{"id":1301,"date":"2016-10-21T22:49:41","date_gmt":"2016-10-21T22:49:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1301"},"modified":"2017-04-04T22:33:59","modified_gmt":"2017-04-04T22:33:59","slug":"classify-a-real-number","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/classify-a-real-number\/","title":{"raw":"Classify a Real Number","rendered":"Classify a Real Number"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Write integers as rational numbers<\/li>\r\n \t<li>Identify rational numbers<\/li>\r\n \t<li>Classify real numbers into sets<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1, 2, 3, ...} 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 plus zero: {0, 1, 2, 3,...}.\r\n\r\nThe set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: {...-3, -2, -1, 0, 1, 2, 3,...}. 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\\{\\frac{m}{n}|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 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.\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]\\frac{15}{8}=1.875[\/latex], or<\/li>\r\n \t<li>a repeating decimal: [latex]\\frac{4}{11}=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: Writing Integers as Rational Numbers<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol>\r\n \t<li>7<\/li>\r\n \t<li>0<\/li>\r\n \t<li>\u20138<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"534535\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"534535\"]\r\nWrite a fraction with the integer in the numerator and 1 in the denominator.\r\n<ol>\r\n \t<li>[latex]7=\\frac{7}{1}[\/latex]<\/li>\r\n \t<li>[latex]0=\\frac{0}{1}[\/latex]<\/li>\r\n \t<li>[latex]-8=-\\frac{8}{1}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol>\r\n \t<li>11<\/li>\r\n \t<li>3<\/li>\r\n \t<li>\u20134<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"755048\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"755048\"]\r\n<ol>\r\n \t<li>[latex]\\frac{11}{1}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{3}{1}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{4}{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: Identifying Rational Numbers<\/h3>\r\nWrite each of the following rational numbers as either a terminating or repeating decimal.\r\n<ol>\r\n \t<li>[latex]-\\frac{5}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{15}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{13}{25}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"549544\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"549544\"]\r\nWrite each fraction as a decimal by dividing the numerator by the denominator.\r\n<ol>\r\n \t<li>[latex]-\\frac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\r\n \t<li>[latex]\\frac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\r\n \t<li>[latex]\\frac{13}{25}=0.52[\/latex],\r\na terminating decimal<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<h2 data-type=\"title\">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 2 or even [latex]\\frac{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 3, 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: Differentiating Rational and Irrational Numbers<\/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>\r\n \t<li>[latex]\\sqrt{25}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{33}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{11}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{17}{34}[\/latex]<\/li>\r\n \t<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"502265\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"502265\"]\r\n<ol>\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]\\frac{33}{9}:[\/latex] Because it is a fraction, [latex]\\frac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\frac{33}{9}=\\frac{{{11}\\cdot{3}}}{{{3}\\cot{3}}}=\\frac{11}{3}=3.\\overline{6}[\/latex]<\/div>\r\nSo, [latex]\\frac{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]\\frac{17}{34}:[\/latex] Because it is a fraction, [latex]\\frac{17}{34}[\/latex] is a rational number. Simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\frac{17}{34}=\\frac{{1}{\\overline{)17}}}{\\underset{2}{\\overline{)34}}}=\\frac{1}{2}=0.5[\/latex]<\/div>\r\nSo, [latex]\\frac{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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92383&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div>\r\n<h2 data-type=\"title\">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 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. 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>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/> The real number line[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Classifying Real Numbers<\/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>\r\n \t<li>[latex]-\\frac{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=\"705558\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"705558\"]\r\n<ol>\r\n \t<li>[latex]-\\frac{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 0.<\/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 0.<\/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 0.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/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>\r\n \t<li>[latex]\\sqrt{73}[\/latex]<\/li>\r\n \t<li>[latex]-11.411411411\\dots [\/latex]<\/li>\r\n \t<li>[latex]\\frac{47}{19}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{\\sqrt{5}}{2}[\/latex]<\/li>\r\n \t<li>[latex]6.210735[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"155954\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"155954\"]\r\n<ol>\r\n \t<li>positive, irrational; right<\/li>\r\n \t<li>negative, rational; left<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n \t<li>negative, irrational; left<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">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[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/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\" data-media-type=\"image\/jpg\" \/> 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 plus 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]\\{\\frac{m}{n}|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, are nonrepeating, and are nonterminating: [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: Differentiating the Sets of Numbers<\/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>\r\n \t<li>[latex]\\sqrt{36}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{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=\"779749\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"779749\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>N<\/strong><\/td>\r\n<td><strong>W<\/strong><\/td>\r\n<td><strong>I<\/strong><\/td>\r\n<td><strong>Q<\/strong><\/td>\r\n<td><strong>Q`<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1. [latex]\\sqrt{36}=6[\/latex]<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. [latex]\\frac{8}{3}=2.\\overline{6}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. [latex]\\sqrt{73}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. \u20136<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. [latex]3.2121121112\\dots[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>\u00a0X<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/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]-\\frac{35}{7}[\/latex]<\/li>\r\n \t<li>[latex]0[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{169}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{24}[\/latex]<\/li>\r\n \t<li>[latex]4.763763763\\dots [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"266197\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"266197\"]\r\n<table summary=\"A table with six rows and six columns. The first entry of the first row is empty, but the rest read: N, W, I, Q, and Q'. (These are the sets of numbers.) The first entry of the second row reads: negative thirty-five over seven. Then the fourth and fifth columns are marked. The first entry of the third row reads: zero. Then the third, fourth, and fifth columns are marked. The first entry of the fourth row reads: square root of one hundred sixty-nine. Then the second, third, fourth, and fifth columns are marked. The first entry of the fifth row reads: square root of twenty-four. Then only the sixth column is marked. The first entry of the sixth row reads: 4.763763763\u2026. Then only the fifth column is marked\">\r\n<thead>\r\n<tr>\r\n<th><em>N<\/em><\/th>\r\n<th><em>W<\/em><\/th>\r\n<th><em>I<\/em><\/th>\r\n<th><em>Q<\/em><\/th>\r\n<th><em>Q'<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>a. [latex]-\\frac{35}{7}[\/latex]<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b. 0<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>c. [latex]\\sqrt{169}[\/latex]<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>d. [latex]\\sqrt{24}[\/latex]<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>e. [latex]4.763763763\\dots[\/latex]<\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=13740&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"100%\"><\/iframe>\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=13741&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"100%\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWatch this video for an overview of the sets of numbers, and how to identify which set a number belongs to.\r\nhttps:\/\/youtu.be\/htP2goe31MM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Write integers as rational numbers<\/li>\n<li>Identify rational numbers<\/li>\n<li>Classify real numbers into sets<\/li>\n<\/ul>\n<\/div>\n<p>The numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1, 2, 3, &#8230;} 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 plus zero: {0, 1, 2, 3,&#8230;}.<\/p>\n<p>The set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: {&#8230;-3, -2, -1, 0, 1, 2, 3,&#8230;}. 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\\{\\frac{m}{n}|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 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.<\/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]\\frac{15}{8}=1.875[\/latex], or<\/li>\n<li>a repeating decimal: [latex]\\frac{4}{11}=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: Writing Integers as Rational Numbers<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol>\n<li>7<\/li>\n<li>0<\/li>\n<li>\u20138<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q534535\">Solution<\/span><\/p>\n<div id=\"q534535\" class=\"hidden-answer\" style=\"display: none\">\nWrite a fraction with the integer in the numerator and 1 in the denominator.<\/p>\n<ol>\n<li>[latex]7=\\frac{7}{1}[\/latex]<\/li>\n<li>[latex]0=\\frac{0}{1}[\/latex]<\/li>\n<li>[latex]-8=-\\frac{8}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol>\n<li>11<\/li>\n<li>3<\/li>\n<li>\u20134<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q755048\">Solution<\/span><\/p>\n<div id=\"q755048\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\frac{11}{1}[\/latex]<\/li>\n<li>[latex]\\frac{3}{1}[\/latex]<\/li>\n<li>[latex]-\\frac{4}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Rational Numbers<\/h3>\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<ol>\n<li>[latex]-\\frac{5}{7}[\/latex]<\/li>\n<li>[latex]\\frac{15}{5}[\/latex]<\/li>\n<li>[latex]\\frac{13}{25}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q549544\">Solution<\/span><\/p>\n<div id=\"q549544\" class=\"hidden-answer\" style=\"display: none\">\nWrite each fraction as a decimal by dividing the numerator by the denominator.<\/p>\n<ol>\n<li>[latex]-\\frac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\n<li>[latex]\\frac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\n<li>[latex]\\frac{13}{25}=0.52[\/latex],<br \/>\na terminating decimal<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2 data-type=\"title\">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 2 or even [latex]\\frac{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 3, 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: Differentiating Rational and Irrational Numbers<\/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>\n<li>[latex]\\sqrt{25}[\/latex]<\/li>\n<li>[latex]\\frac{33}{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{11}[\/latex]<\/li>\n<li>[latex]\\frac{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=\"q502265\">Solution<\/span><\/p>\n<div id=\"q502265\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\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]\\frac{33}{9}:[\/latex] Because it is a fraction, [latex]\\frac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\n<div style=\"text-align: center;\">[latex]\\frac{33}{9}=\\frac{{{11}\\cdot{3}}}{{{3}\\cot{3}}}=\\frac{11}{3}=3.\\overline{6}[\/latex]<\/div>\n<p>So, [latex]\\frac{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]\\frac{17}{34}:[\/latex] Because it is a fraction, [latex]\\frac{17}{34}[\/latex] is a rational number. Simplify and divide.\n<div style=\"text-align: center;\">[latex]\\frac{17}{34}=\\frac{{1}{\\overline{)17}}}{\\underset{2}{\\overline{)34}}}=\\frac{1}{2}=0.5[\/latex]<\/div>\n<p>So, [latex]\\frac{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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92383&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div>\n<h2 data-type=\"title\">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 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. 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>.<\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\">The real number line<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Classifying Real Numbers<\/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>\n<li>[latex]-\\frac{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=\"q705558\">Solution<\/span><\/p>\n<div id=\"q705558\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-\\frac{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 0.<\/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 0.<\/li>\n<li>[latex]0.615384615384\\dots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/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>\n<li>[latex]\\sqrt{73}[\/latex]<\/li>\n<li>[latex]-11.411411411\\dots[\/latex]<\/li>\n<li>[latex]\\frac{47}{19}[\/latex]<\/li>\n<li>[latex]-\\frac{\\sqrt{5}}{2}[\/latex]<\/li>\n<li>[latex]6.210735[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155954\">Solution<\/span><\/p>\n<div id=\"q155954\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>positive, irrational; right<\/li>\n<li>negative, rational; left<\/li>\n<li>positive, rational; right<\/li>\n<li>negative, irrational; left<\/li>\n<li>positive, rational; right<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2 data-type=\"title\">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<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/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\" data-media-type=\"image\/jpg\" \/><\/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 plus 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]\\{\\frac{m}{n}|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, are nonrepeating, and are nonterminating: [latex]\\{h|h\\text{ is not a rational number}\\}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Differentiating the Sets of Numbers<\/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>\n<li>[latex]\\sqrt{36}[\/latex]<\/li>\n<li>[latex]\\frac{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=\"q779749\">Solution<\/span><\/p>\n<div id=\"q779749\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td><strong>N<\/strong><\/td>\n<td><strong>W<\/strong><\/td>\n<td><strong>I<\/strong><\/td>\n<td><strong>Q<\/strong><\/td>\n<td><strong>Q`<\/strong><\/td>\n<\/tr>\n<tr>\n<td>1. [latex]\\sqrt{36}=6[\/latex]<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>2. [latex]\\frac{8}{3}=2.\\overline{6}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3. [latex]\\sqrt{73}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>4. \u20136<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>5. [latex]3.2121121112\\dots[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>\u00a0X<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/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]-\\frac{35}{7}[\/latex]<\/li>\n<li>[latex]0[\/latex]<\/li>\n<li>[latex]\\sqrt{169}[\/latex]<\/li>\n<li>[latex]\\sqrt{24}[\/latex]<\/li>\n<li>[latex]4.763763763\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266197\">Solution<\/span><\/p>\n<div id=\"q266197\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"A table with six rows and six columns. The first entry of the first row is empty, but the rest read: N, W, I, Q, and Q'. (These are the sets of numbers.) The first entry of the second row reads: negative thirty-five over seven. Then the fourth and fifth columns are marked. The first entry of the third row reads: zero. Then the third, fourth, and fifth columns are marked. The first entry of the fourth row reads: square root of one hundred sixty-nine. Then the second, third, fourth, and fifth columns are marked. The first entry of the fifth row reads: square root of twenty-four. Then only the sixth column is marked. The first entry of the sixth row reads: 4.763763763\u2026. Then only the fifth column is marked\">\n<thead>\n<tr>\n<th><em>N<\/em><\/th>\n<th><em>W<\/em><\/th>\n<th><em>I<\/em><\/th>\n<th><em>Q<\/em><\/th>\n<th><em>Q&#8217;<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>a. [latex]-\\frac{35}{7}[\/latex]<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>b. 0<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>c. [latex]\\sqrt{169}[\/latex]<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>d. [latex]\\sqrt{24}[\/latex]<\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>e. [latex]4.763763763\\dots[\/latex]<\/td>\n<td>X<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=13740&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"100%\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=13741&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"100%\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Watch this video for an overview of the sets of numbers, and how to identify which set a number belongs to.<br \/>\n<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><\/div>\n<\/div>\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-1301\">\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>Question ID 92383. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 13740. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 13741. <strong>Authored by<\/strong>: Sousa, James. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><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. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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