{"id":1457,"date":"2021-11-06T21:27:09","date_gmt":"2021-11-06T21:27:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1457"},"modified":"2026-03-27T15:39:14","modified_gmt":"2026-03-27T15:39:14","slug":"1-1-2-sets-of-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-1-2-sets-of-numbers\/","title":{"raw":"1.1.2: Sets of Numbers","rendered":"1.1.2: Sets of Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Define natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers in terms of sets.<\/li>\r\n \t<li>Use interval notation to define sets of numbers<\/li>\r\n \t<li>Use set-builder notation to define sets of numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>KEY words<\/h1>\r\n<ul>\r\n \t<li><strong>Natural numbers<\/strong>:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]\\mathbb{N}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0[latex]=\\{1, 2, 3, ...\\}[\/latex]<\/span><\/li>\r\n \t<li><strong>Whole numbers<\/strong>: <span style=\"font-size: 1rem; text-align: initial;\">[latex]\\mathbb{W}[\/latex] [latex]=\\{0,1, 2, 3, ...\\}[\/latex]<\/span><\/li>\r\n \t<li><strong>Integers<\/strong>: \u00a0[latex]\\mathbb{Z}[\/latex]<span style=\"font-size: 1rem; text-align: initial;\"> [latex]=\\{... -3, -2, -1, 0,1, 2, 3, ...\\}[\/latex]<\/span><\/li>\r\n \t<li><strong>Rational numbers<\/strong>t: [latex]\\mathbb{Q}[\/latex] [latex]=\\,\\left\\{\\dfrac{m}{n}\\normalsize \\;\\large\\vert\\;\\normalsize\\,m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex]<\/li>\r\n \t<li><strong>Irrational numbers<\/strong>: the set of numbers that cannot be written as rational numbers<\/li>\r\n \t<li><strong>Real numbers<\/strong>: [latex]\\mathbb{R}[\/latex] = the union of the set of rational numbers and the set of irrational numbers<\/li>\r\n \t<li><strong>Interval notation<\/strong>: shows highest and lowest values in an interval inside brackets or parentheses<\/li>\r\n \t<li><strong>Set-builder notation<\/strong>: defines a set inside braces using variables, words, inequalities, etc.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Number Systems<\/h2>\r\nThe number system that we use today is called the Real Numbers. It is divided into subsets. We will define each subset and then further define the Real Numbers.\r\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\r\nThe <em><strong>natural numbers<\/strong><\/em> (sometimes called counting numbers) are: [latex]1, 2, 3[\/latex], and so on.\u00a0 These are\u00a0numbers we use for counting, or enumerating items. \u00a0<span style=\"font-size: 1em;\">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.\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">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\u00a0[latex]\\mathbb{N}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0[latex]=\\{1, 2, 3, ...\\}[\/latex] where the ellipsis (\u2026) indicates that the numbers continue following the same pattern to infinity.<\/span>\r\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Whole numbers<\/span><\/h3>\r\nThe set of <em><strong>whole numbers<\/strong><\/em> includes all natural numbers as well as [latex]0[\/latex].\u00a0<span style=\"font-size: 1rem; text-align: initial;\">We describe them in set notation as [latex]\\mathbb{W}[\/latex] [latex]=\\{0,1, 2, 3, ...\\}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\r\nWhen the opposites of the natural numbers are combined with the set of whole numbers, the result is defined as the set of <em><strong>integers<\/strong><\/em>,\u00a0[latex]\\mathbb{Z}[\/latex]<span style=\"font-size: 1rem; text-align: initial;\"> [latex]=\\{... -3, -2, -1, 0,1, 2, 3, ...\\}[\/latex]<\/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<h3><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h3>\r\n<div class=\"hatnote relarticle mainarticle\">\r\n\r\nNumbers that can be written as one integer divided by another integer in the form [latex]\\frac{p}{q}[\/latex] are known as <em><strong>fractions<\/strong><\/em>. Fractions represent part of a whole. The bottom line (denominator) tells us how many equal parts the whole (1) is divided into, and the top line (numerator) tells us how many parts are being used. For example, the fraction [latex]\\frac{3}{4}[\/latex], read \"three-fourths\" or \"three-quarters\", represents 3 out of 4 equal parts of a whole. When fractions are combined with the set of integers, the result is defined as the set of <em><strong>rational numbers<\/strong><\/em>, [latex]\\mathbb{Q}[\/latex].\u00a0A rational number is any number that can be written as a <em><strong>ratio<\/strong><\/em> of two integers. A ratio is just the comparison of two numbers, the numerator and denominator of the fraction.\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]{\\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 integer numerator and denominator.\u00a0\u00a0The set of <strong>rational numbers<\/strong> is written as \u00a0[latex]\\mathbb{Q}[\/latex] [latex]=\\,\\left\\{\\dfrac{p}{q}\\normalsize \\;\\large\\vert\\;\\normalsize\\,p\\text{ and }{q}\\text{ are integers and }{q}\\ne{ 0 }\\right\\}[\/latex]. This is referred to as <em><strong>set builder notation<\/strong><\/em>, and is read, \" the set of fractions where the numerator and denominator are integers,\u00a0and the denominator is never [latex]0[\/latex]\". The vertical line [latex]\\large\\vert[\/latex] is read \"where\" or \"such that\". Recall that letters like [latex]p[\/latex] and [latex]q[\/latex] are called\u00a0<em><strong>variables<\/strong>.<\/em> In this context the variables are defined as integer values with [latex]p\\neq 0[\/latex].\r\n\r\nRational numbers are defined as fractions, so the fractions,\u00a0[latex]\\frac{4}{5} ,-\\frac{7}{8} ,\\frac{13}{4}[\/latex] and [latex]-\\frac{20}{3}[\/latex] that all have\u00a0numerators and each denominators that are integers are rational numbers.\r\n\r\nThe definition of rational numbers tells us that all fractions are rational, but what about\u00a0natural numbers, whole numbers, and integers?\r\n<h4>Are integers rational?<\/h4>\r\nTo 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=\\frac{3}{1}\\normalsize ,\\space-8=\\frac{-8}{1} ,\\space0=\\frac{0}{1}[\/latex]<\/p>\r\nSince any integer can be written as the ratio of two integers, all integers are rational numbers. And since natural numbers and whole numbers are subsets of the integers, they, too, are rational.\r\n<div class=\"hatnote relarticle mainarticle\">\r\n<h4>Can rational numbers be written 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=\\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\nBy definition, every fraction is a rational number. Look at the decimal form of some fractions.\r\n<p style=\"padding-left: 60px;\">Ratio of Integers [latex]\\frac{4}{5} ,\\frac{7}{8} ,\\frac{13}{4} ,\\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 us? <strong>Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats.<\/strong>\r\n\r\nBecause they are fractions, any rational number can also be expressed in decimal form by dividing the numerator of the fraction by the denominator. Any rational number can be represented as either:\r\n\r\n<\/div>\r\n<div>\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<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<h4>Are all decimals 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, so any decimal that is also an integer must also be a rational number. So, clearly, some decimals are rational.\r\n\r\nA decimal number has an integer part before the decimal point, while numbers after the decimal point are parts of a whole. For example, the decimal [latex]12.645[\/latex] consists of 12 wholes and 645 thousandths. We know it is thousandths because there are three places after the decimal point. Consequently, [latex]12.645[\/latex] can be written as the mixed number [latex]12\\frac{645}{1000}[\/latex] or as the fraction [latex]\\frac{12,645}{1000}[\/latex]. So,\u00a0[latex]12.645[\/latex] 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.<span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1em;\">Also, any decimal that never ends but repeats such as [latex]0.3333...[\/latex] is a rational number. If you use your calculator to divide 1 by 3 you will find that \u00a0[latex]\\frac{1}{3} = 0.3333...[\/latex]. When a number repeats, we can use ellipses\u00a0[latex](...)[\/latex] or a bar over the repeating digit(s) [latex]0.\\bar{3}[\/latex] to show that the pattern repeats forever.<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite each number as the ratio of two integers:\r\n<ol>\r\n \t<li>[latex]-15[\/latex]<\/li>\r\n \t<li>[latex]6.81[\/latex]<\/li>\r\n \t<li>[latex]-3\\frac{6}{7}[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469454543\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\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]\\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<th>2.<\/th>\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\\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]\\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<th>3.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3\\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]-\\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<\/div>\r\n<div>\r\n<h2>Irrational Numbers<\/h2>\r\nNot all numbers are rational. Such numbers are said to be <em>irrational<\/em> because they cannot be written as fractions. <em><strong>Irrational numbers<\/strong><\/em> 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 [latex]\\{\\;x \\;\\;\\large | \\; \\normalsize x \\text{ is not a rational number}\\}[\/latex].\r\n\r\nExamples of irrational numbers are [latex]pi[\/latex], which is used with circles, and\u00a0[latex]e[\/latex], which is used in growth problems.\u00a0Similarly, the decimal representations of square roots of numbers that are not perfect squares are irrational numbers. Approximations of these numbers can be found and used, but the numbers themselves are decimals that never repeat and never end.\r\n\r\nFor 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\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<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<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<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<\/div>\r\n<h2>Real numbers<\/h2>\r\nAny number\u00a0is\u00a0either rational or irrational. It cannot be both. It can either be written as a fraction or it cannot. The sets of rational and irrational numbers together make up the set of <em><strong>real numbers<\/strong><\/em>, [latex]\\mathbb{R}[\/latex]. This means that the set of irrational numbers is the complement of the set of rational numbers in the set of real numbers.\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 often represented using decimal numbers.\u00a0 Like 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 <em><strong>real number line<\/strong><\/em> 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\nWe have seen that all natural 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\n[caption id=\"attachment_1461\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-1461 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-1024x648.png\" alt=\"Four nested rectangles represent number sets. The innermost rectangle is labeled \u201cNatural Numbers,\u201d enclosed by a larger rectangle labeled \u201cWhole Numbers,\u201d which is enclosed by another rectangle labeled \u201cIntegers.\u201d This is further enclosed by the largest rectangle labeled \u201cRational Numbers.\u201d A separate rectangle labeled \u201cIrrational Numbers\u201d is placed next to, but not enclosing or enclosed by, the \u201cRational Numbers\u201d rectangle.\" width=\"1024\" height=\"648\" \/> The set of irrational numbers is the complement of the set of rational numbers in the set of real numbers. The set of natural numbers is a subset of the set of whole numbers is a subset of the set of integers is a subset of the set of rational numbers is a subset of the set of real numbers.[\/caption]\r\n<h2>Interval Notation<\/h2>\r\nAnother commonly used method for describing sets of numbers is called\u00a0<strong>interval notation.<\/strong> With this convention, sets are built with parentheses or brackets, each having a distinct meaning. Interval notation is used to denote an interval of numbers. For example, the interval [latex](2,3)[\/latex] represents the interval of numbers that are greater than two and less than three.\r\n\r\nThe main concept is that parentheses represent solutions greater than or less than the number, and brackets represent numbers that are greater than or equal to or less than or equal to the number. For example, the interval [latex][-2,3)[\/latex] represents the interval greater than or equal to negative two to less than three.\r\n\r\nParentheses are used to represent infinity or negative infinity, as infinity is not a number in the usual sense of the word. For example [latex](-\\infty, -3][\/latex] is all numbers less than and including -3. However, the interval [latex](-\\infty, -3)[\/latex] is all numbers less than -3, not including -3 itself.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to -2.\r\n\r\n[reveal-answer q=\"RB0001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"RB0001\"]\r\n\r\nUse a bracket to the left of -2 and parentheses after infinity: [latex][-2, \\infty)[\/latex]. The bracket indicates that -2 is included in the set with all numbers that are larger than -2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Representing an Interval on a Number Line<\/h2>\r\nIntervals can be graphed on a number line. Graphs of number lines and intervals can be very helpful in visualizing the interval. For example the interval [latex][-3, 4)[\/latex] can be represented by the following graph:\r\n\r\n<img class=\"aligncenter wp-image-1278\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20182406\/Interval-Image-1-300x77.jpg\" alt=\"A number line marked from -10 to 10 with an highlighted interval from -3 (a solid circle) to 4 (an empty circle).\" width=\"436\" height=\"112\" \/>\r\n\r\nNote that the closed circle is used to represent the inclusion of that point in the set, and the open point is used to demonstrate that the point is not included in the set.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse a real number line to describe the interval [latex](-2,\\,6][\/latex].\r\n\r\n[reveal-answer q=\"H000011\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H000011\"]\r\n\r\nThe parentheses ( next to the -2 indicates that -2 is not included in the interval so an open point is used at -2. The bracket ] next to the 6 indicates that 6 is included in the interval so a closed point is used at 6.\r\n\r\n<img class=\"aligncenter wp-image-1463 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-300x44.png\" alt=\"A number line marked from -3 to 7 with an highlighted interval from -2 (an empty circle) to 6 (a solid circle).\" width=\"300\" height=\"44\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse a real number line to describe the interval [latex][3,\\,5)[\/latex].\r\n\r\n[reveal-answer q=\"H00012\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H00012\"]\r\n\r\n<img class=\"aligncenter wp-image-1464 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-300x48.png\" alt=\"A number line marked from -3 to 7 with an highlighted interval from 3 (a solid circle) to 5 (an empty circle).\" width=\"300\" height=\"48\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Set-Builder Notation<\/h2>\r\nAnother way to represent an interval of real numbers is to use\u00a0<em><strong>set-builder notation<\/strong><\/em>. An example of set-builder notation is the set of real numbers that are greater than 5: [latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\gt\\,5\\right\\}[\/latex]. This\u00a0<span style=\"font-size: 1em;\">is read, \" the set of all real numbers, [latex]x[\/latex], where\u00a0[latex]x[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is greater than 5\".\u00a0<\/span><span style=\"font-size: 1em; text-align: initial;\">The vertical line [latex]\\large\\vert[\/latex] is read \"where\" or \"such that\".<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse set builder notation to describe the real numbers that lie in the interval [latex](-2,\\,6][\/latex].\r\n\r\n[reveal-answer q=\"H000112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H000112\"]\r\n\r\nThe [latex](-2[\/latex] in the interval [latex](-2,\\,6][\/latex] tells us that the numbers are greater than -2 and the bracket [latex]6][\/latex] tells us that the numbers are also less than or equal to 6.\r\n\r\n[latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\gt\\,-2\\text{ and }x\\le\\,6\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse set builder notation to describe the real numbers that lie in the interval [latex][3,\\,5)[\/latex].\r\n\r\n[reveal-answer q=\"H00013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H00013\"]\r\n\r\n[latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\ge\\,3\\text{ and }x\\lt\\,5\\right\\}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Define natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers in terms of sets.<\/li>\n<li>Use interval notation to define sets of numbers<\/li>\n<li>Use set-builder notation to define sets of numbers<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>KEY words<\/h1>\n<ul>\n<li><strong>Natural numbers<\/strong>:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]\\mathbb{N}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0[latex]=\\{1, 2, 3, ...\\}[\/latex]<\/span><\/li>\n<li><strong>Whole numbers<\/strong>: <span style=\"font-size: 1rem; text-align: initial;\">[latex]\\mathbb{W}[\/latex] [latex]=\\{0,1, 2, 3, ...\\}[\/latex]<\/span><\/li>\n<li><strong>Integers<\/strong>: \u00a0[latex]\\mathbb{Z}[\/latex]<span style=\"font-size: 1rem; text-align: initial;\"> [latex]=\\{... -3, -2, -1, 0,1, 2, 3, ...\\}[\/latex]<\/span><\/li>\n<li><strong>Rational numbers<\/strong>t: [latex]\\mathbb{Q}[\/latex] [latex]=\\,\\left\\{\\dfrac{m}{n}\\normalsize \\;\\large\\vert\\;\\normalsize\\,m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex]<\/li>\n<li><strong>Irrational numbers<\/strong>: the set of numbers that cannot be written as rational numbers<\/li>\n<li><strong>Real numbers<\/strong>: [latex]\\mathbb{R}[\/latex] = the union of the set of rational numbers and the set of irrational numbers<\/li>\n<li><strong>Interval notation<\/strong>: shows highest and lowest values in an interval inside brackets or parentheses<\/li>\n<li><strong>Set-builder notation<\/strong>: defines a set inside braces using variables, words, inequalities, etc.<\/li>\n<\/ul>\n<\/div>\n<h2>Number Systems<\/h2>\n<p>The number system that we use today is called the Real Numbers. It is divided into subsets. We will define each subset and then further define the Real Numbers.<\/p>\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\n<p>The <em><strong>natural numbers<\/strong><\/em> (sometimes called counting numbers) are: [latex]1, 2, 3[\/latex], and so on.\u00a0 These are\u00a0numbers we use for counting, or enumerating items. \u00a0<span style=\"font-size: 1em;\">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.\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">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\u00a0[latex]\\mathbb{N}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0[latex]=\\{1, 2, 3, ...\\}[\/latex] where the ellipsis (\u2026) indicates that the numbers continue following the same pattern to infinity.<\/span><\/p>\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Whole numbers<\/span><\/h3>\n<p>The set of <em><strong>whole numbers<\/strong><\/em> includes all natural numbers as well as [latex]0[\/latex].\u00a0<span style=\"font-size: 1rem; text-align: initial;\">We describe them in set notation as [latex]\\mathbb{W}[\/latex] [latex]=\\{0,1, 2, 3, ...\\}[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\n<p>When the opposites of the natural numbers are combined with the set of whole numbers, the result is defined as the set of <em><strong>integers<\/strong><\/em>,\u00a0[latex]\\mathbb{Z}[\/latex]<span style=\"font-size: 1rem; text-align: initial;\"> [latex]=\\{... -3, -2, -1, 0,1, 2, 3, ...\\}[\/latex]<\/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<h3><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h3>\n<div class=\"hatnote relarticle mainarticle\">\n<p>Numbers that can be written as one integer divided by another integer in the form [latex]\\frac{p}{q}[\/latex] are known as <em><strong>fractions<\/strong><\/em>. Fractions represent part of a whole. The bottom line (denominator) tells us how many equal parts the whole (1) is divided into, and the top line (numerator) tells us how many parts are being used. For example, the fraction [latex]\\frac{3}{4}[\/latex], read &#8220;three-fourths&#8221; or &#8220;three-quarters&#8221;, represents 3 out of 4 equal parts of a whole. When fractions are combined with the set of integers, the result is defined as the set of <em><strong>rational numbers<\/strong><\/em>, [latex]\\mathbb{Q}[\/latex].\u00a0A rational number is any number that can be written as a <em><strong>ratio<\/strong><\/em> of two integers. A ratio is just the comparison of two numbers, the numerator and denominator of the fraction.<\/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]{\\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 integer numerator and denominator.\u00a0\u00a0The set of <strong>rational numbers<\/strong> is written as \u00a0[latex]\\mathbb{Q}[\/latex] [latex]=\\,\\left\\{\\dfrac{p}{q}\\normalsize \\;\\large\\vert\\;\\normalsize\\,p\\text{ and }{q}\\text{ are integers and }{q}\\ne{ 0 }\\right\\}[\/latex]. This is referred to as <em><strong>set builder notation<\/strong><\/em>, and is read, &#8221; the set of fractions where the numerator and denominator are integers,\u00a0and the denominator is never [latex]0[\/latex]&#8220;. The vertical line [latex]\\large\\vert[\/latex] is read &#8220;where&#8221; or &#8220;such that&#8221;. Recall that letters like [latex]p[\/latex] and [latex]q[\/latex] are called\u00a0<em><strong>variables<\/strong>.<\/em> In this context the variables are defined as integer values with [latex]p\\neq 0[\/latex].<\/p>\n<p>Rational numbers are defined as fractions, so the fractions,\u00a0[latex]\\frac{4}{5} ,-\\frac{7}{8} ,\\frac{13}{4}[\/latex] and [latex]-\\frac{20}{3}[\/latex] that all have\u00a0numerators and each denominators that are integers are rational numbers.<\/p>\n<p>The definition of rational numbers tells us that all fractions are rational, but what about\u00a0natural numbers, whole numbers, and integers?<\/p>\n<h4>Are integers rational?<\/h4>\n<p>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=\\frac{3}{1}\\normalsize ,\\space-8=\\frac{-8}{1} ,\\space0=\\frac{0}{1}[\/latex]<\/p>\n<p>Since any integer can be written as the ratio of two integers, all integers are rational numbers. And since natural numbers and whole numbers are subsets of the integers, they, too, are rational.<\/p>\n<div class=\"hatnote relarticle mainarticle\">\n<h4>Can rational numbers be written 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=\\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>By definition, every fraction is a rational number. Look at the decimal form of some fractions.<\/p>\n<p style=\"padding-left: 60px;\">Ratio of Integers [latex]\\frac{4}{5} ,\\frac{7}{8} ,\\frac{13}{4} ,\\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 us? <strong>Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats.<\/strong><\/p>\n<p>Because they are fractions, any rational number can also be expressed in decimal form by dividing the numerator of the fraction by the denominator. Any rational number can be represented as either:<\/p>\n<\/div>\n<div>\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<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<h4>Are all decimals 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, so any decimal that is also an integer must also be a rational number. So, clearly, some decimals are rational.<\/p>\n<p>A decimal number has an integer part before the decimal point, while numbers after the decimal point are parts of a whole. For example, the decimal [latex]12.645[\/latex] consists of 12 wholes and 645 thousandths. We know it is thousandths because there are three places after the decimal point. Consequently, [latex]12.645[\/latex] can be written as the mixed number [latex]12\\frac{645}{1000}[\/latex] or as the fraction [latex]\\frac{12,645}{1000}[\/latex]. So,\u00a0[latex]12.645[\/latex] 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.<span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1em;\">Also, any decimal that never ends but repeats such as [latex]0.3333...[\/latex] is a rational number. If you use your calculator to divide 1 by 3 you will find that \u00a0[latex]\\frac{1}{3} = 0.3333...[\/latex]. When a number repeats, we can use ellipses\u00a0[latex](...)[\/latex] or a bar over the repeating digit(s) [latex]0.\\bar{3}[\/latex] to show that the pattern repeats forever.<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write each number as the ratio of two integers:<\/p>\n<ol>\n<li>[latex]-15[\/latex]<\/li>\n<li>[latex]6.81[\/latex]<\/li>\n<li>[latex]-3\\frac{6}{7}[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469454543\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\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]\\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<th>2.<\/th>\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\\frac{81}{100}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Then convert it to an improper fraction.<\/td>\n<td>[latex]\\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<th>3.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3\\frac{6}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert the mixed number to an improper fraction.<\/td>\n<td>[latex]-\\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<\/div>\n<div>\n<h2>Irrational Numbers<\/h2>\n<p>Not all numbers are rational. Such numbers are said to be <em>irrational<\/em> because they cannot be written as fractions. <em><strong>Irrational numbers<\/strong><\/em> 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 [latex]\\{\\;x \\;\\;\\large | \\; \\normalsize x \\text{ is not a rational number}\\}[\/latex].<\/p>\n<p>Examples of irrational numbers are [latex]pi[\/latex], which is used with circles, and\u00a0[latex]e[\/latex], which is used in growth problems.\u00a0Similarly, the decimal representations of square roots of numbers that are not perfect squares are irrational numbers. Approximations of these numbers can be found and used, but the numbers themselves are decimals that never repeat and never end.<\/p>\n<p>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<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<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<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<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<\/div>\n<h2>Real numbers<\/h2>\n<p>Any number\u00a0is\u00a0either rational or irrational. It cannot be both. It can either be written as a fraction or it cannot. The sets of rational and irrational numbers together make up the set of <em><strong>real numbers<\/strong><\/em>, [latex]\\mathbb{R}[\/latex]. This means that the set of irrational numbers is the complement of the set of rational numbers in the set of real numbers.<\/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 often represented using decimal numbers.\u00a0 Like 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 <em><strong>real number line<\/strong><\/em> 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>We have seen that all natural 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<div id=\"attachment_1461\" style=\"width: 1034px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1461\" class=\"wp-image-1461 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-1024x648.png\" alt=\"Four nested rectangles represent number sets. The innermost rectangle is labeled \u201cNatural Numbers,\u201d enclosed by a larger rectangle labeled \u201cWhole Numbers,\u201d which is enclosed by another rectangle labeled \u201cIntegers.\u201d This is further enclosed by the largest rectangle labeled \u201cRational Numbers.\u201d A separate rectangle labeled \u201cIrrational Numbers\u201d is placed next to, but not enclosing or enclosed by, the \u201cRational Numbers\u201d rectangle.\" width=\"1024\" height=\"648\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-1024x648.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-300x190.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-768x486.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-65x41.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-225x142.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set-350x222.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Real-Numbers-Set.png 1558w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p id=\"caption-attachment-1461\" class=\"wp-caption-text\">The set of irrational numbers is the complement of the set of rational numbers in the set of real numbers. The set of natural numbers is a subset of the set of whole numbers is a subset of the set of integers is a subset of the set of rational numbers is a subset of the set of real numbers.<\/p>\n<\/div>\n<h2>Interval Notation<\/h2>\n<p>Another commonly used method for describing sets of numbers is called\u00a0<strong>interval notation.<\/strong> With this convention, sets are built with parentheses or brackets, each having a distinct meaning. Interval notation is used to denote an interval of numbers. For example, the interval [latex](2,3)[\/latex] represents the interval of numbers that are greater than two and less than three.<\/p>\n<p>The main concept is that parentheses represent solutions greater than or less than the number, and brackets represent numbers that are greater than or equal to or less than or equal to the number. For example, the interval [latex][-2,3)[\/latex] represents the interval greater than or equal to negative two to less than three.<\/p>\n<p>Parentheses are used to represent infinity or negative infinity, as infinity is not a number in the usual sense of the word. For example [latex](-\\infty, -3][\/latex] is all numbers less than and including -3. However, the interval [latex](-\\infty, -3)[\/latex] is all numbers less than -3, not including -3 itself.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to -2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qRB0001\">Show Solution<\/span><\/p>\n<div id=\"qRB0001\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket to the left of -2 and parentheses after infinity: [latex][-2, \\infty)[\/latex]. The bracket indicates that -2 is included in the set with all numbers that are larger than -2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Representing an Interval on a Number Line<\/h2>\n<p>Intervals can be graphed on a number line. Graphs of number lines and intervals can be very helpful in visualizing the interval. For example the interval [latex][-3, 4)[\/latex] can be represented by the following graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1278\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20182406\/Interval-Image-1-300x77.jpg\" alt=\"A number line marked from -10 to 10 with an highlighted interval from -3 (a solid circle) to 4 (an empty circle).\" width=\"436\" height=\"112\" \/><\/p>\n<p>Note that the closed circle is used to represent the inclusion of that point in the set, and the open point is used to demonstrate that the point is not included in the set.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use a real number line to describe the interval [latex](-2,\\,6][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH000011\">Show Solution<\/span><\/p>\n<div id=\"qH000011\" class=\"hidden-answer\" style=\"display: none\">\n<p>The parentheses ( next to the -2 indicates that -2 is not included in the interval so an open point is used at -2. The bracket ] next to the 6 indicates that 6 is included in the interval so a closed point is used at 6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1463 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-300x44.png\" alt=\"A number line marked from -3 to 7 with an highlighted interval from -2 (an empty circle) to 6 (a solid circle).\" width=\"300\" height=\"44\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-300x44.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-768x113.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-1024x151.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-225x33.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26-350x52.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Number-line-26.png 1222w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use a real number line to describe the interval [latex][3,\\,5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH00012\">Show Solution<\/span><\/p>\n<div id=\"qH00012\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1464 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-300x48.png\" alt=\"A number line marked from -3 to 7 with an highlighted interval from 3 (a solid circle) to 5 (an empty circle).\" width=\"300\" height=\"48\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-300x48.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-768x124.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-1024x165.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-225x36.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35-350x57.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/11\/Line-35.png 1226w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Set-Builder Notation<\/h2>\n<p>Another way to represent an interval of real numbers is to use\u00a0<em><strong>set-builder notation<\/strong><\/em>. An example of set-builder notation is the set of real numbers that are greater than 5: [latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\gt\\,5\\right\\}[\/latex]. This\u00a0<span style=\"font-size: 1em;\">is read, &#8221; the set of all real numbers, [latex]x[\/latex], where\u00a0[latex]x[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is greater than 5&#8243;.\u00a0<\/span><span style=\"font-size: 1em; text-align: initial;\">The vertical line [latex]\\large\\vert[\/latex] is read &#8220;where&#8221; or &#8220;such that&#8221;.<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use set builder notation to describe the real numbers that lie in the interval [latex](-2,\\,6][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH000112\">Show Solution<\/span><\/p>\n<div id=\"qH000112\" class=\"hidden-answer\" style=\"display: none\">\n<p>The [latex](-2[\/latex] in the interval [latex](-2,\\,6][\/latex] tells us that the numbers are greater than -2 and the bracket [latex]6][\/latex] tells us that the numbers are also less than or equal to 6.<\/p>\n<p>[latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\gt\\,-2\\text{ and }x\\le\\,6\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use set builder notation to describe the real numbers that lie in the interval [latex][3,\\,5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH00013\">Show Solution<\/span><\/p>\n<div id=\"qH00013\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{x\\in\\mathbb{R}\\;\\large\\vert\\;\\normalsize\\,x\\ge\\,3\\text{ and }x\\lt\\,5\\right\\}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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-1457\">\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>Real numbers diagram; Example 6 and Interval notation Try It; Set-builder Notation. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Adapted &amp; revised: Lumen Learning. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":370291,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Adapted & revised: Lumen Learning\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Real numbers diagram; Example 6 and Interval notation Try It; Set-builder Notation\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1457","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1457","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":18,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1457\/revisions"}],"predecessor-version":[{"id":3206,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1457\/revisions\/3206"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1457\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1457"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1457"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1457"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1457"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}