{"id":8115,"date":"2023-09-20T17:47:24","date_gmt":"2023-09-20T17:47:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8115"},"modified":"2025-08-28T04:00:34","modified_gmt":"2025-08-28T04:00:34","slug":"irrational-numbers-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/irrational-numbers-learn-it-1\/","title":{"raw":"Irrational Numbers: Learn It 1","rendered":"Irrational Numbers: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize irrational numbers in a list of numbers<\/li>\r\n\t<li>Simplify irrational numbers to their lowest terms<\/li>\r\n\t<li>Add, subtract, multiple and divide irrational numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining and Identifying Numbers That Are Irrational<\/h2>\r\n<p>We defined rational numbers in the last section 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.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point.<\/p>\r\n<\/section>\r\n<p>So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>irrational number<\/h3>\r\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>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Let's summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number,<\/p>\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>\r\n<section class=\"textbox example\">\r\n<p>Identify each of the following as rational or irrational:<\/p>\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<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>[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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]12702[\/ohm2_question]<\/p>\r\n<\/section>\r\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.<\/p>\r\n<center>\r\n[caption id=\"attachment_8629\" align=\"aligncenter\" width=\"276\" class=\"align= \"]<img class=\"wp-image-8629 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a-276x300.png\" alt=\"A diagram of a circle with the words radius, diameter, and circumference.\" width=\"276\" height=\"300\" \/> Figure 1. Circle with radius, diameter, and circumference labeled[\/caption]\r\n<\/center><center><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Any multiple or power of [latex]\\pi[\/latex] is an irrational number.<\/p>\r\n<p>Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the <strong>rational part<\/strong>, and the irrational number the <strong>irrational part<\/strong>. It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number.<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize irrational numbers in a list of numbers<\/li>\n<li>Simplify irrational numbers to their lowest terms<\/li>\n<li>Add, subtract, multiple and divide irrational numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Identifying Numbers That Are Irrational<\/h2>\n<p>We defined rational numbers in the last section 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.<\/p>\n<section class=\"textbox recall\">\n<p>Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point.<\/p>\n<\/section>\n<p>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\">\n<p>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\">\n<p>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<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>[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<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm12702\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12702&theme=lumen&iframe_resize_id=ohm12702&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\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.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_8629\" aria-describedby=\"caption-attachment-8629\" style=\"width: 276px\" class=\"wp-caption aligncenter align=\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-8629 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a-276x300.png\" alt=\"A diagram of a circle with the words radius, diameter, and circumference.\" width=\"276\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a-276x300.png 276w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a-65x71.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a-225x245.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/04143308\/dd6dcc75dd57d774b2e9856d9d13802f3462f72a.png 331w\" sizes=\"(max-width: 276px) 100vw, 276px\" \/><figcaption id=\"caption-attachment-8629\" class=\"wp-caption-text\">Figure 1. Circle with radius, diameter, and circumference labeled<\/figcaption><\/figure>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<p>&nbsp;<\/p>\n<p>Any multiple or power of [latex]\\pi[\/latex] is an irrational number.<\/p>\n<p>Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the <strong>rational part<\/strong>, and the irrational number the <strong>irrational part<\/strong>. It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number.<\/p>\n","protected":false},"author":15,"menu_order":19,"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 https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\"},{\"type\":\"cc-attribution\",\"description\":\"Contemporary Mathematics\",\"author\":\"Donna Kirk\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/3-5-irrational-numbers\",\"project\":\"3.5 Irrational Numbers\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":53,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"Lynn Marecek & MaryAnne Anthony-Smith","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction"},{"type":"cc-attribution","description":"Contemporary Mathematics","author":"Donna Kirk","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/3-5-irrational-numbers","project":"3.5 Irrational Numbers","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction"}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8115"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":26,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8115\/revisions"}],"predecessor-version":[{"id":15808,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8115\/revisions\/15808"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/53"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8115\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8115"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8115"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8115"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}