Number System. p, where q 0, where p and q are integers, then the q. number is called rational number. It is denoted by Q.

Transcription

1 CHAPTER Number System We are Starting from a Point but want to Make it a Circle of Infinite Radius We use numbers in our daily life for counting, measuring, etc. Let us recall the types of numbers. Natural Numbers: Counting numbers are called Natural numbers, e.g.,,,,,... are Natural Numbers. It is denoted by N. There are infinite natural numbers and the smallest natural number is. Whole Numbers: Integers: The natural numbers along with zero form the system of whole numbers, e.g., 0,,,,,. It is denoted W. There is no largest whole number and the smallest whole number is 0. The number system consisting of natural numbers, their negative and zero is called integers, e.g.,, -, -, 0,,,, +. It is denoted by Z or I. The smallest and the largest integers cannot be determined. Even numbers: Natural numbers which are divisible by are called even numbers, e.g.,,, 6, 8. Smallest even number is. There is no largest even number. Odd Numbers: Natural numbers which are not divisible by are called odd number. e.g,,,,,. Smallest odd number is. There is no largest odd number Prime Numbers: Natural numbers which have exactly two factors, i.e., and the number itself are called prime numbers, e.g.,,,,,.... Rational numbers: If a number can be expressed in the form of number is called rational number. It is denoted by Q. For Example, Irrational Numbers: 0-690/ p, where q 0, where p and q are integers, then the q 9 6 8,,, etc. p The numbers which cannot be expressed in the form of, where p and q are integers and q 0, is q called irrational number. It is denoted by I. For example,,,,,, are irrational numbers? Important Point: Every positive irrational number has a negative irrational number corresponding to it.

2 6,, 6, 6 Sometimes, product of two irrational numbers is a rational number For example: (i) () The Number Line: The number line is a straight line between negative infinity on the left to positive infinity on the right. On number line integers are placed at a equal distance Real Numbers: A set of rational and irrational numbers is called the set of real numbers. It is denoted by R. Therefore, Real numbers = Rational numbers + Irrational numbers. In other words, all numbers that can be represented on the number line are called real numbers. R + : Positive real numbers and R : Negative real numbers. TYPES OF NUMBERS Real numbers (R) Rational numbers (Q) Imaginary numbers Z ( a ib), a, br Irrational numbers (T) The number which cannot be Expressed as form Natural numbers(n) Whole numbers (W) Integer(I) Even Odd number number Positive Integers Negative Integers Non-negative Integers Non-positive Integers Prime numbers Composite numbers {,,, } {-, -, -, - } {0,,,, } {0, -, -, -, } Important Points: is neither prime nor composite. is an odd integer. 0 is neither positive nor negative. is smallest even prime number. All prime numbers (except ) are odd /

3 Fraction: A fraction is a quantity which expresses a part of a whole. Numerator Fraction = Deno min ator Types of Fractions: (c) (d) Proper Fraction: If numerator is less than its denominator, then it is called a proper fraction. 6 For example:,. 8 Improper Fraction : If numerator is greater than or equal to its denominator, then it is called an improper fraction. For example:, 8,. Mixed fraction: It consists of an integer a proper fraction. For example:,,. 9 Mixed fraction can always be changed into improper fraction advice versa For example: and Equivalent fractions / Equal fractions: Fractions with the same value. 9 For example: Important Points:, 6 8,, 6 9 Value of fraction is not change by multiplying or dividing the numerator or denominator by the same number. For example: (i) 0 So, So, (e) (f) If in a fraction, its numerator and denominator are of equal value then fraction is equal to unity i.e.. Like fractions: Fraction with same denominators. For example:, 9,, Unlike Fractions: Fractions with different denominators. For example: Important Point:, 9,, 8 9 Unlike fractions can be converted into like fractions /

4 For example: and and 0 (g) Decimal fraction: The fractions whose denominators are of the powers of For example: 0., 0.09 (h) Vulgar fraction: Denominators are not the power of 0. For example:, 9, 9 We can also study real numbers by studying classification of decimal expansion. Decimal Expansion of real numbers Terminating Non-Terminating Recurring Non-recurring Pure Recurring Mixed Recurring Terminating (or finite decimal fractions) : For example: 0. 8,. 8 Non-terminating decimal fractions : There are two types of Non-terminating decimal fractions : (i) Non-terminating periodic fractions or non-terminating recurring (repeating) decimal factions : 0 For example: Non-terminating non-periodic fraction or non-terminating non-recurring fractions : For example.969 Important point: The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. 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 /

5 For example: = = We often take as an approximate value of, but. Rational Numbers To find out rational numbers between two Rational If r and s be two rational numbers then r s is between r and s. r s Then find the rational number between r and r s r, i.e.. Example : Find two rational numbers between and. A rational number between and = A rational number between and =. So, two rational numbers between and is and. Example : Write five rational numbers between and \ 0 0 Five rational numbers between and are Example : Find four rational numbers between and and Four rational number between and,,,, and are,, and /

6 IX ACADEMIC QUESTIONS Subjective Assignment-. Is zero a rational number? If yes, write it in the form of p/q.. Find six rational numbers between (i) and and (iii) 9 and 0 (iv) and 6. Find rational numbers between and.. Find five rational numbers between and.. Find six rational numbers between - and. 6. Find four rational numbers between and.. Find six numbers between and 8. Find nine rational numbers between 0 and Find three rational numbers lying between and 0. Find five rational numbers lying between and. Find a rational numbers lying between and. State true or false for the following. Justify your answer (i) (iii) (iv) (v) Every whole number is a natural number. Every integer is a rational number. Every rational number is an integer. Every integer is a whole number. Every natural number is a whole number.. State true or false. Justify by giving example. (i) (iii) (iv) (v) (vi) Every real number is an irrational number. Every real no. is either rational or irrational is an irrational number Irrational numbers cannot be represented by points on the number line. The sum of two rational numbers is rational. The sum of two irrational numbers is an irrational. (vii) The product of two rational numbers is rational. (viii) The product of two irrational numbers is an irrational. (ix) (x) The sum of a rational number and an irrational number is an irrational. The product of a non-zero rational number and an irrational number is a rational number /

7 . Represent 6 on the number line.. Represent on the number line. 6. Represent on the number line. CONVERSION OF A RATIONAL NUMBER INTO DECIMAL Terminating Decimals (Finite Decimals) Let us convert into decimal from. (Rational number) 0. (Terminating decimal) This is a terminating decimal form. Hence, we conclude that terminating decimals are rational numbers, since the process of division terminates (comes to an end). Here, the division stops at a point, where there is no remainder. Non-Terminating and repeating decimals (Recurring Decimals) Let us convert into decimal form is a non-terminating and repeating decimal and is represented as 0.6. Non terminating and repeating decimals are also known as recurring decimals. is a rational number whose decimal expansion is a non-terminating and repeating decimal (recurring decimal). Q. Write the following in decimal form and say what kind of decimal expansion each has: (i) 6 (NCERT) 00 (NCERT) (iii) (NCERT) 8 (iv) (NCERT) (v) (NCERT) (vi) 8 (NCERT) (vii) 0 (viii) Conversion of terminating decimals into p q form Steps:. Assume the given recurring decimal as x.. In the given number, check the number of digits which have bar on their heads.. If only one digit has bar on its head, multiply the number by 0; if two digits have bar on its head, multiply the number by 00; similarly if three digits have bars on its head, multiply the number by 00; similarly if three digits have bars on its head, multiply the number by 000 and so on. When 0-690/

8 we multiple the number by 0, the decimal will shift one place to the right i.e., if x = 0.6 then 0x = 6.6. Subtract the numbers obtained in step and that obtained in step.. Simplify the equation obtained in step and calculate x. 6. Write the value of x in simplest form. Example:.8: Express each of the following in p q form. (i) (iii) 0.9 Solution: (i) Let x = 0. (i) Multiplying both sides by 0, we get 0x =. Subtracting (i) and, we get x x x. 0. 9x = Let x = 0. (i) Multiplying both sides by 00, we get 00x =. 99x = x = = 99 (iii) Let x 0.9 (i) Multiplying both sides by 000, we get 000x = 9. 9 Subtracting (i) and, we get 000x x = x = 9 9 x = /

9 IX ACADEMIC QUESTIONS Subjective Assignment- Convert the following decimal numbers in the form p q.. (i) (iii) (i) 0.6 (NCERT) 0. (NCERT) (iii) 0.00 (NCERT). (i) (iii).6. Express p (. 0.) in form q. If 0x 0. 0., find the value of x. 6. If x , find the value of x. 9 Irrational Number A number which cannot be expressed in the form of p q irrational number. Example:,,, 6, are irrational numbers. A non terminating and non-repeating decimal is known as irrational numbers., where p and q are integers and q 0, is known as Examples: All the above decimal numbers are non-terminating and non-repeating decimal and therefore these are irrational numbers. Properties of Irrational Numbers: (i) The sum of two irrational numbers need not be an irrational number. Example: Sum of and ( ) 0, which is a rational numbers Sum of ( ) and ( ), which is a rational number. The difference of two irrational numbers need not be an irrational number /

10 Example: Difference of ( 6) and 6 6 6, which is a rational number. (iii) The product of two irrational numbers need not be an irrational number. Examples: Product of and is a rational number. (iv) The division of two irrational numbers need not be an irrational number. Examples: Division of and =, is a rational number. (v) Division of 8 and 8 =, is a rational number. The sum of a rational and irrational number is irrational number. Examples: Sum of and = ( ), which is an irrational number. (vi) The difference of a rational and an irrational number is irrational number. Examples: Difference of and ( ), which is an irrational number. (vii) The product of a rational and an irrational number is irrational number. Examples: The product of and, which is an irrational number. (viii) The division of a rational and an irrational number is irrational numbers. Examples: The division of and Addition and Subtraction of Irrational Numbers: Example: Add ( ) and ( )., which is an irrational number. Solution: ( ) ( ) = ( ) ( ) = () () /

11 IX ACADEMIC QUESTIONS Subjective Assignment-. Add ( 6) and ( 6). Add ( ) and ( ). Add ( ) and ( ). Add ( )and( ). 6 and a b ab Multiplication of Irrational Numbers To multiply irrational numbers, multiply the rational parts and irrational parts separately. For example: Multiply and, write ( ) and ( ) separately. Division of Irrational Numbers: () ( ) 0 6 To divide irrational numbers, divide the rational parts and irrational parts separately. For example: Divide 0 6 and, write (0 ) and ( 6 ) /

12 IX ACADEMIC QUESTIONS Subjective Assignment-. Multiply ( ) by ( ).. Simplify. Simplify ( )( )... 6 ( ) ( )( ). ( 6 )( 6 ). 8 Rationalisation of the Denominator of an Irrational number having two terms in the denominator, multiply the numerator and denominator of the number by the conjugate of its denominator. For example: To rationalize a a b a b a b a b a b a b ( a) ( b) a b b, multiply the numerator and denominator by ( a b). Example: Rationalise the denominator of. Solution: Multiplying numerator and denominator by ( ) = = = ( ) ( ) = /

13 IX ACADEMIC QUESTIONS Subjective Assignment-. Rationalise the denominators of the following real numbers: (i) (v) (vi) (vii) (iii). Simplify the following by rationalizing the denominator: (iv) (i) (iii) 6 6. If. and.6, find the value of. Rationalise the denominator of the following: (iv) 8 8 (i) 9 y x y x. Rationalise the denominator of 6. Rationalise the denominator of. Simplify each of the following:. (i) (iii) 8. Find the values of a and b in each of the following: (i) a b a b a b (iv) a b 6 8 (v) a b 8 (vi) a b 0-690/

14 9. Find the values of a and b a b 0. Show that. Prove that 0. If x =, find the value of. If x =, find the value of Laws of Exponents for Real Numbers If a, n and m are natural numbers, then (i) a m a n = a m n (iii) a a m n Examples: mn x. x x (a m ) n = a mn. x a, where m > n. If a, b, m are natural numbers then (iv) a m b m = (ab) m (i) = + = ( ) = = 6 (iv) Laws of Radicals () 6 If n is a positive integer and a and b are positive rational numbers, then n n n n (i) n n ( a ) a n a n n n n /n n a b a b (ab) ab m n n n n m mn mn (iii) m m (iv) n n a a a a a a /n n a a a a n /n b b b b n m p m pm m p n n p (v) pm n m n (a ) a a (a ) a 0-690/

15 Examples: (i) / ( ) / / (iii) (v) / / / () 6 (iv) / /6 6 ( ) / / / / 0 0 (vi) (a ) a a (a ) a 0-690/

16 IX ACADEMIC QUESTIONS Subjective. Find: (i). Find: (i) 6 9 (iii) (iii) Assignment-6 (NCERT) 6 (iv) (NCERT). Find: (i) (iii) (iv) 8 (NCERT). Find the value of x in each of the following: (i) x x 0 x x (iii) x x. If x x 00, find the value of x. 6. Prove that If 0, then find the value of x. x 8. Prove that ab x x ba 9. Prove that a ab b bc c ca x x x b c a x x x 0. If x = y = z, then show that 0. x y z 0-690/

17 XI SCIENCE & DIP. ENTRANCE Multiple Choice Questions Assignment... can be expressed in rational form as 99 (c) (d) 99. The sum of two rational numbers is. number Rational Irrational (c) Either a& or (d) Natural. 0. can be expressed in rational form as (c) 0 (d) 900. The fraction ( 6) ( ) is equal to (c) (d). Which of the following is a pure surd? (c) (d) 8 6. If x 0, then x x x x x x x (c) 8 x (d) 8 x. 6x 8y 6 / y x y 6x y (c) 8x y (d) x 8. Set of natural numbers is a subset of Set of even numbers (c) Set of composite numbers Set of odd numbers (d) Set of real numbers 0-690/

18 9. For an integer n, a student states the following: I. If n is odd, (n + ) is even. II. If n is even, (n ) is odd. III. If n is even (n ) is irrational Which of the above statements would be true? I and III I and II (c) I, II and III (d) II and III 0. The irrational number between and is (c) (d). If =.6 and 0 =.6, then the value of (c).98 (d) 6.98 is. ( 0 ) ( 6 ) ( ) (c) (d). The rationalizing factor of a b c is a b c a b c (c) a b c (d) a b c. Two irrational numbers between and are,6 6, (c) 8 6, (d), 8. Which of the following sets of fraction is in ascending order?, 9, 9 9,, 9 (c),, 9 9 (d) 9,, 9 6. / ( ) is not equal to / ( 6 ) (c) ( ) / ( 6) (d) / (9 6). The value of... is 9900 Less than (c) Greater than Equal to (d) Equal to /

19 8. The numerator of a a b a a b a a b a a b a b (c) a b (d) a b is 9. The ascending order of the surds, 6, 9 is 9 6,,,, (c), 6, 9 (d) ,, 0. Rational number between and is (c). (d).8. The greater between and 6 is 6 (c) Both are equal (d) Cannot compare. Which of the expressions is the same as? ( ) (c) (d). If cad m, a b then b equals m(a b) ca cab ma m (c) c (d) ma m ca qr rp pq q r p x x x. The value of r p q x x x is equal to p q r x 0 (c) pq qr rp x (d). The value of 6 6 equals (c) (d) 6. The rationalizing factor of is (c) (d) 0-690/

20 . If x = then x x 8 8 (c) 8 8 (d) 8. The rational number between and is (c) (d) 9. Simplify : n n n ( ) n ( ) n (c) n (d) If a 9, then a 0 a equals a ( c) 8 (d). If both a and b are rational numbers, then a and b from a b, are 9 9 a,b 9 9 a,b (c) 8 a,b (d) 0 a,b. a b c d where d, c, b, a are consecutive natural numbers. Then which of the following is true? a b c d a b c d (c) a c b d (d) c d a b. 6 equals (c) 6 (d) ( ). The value of of is 9 (c) 6 (d) /

21 . Which one of the following is correct? ( )( ) ( )( 6) ( )( ) (c) ( )( 6) ( )( ) ( )( 6) ( )( ) (d) ( )( 6) 6. If x = x 00, then the value of x is (c) (d). If x =, then the value of x + x and x is x, 8, 8 (c), 8 (d), 8 8. If = x, then x is (c) 6 (d) 9. The 00 th root of 0 (0 ) 0 is (c) 0 ( 0 ) ( 0) (d) 0( (0)) 0 0. Which of the following numbers has the terminal decimal representation? (c) (d) is equal to 8/99 8/99 (c) /99 (d) /99. If x and y, then x + y = 8 00 (c) 08 (d) 9. If x and y, then x y 8x y 9 00 (c) 86 (d) 86. If a and b, then a b a b 8 (c) (d) 0-690/

22 . If x and y, then value of xy + x y is (c) 6 (d) 9 6. If x, then value of 80 8 x x (c) (d). If x, then value of x x is 0 6 (c) 0 (d) Passage: Given that =., =., =.6, 0 =.6 and =. 8. Evaluate (c).08 (d) Evaluate (c). (d) Find the value of (c).6 (d) /

23 ANSWER Assignment : Yes,,,, etc (iii),,,,, ,,,, ,,,,, ,, (i),,,,, (iv),,,,, 6.,,,, ,,,,,,,, ,,,, (i) (iii) (iv) are false and and (v) is True., (iii) (iv) (v) (vi) and (ix) are True and (i) (viii) (x) are false. Assignment. (i) /0 /000 (iii) 0/00. (i) / /90 (iii) /999. (i) /99 /9900 (iii) 99/. /9. / ,,,,, 6 8 9,,,, ,,, Assignment ab a b. Assignment / /

24 Assignment. (i) (v) (vi) 8. (i) (i) (iii) (vii) 6 (iii) ( ) (iv) (iv) x y x (i) 0 (iii) a = 8, b = a =, b = (iii) a, b (iv) a =, b = 6 (v) a =, b = (vi) a =, b = 9. a = 0, b = 8.. Assignment 6. (i) 8 (iii). (i) (iii) 8 (iv). (i) (iii). (i) x = x = (iii) x =. x =. x (iv) Assignment.b.c.b.d.a 6.d.a 8.d 9.b 0.c.a.a.a.c.a 6.d.b 8.d 9.a 0.c.b.c.d.d.d 6.b.b 8.a 9.d 0.c.a.b.a.c.a 6.b.c 8.a 9. b 0.c.b.a.c.b.c 6.c.a 8.d 9.d 0.b 0-690/