{"id":24143,"date":"2022-02-15T23:26:01","date_gmt":"2022-02-15T17:56:01","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=24143"},"modified":"2024-04-30T15:13:19","modified_gmt":"2024-04-30T09:43:19","slug":"cbse-class-10-maths-notes-chapter-1-real-numbers","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/","title":{"rendered":"Real Numbers Class 10 Notes Maths Chapter 1"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Real_Numbers_Class_10_Notes\" title=\"Real Numbers Class 10 Notes\">Real Numbers Class 10 Notes<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Euclids_Division_Lemma\" title=\"Euclid\u2019s Division Lemma\">Euclid\u2019s Division Lemma<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Euclids_Division_Algorithm\" title=\"Euclid\u2019s Division Algorithm\">Euclid\u2019s Division Algorithm<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Real_Numbers\" title=\"Real Numbers\">Real Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Revisiting_Irrational_Numbers\" title=\"Revisiting Irrational Numbers\">Revisiting Irrational Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#FAQs_Real_Numbers_Class_10_Notes\" title=\"FAQs Real Numbers Class 10 Notes\">FAQs Real Numbers Class 10 Notes<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#What_are_real_numbers_in_class_10th\" title=\"What are real numbers in class 10th?\">What are real numbers in class 10th?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#What_are_the_types_of_real_numbers\" title=\"What are the types of real numbers?\">What are the types of real numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/#Is_zero_a_real_number\" title=\"Is zero a real number?\">Is zero a real number?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/ncert-solutions\/class-10\/maths\/\" target=\"_blank\" rel=\"noopener\"><strong>CBSE Class 10 Maths<\/strong><\/a> Chapter 1 Real Numbers Notes are here to help you understand the basics. Real numbers are the ones we commonly use, like positive and negative whole numbers, fractions, and numbers with decimals. Basically, if you can find a number in the real world, it&#8217;s a real number. Think about it: we use numbers every day, whether it&#8217;s counting things, measuring temperature with positive or negative whole numbers, or working with fractions for parts of a whole. We&#8217;ll cover what real numbers are, how to divide them using Euclid\u2019s division algorithm, the fundamental theorem of arithmetic, ways to find the lowest common multiple (LCM) and highest common factor (HCF), and we&#8217;ll explain rational and irrational numbers with examples.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Real_Numbers_Class_10_Notes\"><\/span>Real Numbers Class 10 Notes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Get your free PDF download of CBSE Class 10 Maths Notes for Chapter 1 on Real Numbers. These notes are designed for quick revision. Included are NCERT Class 10 Maths Notes for Chapter 1, covering the topic of Real Numbers. In the updated <strong><a href=\"https:\/\/infinitylearn.com\/surge\/cbse\/\" target=\"_blank\" rel=\"noopener\">CBSE<\/a><\/strong> Exam Pattern, expect <strong><a href=\"https:\/\/infinitylearn.com\/surge\/mcqs\/\" target=\"_blank\" rel=\"noopener\">Multiple Choice Questions (MCQs) <\/a><\/strong>in Class 10 Maths, which contribute 20 marks.<\/p>\n<p><strong>R = Real Numbers: <\/strong>All rational and irrational numbers are called real numbers.<\/p>\n<p><strong>I = Integers: <\/strong>All numbers from (\u2026-3, -2, -1, 0, 1, 2, 3\u2026) are called integers.<\/p>\n<p><strong>Q = Rational Numbers: <\/strong>Real numbers of the form \\(\\frac { p }{ q }\\), q \u2260 0, p, q \u2208 I are rational numbers.<\/p>\n<ul>\n<li>All integers can be expressed as rational, for example, 5 = \\(\\frac { 5 }{ 1 }\\)<\/li>\n<li>Decimal expansion of rational numbers terminating or non-terminating recurring.<\/li>\n<\/ul>\n<p><strong>Q\u2019 = Irrational Numbers: <\/strong>Real numbers which cannot be expressed in the form \\(\\frac { p }{ q }\\) and whose decimal expansions are non-terminating and non-recurring.<\/p>\n<ul>\n<li>Roots of primes like \u221a2, \u221a3, \u221a5 etc. are irrational<\/li>\n<\/ul>\n<p><strong>N = Natural Numbers: <\/strong>Counting numbers are called natural numbers. N = {1, 2, 3, \u2026}<\/p>\n<p><strong>W = Whole Numbers: <\/strong>Zero along with all natural numbers are together called whole numbers. {0, 1, 2, 3,\u2026}<\/p>\n<p><strong>Even Numbers: <\/strong>Natural numbers of the form 2n are called even numbers. (2, 4, 6, \u2026}<\/p>\n<p><strong>Odd Numbers: <\/strong>Natural numbers of the form 2n -1 are called odd numbers. {1, 3, 5, \u2026}<\/p>\n<ul>\n<li>Why can\u2019t we write the form as 2n+1?<\/li>\n<\/ul>\n<p><strong>Remember this!<\/strong><\/p>\n<ul>\n<li>All Natural Numbers are whole numbers.<\/li>\n<li>All Whole Numbers are Integers.<\/li>\n<li>All Integers are Rational Numbers.<\/li>\n<li>All Rational Numbers are Real Numbers.<\/li>\n<\/ul>\n<p><strong>Prime Numbers: <\/strong>The natural numbers greater than 1 which are divisible by 1 and the number itself are called prime numbers, Prime numbers have two factors i.e., 1 and the number itself For example, 2, 3, 5, 7 &amp; 11 etc.<\/p>\n<ul>\n<li>1 is not a prime number as it has only one factor.<\/li>\n<\/ul>\n<p><strong>Composite Numbers: <\/strong>The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers. For example, 4, 6, 8, 9, 10 etc.<br \/>\nNote: 1 is neither prime nor a composite number.<\/p>\n<p><strong>Since remainder is zero, divisor (8) is HCF.<\/strong><br \/>\nAlthough Euclid\u2019s Division lemma is stated for only positive integers, it can be extended for all integers except zero, i.e., b \u2260 0.<\/p>\n<p><strong>1. Algorithm to locate HCF and LCM of two or more positive integers:<\/strong><\/p>\n<p><strong>Step I: <\/strong>Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitude of primes.<br \/>\n<strong>Step II: <\/strong>To find HCF, identify common prime factor and find the least powers and multiply them to get HCF.<br \/>\n<strong>Step III: <\/strong>To find LCM, find the greatest exponent and then multiply them to get the LCM.<\/p>\n<p><strong>2. To prove Irrationality of numbers:<\/strong><\/p>\n<ul>\n<li>The sum or difference of a rational and an irrational number is irrational.<\/li>\n<li>The product or quotient of a non-zero rational number and an irrational number is irrational.<\/li>\n<\/ul>\n<p><strong>3. To determine the nature of the decimal expansion of rational numbers:<\/strong><\/p>\n<ul>\n<li>Let x = p\/q, p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of\u2019q\u2019 is of the form 2<sup>m<\/sup>5<sup>n<\/sup>, m and n are non-negative integers.<\/li>\n<li>Let x = p\/q be a rational number such that the prime factorization of \u2018q\u2019 is not of the form 2<sup>m<\/sup>5<sup>n<\/sup>, \u2018m\u2019 and \u2018n\u2019 being non-negative integers, then x has a non-terminating repeating decimal expansion.<\/li>\n<\/ul>\n<p><strong>Alert!<\/strong><\/p>\n<ul>\n<li>2<sup>3<\/sup> can be written as: 2<sup>3<\/sup> = 2<sup>3<\/sup>5<sup>0<\/sup><\/li>\n<li>5<sup>2<\/sup> can be written as: 5<sup>2<\/sup> = 2<sup>0<\/sup>5<sup>2<\/sup><\/li>\n<\/ul>\n<div class=\"featured-snippet\">\n<h3><span class=\"ez-toc-section\" id=\"Euclids_Division_Lemma\"><\/span>Euclid\u2019s Division Lemma<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Euclid\u2019s Division Lemma&#8221; in Class 10 Maths is a fundamental theorem used for proving the existence and uniqueness of the quotient and remainder when one integer is divided by another. It states that for any two positive integers &#8216;a&#8217; and &#8216;b&#8217;, there exist unique integers &#8216;q&#8217; (quotient) and &#8216;r&#8217; (remainder) such that<\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\ud835\udc4e<\/mi><mo>=<\/mo><mi>\ud835\udc4f<\/mi><mi>\ud835\udc5e<\/mi><mo>+<\/mo><mi>\ud835\udc5f<\/mi><\/mrow><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mord mathnormal\">q<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><\/span><\/span>, where<\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>0<\/mn><mo>\u2264<\/mo><mi>\ud835\udc5f<\/mi><mo>&lt;<\/mo><mi>\ud835\udc4f<\/mi><\/mrow><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span>.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Euclids_Division_Algorithm\"><\/span>Euclid\u2019s Division Algorithm<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Euclid\u2019s Division Algorithm in Class 10 is a method used to find the highest common factor (HCF) of two numbers by repeatedly dividing the larger number by the smaller one and taking the remainder until we get a zero remainder.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Real_Numbers\"><\/span>Real Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Real numbers refer to all the numbers that we use in everyday life, including positive and negative whole numbers, fractions, decimals, and irrational numbers like square roots. They are the numbers we can see on a number line and use in various mathematical operations.<\/p>\n<ul>\n<li>Any real number can be plotted on the number line.<\/li>\n<li>Real numbers constitute the union of all rational and irrational numbers.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"alignleft wp-image-716485 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/real-number-copy.jpg\" alt=\"Real Numbers\" width=\"750\" height=\"466\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/real-number-copy.jpg 750w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/real-number-copy-300x186.jpg 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/real-number-copy-150x93.jpg 150w\" sizes=\"(max-width: 750px) 100vw, 750px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Revisiting_Irrational_Numbers\"><\/span>Revisiting Irrational Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Revisiting Irrational Numbers&#8221; in Class 10 likely refers to a topic in mathematics where students review and deepen their understanding of irrational numbers, which are numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. This could involve exploring properties of irrational numbers, operations involving them, and their significance in real-world contexts.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"FAQs_Real_Numbers_Class_10_Notes\"><\/span>FAQs Real Numbers Class 10 Notes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_real_numbers_in_class_10th\"><\/span>What are real numbers in class 10th?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tReal numbers in class 10th are the numbers that include all the positive and negative whole numbers, fractions, and decimals. They're basically numbers we use every day.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_types_of_real_numbers\"><\/span>What are the types of real numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tReal numbers are classified into rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot and include square roots of non-perfect squares.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_zero_a_real_number\"><\/span>Is zero a real number?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, zero is a real number. It's a whole number and also considered neither positive nor negative.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are real numbers in class 10th?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Real numbers in class 10th are the numbers that include all the positive and negative whole numbers, fractions, and decimals. They're basically numbers we use every day.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the types of real numbers?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Real numbers are classified into rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot and include square roots of non-perfect squares.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is zero a real number?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, zero is a real number. It's a whole number and also considered neither positive nor negative.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>CBSE Class 10 Maths Chapter 1 Real Numbers Notes are here to help you understand the basics. Real numbers are [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Real Numbers Class 10 Notes","_yoast_wpseo_title":"Real Numbers Class 10 Notes Maths Chapter 1 Notes","_yoast_wpseo_metadesc":"Real Numbers Class 10 notes are provided here along with examples & practice questions. real numbers for class 10 in a simple and concise manner.","custom_permalink":"math\/notes\/real-numbers-class-10-notes-maths-chapter-1\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Real Numbers Class 10 Notes Maths Chapter 1 Notes<\/title>\n<meta name=\"description\" content=\"Real Numbers Class 10 notes are provided here along with examples &amp; practice questions. real numbers for class 10 in a simple and concise 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