RATIONAL AND IRRATIONAL NUMBERS

Transcription

1 Q.. RATIONAL AND IRRATIONAL NUMBERS Without actual division find which of the following rationals are terminating decimal: (i) 9 5 (ii) 7 (iii) 5 (iv) 7 78 Ans. (i) In 9, the prime factors of denominator 5 are 5, 5. Thus it is terminating decimal. 5 (ii) In 7, the prime factors of denominator are, and. Thus it is not terminating decimal. (iii) In, the prime factors of denominator 5 are 5, 5 and 5. Thus it is 5 terminating decimal. (iv) In 7, the prime factors of denominator 78 are, and. Thus it is not 78 terminating decimal. Q.. Represent each of the following as a decimal number. (i) 4 5 Ans. (i) In (ii) 5 (iii), using long division method: 5 Hence, Math Class IX Question Bank

2 (iii) In Q.. 5, using long division method: Hence, (iv) In 5 55, using long division method: Hence, p q, Express each of the following as a rational number in the form of where q 0. (i) 0.6 (ii) 0.4 (iii) 0.7 (iv) 0.04 Ans. (i) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Subtracting eqn. (i) from eqn. (ii), we get 0x x x 6 x Hence, required fraction. 9 (ii) Let x (i) Multiplying both sides of eqn. (i) by 00, we get 00x (ii) Subtracting eqn. (i) from eqn. (ii), we get Math Class IX Question Bank

3 00x x x 4 Hence, required fraction 4 99x 4 x p q (iii) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 00, we get 000x (iii) Subtracting eqn. (ii) from (iii), we get 000x x x 5 990x 5 x 990 p 5 Hence, required fraction. q (iv) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 000, we get 0000x (iii) Subtracting eqn. (ii) from (iii), we get 0000x x x 0 Hence, required fraction x 0 x Math Class IX Question Bank

4 Q.4. Express each of the following as a vulgar fraction. (i).46 (ii) 4.4 Ans. Let x (i) Multiplying both sides of eqn. (i) by 000, we get 000x (ii) Subtracting eqn. (i) from eqn. (ii), we get 000x Q.5. x x 4 999x 4 x 999 Hence, required vulgar fraction (ii) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 00, we get 000x (iii) Subtracting eqn. (ii) from eqn. (iii), we get 000x x x x 48 x Hence, required vulgar fraction. 0 Insert one rational number between: (i) and 7 (ii) 8 and Ans. If a and b are two rational numbers, then between these two numbers, one rational number will be ( a + b ). Required rational number between 5 and < < Math Class IX 4 Question Bank

5 (ii) Required rational number between 8 and 8.04 ( ) (6.04) < 8.0 < 8.04 Q.6. Insert two rational numbers between and Ans. and and < + < < < < < < < < < + < < < < < < < < < < Hence, required rational numbers are 9 40 and Q.7. Insert three rational numbers between (i) 4 and 5 (ii) and 5 (iii) 4 and 4.5 (iv) and (v) and Ans. (i) The given numbers are 4 and 5. As, 4 < < < 5 4 < < 5 4 < 4.5 < 5...(i) 9 9 Again, 4 < 4 + < 4 < 4.5 < (ii) Again, 4.5 < < ( ) < < 4.75 < 5...(iii) From eqn. (i), (ii) and (iii), we get 4 < 4.5 < 4.5 < 4.75 < 5. Thus, required rational numbers between 4 and 5 are 4.5, 4.75 and 4.5. Math Class IX 5 Question Bank

6 (ii) The given numbers are and 5 As, < < + < 5 < < < < < < Again, < + < < < < <...(ii) 40 5 Again, < < + < 0 5 < < < <...(iii) 40 5 From eqn. (i), (ii) and (iii), we get < < < < Thus, required rational numbers between and are, and. 40 (iii) The given numbers are 4 and 4.5 As 4 < < ( ) < < 4.5 < (i) 4 < ( ) < < 4.5 < (ii) Again, 4.5 < < ( ) < 4.75 < (iii) From eqn. (i), (ii) and (iii), we have 4 < 4.5 < 4.5 < 4.75 < 4.5 Thus, required rational numbers between 4 and 4.5 are 4.5, 4.5 and Math Class IX 6 Question Bank

7 (iv) The given numbers are As 7 < 7 and i.e., and. 7 7 < + < 7 8 < < < < 7 < <...(i) Again, 7 7 < + < 7 8 < <...(ii) Again, < 0 < + < < < 0 < <...(iii) From eqn. (i), (ii) and (iii), we get < < < <. Thus required rational numbers between and i.e., 7 and are 8, and 0. Q.8. Find the decimal representation of 7 and. Deduce from the decimal 7 representation of, without actual calculation, the decimal representation of, 4, and Math Class IX 7 Question Bank

8 Ans. Decimal representation of 7 using long division method Thus decimal representation of Decimal representation of Decimal representation of Decimal representation of Decimal representation of Decimal representation of Q.9. State, whether the following numbers are rational or irrational: (i) ( + ) (ii) ( )( 5 5 ) Ans. (i) ( + ) Hence, it is an irrational number. (iii) ( ) ( ) ( ) ( ) [Using ( a + b)( a b) a b ] Hence, it is a rational number. Q.0. Given 4 set { 6, 5, 4,,, 0,,,, 8,.0,π, 8.47} From the given set find: (i) Set of rational numbers (ii) Set of irrational numbers (iii) Set of integers (iv) Set of non-negative integers Ans. The given universal set is 4 { 6, 5, 4,,, 0,,,, 8,.0, π, 8.47} universal Math Class IX 8 Question Bank

9 (i) Set of rational numbers 4 { 6, 5, 4,,, 0,,,,.0, 8.47} (ii) Set of irrational numbers { 8, π} (iii) Set of integers { 6, 4, 0, } (iv) Set of non-negative integers {0, } Q.. Use division method to show that and 5 are irrational numbers. Ans It is non-terminating and non-recurring decimals. is an irrational number. It is non terminating and non recurring decimals. 5 is an irrational number. Math Class IX 9 Question Bank

10 Q.4. Show that 5 is not a rational number. p Ans. Let 5 is a rational number and let 5. q Where p and q have no common factor and q 0. Squaring both sides, we get ( ) p p 5 5 p 5q q q p is a multiple of 5 p is also multiple of 5 Let p 5m for some positive integer m....(i) p 5m...(ii) From eqn. (i) and (ii), we get 5q 5m q 5m q is multiple of 5 q is multiple of 5 Thus, p and q both are multiple of 5. This shows that 5 is a common factor of p and q. This contradicts the hypothesis that p and q have no common factor, other than. 5 is not a rational number. Q.. Show that: (i) ( + 7 ) is an irrational number (ii) ( + 5 ) is an irrational number. Ans. (i) Let ( + 7 ) is a rational number. Then square of given number i.e., ( + 7 ) is rational. ( + 7 ) is rational ( ) + ( 7 ) ( 0 + ) is rational But, ( 0 + ) being the sum of a rational and irrational is irrational. This contradiction arises by assuming that ( + 7 ) is rational number. Hence, + 7 is an irrational number. Math Class IX 0 Question Bank

11 (ii) Let ( + 5) is a rational number. Then square of given number i.e., ( + 5 ) is rational. ( + 5 ) is rational ( ) + ( 5 ) ( 8 + 5) rational. But, ( 8 + 5) being the sum of a rational and irrational it is irrational. This contradiction arises by assuming that ( + 5 ) is rational. Hence, ( + 5 ) is irrational number. Q.4. Use method of contradiction to show that Ans. (i) Now, Let is a rational number p (where q 0) q Then, ( ) p q p q p q p is divisible by as p is divisible by. Let p r Then p q is divisible by. 9r (On squaring both sides) q 9r q r is an irrational number....(i) r is also divisible by. q is divisible by....(ii) From (i) and (ii), we get p is divisible by. q Math Class IX Question Bank

12 p and q have as their common factor but p q is a rational number i.e. p and q have no common factor. p q is not rational. So is not rational. Hence, is irrational number. Q.5. Insert three irrational numbers between 0 and. Ans. Three irrational numbers between 0 and can be 0 < < < < Q.6. Rationalise the denominator and simplify. Ans. (i) (ii) (i) (iii) (v) (ii) (iv) (vi) Multiplying numerator and denominator by + 5, we get ( ) ( ) { ( a + b)( a b) a b } ( ) ( 5) 5 + Multiplying numerator and denominator by 5, we get ( ) ( ) ( ) ( ) ( ) ( ) 6( 5 ) 6( 5 ) { ( a + b)( a b) a b } ( ) 5 5 Math Class IX Question Bank

13 (iii) 5 Multiplying numerator and denominator by 5 +, we get ( 5 ) ( ) ( ) ( ) ( ) (iv) (v) (7 + 5) ( 5) (7 5) ( + 5) ( + 5) ( 5) () ( 5) Multiplying numerator and denominator by ( 5 + ), we get ( 5 + ) ( 5 + ) ( 5 + ) ( 5 + ) () ( 5 + ) 5 (5 5) Multiplying numerator and denominator by (7 5) (7 5) (5) , we get Math Class IX Question Bank

14 (vi) ( ) Multiplying numerator and denominator by , we get [( 6 + 5) + ] [( 6 + 5) ][ 6 + 5) + ] ( 6 + 5) ( ) Multiplying numerator and denominator by ( ) , we get Q.7. If a + b, + Ans. a + b + find the value of a and b. Multiplying both sides numerator and denominator of L.H.S. by ( ), we get ( ) ( ) + ( + ) ( ) Math Class IX 4 Question Bank

15 + 4 But a + b, + Comparing both sides a and b so a + b Q.8. If + a + b, find the value of a and b. - Ans. + a + b Multiplying numerator and denominator of L.H.S. by +, we get + ( + ) ( + ) ( ) ( + ) ( + ) () ( ) But + a + b, Comparing both sides, a, b 7 7 Q.9. Simplify: Ans so + 6 a + b (i) + + (ii) (iii) (i) + + By rationalising the denominator of each term, we get () () ( ) ( ) Math Class IX 5 Question Bank

16 (ii) By rationalising the denominator of each term, we get ( 6) ( ) ( 6) ( ) ( 6) (iii) By rationalising the denominator of each term, we get ( ) ( ) (5 ) ( ) ( ) + (5 ) Math Class IX 6 Question Bank

17 Q.0. If x Ans. (i) (ii) and y 5 + ; find : x + y + xy 5 (i) x (ii) y (iii) xy (iv) x + y + xy x ( 5 ) x Squaring both sides, we get x (9 4 5) 8+ 6(5) y y Squaring both sides, we get y ( ) (iii) xy (9 4 5) ( ) 8 80 (iv) x + y + xy Q.. Write down the values of: Ans. (i) (ii) (i) (iii) (i) ( 6 ) (iv) (5 + ) ( 5 + 6) ( ) (5 + ) (5) + ( ) + (5) ( ) [using ( a + b) a + b + ab] (iii) ( 6 ) ( 6) + () 6 [using ( a b) a + b ab] (iv) ( 5 + 6) ( 5) + ( 6) [using ( a + b) a + b + ab] Math Class IX 7 Question Bank

18 Q.. Rationalize the denominator of: (i) (iii) (ii) (iv) Ans. (i) + Multiplying numerator and denominator by, we get ( ) + ( ) ( ) ( ) + ( ) (ii) 7 5 Multiplying numerator and denominator by 7 + 5, we get ( 7 + 5) ( 7) ( 5) ( 7) + ( 5) (6 + 5) (iii) 5 Multiplying numerator and denominator by 5 +, we get ( 5 + ) ( 5) ( ) ( 5) + ( ) Math Class IX 8 Question Bank

19 (4 + 5) (iv) Multiplying numerator and denominator by 5 +, we get ( 5 + ) ( 5) ( ) ( 5) + ( ) ( ) Q.. Find the values of a and b in each of the following: + (i) a + b (iii) a b 7 (ii) a 7 + b (iv) a + b (v) a b + 5 Ans. (i) + a + b (vi) a + b 0 5 Multiplying numerator and denominator of L.H.S. by ( + ), we get ( + ) + 4 () + ( ) + {using ( a + b) a + ab + b } But + a + b. So, a + b Comparing both sides we get: a 7 and b 4 Math Class IX 9 Question Bank

20 7 (ii) a 7 + b 7 + Multiplying numerator and denominator of L.H.S. by 7, we get ( 7 ) ( 7) + () 7 {using ( a b) a ab + b } But a 7 + b. So, 4 7 a 7 + b. 7 + Comparing both sides, we get 4 7 a 7 and b 4 a and b (iii) a b Multiplying numerator and denominator of L.H.S. by +, we get ( ) ( ) + + Also a b. So, + a b Comparing both sides, we get a and b Math Class IX 0 Question Bank

21 (iv) 5 + a + b 5 Multiplying numerator and denominator of L.H.S. by 5 +, we get (5 + ) (5) ( ) (5) + ( ) + 5 {using ( a + b) a + ab + b } Also, 5 + a + b. So, a + b Comparing both sides, we get : 4 0 a and b (v) a b + 5 Multiplying numerator and denominator of L.H.S. by 5, we get : ( ) (5 ) But a b. So, a b Comparing both sides, we get : 7 a and b 9 57 Math Class IX Question Bank

22 (vi) a + b 0 5 Multiplying numerator and denominator of L.H.S. by 5 +, we get ( 5) ( ) Also, a + b 0. So, a + b Comparing both sides, we get : 7 7 a and b 7 7 Math Class IX Question Bank