{"id":14358,"date":"2022-12-20T10:56:09","date_gmt":"2022-12-20T10:56:09","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=14358"},"modified":"2023-08-03T06:03:29","modified_gmt":"2023-08-03T06:03:29","slug":"irrational-numbers","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers","title":{"rendered":"Irrational Numbers"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-ec7179a8-cd91-40c3-bd8a-232ee5db4da3\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers#1-what-are-irrational-numbers>What Are Irrational Numbers?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers#5-irrational-numbers-symbol>Irrational Numbers Symbol<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers#17-solved-examples>Solved Examples<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers#18-practice-problems>Practice Problems<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers#19-frequently-asked-questions>Frequently Asked Questions<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-irrational-numbers-introduction\">Irrational Numbers &#8211; Introduction<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">We use numbers in daily life for a variety of reasons. Also, we use different types of numbers for different purposes, such as natural numbers for counting, fractions for describing portions or parts of a whole, decimals for precision, etc. Today we will explore \u2018Irrational Numbers\u2019 in math, their applications, examples, operations. Let\u2019s begin!<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/10-and-100-more-than-the-same-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_1_10_100_more_same_num_pt.png\" alt=\"10 and 100 More than the Same Number Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">10 and 100 More than the Same Number Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_make_10_pt.png\" alt=\"Add 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-10-to-a-3-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_10_number_pt.png\" alt=\"Add 10 to a 3-Digit Number Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 10 to a 3-Digit Number Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-100-to-a-3-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_100_horizontal_pt.png\" alt=\"Add 100 to a 3-Digit Number Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 100 to a 3-Digit Number Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-and-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_2d_1d_match_pt.png\" alt=\"Add 2-Digit and 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit and 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-numbers-by-regrouping\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_regrp_2d_2d_vertical_pt.png\" alt=\"Add 2-Digit Numbers By Regrouping Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit Numbers By Regrouping Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_pt.png\" alt=\"Add 3 Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-in-any-order\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_shifty_bridges_4_gm.png\" alt=\"Add 3 Numbers in Any Order Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers in Any Order Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-groups-of-objects\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_using_model_pt.png\" alt=\"Add 3 Numbers Using Groups of Objects Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Using Groups of Objects Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-model\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_using_models_pt.png\" alt=\"Add 3 Numbers using Model Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers using Model Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-what-are-irrational-numbers\">What Are Irrational Numbers?<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Irrational numbers are the type of real numbers that cannot be expressed in the rational form $\\frac{p}{q}$, where $p, q$ are integers and $q \\neq 0$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">In simple words, all the real numbers that are not rational numbers are irrational.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">We see numbers everywhere around us and use them on a daily basis. Let\u2019s quickly revise.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Natural Numbers $= N = {1, 2, 3, 4, . . .}$&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Whole Numbers $= W = {0, 1, 2, 3, 4, . . .}$&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Integers $= Z = {. . . &#8211; 3,-2, -1, 0, 1, 2, 3, . . .}$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Rational Numbers $= Q$<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">They include all the numbers of the form $\\frac{p}{q}$, where $p, q$ are integers and $q \\neq 0$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Decimal expansions for rational numbers can be either terminating or repeating decimals.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Examples: $\\frac{1}{2} , \\frac{11}{3}, \\frac{5}{1}$, 3.25, 0.252525 . . .<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Irrational Numbers $= P$<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Irrational numbers are the type of real numbers that cannot be expressed in the form $\\frac{p}{q}, q \\neq 0$. These numbers include non-terminating, non-repeating decimals<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Real Numbers $= R$<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Rational and irrational numbers together make real numbers.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"661\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-1.png\" alt=\"Irrational numbers and rational numbers together form real numbers\" class=\"wp-image-14384\" title=\"Irrational numbers and rational numbers together form real numbers\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-1.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-1-281x300.png 281w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/10-and-100-more-than-a-3-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/10-and-100-more-than-a-3-digit-number.jpeg\" alt=\"10 and 100 More than a 3-digit Number\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/100-more-and-100-less-than-a-3-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/100-more-and-100-less-than-a-3-digit-number.jpeg\" alt=\"100 More and 100 Less than a 3-digit Number\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/100-more-than-a-3-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/100-more-than-a-3-digit-number.jpeg\" alt=\"100 More than a 3-digit Number\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers.jpeg\" alt=\"Add & Subtract Ones & 2-Digit Numbers Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Find the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Match the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Addition Tables\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-wheel\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" 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to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-irrational-numbers-definition\">Irrational Numbers Definition<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Irrational numbers can be defined as real numbers that cannot be expressed in the form of $\\frac{p}{q}$, where p and q are integers and the denominator $q \\neq 0$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Example:&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The decimal expansion of an irrational number is non-terminating and non-recurring\/non-repeating. So, all non-terminating and non-recurring decimal numbers are \u201cirrational numbers.\u201d<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example:<\/strong> Suppose a square has an area of 5 square meters. Calculate the length of each side.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"285\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-2.png\" alt=\"5 square meters\" class=\"wp-image-14386\" title=\"5 square meters\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-2.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-2-300x138.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The area of the square = 5 sq. meters<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The length of each side is $\\sqrt{5}$&nbsp; meter $=$ 2.23606797749 . . . meter<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">This answer is in irrational form since we cannot express $\\sqrt{5}$ in rational form. Also, the decimal expansion is non-terminating, non-repeating.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-irrational-numbers-examples\">Irrational Numbers Examples<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The following are examples of a few specific irrational numbers that are commonly used.&nbsp;<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">In math, we know \u201cpi\u201d as the circumference to diameter ratio. Is pi an irrational number?&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Yes! Pi or is an irrational number. The decimal expansion of $\u03c0 =$ 3.14159265 . . . is neither terminating nor repeating decimal.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">&nbsp;(Understand that we use pi as 3.14 or $\\frac{22}{7}$ to make calculations easier.)<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">$\\sqrt{2}$<strong> <\/strong>is an irrational number.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Euler\u2019s number <strong>e<\/strong> is an irrational number, where $e$ $=$ 2.718281 . . .<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Golden ratio, $\\varphi =$ 1.61803398874989 . . .<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Square root of non-perfect squares like $\\sqrt{26}, \\sqrt{63}$, etc.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Square root of a prime numbers like $\\sqrt{2}, \\sqrt{3}$, etc.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">All non-terminating and non-recurring decimals.<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"436\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-3.png\" alt=\"Examples of irrational numbers in decimals\" class=\"wp-image-14387\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-3.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-3-300x211.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-irrational-numbers-list\">Irrational Numbers List<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Here\u2019s a list of some common and frequently used irrational numbers.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Pi or $\\Pi=$ 3.14159265358979\u2026<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Euler\u2019s Number e $=$ 2.71828182845904\u2026<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Golden ratio $\\Theta =$ 1.61803398874989\u2026.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">$\\sqrt{2}$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-irrational-numbers-symbol\">Irrational Numbers Symbol<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Generally, we use the symbol \u201cP\u201d to represent an irrational number, since&nbsp; the set of real numbers is denoted by R and the set of rational numbers is denoted by Q. We can also represent irrational numbers using the set difference of the real minus rationals, in a way $\\text{R} \u2013 \\text{Q}$ or $\\frac{R}{Q}$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-are-all-irrational-numbers-real-numbers\">Are all Irrational Numbers Real Numbers?<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Rational numbers and irrational numbers together form real numbers. So, all irrational numbers are considered to be real numbers. The real numbers which are not rational numbers are irrational numbers. Irrational numbers cannot be expressed as the ratio of two numbers. However, every real number is not an irrational number.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-properties-of-irrational-numbers\">Properties of Irrational Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The irrational numbers, being a type of real numbers, follow all the properties of real numbers. The following are the properties of irrational numbers:&nbsp;<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">When we add an irrational number and a rational number, it will always give an irrational number.&nbsp; Example: $\\sqrt{3} + \\frac{2}{5}$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">When we multiply an irrational number with a non-zero rational number, it will result in an irrational number. Example: $\\frac{2}{5} \\times \\sqrt{3}$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The LCM (Least Common Multiple) of any two irrational numbers may or may not exist.&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The addition or the multiplication of two irrational numbers may be rational.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">For example, $\\sqrt{3} \\times \\sqrt{3} = 3$. Here, $\\sqrt{3}$ is an irrational number. If we multiply it twice, then the final product obtained is a rational number, 3.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Also, consider the example of addition of two irrational numbers that gives a rational number. $\\sqrt{3} + (2$ $-$ $\\sqrt{3} ) = 2$&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">It means the set of irrational numbers is not closed under the multiplication process and addition process, unlike the set of rational numbers.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-operations-on-two-irrational-numbers\">Operations on Two Irrational Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">We can do some operations on two or more irrational numbers like addition, subtraction, multiplication, and division.&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"9-addition\">Addition<\/h3>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Addition of two irrational numbers may or may not be irrational.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 1:<\/strong> $\\sqrt{2}$ and $3\\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$\\sqrt{2} + \\sqrt{3}2 = \\sqrt{2} + 3\\sqrt{2} = 3\\sqrt{2}$ is an irrational number.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 2:<\/strong> $6 + \\sqrt{2}$ and $2$ $-$ $\\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$6 + \\sqrt{2} + 2$ $-$ $\\sqrt{2} = (6 + 2) + (\\sqrt{2}$ $-$ $\\sqrt{2}) = 8$ is a rational number.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"10-subtraction\">Subtraction<\/h3>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Subtraction of two irrational numbers may or may not be irrational.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 1:<\/strong> $5 + \\sqrt{3}$ and $2$ $-$ $3\\sqrt{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$5 + \\sqrt{3}$ $-$ $(2$ $-$ $3\\sqrt{3}) = (5$ $-$ $2) + (\\sqrt{3}$ $-$ $3\\sqrt{3}) = 3$ $-$ $2\\sqrt{3}$ is an irrational number.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 2:<\/strong> $5 + 3\\sqrt{3}$ and $4 + 3\\sqrt{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$5 + 3\\sqrt{3}$ $-$ $( 4 + 3\\sqrt{3}) = ( 5$ $-$ $4 ) + ( 3\\sqrt{3}$ $-$ $3\\sqrt{3}) = 1$ is a rational number.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"11-multiplication\">Multiplication<\/h3>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The product of two irrational numbers may or may not be irrational.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 1:<\/strong> $2\\sqrt{3}$ and $6\\sqrt{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$2\\sqrt{3} \\times 6\\sqrt{3} =2 \\times 3 \\times 6 = 36$ is a rational number.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 2:<\/strong> $6+\\sqrt{2}$ and $2$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$(6 + \\sqrt{2}) \\times 2 = (6 \\times 2) + (\\sqrt{2} \\times 2) = 12 + \\sqrt{2}2$ is an irrational number.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"12-division\">Division<\/h3>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The division of two irrational numbers may or may not be irrational.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 1:<\/strong> $6 \\sqrt{2} \\div \\sqrt{2}$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$\\frac{6\\sqrt{2}}{\\sqrt{2}} = 6$ is a rational number.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Example 2:<\/strong> $(5 &#8211; \\sqrt{2} ) \\div \\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$\\frac{5-\\sqrt{2}}{2} = \\frac{5}{\\sqrt{2}}$ $-$ $\\frac{\\sqrt{2}}{\\sqrt{2}} = \\frac{5} {\\sqrt{2}}$ $- 1$ is an irrational number.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"13-how-to-find-an-irrational-number-between-two-numbers\">How to Find an Irrational Number between Two Numbers?<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Let us find the irrational numbers between 3 and 4.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">We know, square root of 9 is 3; $\\sqrt{9} = \\sqrt{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">and the square root of 16 is 4; $\\sqrt16 = \\sqrt4$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Therefore, $\\sqrt{10},\\sqrt{11}, \\sqrt{12}$, etc., are irrational numbers between 3 and 4. These are not perfect squares and cannot be simplified further. Also, all the non-repeating, non-terminating decimals between 3 and 4 like 3.12537 . . . are irrational.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"14-rational-numbers-vs-irrational-numbers\">Rational Numbers vs. Irrational Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The table illustrates the difference between rational numbers and irrational numbers.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"318\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-4.png\" alt=\"Rational numbers vs. Irrational numbers\" class=\"wp-image-14388\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-4.png 600w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-4-300x159.png 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"15-fun-facts\">Fun Facts<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">$\\sqrt{2}$<strong> is the first invented irrational number!<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Hippasus, the Greek mathematician and the student of the great mathematician Pythagoras proved that \u201croot two\u201d could never be expressed as a fraction.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"377\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-5.png\" alt=\"Square root of two as a hypotenuse of a right triangle\" class=\"wp-image-14389\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-5.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2022\/12\/What-are-Irrational-Numbers-5-300x182.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">It was invented while calculating the length of the isosceles right-angled triangle using Pythagoras\u2019 theorem.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$\\text{AC}^2 = \\text{AB}^2 + \\text{BC}^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$1^2 + 1^ 2 = \\text{AC}^2 \\Rightarrow \\text{AC} = \\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">$\\sqrt{2}$ lies between the rational numbers 1 and 2 as the value of $\\sqrt{2}$ is 1.41421 . . .<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>Pi day!<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">14 March or 3\/14 is celebrated as Pi day since the date matches the first three digits of the decimal expansion of pi.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"16-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">In this article, we learned about irrational numbers. An irrational number is a number that cannot be expressed in the form of a fraction or ratio. To read more such informative articles on other concepts, do visit our <a href=\"https:\/\/www.splashlearn.com\/\">website<\/a>. We, at SplashLearn, are on a mission to make learning fun and interactive for all students.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"17-solved-examples\">Solved Examples<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>1. Find two irrational numbers between 3.14 and 3.2.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Solution:<\/strong> The decimal expansion of an irrational number is non-terminating and non-repeating. The two irrational numbers between 3.14 and 3.2 can be 3.15155155515555 . . . and 3.19876543 . . .<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>2. Identify rational and irrational numbers from the following numbers.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">&nbsp;$\\sqrt{5}$, 2, $\\sqrt{11}$, 3.56, 1.3333 . . ., 100, 4.5346782 . . .<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Rational numbers:&nbsp; 2, 3.56, 100, 1.3333 . . .<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Irrational numbers: $\\sqrt{5}, \\sqrt{11}$, 4.5346782 . . .<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>3. Add <\/strong>$(2 + \\sqrt{3} )$<strong> and <\/strong>$( 3$ $- \\sqrt{3} )$<strong>. Is the sum irrational?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">&nbsp;$( 2 + \\sqrt{3} ) + ( 3$ $-\\sqrt{3} ) = 2 + 3 +\\sqrt{3} &#8211; \\sqrt{3} = 5$<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The sum is a rational number.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>4. Compare irrational numbers <\/strong>$\\sqrt{12}$<strong> and <\/strong>$\\sqrt{21}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>Solution:<\/strong> Since $12 \\lt 21$, we can say that $\\sqrt{12} &lt; \\sqrt{21}$.<\/p>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><strong>5. Which decimals are irrational numbers?Solution:<\/strong> The decimals that are non-repeating and non-recurring are irrational numbers.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"18-practice-problems\">Practice Problems<\/h2>\n\n\n\n<p class=\"eplus-wrapper wp-block-paragraph\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Irrational Numbers<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Which of the following is an irrational number?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\sqrt{21}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\sqrt{9}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">36<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\frac{2}{3}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\\sqrt{21}$<br\/>$\\sqrt{9} = 3$. So, it is a rational number. 36 and $\\frac{2}{3}$ can be expressed in the form of $\\frac{p}{q}$, where p and q are integers and $q \\neq 0$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Find the value of $(\\sqrt{2} + \\sqrt{3}) (\\sqrt{2} - \\sqrt{3})$.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">2<\/div><div class=\"spq_answer_block\" data-value=\"1\">1<\/div><div class=\"spq_answer_block\" data-value=\"2\">$-1$<\/div><div class=\"spq_answer_block\" data-value=\"3\">0<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $-1$<br\/>$(\\sqrt{2}+\\sqrt{3})(\\sqrt{2} - \\sqrt{3}) = (\\sqrt{2})^2 - (\\sqrt{3})^2 = 2 - 3 = -1$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">On multiplying 0 and irrational number, we always get:<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">an irrational number<\/div><div class=\"spq_answer_block\" data-value=\"1\">0<\/div><div class=\"spq_answer_block\" data-value=\"2\">a rational number except 0<\/div><div class=\"spq_answer_block\" data-value=\"3\">None of these<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 0<br\/>The product of any number with 0 is always 0.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">On dividing $16\\sqrt{3}$ by $2\\sqrt{3}$, we get:<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">8<\/div><div class=\"spq_answer_block\" data-value=\"1\">$8\\sqrt{3}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\frac{8}{\\sqrt{3}}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$8 + \\sqrt{3}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 8<br\/>$\\frac{16\\sqrt{3}}{2\\sqrt{3}} = \\frac{16}{2} = 8$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">The decimal expansion of $\\sqrt{6}$ is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">terminating, non repeating<\/div><div class=\"spq_answer_block\" data-value=\"1\">non-terminating, repeating<\/div><div class=\"spq_answer_block\" data-value=\"2\">non-terminating, non-repeating<\/div><div class=\"spq_answer_block\" data-value=\"3\">terminating, repeating<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: non-terminating, non-repeating<br\/>$\\sqrt{6}$ is an irrational number and hence the decimal expansion of $\\sqrt{6}$ is non-terminating and non-repeating.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Irrational Numbers\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Irrational Numbers\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is an irrational number?\",\n                    \"text\": \"Which of the following is an irrational number?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\sqrt{9}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"36\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{2}{3}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\sqrt{21}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\sqrt{9} = 3$$. So, it is a rational number. 36 and $$\\\\frac{2}{3}$$ can be expressed in the form of $$\\\\frac{p}{q}$$, where p and q are integers and $$q \\\\neq 0$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Find the value of $$(\\\\sqrt{2} + \\\\sqrt{3}) (\\\\sqrt{2} - \\\\sqrt{3})$$.\",\n                    \"text\": \"Find the value of $$(\\\\sqrt{2} + \\\\sqrt{3}) (\\\\sqrt{2} - \\\\sqrt{3})$$.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"2\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$-1$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$(\\\\sqrt{2}+\\\\sqrt{3})(\\\\sqrt{2} - \\\\sqrt{3}) = (\\\\sqrt{2})^2 - (\\\\sqrt{3})^2 = 2 - 3 = -1$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"On multiplying 0 and irrational number, we always get:\",\n                    \"text\": \"On multiplying 0 and irrational number, we always get:\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The product of any number with 0 is always 0.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"an irrational number\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The product of any number with 0 is always 0.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a rational number except 0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The product of any number with 0 is always 0.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"None of these\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The product of any number with 0 is always 0.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"0\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The product of any number with 0 is always 0.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The product of any number with 0 is always 0.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"On dividing $$16\\\\sqrt{3}$$ by $$2\\\\sqrt{3}$$, we get:\",\n                    \"text\": \"On dividing $$16\\\\sqrt{3}$$ by $$2\\\\sqrt{3}$$, we get:\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$8\\\\sqrt{3}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{8}{\\\\sqrt{3}}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$8 + \\\\sqrt{3}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"8\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\frac{16\\\\sqrt{3}}{2\\\\sqrt{3}} = \\\\frac{16}{2} = 8$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The decimal expansion of $$\\\\sqrt{6}$$ is\",\n                    \"text\": \"The decimal expansion of $$\\\\sqrt{6}$$ is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"terminating, non repeating\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"non-terminating, repeating\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"terminating, repeating\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"non-terminating, non-repeating\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\sqrt{6}$$ is an irrational number and hence the decimal expansion of $$\\\\sqrt{6}$$ is non-terminating and non-repeating.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"19-frequently-asked-questions\">Frequently Asked Questions<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6a6140a2-44c9-403f-91e2-e1ccf30178a0\"><strong>Why are irrational numbers included in the set of real numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-6a6140a2-44c9-403f-91e2-e1ccf30178a0\">\n\n<p class=\"eplus-wrapper wp-block-paragraph\">The numbers we can express in the form of decimals are known as real numbers. We can express the irrational numbers in terms of decimals and we can also represent them on a real number line, hence irrational numbers are in the set of real numbers.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6a6140a2-44c9-403f-91e2-e1ccf30178a0\"><strong>What is another name for irrational numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-6a6140a2-44c9-403f-91e2-e1ccf30178a0\">\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Another name for irrational numbers is surds.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6a6140a2-44c9-403f-91e2-e1ccf30178a0\"><strong>Are all real numbers irrational?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-6a6140a2-44c9-403f-91e2-e1ccf30178a0\">\n\n<p class=\"eplus-wrapper wp-block-paragraph\">No, real numbers are divided into rational and irrational numbers. Every irrational number is a real number, but every real number is not an irrational number.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6a6140a2-44c9-403f-91e2-e1ccf30178a0\"><strong>What is the difference between integers and irrational numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-6a6140a2-44c9-403f-91e2-e1ccf30178a0\">\n\n<p class=\"eplus-wrapper wp-block-paragraph\">Integers are a type of rational numbers and can be expressed in the form of fraction. For example: \u2013 2, 9, etc.&nbsp;On the other hand, irrational numbers are a type of numbers that cannot be expressed in the form of a ratio or fraction. For example: $\\sqrt{3}$, \ud835\uded1<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-6a6140a2-44c9-403f-91e2-e1ccf30178a0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6a6140a2-44c9-403f-91e2-e1ccf30178a0\"><strong>How many irrational numbers are there between two rational numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-6a6140a2-44c9-403f-91e2-e1ccf30178a0\">\n\n<p class=\"eplus-wrapper wp-block-paragraph\">There are infinite irrational numbers between any two rational numbers.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Irrational Numbers &#8211; Introduction We use numbers in daily life for a variety of reasons. Also, we use different types of numbers for different purposes, such as natural numbers for counting, fractions for describing portions or parts of a whole, decimals for precision, etc. Today we will explore \u2018Irrational Numbers\u2019 in math, their applications, examples, &#8230; <a title=\"Irrational Numbers\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/number\/irrational-numbers\" aria-label=\"More on Irrational Numbers\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-14358","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/14358","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=14358"}],"version-history":[{"count":25,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/14358\/revisions"}],"predecessor-version":[{"id":32640,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/14358\/revisions\/32640"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=14358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=14358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=14358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}