{"id":26525,"date":"2022-01-17T18:07:59","date_gmt":"2022-01-17T12:37:59","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=26525"},"modified":"2024-03-07T18:16:53","modified_gmt":"2024-03-07T12:46:53","slug":"number-systems-class-9-notes-maths-chapter-1","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-materials\/number-systems\/class-9-notes\/maths\/chapter-1\/","title":{"rendered":"Number Systems Class 9 Notes Maths Chapter 1"},"content":{"rendered":"<h2>Number Systems Class 9 Notes Maths Chapter 1<\/h2>\n<p>The number system involves the representation of numbers on a linear scale using specific symbols and rules. A number line, which features a defined interval between numbers, serves as a visual tool for depicting numerical values. This system is employed for a broad range of mathematical computations, encompassing intricate scientific calculations to the simple task of counting the remaining chocolates in a box. In this article, we will delve into the concepts of Class 9 Maths Chapter 1 on the Number System.<\/p>\n<p>Also Check out our <a href=\"https:\/\/infinitylearn.com\/surge\/cbse-notes-class-9\/\"><strong>CBSE Notes for Class 9<\/strong><\/a><\/p>\n<p><strong>NCERT Solutions for Class 9 All Subjects<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/ncert-solutions\/class-9\/maths\/\"><strong>NCERT Solutions for Class 9 Maths<\/strong><\/a><\/li>\n<li><strong><a href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/ncert-solutions\/class-9\/science\/\">NCERT Solutions for Class 9 Science<\/a><\/strong><\/li>\n<li><strong><a href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/ncert-solutions\/class-9\/social-science\/\">NCERT Solutions for Class 9 Social Science<\/a><\/strong><\/li>\n<li><strong><a href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/ncert-solutions\/class-9\/english\/\">NCERT Solutions for Class 9 English<\/a><\/strong><\/li>\n<\/ul>\n<p>1. <strong>Natural Numbers<\/strong>: Numbers which start from one (1) are known as natural numbers. The collection of all natural numbers is denoted by N.<br \/>\nN = {1, 2, 3, 4, \u2026\u2026.}<\/p>\n<p>2. <strong>Whole Numbers:<\/strong> Numbers which start from zero (0) are known as whole numbers. The collection of all whole numbers is denoted by W.<br \/>\nor<br \/>\nIf \u20180\u2019 is included in the collection of natural numbers, then the collection are known as whole numbers.<br \/>\nW = {0, 1, 2, 3, \u2026}<\/p>\n<p>3. <strong>Integers:<\/strong> The collection of all whole number (natural numbers + zero) and negative of natural numbers are called integers.<br \/>\nIt is denoted by Z.<br \/>\nZ = {\u2026, -3, -2, -1, 0, 1, 2, 3, \u2026}<\/p>\n<p>4. <strong>Rational Number:<\/strong> A number r is called a rational number, if it can be written in the form \\(\\frac { p }{ q }\\), where p and q are integers and q \u2260 0. The collection of rational numbers is denoted by Q.<br \/>\ne.g., \\(\\frac { 1 }{ 2 }\\) , \\(\\frac { 3 }{ 5 }\\) ,\u2026\u2026.<\/p>\n<p>5. <strong>Irrational Number:<\/strong> A number S is called an irrational number, if it can not be written in the form \\(\\frac { p }{ q }\\), where p and q are integers and q \u2260 0, and its decimal representation is non-terminating and non-repeating.<br \/>\ne.g., \u221a2, \u221a5, \u03c0, \u2026<\/p>\n<p>6. <strong>Equivalent Rational Number:<\/strong> The rational number whose numerator and denominator both are equal or they are reducible to equal.<br \/>\ne.g.,<strong> 1\/2=2\/4=10\/20=25\/50=47\/94<\/strong><br \/>\nNote: There are infinitely many rational numbers between any two given rational numbers. Symbol \u221a indicate the square root of the number<br \/>\ne.g., \u221a4 = 2, though both 2 and -2 are square roots of 4.<\/p>\n<p>7. <strong>Real Numbers:<\/strong> The collection of all rational numbers and irrational numbers together make up what we call the collection of real numbers, which is denoted by R. Therefore, a real number is either rational or irrational.<\/p>\n<p><strong>Note:<\/strong> Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.<\/p>\n<p>8. <strong>Real Numbers and their Decimal Expansions:<\/strong> For all rationals of the form \\(\\frac { p }{ q }\\) (q \u2260 0). On a division of p by q, two main things happen\u2014either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.<\/p>\n<p><strong>Case I.<\/strong> The remainder becomes zero<br \/>\nIn this case, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.<br \/>\ne.g., \\(\\frac { 7 }{ 8 }\\) , \\(\\frac { 1 }{ 2 }\\) , \\(\\frac { 3 }{ 4 }\\), etc.<\/p>\n<p><strong>Case II.<\/strong> The remainder never becomes zero<br \/>\nIn this case, we have a repeating block of digits in the quotient, this expansion is called non-terminating recurring.<br \/>\ne.g., \\(\\frac { 2 }{ 3 }\\) = 0.6666\u2026..<br \/>\n\\(\\frac { 22 }{ 7 }\\) = 3.142857142857\u2026\u2026..<br \/>\nThe repeated digits are written as \\(\\frac { 2 }{ 3 }\\) = \\(0.\\bar { 6 }\\)<br \/>\n\\(\\frac { 22 }{ 7 }\\) = \\(3.\\bar { 142857 }\\)<\/p>\n<p><strong>Note:<\/strong><br \/>\nThe decimal expansion of rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.<\/p>\n<p>The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.<br \/>\nS = 0.10110111011110\u2026 (non-terminating and non-recurring).<\/p>\n<p>9. <strong>Operations on Real Numbers:<\/strong> Rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers. We still get a rational number (i.e., rational numbers are \u2018closed\u2019 with respect to addition, subtraction, multiplication and division).<\/p>\n<p>Irrational numbers also satisfy the commutative, associated and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always.<br \/>\ne.g., \u221a5 + (-\u221a5) = 0<br \/>\n\\(\\frac { \\surd 15 }{ \\surd 15 }\\) = 1 are rational.<br \/>\n\u221a3 is irrational.<br \/>\nHence, (5 + \u221a3) is also irrational (\u221a3 has a non-terminating, non-recurring decimal expansion).<\/p>\n<p><strong>Note:<\/strong><br \/>\nThe sum or difference of a rational number and an irrational number is irrational.<br \/>\nThe product or quotient of a non-zero rational number with an irrational number is irrational.<br \/>\nIf we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.<\/p>\n<p>10. <strong>Radicand:<\/strong> If \\(\\sqrt [ n ]{ a }\\) is a surd then n is known as order of surd and a is known as radicand.<\/p>\n<p>11. <strong>Laws of Radicals:<\/strong> Let a and b be positive real numbers. Then,<\/p>\n<ul>\n<li>\u221aab = \u221aa \u221ab<\/li>\n<li>\\(\\sqrt { \\frac { a }{ b } } =\\frac { \\surd a }{ \\surd b }\\)<\/li>\n<li>(\u221aa + \u221ab) (\u221aa \u2013 \u221ab) = (a \u2013 b)<\/li>\n<li>(a + \u221ab) (a \u2013 \u221ab) = a<sup>2<\/sup> \u2013 b<\/li>\n<li>(\u221aa + \u221ab) (\u221ac + \u221ad) = \u221aac + \u221aad + \u221abc + \u221abd<\/li>\n<li>(\u221aa + \u221ab)<sup>2<\/sup> = a + 2\u221aab + b<\/li>\n<\/ul>\n<p>12. <strong>Rationalising the Denominator:<\/strong> When the denominator of an expression contains a term with a square root (or a number under a radical sign), the process of converting it to an equivalent expression whose denominator is a rational number is called rationalising the denominator.<br \/>\nTo rationalise the denominator of\\(\\frac { 1 }{ \\surd a+b }\\), it is multiplied by \\(\\frac { \\surd a-b }{ \\surd a-b }\\) , where a and b are integers.<\/p>\n<p>13.<strong> Laws of Exponents<\/strong><\/p>\n<ul>\n<li>a<sup>m<\/sup> . a<sup>n<\/sup> = a<sup>m+n<\/sup><\/li>\n<li>(a<sup>m<\/sup>)<sup>n<\/sup> = a<sup>mn<\/sup><\/li>\n<li>\\(\\frac { { a }^{ m } }{ { a }^{ n } } ={ a }^{ m-n }\\), m &gt; n<\/li>\n<li>a<sup>m<\/sup> b<sup>m<\/sup> = (ab)<sup>m<\/sup>\u200b<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Number Systems Class 9 Notes Maths Chapter 1 The number system involves the representation of numbers on a linear scale [&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":"Number Systems Class 9 Notes Maths Chapter 1","_yoast_wpseo_title":"Number Systems Class 9 Notes Maths Chapter 1","_yoast_wpseo_metadesc":"CBSE Class 9 Maths Notes Chapter 1 Number Systems ... 1. 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