Insert a rational number between 5 9 and 7 13 and arrange in ascending order.

To find rational number between 5 9 and 7 13 , add the numbers and divide by 2 5 9 + 7 13 2 = 65 + 63 117 2 = 128 117 × 2 = 64 117 Now 7 13 < 64 117 < 5 9 It is in increasing order

insert four rational numbers between 4 5 and 5 6

LCM of 5 , 6 = 30 s o 4 5 = 4 × 6 5 × 6 = 24 30 = 48 60 = 72 90 = 96 120 = 120 150 , 5 6 = 5 × 5 6 × 5 = 25 30 = 50 60 = 75 90 = 100 120 = 125 150 Therefore four rational numbers between 4 5 and 5 6 are 121 150 , 122 150 , 123 150 , 124 150

Prove that the following number is irrational 8

Suppose 8 is a rational number let 8 = p q , where p and q are integers , q > 0 and p , q have no common factor Squaring both sides we get , 8 = p 2 q 2 ⇒ p 2 = 8 q 2 therefore 8 q 2 is divisible by 8 p 2 is also divisible by 8 ⇒ p is divisible by 8 let p = 8 k where k is an integer, squaring both sides , p 2 = 64 k 2 , 64 k 2 = 8 q 2 therefore 64 k 2 is divisible by 8 p 2 is divisible by 8 means p is divisible by 8 hence p , q are both divisible by 8 so our supposition is wrong and 8 is an irrational number