{"id":455,"date":"2024-04-19T01:49:42","date_gmt":"2024-04-19T01:49:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=455"},"modified":"2025-07-11T21:54:00","modified_gmt":"2025-07-11T21:54:00","slug":"introduction-to-real-numbers-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-1\/","title":{"raw":"Introduction to Real Numbers: Learn It 1","rendered":"Introduction to Real Numbers: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Categorize real numbers as counting numbers, whole numbers, rational numbers, irrational numbers, or integers<\/li>\r\n \t<li><span data-sheets-root=\"1\">Recognize and use the properties of real numbers<\/span><\/li>\r\n \t<li><span data-sheets-root=\"1\">Simplify and evaluate an algebraic equation<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Classifying a Real Number<\/h2>\r\n<h3>Categories of Real Numbers<\/h3>\r\n<h4>Whole, Counting Numbers, and Integers<\/h4>\r\nIn elementary mathematics, we frequently use the most fundamental set of numbers, which we typically employ for counting objects: [latex]1, 2, 3, 4, 5, ...[\/latex] and so forth. These numbers are referred to as the <strong>counting numbers<\/strong>. The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the <strong>whole numbers<\/strong>.\r\n\r\n<strong>Integers<\/strong> are counting numbers, their opposites, and zero.\r\n\r\n&nbsp;\r\n\r\n<center>[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots [\/latex]<\/center><section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>whole, counting numbers, and integers<\/h3>\r\n<strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.\r\n\r\n&nbsp;\r\n\r\n<center>[latex]1,2,3,4,5\\dots[\/latex]<\/center>&nbsp;\r\n\r\n<strong>Whole numbers<\/strong> are the counting numbers and zero.\r\n\r\n&nbsp;\r\n\r\n<center>[latex]0,1,2,3,4,5\\dots[\/latex]<\/center><strong>Integers<\/strong> are counting numbers, their opposites, and zero.\r\n\r\n&nbsp;\r\n\r\n<center>[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots [\/latex]<\/center><\/div>\r\n<\/section>\r\n<h4>Rational Numbers<\/h4>\r\nWhat type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A <strong>rational number<\/strong> is a number that can be written as a ratio of two integers.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>rational number<\/h3>\r\nA <strong>rational number<\/strong> is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne 0[\/latex].\r\n\r\n&nbsp;\r\n\r\nAll fractions, both positive and negative, are rational numbers.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\">We can write each of the following as a rational number.\r\n<ol>\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\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\r\n<\/section><section class=\"textbox connectIt\"><strong><strong>Are Integers Rational Numbers?\r\n<\/strong><\/strong>\r\n\r\n<hr \/>\r\n\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=\\Large\\frac{3}{1}\\normalsize ,\\space-8=\\Large\\frac{-8}{1}\\normalsize ,\\space0=\\Large\\frac{0}{1}[\/latex]<\/p>\r\nSince any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.\r\n\r\nWhat about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer [latex]-8[\/latex] could be written as the decimal [latex]-8.0[\/latex]. So, clearly, some decimals are rational.\r\n\r\nThink about the decimal [latex]7.3[\/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[\/latex] means [latex]7\\Large\\frac{3}{10}[\/latex], we can write it as an improper fraction, [latex]\\Large\\frac{73}{10}[\/latex]. So [latex]7.3[\/latex] is the ratio of the integers [latex]73[\/latex] and [latex]10[\/latex]. It is a rational number. 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.\r\n\r\n<\/section>\r\n<h4>Irrational Numbers<\/h4>\r\nWe defined rational numbers as numbers that could be expressed as a fraction of two integers. <strong>Irrational numbers<\/strong> are numbers that cannot be expressed as a fraction of two integers. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>irrational number<\/h3>\r\nAn <strong>irrational number<\/strong> 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<\/section><section class=\"textbox proTip\">Let's summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number,\r\n<ul id=\"fs-id1460638\">\r\n \t<li>stops or repeats, the number is rational.<\/li>\r\n \t<li>does not stop and does not repeat, the number is irrational.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Identify each of the following as rational or irrational:\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]0.58\\overline{3}[\/latex]<\/li>\r\n \t<li>[latex]0.475[\/latex]<\/li>\r\n \t<li>[latex]3.605551275\\dots [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]0.58\\overline{3}[\/latex] The 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.<\/li>\r\n \t<li>[latex]0.475[\/latex] This decimal stops after the [latex]5[\/latex], so it is a rational number.<\/li>\r\n \t<li>[latex]3.605551275\\dots[\/latex] The 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.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter [latex]\\pi[\/latex], which is the ratio of the circumference of the diameter of the circle. Any multiple or power of [latex]\\pi[\/latex] is an irrational number.\r\n<h2>Real Numbers<\/h2>\r\nWe have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of <strong>real numbers<\/strong>.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>real numbers<\/h3>\r\n<strong>Real numbers<\/strong> are numbers that are either rational or irrational.\r\n\r\n<\/div>\r\n<\/section>This diagram illustrates the relationships between the different types of real numbers.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"654\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222311\/CNX_BMath_Figure_07_01_001.png\" alt=\"The image shows a large rectangle labeled real numbers. Inside this rectangle, there are two more rectangles labeled rational numbers and irrational numbers. Inside the rational numbers rectangle, there is a rectangle labeled integers. Inside of that rectangle, there is another rectangle labeled whole numbers. Inside of that rectangle is a square labeled counting numbers.\" width=\"654\" height=\"396\" \/> Visual Representation of the Real Number System[\/caption]\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>.\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\" \/> The real number line[\/caption]\r\n\r\n<section class=\"textbox proTip\">Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal\u201d, and, if so, what could they be?For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers.An imaginary number is a number that, when squared, has a negative result, typically expressed as a real number multiplied by the imaginary unit [latex]i[\/latex], where [latex]i^2=\u22121[\/latex]. We won't discuss imaginary numbers in this section, but you may encounter them in other places in this course.<\/section><section class=\"textbox example\">Determine whether each of the numbers in the following list is a\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>whole number<\/li>\r\n \t<li>integer<\/li>\r\n \t<li>rational number<\/li>\r\n \t<li>irrational number<\/li>\r\n \t<li>real number<\/li>\r\n<\/ol>\r\n<center>[latex]-7[\/latex], [latex]\\frac{14}{5}[\/latex], [latex]8[\/latex], [latex]\\sqrt{5}[\/latex], [latex]5.9[\/latex], [latex]-\\sqrt{64}[\/latex]<\/center>[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>The whole numbers are [latex]0,1,2,3\\dots[\/latex] The number [latex]8[\/latex] is the only whole number given.<\/li>\r\n \t<li>The integers are the whole numbers, their opposites, and [latex]0[\/latex]. From the given numbers, [latex]-7[\/latex] and [latex]8[\/latex] are integers. Also, notice that [latex]64[\/latex] is the square of [latex]8[\/latex] so [latex]-\\sqrt{64}=-8[\/latex]. So the integers are [latex]-7,8,-\\sqrt{64}[\/latex].<\/li>\r\n \t<li>Since all integers are rational, the numbers [latex]-7,8,[\/latex] and [latex]-\\sqrt{64}[\/latex] are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so [latex]\\Large\\frac{14}{5}[\/latex] and [latex]5.9[\/latex] are rational.<\/li>\r\n \t<li>The number [latex]5[\/latex] is not a perfect square, so [latex]\\sqrt{5}[\/latex] is irrational.<\/li>\r\n \t<li>All of the numbers listed are real.<\/li>\r\n<\/ol>\r\nWe'll summarize the results in a table.\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<td style=\"width: 22%;\">Number<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<td>[latex]\\frac{14}{5}[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]\\sqrt{5}[\/latex]<\/td>\r\n<td>[latex]5.9[\/latex]<\/td>\r\n<td style=\"width: 17%;\">[latex]-\\sqrt{64}[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>Whole<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Integer<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rational<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Irrational<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Real<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]12705[\/ohm2_question]<\/section>\r\n<h4>Sets of Numbers as Subsets<\/h4>\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\" \/> 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<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>sets of numbers<\/h3>\r\n<ul>\r\n \t<li>The set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].<\/li>\r\n \t<li>The set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].<\/li>\r\n \t<li>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].<\/li>\r\n \t<li>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].<\/li>\r\n \t<li>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].<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Categorize real numbers as counting numbers, whole numbers, rational numbers, irrational numbers, or integers<\/li>\n<li><span data-sheets-root=\"1\">Recognize and use the properties of real numbers<\/span><\/li>\n<li><span data-sheets-root=\"1\">Simplify and evaluate an algebraic equation<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Classifying a Real Number<\/h2>\n<h3>Categories of Real Numbers<\/h3>\n<h4>Whole, Counting Numbers, and Integers<\/h4>\n<p>In elementary mathematics, we frequently use the most fundamental set of numbers, which we typically employ for counting objects: [latex]1, 2, 3, 4, 5, ...[\/latex] and so forth. These numbers are referred to as the <strong>counting numbers<\/strong>. The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the <strong>whole numbers<\/strong>.<\/p>\n<p><strong>Integers<\/strong> are counting numbers, their opposites, and zero.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>whole, counting numbers, and integers<\/h3>\n<p><strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]1,2,3,4,5\\dots[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Whole numbers<\/strong> are the counting numbers and zero.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]0,1,2,3,4,5\\dots[\/latex]<\/div>\n<p><strong>Integers<\/strong> are counting numbers, their opposites, and zero.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots[\/latex]<\/div>\n<\/div>\n<\/section>\n<h4>Rational Numbers<\/h4>\n<p>What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A <strong>rational number<\/strong> is a number that can be written as a ratio of two integers.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>rational number<\/h3>\n<p>A <strong>rational number<\/strong> is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne 0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>All fractions, both positive and negative, are rational numbers.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">We can write each of the following as a rational number.<\/p>\n<ol>\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<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<\/section>\n<section class=\"textbox connectIt\"><strong><strong>Are Integers Rational Numbers?<br \/>\n<\/strong><\/strong><\/p>\n<hr \/>\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=\\Large\\frac{3}{1}\\normalsize ,\\space-8=\\Large\\frac{-8}{1}\\normalsize ,\\space0=\\Large\\frac{0}{1}[\/latex]<\/p>\n<p>Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.<\/p>\n<p>What about decimals? Are they rational? Let&#8217;s look at a few to see if we can write each of them as the ratio of two integers. We&#8217;ve already seen that integers are rational numbers. The integer [latex]-8[\/latex] could be written as the decimal [latex]-8.0[\/latex]. So, clearly, some decimals are rational.<\/p>\n<p>Think about the decimal [latex]7.3[\/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[\/latex] means [latex]7\\Large\\frac{3}{10}[\/latex], we can write it as an improper fraction, [latex]\\Large\\frac{73}{10}[\/latex]. So [latex]7.3[\/latex] is the ratio of the integers [latex]73[\/latex] and [latex]10[\/latex]. It is a rational number. In general, any decimal that ends after a number of digits such as [latex]7.3[\/latex] or [latex]-1.2684[\/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.<\/p>\n<\/section>\n<h4>Irrational Numbers<\/h4>\n<p>We defined rational numbers as numbers that could be expressed as a fraction of two integers. <strong>Irrational numbers<\/strong> are numbers that cannot be expressed as a fraction of two integers. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>irrational number<\/h3>\n<p>An <strong>irrational number<\/strong> 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<\/section>\n<section class=\"textbox proTip\">Let&#8217;s summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number,<\/p>\n<ul id=\"fs-id1460638\">\n<li>stops or repeats, the number is rational.<\/li>\n<li>does not stop and does not repeat, the number is irrational.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Identify each of the following as rational or irrational:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]0.58\\overline{3}[\/latex]<\/li>\n<li>[latex]0.475[\/latex]<\/li>\n<li>[latex]3.605551275\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]0.58\\overline{3}[\/latex] The 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.<\/li>\n<li>[latex]0.475[\/latex] This decimal stops after the [latex]5[\/latex], so it is a rational number.<\/li>\n<li>[latex]3.605551275\\dots[\/latex] The 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.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter [latex]\\pi[\/latex], which is the ratio of the circumference of the diameter of the circle. Any multiple or power of [latex]\\pi[\/latex] is an irrational number.<\/p>\n<h2>Real Numbers<\/h2>\n<p>We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of <strong>real numbers<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>real numbers<\/h3>\n<p><strong>Real numbers<\/strong> are numbers that are either rational or irrational.<\/p>\n<\/div>\n<\/section>\n<p>This diagram illustrates the relationships between the different types of real numbers.<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 654px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222311\/CNX_BMath_Figure_07_01_001.png\" alt=\"The image shows a large rectangle labeled real numbers. Inside this rectangle, there are two more rectangles labeled rational numbers and irrational numbers. Inside the rational numbers rectangle, there is a rectangle labeled integers. Inside of that rectangle, there is another rectangle labeled whole numbers. Inside of that rectangle is a square labeled counting numbers.\" width=\"654\" height=\"396\" \/><figcaption class=\"wp-caption-text\">Visual Representation of the Real Number System<\/figcaption><\/figure>\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>.<\/p>\n<figure style=\"width: 487px\" 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\" \/><figcaption class=\"wp-caption-text\">The real number line<\/figcaption><\/figure>\n<section class=\"textbox proTip\">Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal\u201d, and, if so, what could they be?For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers.An imaginary number is a number that, when squared, has a negative result, typically expressed as a real number multiplied by the imaginary unit [latex]i[\/latex], where [latex]i^2=\u22121[\/latex]. We won&#8217;t discuss imaginary numbers in this section, but you may encounter them in other places in this course.<\/section>\n<section class=\"textbox example\">Determine whether each of the numbers in the following list is a<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>whole number<\/li>\n<li>integer<\/li>\n<li>rational number<\/li>\n<li>irrational number<\/li>\n<li>real number<\/li>\n<\/ol>\n<div style=\"text-align: center;\">[latex]-7[\/latex], [latex]\\frac{14}{5}[\/latex], [latex]8[\/latex], [latex]\\sqrt{5}[\/latex], [latex]5.9[\/latex], [latex]-\\sqrt{64}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>The whole numbers are [latex]0,1,2,3\\dots[\/latex] The number [latex]8[\/latex] is the only whole number given.<\/li>\n<li>The integers are the whole numbers, their opposites, and [latex]0[\/latex]. From the given numbers, [latex]-7[\/latex] and [latex]8[\/latex] are integers. Also, notice that [latex]64[\/latex] is the square of [latex]8[\/latex] so [latex]-\\sqrt{64}=-8[\/latex]. So the integers are [latex]-7,8,-\\sqrt{64}[\/latex].<\/li>\n<li>Since all integers are rational, the numbers [latex]-7,8,[\/latex] and [latex]-\\sqrt{64}[\/latex] are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so [latex]\\Large\\frac{14}{5}[\/latex] and [latex]5.9[\/latex] are rational.<\/li>\n<li>The number [latex]5[\/latex] is not a perfect square, so [latex]\\sqrt{5}[\/latex] is irrational.<\/li>\n<li>All of the numbers listed are real.<\/li>\n<\/ol>\n<p>We&#8217;ll summarize the results in a table.<\/p>\n<table>\n<thead>\n<tr valign=\"top\">\n<td style=\"width: 22%;\">Number<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<td>[latex]\\frac{14}{5}[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]\\sqrt{5}[\/latex]<\/td>\n<td>[latex]5.9[\/latex]<\/td>\n<td style=\"width: 17%;\">[latex]-\\sqrt{64}[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Whole<\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Integer<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rational<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Irrational<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Real<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm12705\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12705&theme=lumen&iframe_resize_id=ohm12705&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h4>Sets of Numbers as Subsets<\/h4>\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<figure style=\"width: 731px\" 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\" \/><figcaption 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<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>sets of numbers<\/h3>\n<ul>\n<li>The set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].<\/li>\n<li>The set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].<\/li>\n<li>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].<\/li>\n<li>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].<\/li>\n<li>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].<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":12,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/455"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":39,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/455\/revisions"}],"predecessor-version":[{"id":7517,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/455\/revisions\/7517"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/455\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=455"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=455"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=455"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}