{"id":15988,"date":"2021-04-01T23:47:50","date_gmt":"2021-04-01T23:47:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/classifying-real-numbers\/"},"modified":"2021-06-30T14:13:36","modified_gmt":"2021-06-30T14:13:36","slug":"classifying-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/classifying-real-numbers\/","title":{"raw":"Classifying Real Numbers","rendered":"Classifying Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify rational numbers and irrational numbers<\/li>\r\n \t<li>Classify different types of real numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). In this section we will further define real numbers.\r\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\r\nThe most familiar numbers are the natural numbers: [latex]1, 2, 3[\/latex], and so on.\u00a0 These are\u00a0numbers we use for counting, or enumerating items.\u00a0 The mathematical symbol for the set of all natural numbers is written as [latex]\\mathbb{N}[\/latex].\u00a0\u00a0We describe them in set notation as [latex]\\{1, 2, 3, ...\\}[\/latex] where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em>counting numbers<\/em>. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers.\r\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Whole numbers<\/span><\/h3>\r\nThe set of whole numbers includes all natural numbers as well as\u00a0 [latex]0[\/latex]:\u00a0 [latex]\\{0, 1, 2, 3,...\\}[\/latex].\r\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\r\nWhen the set of negative numbers is combined with the set of natural numbers (including\u00a00), the result is defined as the set of integers,\u00a0[latex]\\mathbb{Z}[\/latex].\u00a0\u00a0The set of\u00a0<strong>integers<\/strong>\u00a0adds the opposites of the natural numbers to the set of whole numbers:\u00a0<span id=\"MathJax-Element-4-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; padding: 1px 0px; margin: 0px; font-size: 17.44px; vertical-align: baseline; background: #ffffff; border: 0px; line-height: 0; text-indent: 0px; text-align: left; font-style: normal; font-weight: 400; letter-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: #373d3f;\" role=\"presentation\"><span id=\"MJXc-Node-37\" class=\"mjx-math\"><span id=\"MJXc-Node-38\" class=\"mjx-mrow\"><span id=\"MJXc-Node-39\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">{<\/span><\/span><span id=\"MJXc-Node-40\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2026<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-42\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-43\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-44\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-45\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-47\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-48\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-49\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-50\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-51\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-52\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-53\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-54\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-55\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-56\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-58\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2026<\/span><\/span><span id=\"MJXc-Node-59\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">}<\/span><\/span><\/span><\/span><\/span>. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{\\text{negative integers}}\\hfill &amp; {\\text{zero}}\\hfill &amp; {\\text{positive integers}}\\\\{\\dots ,-3,-2,-1,}\\hfill &amp; {0,}\\hfill &amp; {1,2,3,\\dots }\\end{array}[\/latex]<\/p>\r\n\r\n<h2><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h2>\r\n<div class=\"hatnote relarticle mainarticle\">\r\n\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 0[\/latex].\r\n\r\n<\/div>\r\nA rational number,\u00a0[latex]\\mathbb{Q}[\/latex], is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.\u00a0\u00a0The set of <strong>rational numbers<\/strong> is written as [latex]\\left\\{\\dfrac{m}{n}\\normalsize |m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex]. Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never [latex]0[\/latex].\r\n\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<h4>Integers are rational<\/h4>\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<h4>Decimals are rational<\/h4>\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\n<h4>Rational numbers as decimals<\/h4>\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&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\nBecause they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:\r\n<ol>\r\n \t<li>a terminating decimal: [latex]\\dfrac{15}{8}\\normalsize =1.875[\/latex], or<\/li>\r\n \t<li>a repeating decimal: [latex]\\dfrac{4}{11}\\normalsize =0.36363636\\dots =0.\\overline{36}[\/latex]<\/li>\r\n<\/ol>\r\nWe use a line drawn over the repeating block of numbers instead of writing the group multiple times.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]7[\/latex]<\/li>\r\n \t<li>[latex]0[\/latex]<\/li>\r\n \t<li>[latex]\u20138[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"725771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725771\"]\r\n\r\nWrite a fraction with the integer in the numerator and 1 in the denominator.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]7=\\dfrac{7}{1}[\/latex]<\/li>\r\n \t<li>[latex]0=\\dfrac{0}{1}[\/latex]<\/li>\r\n \t<li>[latex]-8=-\\dfrac{8}{1}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite each of the following rational numbers as either a terminating or repeating decimal.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\dfrac{5}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{15}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{13}{25}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"88918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"88918\"]\r\n\r\nWrite each fraction as a decimal by dividing the numerator by the denominator.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\dfrac{5}{7}\\normalsize =-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\r\n \t<li>[latex]\\dfrac{15}{5}\\normalsize =3[\/latex] (or 3.0), a terminating decimal<\/li>\r\n \t<li>[latex]\\dfrac{13}{25}\\normalsize=0.52[\/latex],\u00a0a terminating decimal<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<h2>Irrational Numbers<\/h2>\r\nAre there any decimals that do not stop or repeat? Yes.\u00a0 At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not [latex]2[\/latex] or even [latex]\\dfrac{3}{2}[\/latex], but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than [latex]3[\/latex], but still not a rational number. \u00a0The 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 ([latex]\\pi =\\text{3.141592654.......}[\/latex]).\u00a0 Such numbers are said to be <em>irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong>irrational numbers<\/strong>. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as {h | h is not a rational number}.\r\n\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\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\sqrt{25}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{33}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{11}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{17}{34}[\/latex]<\/li>\r\n \t<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"644924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"644924\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\sqrt{25}:[\/latex] This can be simplified as [latex]\\sqrt{25}=5[\/latex]. Therefore, [latex]\\sqrt{25}[\/latex] is rational.<\/li>\r\n \t<li>[latex]\\dfrac{33}{9}:[\/latex] Because it is a fraction, [latex]\\dfrac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{33}{9}\\normalsize =\\dfrac{{{11}\\cdot{3}}}{{{3}\\cdot{3}}}\\normalsize =\\dfrac{11}{3}\\normalsize =3.\\overline{6}[\/latex]<\/div>\r\nSo, [latex]\\dfrac{33}{9}[\/latex] is rational and a repeating decimal.<\/li>\r\n \t<li>[latex]\\sqrt{11}:[\/latex] This cannot be simplified any further. Therefore, [latex]\\sqrt{11}[\/latex] is an irrational number.<\/li>\r\n \t<li>[latex]\\dfrac{17}{34}:[\/latex] Because it is a fraction, [latex]\\dfrac{17}{34}[\/latex] is a rational number. Simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{17}{34}\\normalsize =\\dfrac{1}{2}\\normalsize=0.5[\/latex]<\/div>\r\nSo, [latex]\\dfrac{17}{34}[\/latex] is rational and a terminating decimal.<\/li>\r\n \t<li>0.3033033303333... is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<\/div>\r\n<\/div>\r\n<h2>Real numbers<\/h2>\r\nGiven any number <em>n<\/em>, we know that <em>n<\/em> is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of <strong>real numbers<\/strong>.\r\n<div class=\"textbox shaded\">\r\n<h3>Real Numbers<\/h3>\r\nReal numbers are numbers that are either rational or irrational.\r\n\r\n<\/div>\r\nThe real numbers\u00a0include all the measuring numbers. The symbol for the real numbers is [latex]\\mathbb{R}[\/latex]. Real numbers are usually represented by using decimal numerals.\u00a0 As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.\r\n\r\nThe real numbers can be visualized on a horizontal number line with an arbitrary point chosen as [latex]0[\/latex], with negative numbers to the left of [latex]0[\/latex] and positive numbers to the right of [latex]0[\/latex]. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of [latex]0[\/latex]. Any real number corresponds to a unique position on the number line.\u00a0 The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the <strong>real number line<\/strong> as shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200208\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/> The real number line.[\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\dfrac{10}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{289}[\/latex]<\/li>\r\n \t<li>[latex]-6\\pi[\/latex]<\/li>\r\n \t<li>[latex]0.615384615384\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"303752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303752\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\dfrac{10}{3}[\/latex] is negative and rational. It lies to the left of 0 on the number line.<\/li>\r\n \t<li>[latex]\\sqrt{5}[\/latex] is positive and irrational. It lies to the right of [latex]0[\/latex].<\/li>\r\n \t<li>[latex]-\\sqrt{289}=-\\sqrt{{17}^{2}}=-17[\/latex] is negative and rational. It lies to the left of 0.<\/li>\r\n \t<li>[latex]-6\\pi [\/latex] is negative and irrational. It lies to the left of [latex]0[\/latex].<\/li>\r\n \t<li>[latex]0.615384615384\\dots [\/latex] is a repeating decimal so it is rational and positive. It lies to the right of [latex]0[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.\u00a0\u00a0Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.\r\n\r\nThis diagram illustrates the relationships between the different types of real numbers.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222311\/CNX_BMath_Figure_07_01_001.png\" alt=\"The image shows a large rectangle labeled \" \/>\r\n\r\nHere is another visualization of the subsets of the real numbers.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200210\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4. \" width=\"731\" height=\"352\" \/> Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sets of Numbers<\/h3>\r\nThe set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>whole numbers<\/strong> is the set of natural numbers and zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\dfrac{m}{n}\\normalsize |m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].\r\n\r\nThe set of <strong>irrational numbers<\/strong> is the set of numbers that are not rational. They are nonrepeating and nonterminating decimals: [latex]\\{h|h\\text{ is not a rational number}\\}[\/latex].\r\n\r\n<\/div>\r\nDoes the term \"real numbers\" seem strange to you? Are there any numbers that are not \"real\", and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of <em style=\"font-size: 16px;\">imaginary numbers.<\/em><span style=\"font-size: 16px;\"> You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether each of the numbers in the following list is a 1. whole number, 2. integer, 3. rational number, 4. irrational number, and 5. real number.\r\n\r\n[latex]-7,\\Large\\frac{14}{5}\\normalsize ,8,\\sqrt{5},5.9,-\\sqrt{64}[\/latex]\r\n\r\nSolution:\r\n1. The whole numbers are [latex]0,1,2,3\\dots[\/latex] The number [latex]8[\/latex] is the only whole number given.\r\n\r\n2. The integers are the whole numbers, their opposites, and [latex]0[\/latex]. From the given numbers, [latex]-7[\/latex] and [latex]8[\/latex] are integers. Also, notice that [latex]64[\/latex] is the square of [latex]8[\/latex] so [latex]-\\sqrt{64}=-8[\/latex]. So the integers are [latex]-7,8,-\\sqrt{64}[\/latex].\r\n\r\n3. Since all integers are rational, the numbers [latex]-7,8,\\text{and}-\\sqrt{64}[\/latex] are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so [latex]\\Large\\frac{14}{5}\\normalsize\\text{and}5.9[\/latex] are rational.\r\n\r\n4. The number [latex]5[\/latex] is not a perfect square, so [latex]\\sqrt{5}[\/latex] is irrational.\r\n\r\n5. All of the numbers listed are real.\r\n\r\nWe'll summarize the results in a table.\r\n<table id=\"fs-id1343988\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Number<\/th>\r\n<th>Whole<\/th>\r\n<th>Integer<\/th>\r\n<th>Rational<\/th>\r\n<th>Irrational<\/th>\r\n<th>Real<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]-7[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\Large\\frac{14}{5}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\sqrt{5}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]5.9[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]-\\sqrt{64}[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nClassify each number as being a natural number (<em>N<\/em>), whole number (<em>W<\/em>), integer (<em>I<\/em>), rational number (<em>Q<\/em>), and\/or irrational number (<em>Q'<\/em>).\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\sqrt{36}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{8}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{73}[\/latex]<\/li>\r\n \t<li>[latex]-6[\/latex]<\/li>\r\n \t<li>[latex]3.2121121112\\dots [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"400826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"400826\"]\r\n<table style=\"width: 20%;\" summary=\"A table with six rows and six columns. The first entry in the first row is blank, but the rest of the entries read: N, W, I, Q, and Q'. (These are the sets of numbers from before.) The first entry in the second row reads: square root of thirty-six equals six. Then the second, third, fourth, and fifth columns are marked. The first entry in the third row reads: eight over three equals 2.6 with the 6 repeating forever. Then only the fifth column is marked. The first entry in the fourth row reads: square root of seventy-three. Then only the sixth column is marked. The first entry in the fifth row reads: negative six. Then the fourth and fifth columns are marked. The first entry in the sixth row reads: 3.2121121112\u2026. Then only the sixth column is marked.\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 39.8165%;\"><\/th>\r\n<th style=\"width: 39.8165%;\"><\/th>\r\n<th style=\"width: 4.22018%;\"><em>N<\/em><\/th>\r\n<th style=\"width: 4.0367%;\"><em>W<\/em><\/th>\r\n<th style=\"width: 3.48624%;\"><em>I<\/em><\/th>\r\n<th style=\"width: 4.95412%;\"><em>Q<\/em><\/th>\r\n<th style=\"width: 6.05505%;\"><em>Q'<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 39.8165%;\">1.<\/td>\r\n<td style=\"width: 39.8165%;\">[latex]\\sqrt{36}=6[\/latex]<\/td>\r\n<td style=\"width: 4.22018%;\">X<\/td>\r\n<td style=\"width: 4.0367%;\">X<\/td>\r\n<td style=\"width: 3.48624%;\">X<\/td>\r\n<td style=\"width: 4.95412%;\">X<\/td>\r\n<td style=\"width: 6.05505%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8165%;\">2.<\/td>\r\n<td style=\"width: 39.8165%;\">[latex]\\dfrac{8}{3}\\normalsize =2.\\overline{6}[\/latex]<\/td>\r\n<td style=\"width: 4.22018%;\"><\/td>\r\n<td style=\"width: 4.0367%;\"><\/td>\r\n<td style=\"width: 3.48624%;\"><\/td>\r\n<td style=\"width: 4.95412%;\">X<\/td>\r\n<td style=\"width: 6.05505%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8165%;\">3.<\/td>\r\n<td style=\"width: 39.8165%;\">[latex]\\sqrt{73}[\/latex]<\/td>\r\n<td style=\"width: 4.22018%;\"><\/td>\r\n<td style=\"width: 4.0367%;\"><\/td>\r\n<td style=\"width: 3.48624%;\"><\/td>\r\n<td style=\"width: 4.95412%;\"><\/td>\r\n<td style=\"width: 6.05505%;\">X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8165%;\">4.<\/td>\r\n<td style=\"width: 39.8165%;\">[latex]\u20136[\/latex]<\/td>\r\n<td style=\"width: 4.22018%;\"><\/td>\r\n<td style=\"width: 4.0367%;\"><\/td>\r\n<td style=\"width: 3.48624%;\">X<\/td>\r\n<td style=\"width: 4.95412%;\">X<\/td>\r\n<td style=\"width: 6.05505%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8165%;\">5.<\/td>\r\n<td style=\"width: 39.8165%;\">[latex]3.2121121112\\dots[\/latex]<\/td>\r\n<td style=\"width: 4.22018%;\"><\/td>\r\n<td style=\"width: 4.0367%;\"><\/td>\r\n<td style=\"width: 3.48624%;\"><\/td>\r\n<td style=\"width: 4.95412%;\"><\/td>\r\n<td style=\"width: 6.05505%;\">X<\/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\u00a0it<\/h3>\r\n[ohm_question]149621[\/ohm_question]\r\n\r\n<\/div>\r\nThe following mini-lesson provides more examples of how to classify real numbers.\r\n\r\nhttps:\/\/youtu.be\/htP2goe31MM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify rational numbers and irrational numbers<\/li>\n<li>Classify different types of real numbers<\/li>\n<\/ul>\n<\/div>\n<p>The real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). In this section we will further define real numbers.<\/p>\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\n<p>The most familiar numbers are the natural numbers: [latex]1, 2, 3[\/latex], and so on.\u00a0 These are\u00a0numbers we use for counting, or enumerating items.\u00a0 The mathematical symbol for the set of all natural numbers is written as [latex]\\mathbb{N}[\/latex].\u00a0\u00a0We describe them in set notation as [latex]\\{1, 2, 3, ...\\}[\/latex] where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em>counting numbers<\/em>. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers.<\/p>\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Whole numbers<\/span><\/h3>\n<p>The set of whole numbers includes all natural numbers as well as\u00a0 [latex]0[\/latex]:\u00a0 [latex]\\{0, 1, 2, 3,...\\}[\/latex].<\/p>\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\n<p>When the set of negative numbers is combined with the set of natural numbers (including\u00a00), the result is defined as the set of integers,\u00a0[latex]\\mathbb{Z}[\/latex].\u00a0\u00a0The set of\u00a0<strong>integers<\/strong>\u00a0adds the opposites of the natural numbers to the set of whole numbers:\u00a0<span id=\"MathJax-Element-4-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; padding: 1px 0px; margin: 0px; font-size: 17.44px; vertical-align: baseline; background: #ffffff; border: 0px; line-height: 0; text-indent: 0px; text-align: left; font-style: normal; font-weight: 400; letter-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: #373d3f;\" role=\"presentation\"><span id=\"MJXc-Node-37\" class=\"mjx-math\"><span id=\"MJXc-Node-38\" class=\"mjx-mrow\"><span id=\"MJXc-Node-39\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">{<\/span><\/span><span id=\"MJXc-Node-40\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2026<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-42\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-43\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-44\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-45\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-47\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-48\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-49\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-50\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-51\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-52\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-53\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-54\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-55\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-56\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-58\" class=\"mjx-mo MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2026<\/span><\/span><span id=\"MJXc-Node-59\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">}<\/span><\/span><\/span><\/span><\/span>. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{\\text{negative integers}}\\hfill & {\\text{zero}}\\hfill & {\\text{positive integers}}\\\\{\\dots ,-3,-2,-1,}\\hfill & {0,}\\hfill & {1,2,3,\\dots }\\end{array}[\/latex]<\/p>\n<h2><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h2>\n<div class=\"hatnote relarticle mainarticle\">\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 0[\/latex].<\/p>\n<\/div>\n<p>A rational number,\u00a0[latex]\\mathbb{Q}[\/latex], is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.\u00a0\u00a0The set of <strong>rational numbers<\/strong> is written as [latex]\\left\\{\\dfrac{m}{n}\\normalsize |m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex]. Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never [latex]0[\/latex].<\/p>\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<h4>Integers are rational<\/h4>\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<h4>Decimals are rational<\/h4>\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<h4>Rational numbers as decimals<\/h4>\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<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p>Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:<\/p>\n<ol>\n<li>a terminating decimal: [latex]\\dfrac{15}{8}\\normalsize =1.875[\/latex], or<\/li>\n<li>a repeating decimal: [latex]\\dfrac{4}{11}\\normalsize =0.36363636\\dots =0.\\overline{36}[\/latex]<\/li>\n<\/ol>\n<p>We use a line drawn over the repeating block of numbers instead of writing the group multiple times.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]7[\/latex]<\/li>\n<li>[latex]0[\/latex]<\/li>\n<li>[latex]\u20138[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725771\">Show Solution<\/span><\/p>\n<div id=\"q725771\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write a fraction with the integer in the numerator and 1 in the denominator.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]7=\\dfrac{7}{1}[\/latex]<\/li>\n<li>[latex]0=\\dfrac{0}{1}[\/latex]<\/li>\n<li>[latex]-8=-\\dfrac{8}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\dfrac{5}{7}[\/latex]<\/li>\n<li>[latex]\\dfrac{15}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{13}{25}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q88918\">Show Solution<\/span><\/p>\n<div id=\"q88918\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write each fraction as a decimal by dividing the numerator by the denominator.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\dfrac{5}{7}\\normalsize =-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\n<li>[latex]\\dfrac{15}{5}\\normalsize =3[\/latex] (or 3.0), a terminating decimal<\/li>\n<li>[latex]\\dfrac{13}{25}\\normalsize=0.52[\/latex],\u00a0a terminating decimal<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2>Irrational Numbers<\/h2>\n<p>Are there any decimals that do not stop or repeat? Yes.\u00a0 At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not [latex]2[\/latex] or even [latex]\\dfrac{3}{2}[\/latex], but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than [latex]3[\/latex], but still not a rational number. \u00a0The 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 ([latex]\\pi =\\text{3.141592654.......}[\/latex]).\u00a0 Such numbers are said to be <em>irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong>irrational numbers<\/strong>. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as {h | h is not a rational number}.<\/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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\sqrt{25}[\/latex]<\/li>\n<li>[latex]\\dfrac{33}{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{11}[\/latex]<\/li>\n<li>[latex]\\dfrac{17}{34}[\/latex]<\/li>\n<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q644924\">Show Solution<\/span><\/p>\n<div id=\"q644924\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\sqrt{25}:[\/latex] This can be simplified as [latex]\\sqrt{25}=5[\/latex]. Therefore, [latex]\\sqrt{25}[\/latex] is rational.<\/li>\n<li>[latex]\\dfrac{33}{9}:[\/latex] Because it is a fraction, [latex]\\dfrac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\n<div style=\"text-align: center;\">[latex]\\dfrac{33}{9}\\normalsize =\\dfrac{{{11}\\cdot{3}}}{{{3}\\cdot{3}}}\\normalsize =\\dfrac{11}{3}\\normalsize =3.\\overline{6}[\/latex]<\/div>\n<p>So, [latex]\\dfrac{33}{9}[\/latex] is rational and a repeating decimal.<\/li>\n<li>[latex]\\sqrt{11}:[\/latex] This cannot be simplified any further. Therefore, [latex]\\sqrt{11}[\/latex] is an irrational number.<\/li>\n<li>[latex]\\dfrac{17}{34}:[\/latex] Because it is a fraction, [latex]\\dfrac{17}{34}[\/latex] is a rational number. Simplify and divide.\n<div style=\"text-align: center;\">[latex]\\dfrac{17}{34}\\normalsize =\\dfrac{1}{2}\\normalsize=0.5[\/latex]<\/div>\n<p>So, [latex]\\dfrac{17}{34}[\/latex] is rational and a terminating decimal.<\/li>\n<li>0.3033033303333&#8230; is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<\/div>\n<\/div>\n<h2>Real numbers<\/h2>\n<p>Given any number <em>n<\/em>, we know that <em>n<\/em> is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of <strong>real numbers<\/strong>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Real Numbers<\/h3>\n<p>Real numbers are numbers that are either rational or irrational.<\/p>\n<\/div>\n<p>The real numbers\u00a0include all the measuring numbers. The symbol for the real numbers is [latex]\\mathbb{R}[\/latex]. Real numbers are usually represented by using decimal numerals.\u00a0 As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.<\/p>\n<p>The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as [latex]0[\/latex], with negative numbers to the left of [latex]0[\/latex] and positive numbers to the right of [latex]0[\/latex]. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of [latex]0[\/latex]. Any real number corresponds to a unique position on the number line.\u00a0 The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the <strong>real number line<\/strong> as shown below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200208\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/><\/p>\n<p class=\"wp-caption-text\">The real number line.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\dfrac{10}{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{5}[\/latex]<\/li>\n<li>[latex]-\\sqrt{289}[\/latex]<\/li>\n<li>[latex]-6\\pi[\/latex]<\/li>\n<li>[latex]0.615384615384\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q303752\">Show Solution<\/span><\/p>\n<div id=\"q303752\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\dfrac{10}{3}[\/latex] is negative and rational. It lies to the left of 0 on the number line.<\/li>\n<li>[latex]\\sqrt{5}[\/latex] is positive and irrational. It lies to the right of [latex]0[\/latex].<\/li>\n<li>[latex]-\\sqrt{289}=-\\sqrt{{17}^{2}}=-17[\/latex] is negative and rational. It lies to the left of 0.<\/li>\n<li>[latex]-6\\pi[\/latex] is negative and irrational. It lies to the left of [latex]0[\/latex].<\/li>\n<li>[latex]0.615384615384\\dots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of [latex]0[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.\u00a0\u00a0Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.<\/p>\n<p>This diagram illustrates the relationships between the different types of real numbers.<\/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\/24222311\/CNX_BMath_Figure_07_01_001.png\" alt=\"The image shows a large rectangle labeled\" \/><\/p>\n<p>Here is another visualization of the subsets of the real numbers.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200210\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4.\" width=\"731\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\">Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Sets of Numbers<\/h3>\n<p>The set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>whole numbers<\/strong> is the set of natural numbers and zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\dfrac{m}{n}\\normalsize |m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].<\/p>\n<p>The set of <strong>irrational numbers<\/strong> is the set of numbers that are not rational. They are nonrepeating and nonterminating decimals: [latex]\\{h|h\\text{ is not a rational number}\\}[\/latex].<\/p>\n<\/div>\n<p>Does the term &#8220;real numbers&#8221; seem strange to you? Are there any numbers that are not &#8220;real&#8221;, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of <em style=\"font-size: 16px;\">imaginary numbers.<\/em><span style=\"font-size: 16px;\"> You won&#8217;t encounter imaginary numbers in this course, but you will later on in your studies of algebra.<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine whether each of the numbers in the following list is a 1. whole number, 2. integer, 3. rational number, 4. irrational number, and 5. real number.<\/p>\n<p>[latex]-7,\\Large\\frac{14}{5}\\normalsize ,8,\\sqrt{5},5.9,-\\sqrt{64}[\/latex]<\/p>\n<p>Solution:<br \/>\n1. The whole numbers are [latex]0,1,2,3\\dots[\/latex] The number [latex]8[\/latex] is the only whole number given.<\/p>\n<p>2. The integers are the whole numbers, their opposites, and [latex]0[\/latex]. From the given numbers, [latex]-7[\/latex] and [latex]8[\/latex] are integers. Also, notice that [latex]64[\/latex] is the square of [latex]8[\/latex] so [latex]-\\sqrt{64}=-8[\/latex]. So the integers are [latex]-7,8,-\\sqrt{64}[\/latex].<\/p>\n<p>3. Since all integers are rational, the numbers [latex]-7,8,\\text{and}-\\sqrt{64}[\/latex] are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so [latex]\\Large\\frac{14}{5}\\normalsize\\text{and}5.9[\/latex] are rational.<\/p>\n<p>4. The number [latex]5[\/latex] is not a perfect square, so [latex]\\sqrt{5}[\/latex] is irrational.<\/p>\n<p>5. All of the numbers listed are real.<\/p>\n<p>We&#8217;ll summarize the results in a table.<\/p>\n<table id=\"fs-id1343988\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th>Number<\/th>\n<th>Whole<\/th>\n<th>Integer<\/th>\n<th>Rational<\/th>\n<th>Irrational<\/th>\n<th>Real<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]-7[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\Large\\frac{14}{5}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\sqrt{5}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]5.9[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]-\\sqrt{64}[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Classify each number as being a natural number (<em>N<\/em>), whole number (<em>W<\/em>), integer (<em>I<\/em>), rational number (<em>Q<\/em>), and\/or irrational number (<em>Q&#8217;<\/em>).<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\sqrt{36}[\/latex]<\/li>\n<li>[latex]\\dfrac{8}{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{73}[\/latex]<\/li>\n<li>[latex]-6[\/latex]<\/li>\n<li>[latex]3.2121121112\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400826\">Show Solution<\/span><\/p>\n<div id=\"q400826\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 20%;\" summary=\"A table with six rows and six columns. The first entry in the first row is blank, but the rest of the entries read: N, W, I, Q, and Q'. (These are the sets of numbers from before.) The first entry in the second row reads: square root of thirty-six equals six. Then the second, third, fourth, and fifth columns are marked. The first entry in the third row reads: eight over three equals 2.6 with the 6 repeating forever. Then only the fifth column is marked. The first entry in the fourth row reads: square root of seventy-three. Then only the sixth column is marked. The first entry in the fifth row reads: negative six. Then the fourth and fifth columns are marked. The first entry in the sixth row reads: 3.2121121112\u2026. Then only the sixth column is marked.\">\n<thead>\n<tr>\n<th style=\"width: 39.8165%;\"><\/th>\n<th style=\"width: 39.8165%;\"><\/th>\n<th style=\"width: 4.22018%;\"><em>N<\/em><\/th>\n<th style=\"width: 4.0367%;\"><em>W<\/em><\/th>\n<th style=\"width: 3.48624%;\"><em>I<\/em><\/th>\n<th style=\"width: 4.95412%;\"><em>Q<\/em><\/th>\n<th style=\"width: 6.05505%;\"><em>Q&#8217;<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 39.8165%;\">1.<\/td>\n<td style=\"width: 39.8165%;\">[latex]\\sqrt{36}=6[\/latex]<\/td>\n<td style=\"width: 4.22018%;\">X<\/td>\n<td style=\"width: 4.0367%;\">X<\/td>\n<td style=\"width: 3.48624%;\">X<\/td>\n<td style=\"width: 4.95412%;\">X<\/td>\n<td style=\"width: 6.05505%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8165%;\">2.<\/td>\n<td style=\"width: 39.8165%;\">[latex]\\dfrac{8}{3}\\normalsize =2.\\overline{6}[\/latex]<\/td>\n<td style=\"width: 4.22018%;\"><\/td>\n<td style=\"width: 4.0367%;\"><\/td>\n<td style=\"width: 3.48624%;\"><\/td>\n<td style=\"width: 4.95412%;\">X<\/td>\n<td style=\"width: 6.05505%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8165%;\">3.<\/td>\n<td style=\"width: 39.8165%;\">[latex]\\sqrt{73}[\/latex]<\/td>\n<td style=\"width: 4.22018%;\"><\/td>\n<td style=\"width: 4.0367%;\"><\/td>\n<td style=\"width: 3.48624%;\"><\/td>\n<td style=\"width: 4.95412%;\"><\/td>\n<td style=\"width: 6.05505%;\">X<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8165%;\">4.<\/td>\n<td style=\"width: 39.8165%;\">[latex]\u20136[\/latex]<\/td>\n<td style=\"width: 4.22018%;\"><\/td>\n<td style=\"width: 4.0367%;\"><\/td>\n<td style=\"width: 3.48624%;\">X<\/td>\n<td style=\"width: 4.95412%;\">X<\/td>\n<td style=\"width: 6.05505%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8165%;\">5.<\/td>\n<td style=\"width: 39.8165%;\">[latex]3.2121121112\\dots[\/latex]<\/td>\n<td style=\"width: 4.22018%;\"><\/td>\n<td style=\"width: 4.0367%;\"><\/td>\n<td style=\"width: 3.48624%;\"><\/td>\n<td style=\"width: 4.95412%;\"><\/td>\n<td style=\"width: 6.05505%;\">X<\/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\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm149621\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=149621&theme=oea&iframe_resize_id=ohm149621&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following mini-lesson provides more examples of how to classify real numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Identifying Sets of Real Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?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-15988\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Identifying Sets of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/htP2goe31MM\">https:\/\/youtu.be\/htP2goe31MM<\/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 145962. <strong>Authored by<\/strong>: Harbaugh,Gregg, mb Lippman,David. <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":167848,"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\":\"Identifying Sets of Real Numbers\",\"author\":\"James Sousa 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