers [latex]\\Large\\frac{4}{5}\\normalsize ,\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{20}{3}[\/latex]<\/p>\n<p style=\"padding-left: 60px\">Decimal Forms [latex]0.8,-0.875,3.25,-6.666\\ldots,-6.\\overline{66}[\/latex]<\/p>\n<p>These decimals either stop or repeat.<\/p>\n<p>What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.<\/p>\n<table id=\"fs-id1458671\" class=\"unnumbered\" style=\"width: 85%\" summary=\"The table is labeled\">\n<thead>\n<tr>\n<th colspan=\"3\">Rational Numbers<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td><strong>Fractions<\/strong><\/td>\n<td><strong>Integers<\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Number<\/td>\n<td>[latex]\\Large\\frac{4}{5}\\normalsize ,-\\Large\\frac{7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{-20}{3}[\/latex]<\/td>\n<td>[latex]-2,-1,0,1,2,3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Ratio of Integer<\/td>\n<td>[latex]\\Large\\frac{4}{5}\\normalsize ,\\Large\\frac{-7}{8}\\normalsize ,\\Large\\frac{13}{4}\\normalsize ,\\Large\\frac{-20}{3}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{-2}{1}\\normalsize ,\\Large\\frac{-1}{1}\\normalsize ,\\Large\\frac{0}{1}\\normalsize ,\\Large\\frac{1}{1}\\normalsize ,\\Large\\frac{2}{1}\\normalsize ,\\Large\\frac{3}{1}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Decimal number<\/td>\n<td>[latex]0.8,-0.875,3.25,-6.\\overline{6}[\/latex]<\/td>\n<td>[latex]-2.0,-1.0,0.0,1.0,2.0,3.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Irrational Numbers<\/h3>\n<p>Are there any decimals that do not stop or repeat? Yes. The number [latex]\\pi[\/latex] (the Greek letter pi, pronounced \u2018pie\u2019), which is very important in describing circles, has a decimal form that does not stop or repeat.<\/p>\n<p style=\"padding-left: 60px\">[latex]\\pi =\\text{3.141592654.......}[\/latex]<\/p>\n<p>Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,<\/p>\n<p style=\"padding-left: 60px\">[latex]\\sqrt{5}=\\text{2.236067978.....}[\/latex]<\/p>\n<p>A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.<\/p>\n<div class=\"textbox shaded\">\n<h3>Irrational Number<\/h3>\n<p>An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.<\/p>\n<\/div>\n<p>Let&#8217;s summarize a method we can use to determine whether a number is rational or irrational.<br \/>\nIf the decimal form of a number<\/p>\n<ul id=\"fs-id1460638\">\n<li>stops or repeats, the number is rational.<\/li>\n<li>does not stop and does not repeat, the number is irrational.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify each of the following as rational or irrational:<br \/>\n1. [latex]0.58\\overline{3}[\/latex]<br \/>\n2. [latex]0.475[\/latex]<br \/>\n3. [latex]3.605551275\\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q214538\">Show Solution<\/span><\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. [latex]0.58\\overline{3}[\/latex]<br \/>\nThe bar above the [latex]3[\/latex] indicates that it repeats. Therefore, [latex]0.58\\overline{3}[\/latex] is a repeating decimal, and is therefore a rational number.<\/p>\n<p>2. [latex]0.475[\/latex]<br \/>\nThis decimal stops after the [latex]5[\/latex], so it is a rational number.<\/p>\n<p>3. [latex]3.605551275\\dots[\/latex]<br \/>\nThe ellipsis [latex](\\dots)[\/latex] means that this number does not stop. There is no repeating pattern of digits. Since the number doesn&#8217;t stop and doesn&#8217;t repeat, it is irrational.<\/p>\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=\"ohm145910\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145910&theme=oea&iframe_resize_id=ohm145910&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Let&#8217;s think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify each of the following as rational or irrational:<br \/>\n1. [latex]\\sqrt{36}[\/latex]<br \/>\n2. [latex]\\sqrt{44}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q237122\">Show Solution<\/span><\/p>\n<div id=\"q237122\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. The number [latex]36[\/latex] is a perfect square, since [latex]{6}^{2}=36[\/latex]. So [latex]\\sqrt{36}=6[\/latex]. Therefore [latex]\\sqrt{36}[\/latex] is rational.<br \/>\n2. Remember that [latex]{6}^{2}=36[\/latex] and [latex]{7}^{2}=49[\/latex], so [latex]44[\/latex] is not a perfect square.<br \/>\nThis means [latex]\\sqrt{44}[\/latex] is irrational.<\/p>\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=\"ohm145915\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145915&theme=oea&iframe_resize_id=ohm145915&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;background-color: #ffffff\">In the following video we show more examples of how to determine whether a number is irrational or rational.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine  Rational or Irrational Numbers (Square Roots and Decimals Only)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5lYbSxSBu0Y?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-3672\">\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>Determine Rational or Irrational Numbers (Square Roots and Decimals Only). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5lYbSxSBu0Y\">https:\/\/youtu.be\/5lYbSxSBu0Y<\/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 145910, 145915, 145911. <strong>Authored 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>Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/py-iSJziA_8\">https:\/\/youtu.be\/py-iSJziA_8<\/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":25777,"menu_order":3,"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\":\"Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/py-iSJziA_8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Determine Rational or Irrational Numbers (Square Roots and Decimals Only)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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