MATHEMATICS Chapter 1 Number Systems NUMBER SYSTEMS 01 Number Systems 1 Numbers 1 2 3 which are used for counting are called natural numbers The collection of natural numbers is denoted by N Therefore N 1 2 3 4 5 2 When 0 is included with the natural numbers then the new collection of numbers called is called whole number The collection of whole numbers is denoted by W Therefore W 0 1 2 3 4 5 3 The negative of natural numbers 0 and the natural number together constitutes integers The collection of integers is denoted by I Therefore I 3 2 1 0 1 2 3 4 The numbers which can be represented in the form of p q where q 0 and p and q are integers are called rational numbers Rational numbers are denoted by Q If p and q are co prime then the rational number is in its simplest form 5 All natural numbers whole numbers and integer are rational number 6 Equivalent rational numbers or fractions have same equal values when written in the simplest form 7 Rational number between two numbers x and y x y 2 8 There are infinitely many rational numbers between any two given rational numbers 9 The numbers which are not of the form of p q where q 0 and p and q are integers are called irrational numbers For example 2 7 etc 10 Rational and irrational numbers together constitute are called real numbers The collection of real numbers is denoted by R 11 Irrational number between two numbers x and y xy if x and y both are irrational numbers xy if x is rational number and y is irrational number xy if x y is not a perfect square and x y both are rational numbers 12 Terminating fractions are the fractions which leaves remainder 0 on division 13 Recurring fractions are the fractions which never leave a remainder 0 on division 14 The decimal expansion of rational number is either terminating or non terminating recurring Also a number whose decimal expansion is terminating or non terminating recurring is rational 15 The decimal expansion of an irrational number is non terminating non recurring Also 1 NUMBER SYSTEMS 01 a number whose decimal expansion is non terminating non recurring is irrational 16 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 17 The process of visualization of numbers on the number line through a magnifying glass is known as the process of successive magnification This technique is used to represent a real number with non terminating recurring decimal expansion 18 Irrational numbers like 2 3 5 n for any positive integer n can be represented on number line by using Pythagoras theorem 19 If a 0 is a real number then a b means b2 a and b 0 20 For any positive real number x we have 21 For every positive real number x x can be represented by a point on the number line x 1 2 x 2 x 1 2 2 using the following steps i Obtain the positive real number say x ii Draw a line and mark a point A on it iii Mark a point B on the line such that AB x units iv From B mark a distance of 1 unit on extended AB and name the new point as C v Find the mid point of AC and name that point as O vi Draw a semi circle with centre O and radius OC vii Draw a line perpendicular to AC passing through B and intersecting the semi circle at D viii Length BD is equal to x 2 NUMBER SYSTEMS 01 22 Properties of irrational numbers i The sum difference product and quotient of two irrational numbers need not always be an irrational number ii Negative of an irrational number is an irrational number iii Sum of a rational and an irrational number is irrational iv Product and quotient of a non zero rational and irrational number is always irrational 23 Let a 0 be a real number and n be a positive integer Then an b if bn a and b 0 The symbol is called the radical sign 24 For real numbers a 0 and b 0 i ab a b ii a b a b iii a b a b a b iv a b c d ac bc ad bd v a b a b a2 b vi a b a b 2 ab 2 25 The process of removing the radical sign from the denominator of an expression to convert it to an equivalent expression whose denominator is a rational number is called rationalising the denominator 26 The multiplicating factor used for rationalising the denominator is called the rationalising factor 3 NUMBER SYSTEMS 01 27 If a and b are positive real numbers then Rationalising factor of is a Rationalising factor of is a b Rationalising factor of is a b 1 a 1 a b 1 a b 28 The exponent is the number of times the base is multiplied by itself 29 In the exponential representation am a is called the base and m is called the exponent 30 Laws of exponents If a b are positive real numbers and m n are rational numbers or power then Numbers Natural Numbers Whole Numbers Integers Number Arithmetical value representing a particular quantity The various types of numbers are Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers etc Natural numbers N are positive numbers i e 1 2 3 and so on Whole numbers W are 0 1 2 and so on Whole numbers are all Natural Numbers including 0 Whole numbers do not include any fractions negative numbers or decimals Integers are the numbers that includes whole numbers along with the negative numbers 4 NUMBER SYSTEMS 01 Rational Numbers Irrational Numbers Real Numbers A number r is called a rational number if it can be written in the form p q where p and q are integers and q 0 Any number that cannot be expressed in the form of p q where p and q are integers and q 0 is an irrational number Examples 2 1 010024563 e Any number which can be represented on the number line is a Real Number R It includes both rational and irrational numbers Every point on the number line represents a unique real number Irrational Numbers Representation of Irrational numbers on the Number line Let x be an irrational number To represent it on the number line we will follow the following steps Take any point A Draw a line AB x units Extend AB to point C such that BC 1 unit Find out the mid point of AC and name it O With O as the centre draw a semi Draw a straight line from B which is perpendicular to AC such that it intersects the circle with radius OC semi circle at point D Length of BD x Constructions to Find the root of x 5 …