{"id":8117,"date":"2023-09-20T17:47:36","date_gmt":"2023-09-20T17:47:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8117"},"modified":"2024-10-18T20:56:46","modified_gmt":"2024-10-18T20:56:46","slug":"irrational-numbers-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/irrational-numbers-fresh-take\/","title":{"raw":"Irrational Numbers: Fresh Take","rendered":"Irrational Numbers: Fresh Take"},"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<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Irrational numbers<\/strong> are numbers that cannot be expressed as a fraction of two integers. The decimal representations of irrational numbers neither terminate nor repeat. This is a key feature that distinguishes them from rational numbers. If the decimal stops or repeats, it's rational; otherwise, it's irrational.<\/p>\r\n<\/div>\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]0.1\\overline{5}[\/latex]<\/li>\r\n\t<li>[latex]4\\pi[\/latex]<\/li>\r\n\t<li>[latex]2.65987425\\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]0.1\\overline{5}[\/latex] The bar above the [latex]5[\/latex] indicates that it repeats. Therefore, [latex]0.1\\overline{5}[\/latex] is a repeating decimal, and is therefore a rational number.<\/li>\r\n\t<li>[latex]4\\pi[\/latex] Since [latex]4\\pi[\/latex] is a multiple of pi, it is irrational. In this case, the rational part of the number is [latex]4[\/latex], while the irrational part is [latex]\\pi[\/latex].<\/li>\r\n\t<li>[latex]2.65987425\\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<p>The following video goes into more detail of the difference between rational and irrational numbers.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/cLP7INqs3JM?si=62GtkSX8BIPqNN_g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Introduction+to+rational+and+irrational+numbers+_+Algebra+I+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to rational and irrational numbers | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the following video we show more examples of how to determine whether a number is irrational or rational.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5lYbSxSBu0Y?si=lmYtLc4cy2Fx2YXB\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+Rational+or+Irrational+Numbers+(Square+Roots+and+Decimals+Only).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine Rational or Irrational Numbers (Square Roots and Decimals Only)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Square Roots for Non-Perfect Square Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>In the realm of numbers, square roots are like the keys to hidden treasures. Some keys, like those for perfect squares, fit neatly into locks, revealing whole numbers. Others, the non-perfect squares, open doors to endless decimals that neither terminate nor repeat.<\/p>\r\n<ul>\r\n\t<li><strong>Perfect Squares:<\/strong> These are integers whose square roots are also integers. For example, [latex]81[\/latex] is a perfect square because [latex]\\sqrt{81}=9[\/latex], a whole number. Check the prime factorization. If all prime factors have even powers, you've got a perfect square.<\/li>\r\n\t<li><strong>Non-Perfect Squares:<\/strong> These are numbers whose square roots are irrational, meaning the decimals go on forever without repeating. For example, [latex]\\sqrt{67}[\/latex] is approximately [latex]8.1853...[\/latex], an irrational number.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Identify which of the following numbers are irrational:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]\\sqrt{225}[\/latex]<\/li>\r\n\t<li>[latex]3\\sqrt{5}[\/latex]<\/li>\r\n\t<li>[latex]\\sqrt{80}[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"237122\"]Show Solution[\/reveal-answer] [hidden-answer a=\"237122\"]\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>The square root of [latex]225[\/latex] is [latex]15[\/latex], which is an integer and can be expressed as a fraction [latex]\\frac{15}{1}[\/latex]. Therefore, [latex]\\sqrt{225}[\/latex] is rational.<\/li>\r\n\t<li>The square root of [latex]5[\/latex] is irrational, and multiplying it by [latex]3[\/latex] doesn't change that property. Therefore, [latex]3\\sqrt{5}[\/latex] is irrational.<\/li>\r\n\t<li>The square root of [latex]80[\/latex] can be simplified to [latex]4\\sqrt{5}[\/latex]. Since [latex]\\sqrt{5}[\/latex] is irrational, multiplying it by [latex]4[\/latex] doesn't change its irrationality. Therefore, [latex]\\sqrt{80}[\/latex] is also irrational.<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li><strong>Radicand:<\/strong> The number inside the square root symbol is known as the radicand. For example, in [latex]\\sqrt{a}[\/latex], [latex]a[\/latex] is the radicand.<\/li>\r\n\t<li><strong>Product Rule for Square Roots:<\/strong> The square root of a product of numbers equals the product of the square roots of those numbers. Mathematically, [latex]\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}[\/latex]<\/li>\r\n\t<li><strong>Steps to Simplify Irrational Numbers:<\/strong>\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\r\n\t<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\r\n\t<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\r\n\t<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Rational and Irrational Parts:<\/strong> Once a square root has been simplified, it can be expressed as a rational part multiplied by an irrational part. For example, in [latex]a\\sqrt{b}[\/latex], [latex]a[\/latex] is the rational part and [latex]\\sqrt{b}[\/latex] is the irrational part.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Simplify the irrational number [latex]\\sqrt{733}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\r\n<p>[reveal-answer q=\"214537\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214537\"]<\/p>\r\n<p>Begin by finding the largest perfect square that is a factor of [latex]733[\/latex]. We can do this by writing out the factor pairs of [latex]733[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]1 \\times 733 \\quad 11 \\times 67[\/latex]<\/p>\r\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{733}[\/latex] is already expressed in lowest terms. In this case, [latex]1[\/latex] is the rational part, and [latex]\\sqrt{733}[\/latex] is the irrational part. Though we could write this as [latex]1\\sqrt{733}[\/latex], but the product of [latex]1[\/latex] and any other number is just the number.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Simplify the irrational number [latex]\\sqrt{1815}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\r\n<p>[reveal-answer q=\"214539\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214539\"]<\/p>\r\n<p>Begin by finding the largest perfect square that is a factor of [latex]1815[\/latex]. We can do this by writing out the factor pairs of [latex]1815[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex](1 \\times 1815) \\quad (3 \\times 605) \\quad (5 \\times 363) \\quad (7 \\times 259) \\quad (11 \\times 165) \\quad (13 \\times 139) \\quad (15 \\times 121) \\quad (21 \\times 85) \\quad (33 \\times 55) [\/latex]<\/p>\r\n<p>Among these factors, the largest perfect square is [latex]121[\/latex], so we factor the into [latex]121\u00d715=11^2\u00d715[\/latex]. In the formula, [latex]a=11[\/latex] and [latex]b=15[\/latex]. Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sqrt{11^2 \\times 15}=\\sqrt{11^2} \\times \\sqrt{15}[\/latex]<\/p>\r\n<p>The simplified form of [latex]\\sqrt{1815}[\/latex] is [latex]11\\sqrt{15}[\/latex]. In this example, the [latex]11[\/latex] is the rational part, and the [latex]\\sqrt{15}[\/latex] is the irrational part.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the following video for more on simplifying square roots that are not perfect squares.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/rR3HbrR0m94?si=8UL09GZz2jPMsSVp\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Simplifying+Square+Roots+(not+perfect+squares).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplifying Square Roots (not perfect squares)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Adding and Subtracting Irrational Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<strong>Adding and Subtracting Irrational Numbers:<\/strong> When the irrational parts are the same, you can add or subtract the rational parts and then multiply by the common irrational part.\r\n\r\n<p>To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part.<\/p>\r\n<ul>\r\n\t<li>Let our first irrational number be [latex]a\u00d7x[\/latex], where [latex]a[\/latex] is the rational and [latex]x[\/latex] the irrational parts.<\/li>\r\n\t<li>Let our second irrational number be [latex]b\u00d7x[\/latex], where [latex]b[\/latex] is the rational and [latex]x[\/latex] the irrational parts.<\/li>\r\n\t<li>Then [latex]a\u00d7x\u00b1b\u00d7x=(a\u00b1b)\u00d7x[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Subtract the following irrational numbers.<\/p>\r\n<center>[latex]41\\sqrt{15}\u201323\\sqrt{15}[\/latex]<\/center>\r\n<p>[reveal-answer q=\"214528\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214528\"]<\/p>\r\n<p>Since these two irrational numbers have the same irrational part, [latex]\\sqrt{15}[\/latex], we can subtract without using a calculator. The rational part of the first number is [latex]41[\/latex]. The rational part of the second number is [latex]23[\/latex]. Using the formula yields:<\/p>\r\n<p style=\"text-align: center;\">[latex]41\\sqrt{15}\u201323\\sqrt{15}=(41-23) \\times \\sqrt{15}= 18\\sqrt{15}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Add the following irrational numbers.<\/p>\r\n<center>[latex]4.1 \\pi + 3.2 \\pi[\/latex]<\/center>\r\n<p>[reveal-answer q=\"214521\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214521\"]<\/p>\r\n<p>Since these two irrational numbers have the same irrational part, [latex]\u03c0[\/latex], the addition can be performed without using a calculator. The rational part of the first number is [latex]4.1[\/latex]. The rational part of the second number is [latex]3.2[\/latex]. Using the formula yields<\/p>\r\n<p style=\"text-align: center;\">[latex]41 \\pi + 3.2 \\pi = (4.1+3.2) \\times \\pi = 7.3\\pi[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Multiplying and Dividing Irrational Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<strong>Multiplying and Dividing Irrational Numbers:<\/strong> Here, the irrational parts don't have to be identical but should be similar, like square roots or multiples of pi. The process involves multiplying or dividing the rational parts and then doing the same for the irrational parts.\r\n\r\n<p>When multiplying two square roots, use the following formula.<\/p>\r\n<p style=\"text-align: center;\">For any two positive numbers [latex]a[\/latex] and [latex]b[\/latex], [latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/p>\r\n<p>When dividing two square roots, use the following formula.<\/p>\r\n<p style=\"text-align: center;\">For any two positive numbers [latex]a[\/latex] and [latex]b[\/latex], with [latex]b[\/latex] not equal to [latex]0[\/latex], [latex]\\sqrt{a} \\div \\sqrt{b} = \\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}[\/latex]<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Perform the following operations without a calculator. Simplify if possible.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex](1.2\\sqrt{21})\u00d7(45\\sqrt{14})[\/latex]<\/li>\r\n\t<li>[latex]38 \\pi \\times 2 \\pi[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"214526\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214526\"]<\/p>\r\n<ol>\r\n\t<li>In this multiplication problem, [latex](1.2\\sqrt{21})\u00d7(45\\sqrt{14})[\/latex], notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above.\r\n\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Multiply the rational parts. [latex]1.2\u00d745=54[\/latex]<\/li>\r\n\t<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced.<\/li>\r\n\t<li><strong>Step 3:<\/strong> Multiply the irrational parts. [latex]\\sqrt{21} \\times \\sqrt{14} = \\sqrt{ 3 \\times 12} = \\sqrt{21 \\times 14}[\/latex]<\/li>\r\n\t<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. [latex]\\sqrt{21 \\times 14} = \\sqrt{294} = \\sqrt{2 \\times 147} = \\sqrt{2 \\times 3 \\times 49} = \\sqrt{2} \\times \\sqrt{3} \\times \\sqrt{49} = \\sqrt{2} \\times \\sqrt{3} \\times 7[\/latex]<\/li>\r\n\t<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]54 \\times \\sqrt{2} \\times \\sqrt{3} \\times 7 = 54 \\times 7 \\times \\sqrt{2 \\times 3} = 378 \\sqrt{6}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>In this multiplication problem, [latex]38 \\pi \\times 2 \\pi[\/latex], notice that the irrational parts of these numbers are the same, [latex]\u03c0[\/latex]. Follow the process above.\r\n\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Multiply the rational parts. [latex]38\u00d72=76[\/latex]<\/li>\r\n\t<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.<\/li>\r\n\t<li><strong>Step 3:<\/strong> Multiply the irrational parts. [latex]\u03c0\u00d7\u03c0=\u03c0^2[\/latex]<\/li>\r\n\t<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced.<\/li>\r\n\t<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]76\u03c0^2[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Perform the following operations without a calculator. Simplify if possible.<\/p>\r\n<center>[latex](84\\sqrt{132}) \\div (14\\sqrt{11})[\/latex]<\/center>\r\n<p>[reveal-answer q=\"214527\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214527\"]<\/p>\r\n\r\nIn this division problem, [latex](84\\sqrt{132}) \\div (14\\sqrt{11})[\/latex], notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.\r\n\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Divide the rational parts. [latex]\\frac{84}{14} = 6[\/latex]<\/li>\r\n\t<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. The number 6 is already in lowest terms.<\/li>\r\n\t<li><strong>Step 3:<\/strong> Divide the irrational parts. [latex]\\frac{\\sqrt{132}}{\\sqrt{11}} = \\sqrt{\\frac{132}{11}}[\/latex]<\/li>\r\n\t<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. [latex]\\sqrt{\\frac{132}{11}} = \\sqrt{12} = \\sqrt{4 \\times 3} = 2\\sqrt{3}[\/latex]<\/li>\r\n\t<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]6 \\times 2\\sqrt{3} = 12\\sqrt{3}[\/latex].<\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","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<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Irrational numbers<\/strong> are numbers that cannot be expressed as a fraction of two integers. The decimal representations of irrational numbers neither terminate nor repeat. This is a key feature that distinguishes them from rational numbers. If the decimal stops or repeats, it&#8217;s rational; otherwise, it&#8217;s irrational.<\/p>\n<\/div>\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]0.1\\overline{5}[\/latex]<\/li>\n<li>[latex]4\\pi[\/latex]<\/li>\n<li>[latex]2.65987425\\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]0.1\\overline{5}[\/latex] The bar above the [latex]5[\/latex] indicates that it repeats. Therefore, [latex]0.1\\overline{5}[\/latex] is a repeating decimal, and is therefore a rational number.<\/li>\n<li>[latex]4\\pi[\/latex] Since [latex]4\\pi[\/latex] is a multiple of pi, it is irrational. In this case, the rational part of the number is [latex]4[\/latex], while the irrational part is [latex]\\pi[\/latex].<\/li>\n<li>[latex]2.65987425\\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>The following video goes into more detail of the difference between rational and irrational numbers.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/cLP7INqs3JM?si=62GtkSX8BIPqNN_g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Introduction+to+rational+and+irrational+numbers+_+Algebra+I+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to rational and irrational numbers | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the following video we show more examples of how to determine whether a number is irrational or rational.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5lYbSxSBu0Y?si=lmYtLc4cy2Fx2YXB\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+Rational+or+Irrational+Numbers+(Square+Roots+and+Decimals+Only).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine Rational or Irrational Numbers (Square Roots and Decimals Only)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Square Roots for Non-Perfect Square Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>In the realm of numbers, square roots are like the keys to hidden treasures. Some keys, like those for perfect squares, fit neatly into locks, revealing whole numbers. Others, the non-perfect squares, open doors to endless decimals that neither terminate nor repeat.<\/p>\n<ul>\n<li><strong>Perfect Squares:<\/strong> These are integers whose square roots are also integers. For example, [latex]81[\/latex] is a perfect square because [latex]\\sqrt{81}=9[\/latex], a whole number. Check the prime factorization. If all prime factors have even powers, you&#8217;ve got a perfect square.<\/li>\n<li><strong>Non-Perfect Squares:<\/strong> These are numbers whose square roots are irrational, meaning the decimals go on forever without repeating. For example, [latex]\\sqrt{67}[\/latex] is approximately [latex]8.1853...[\/latex], an irrational number.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Identify which of the following numbers are irrational:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]\\sqrt{225}[\/latex]<\/li>\n<li>[latex]3\\sqrt{5}[\/latex]<\/li>\n<li>[latex]\\sqrt{80}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q237122\">Show Solution<\/button> <\/p>\n<div id=\"q237122\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>The square root of [latex]225[\/latex] is [latex]15[\/latex], which is an integer and can be expressed as a fraction [latex]\\frac{15}{1}[\/latex]. Therefore, [latex]\\sqrt{225}[\/latex] is rational.<\/li>\n<li>The square root of [latex]5[\/latex] is irrational, and multiplying it by [latex]3[\/latex] doesn&#8217;t change that property. Therefore, [latex]3\\sqrt{5}[\/latex] is irrational.<\/li>\n<li>The square root of [latex]80[\/latex] can be simplified to [latex]4\\sqrt{5}[\/latex]. Since [latex]\\sqrt{5}[\/latex] is irrational, multiplying it by [latex]4[\/latex] doesn&#8217;t change its irrationality. Therefore, [latex]\\sqrt{80}[\/latex] is also irrational.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li><strong>Radicand:<\/strong> The number inside the square root symbol is known as the radicand. For example, in [latex]\\sqrt{a}[\/latex], [latex]a[\/latex] is the radicand.<\/li>\n<li><strong>Product Rule for Square Roots:<\/strong> The square root of a product of numbers equals the product of the square roots of those numbers. Mathematically, [latex]\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}[\/latex]<\/li>\n<li><strong>Steps to Simplify Irrational Numbers:<\/strong>\n<ul>\n<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\n<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\n<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\n<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Rational and Irrational Parts:<\/strong> Once a square root has been simplified, it can be expressed as a rational part multiplied by an irrational part. For example, in [latex]a\\sqrt{b}[\/latex], [latex]a[\/latex] is the rational part and [latex]\\sqrt{b}[\/latex] is the irrational part.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Simplify the irrational number [latex]\\sqrt{733}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214537\">Show Solution<\/button> <\/p>\n<div id=\"q214537\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]733[\/latex]. We can do this by writing out the factor pairs of [latex]733[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 733 \\quad 11 \\times 67[\/latex]<\/p>\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{733}[\/latex] is already expressed in lowest terms. In this case, [latex]1[\/latex] is the rational part, and [latex]\\sqrt{733}[\/latex] is the irrational part. Though we could write this as [latex]1\\sqrt{733}[\/latex], but the product of [latex]1[\/latex] and any other number is just the number.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify the irrational number [latex]\\sqrt{1815}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214539\">Show Solution<\/button> <\/p>\n<div id=\"q214539\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]1815[\/latex]. We can do this by writing out the factor pairs of [latex]1815[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex](1 \\times 1815) \\quad (3 \\times 605) \\quad (5 \\times 363) \\quad (7 \\times 259) \\quad (11 \\times 165) \\quad (13 \\times 139) \\quad (15 \\times 121) \\quad (21 \\times 85) \\quad (33 \\times 55)[\/latex]<\/p>\n<p>Among these factors, the largest perfect square is [latex]121[\/latex], so we factor the into [latex]121\u00d715=11^2\u00d715[\/latex]. In the formula, [latex]a=11[\/latex] and [latex]b=15[\/latex]. Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{11^2 \\times 15}=\\sqrt{11^2} \\times \\sqrt{15}[\/latex]<\/p>\n<p>The simplified form of [latex]\\sqrt{1815}[\/latex] is [latex]11\\sqrt{15}[\/latex]. In this example, the [latex]11[\/latex] is the rational part, and the [latex]\\sqrt{15}[\/latex] is the irrational part.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on simplifying square roots that are not perfect squares.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/rR3HbrR0m94?si=8UL09GZz2jPMsSVp\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Simplifying+Square+Roots+(not+perfect+squares).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplifying Square Roots (not perfect squares)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Adding and Subtracting Irrational Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Adding and Subtracting Irrational Numbers:<\/strong> When the irrational parts are the same, you can add or subtract the rational parts and then multiply by the common irrational part.<\/p>\n<p>To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part.<\/p>\n<ul>\n<li>Let our first irrational number be [latex]a\u00d7x[\/latex], where [latex]a[\/latex] is the rational and [latex]x[\/latex] the irrational parts.<\/li>\n<li>Let our second irrational number be [latex]b\u00d7x[\/latex], where [latex]b[\/latex] is the rational and [latex]x[\/latex] the irrational parts.<\/li>\n<li>Then [latex]a\u00d7x\u00b1b\u00d7x=(a\u00b1b)\u00d7x[\/latex].<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Subtract the following irrational numbers.<\/p>\n<div style=\"text-align: center;\">[latex]41\\sqrt{15}\u201323\\sqrt{15}[\/latex]<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214528\">Show Solution<\/button> <\/p>\n<div id=\"q214528\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since these two irrational numbers have the same irrational part, [latex]\\sqrt{15}[\/latex], we can subtract without using a calculator. The rational part of the first number is [latex]41[\/latex]. The rational part of the second number is [latex]23[\/latex]. Using the formula yields:<\/p>\n<p style=\"text-align: center;\">[latex]41\\sqrt{15}\u201323\\sqrt{15}=(41-23) \\times \\sqrt{15}= 18\\sqrt{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Add the following irrational numbers.<\/p>\n<div style=\"text-align: center;\">[latex]4.1 \\pi + 3.2 \\pi[\/latex]<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214521\">Show Solution<\/button> <\/p>\n<div id=\"q214521\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since these two irrational numbers have the same irrational part, [latex]\u03c0[\/latex], the addition can be performed without using a calculator. The rational part of the first number is [latex]4.1[\/latex]. The rational part of the second number is [latex]3.2[\/latex]. Using the formula yields<\/p>\n<p style=\"text-align: center;\">[latex]41 \\pi + 3.2 \\pi = (4.1+3.2) \\times \\pi = 7.3\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Multiplying and Dividing Irrational Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Multiplying and Dividing Irrational Numbers:<\/strong> Here, the irrational parts don&#8217;t have to be identical but should be similar, like square roots or multiples of pi. The process involves multiplying or dividing the rational parts and then doing the same for the irrational parts.<\/p>\n<p>When multiplying two square roots, use the following formula.<\/p>\n<p style=\"text-align: center;\">For any two positive numbers [latex]a[\/latex] and [latex]b[\/latex], [latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/p>\n<p>When dividing two square roots, use the following formula.<\/p>\n<p style=\"text-align: center;\">For any two positive numbers [latex]a[\/latex] and [latex]b[\/latex], with [latex]b[\/latex] not equal to [latex]0[\/latex], [latex]\\sqrt{a} \\div \\sqrt{b} = \\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}[\/latex]<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Perform the following operations without a calculator. Simplify if possible.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex](1.2\\sqrt{21})\u00d7(45\\sqrt{14})[\/latex]<\/li>\n<li>[latex]38 \\pi \\times 2 \\pi[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214526\">Show Solution<\/button> <\/p>\n<div id=\"q214526\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>In this multiplication problem, [latex](1.2\\sqrt{21})\u00d7(45\\sqrt{14})[\/latex], notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above.\n<ul>\n<li><strong>Step 1:<\/strong> Multiply the rational parts. [latex]1.2\u00d745=54[\/latex]<\/li>\n<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced.<\/li>\n<li><strong>Step 3:<\/strong> Multiply the irrational parts. [latex]\\sqrt{21} \\times \\sqrt{14} = \\sqrt{ 3 \\times 12} = \\sqrt{21 \\times 14}[\/latex]<\/li>\n<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. [latex]\\sqrt{21 \\times 14} = \\sqrt{294} = \\sqrt{2 \\times 147} = \\sqrt{2 \\times 3 \\times 49} = \\sqrt{2} \\times \\sqrt{3} \\times \\sqrt{49} = \\sqrt{2} \\times \\sqrt{3} \\times 7[\/latex]<\/li>\n<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]54 \\times \\sqrt{2} \\times \\sqrt{3} \\times 7 = 54 \\times 7 \\times \\sqrt{2 \\times 3} = 378 \\sqrt{6}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>In this multiplication problem, [latex]38 \\pi \\times 2 \\pi[\/latex], notice that the irrational parts of these numbers are the same, [latex]\u03c0[\/latex]. Follow the process above.\n<ul>\n<li><strong>Step 1:<\/strong> Multiply the rational parts. [latex]38\u00d72=76[\/latex]<\/li>\n<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.<\/li>\n<li><strong>Step 3:<\/strong> Multiply the irrational parts. [latex]\u03c0\u00d7\u03c0=\u03c0^2[\/latex]<\/li>\n<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced.<\/li>\n<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]76\u03c0^2[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Perform the following operations without a calculator. Simplify if possible.<\/p>\n<div style=\"text-align: center;\">[latex](84\\sqrt{132}) \\div (14\\sqrt{11})[\/latex]<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214527\">Show Solution<\/button> <\/p>\n<div id=\"q214527\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this division problem, [latex](84\\sqrt{132}) \\div (14\\sqrt{11})[\/latex], notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.<\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Divide the rational parts. [latex]\\frac{84}{14} = 6[\/latex]<\/li>\n<li><strong>Step 2:<\/strong> If necessary, reduce the result of Step 1 to lowest terms. The number 6 is already in lowest terms.<\/li>\n<li><strong>Step 3:<\/strong> Divide the irrational parts. [latex]\\frac{\\sqrt{132}}{\\sqrt{11}} = \\sqrt{\\frac{132}{11}}[\/latex]<\/li>\n<li><strong>Step 4:<\/strong> If necessary, reduce the result from Step 3 to lowest terms. [latex]\\sqrt{\\frac{132}{11}} = \\sqrt{12} = \\sqrt{4 \\times 3} = 2\\sqrt{3}[\/latex]<\/li>\n<li><strong>Step 5:<\/strong> The result is the product of the rational and irrational parts, which is [latex]6 \\times 2\\sqrt{3} = 12\\sqrt{3}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to rational and irrational numbers | Algebra I | Khan Academy\",\"author\":\"Khan Academy\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/cLP7INqs3JM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine Rational or Irrational Numbers (Square Roots and Decimals Only)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5lYbSxSBu0Y\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Simplifying Square Roots (not perfect squares)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/rR3HbrR0m94\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"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":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Introduction to rational and irrational numbers | Algebra I | Khan Academy","author":"Khan Academy","organization":"","url":"https:\/\/youtu.be\/cLP7INqs3JM","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Determine Rational or Irrational Numbers (Square Roots and Decimals Only)","author":"James Sousa (Mathispower4u.com)","organization":"","url":"https:\/\/youtu.be\/5lYbSxSBu0Y","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Ex: Simplifying Square Roots (not perfect squares)","author":"James Sousa (Mathispower4u.com)","organization":"","url":"https:\/\/youtu.be\/rR3HbrR0m94","project":"","license":"arr","license_terms":""},{"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\/8117"}],"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":22,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8117\/revisions"}],"predecessor-version":[{"id":12835,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8117\/revisions\/12835"}],"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\/8117\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8117"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8117"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8117"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8117"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}