{"id":8124,"date":"2023-09-20T17:48:11","date_gmt":"2023-09-20T17:48:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8124"},"modified":"2025-08-28T04:02:30","modified_gmt":"2025-08-28T04:02:30","slug":"real-numbers-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/real-numbers-learn-it-1\/","title":{"raw":"Real Numbers: Learn It 1","rendered":"Real Numbers: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Categorize real numbers into counting, whole, rational, irrational, or integers<\/li>\r\n\t<li>Recognize and use the properties of real numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining and Identifying Real Numbers<\/h2>\r\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>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>real numbers<\/h3>\r\n<p><strong>Real numbers<\/strong> are numbers that are either rational or irrational.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>This diagram illustrates the relationships between the different types of real numbers.<\/p>\r\n<center>\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\" \/> Figure 1. A diagram of the relationships between the different types of real numbers[\/caption]\r\n<\/center>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex] 1,2,3,4,5\u2026[\/latex]<\/li>\r\n\t<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex] 0,1,2,3,4,5\u2026[\/latex]<\/li>\r\n\t<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\r\n\t<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\r\n\t<li><strong>Irrational Numbers:<\/strong> Numbers that can't be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>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? <br \/>\r\n<br \/>\r\nFor centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. <br \/>\r\n<br \/>\r\nAn 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.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Determine whether each of the numbers in the following list is a<\/p>\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>\r\n<p>[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\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\n<p>We'll summarize the results in a table.<\/p>\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>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/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>&nbsp;<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/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>&nbsp;<\/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>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]12705[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Categorize real numbers into counting, whole, rational, irrational, or integers<\/li>\n<li>Recognize and use the properties of real numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Identifying 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<div style=\"text-align: center;\">\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\">Figure 1. A diagram of the relationships between the different types of real numbers<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox recall\">\n<ul>\n<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex]1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex]0,1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\n<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\n<li><strong>Irrational Numbers:<\/strong> Numbers that can&#8217;t be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\">\n<p>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? <\/p>\n<p>For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. <\/p>\n<p>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.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>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<p><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>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Integer<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/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>&nbsp;<\/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>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>[latex]\\quad\\checkmark[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/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\">\n<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><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"Lynn Marecek & MaryAnne Anthony-Smith\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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