{"id":1698,"date":"2016-06-24T22:49:51","date_gmt":"2016-06-24T22:49:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1698"},"modified":"2019-07-24T20:53:24","modified_gmt":"2019-07-24T20:53:24","slug":"read-classify-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-classify-real-numbers\/","title":{"raw":"Classify Real Numbers","rendered":"Classify Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Classify a real number as a natural, whole, integer, rational, or irrational number<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: [latex]1, 2, 3, 4, 5,[\/latex] and so on. We 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. The set of <strong>whole numbers<\/strong> is the set of natural numbers and zero: [latex]\\{0, 1, 2, 3,...\\}[\/latex].\r\n\r\nThe set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: [latex]\\{...-3, -2, -1, 0, 1, 2, 3,...\\}[\/latex]. 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<div 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]<\/div>\r\nThe 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]. We can also see that every natural number, whole number, and integer is a rational number with a denominator of [latex]1[\/latex].\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<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\nAt 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. 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 shown.\r\n<div style=\"text-align: center;\">{h | h is not a rational number}<\/div>\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>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>. 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.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\n<h2>Sets of Numbers as Subsets<\/h2>\r\nBeginning 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&nbsp;\r\n\r\n&nbsp;\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\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<\/div>\r\nhttps:\/\/youtu.be\/htP2goe31MM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Classify a real number as a natural, whole, integer, rational, or irrational number<\/li>\n<\/ul>\n<\/div>\n<p>The numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: [latex]1, 2, 3, 4, 5,[\/latex] and so on. We 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. The set of <strong>whole numbers<\/strong> is the set of natural numbers and zero: [latex]\\{0, 1, 2, 3,...\\}[\/latex].<\/p>\n<p>The set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: [latex]\\{...-3, -2, -1, 0, 1, 2, 3,...\\}[\/latex]. 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<div 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]<\/div>\n<p>The 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]. We can also see that every natural number, whole number, and integer is a rational number with a denominator of [latex]1[\/latex].<\/p>\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<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>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. 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 shown.<\/p>\n<div style=\"text-align: center;\">{h | h is not a rational number}<\/div>\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>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>. 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.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<h2>Sets of Numbers as Subsets<\/h2>\n<p>Beginning 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>&nbsp;<\/p>\n<p>&nbsp;<\/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<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<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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-1698\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra: Classifying a Real Number. <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 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