Write any two irrational numbers between 4 9 and 7 11 .

We have to write any two irrational numbers between 4 9 and 7 11 . The decimal expansions of the given numbers are: 4 9 = 0 . 4444 . . . . = 0 . 44 4 7 11 = 0 . 636363 . . . = 0 . 63 63 Here, the one decimal expansions are non-terminating recurring, while the other is terminating. An irrational number has non-terminating non-recurring decimal expansions. To find two irrational numbers between 4 9 and 7 11 , we find three decimal numbers which are non-terminating non-recurring and lying between them i.e. between 0 . 44 4 and 0 . 63 63 . There are infinitely many irrational numbers. Two such numbers are 0 . 4040040004 . . . and 0 . 6060060006 . . . . Therefore, the two irrational number between 4 9 and 7 11 are 0 . 4040040004 . . . and 0 . 6060060006 . . .

Find a rational number between 2 and 3 .

We have to find the rational number between 2 and 3 . Consider the squares of 2 and 3 . 2 2 = 2 and 3 2 = 3 For any two rational numbers, their average or mid-point is rational. Proceeding similarly, we can generate infinitely many rational numbers. Take any rational number between 2 and 3 , which is perfect square of a rational number. One such number is 2 . 25 , and 2 . 25 = 1 . 5 2 i.e. 2 . 25 = 1 . 5 . As 2 < 2 . 25 < 3 ⇒ 2 < 2 . 25 < 3 ⇒ 2 < 1 . 5 < 3 Hence, 1 . 5 is a rational number between 2 and 3 .

Find two rational numbers between 2 3 and 15 . Hint: 2 3 = 12 ; 12 < 12 . 25 > 12 . 96 < 15 ⇒ 12 < 12 . 25 < 12 . 96 < 15

We have to find the rational number between 2 3 and 15 . Consider the squares of 2 3 and 15 . 2 3 2 = 4 × 3 = 12 and 15 2 = 15 For any two rational numbers, their average or mid-point is rational. Proceeding similarly, we can generate infinitely many rational numbers. Take any rational number between 12 and 15 , which is perfect square of a rational number. Two such numbers are 12 . 25 and 12 . 96 . 12 . 25 = 3 . 5 2 i.e. 12 . 25 = 3 . 5 . 12 . 96 = 3 . 6 2 i.e. 12 . 96 = 3 . 6 As 12 < 3 . 5 < 3 . 6 < 15 ⇒ 12 < 3 . 5 < 3 . 6 < 15 ⇒ 2 3 < 3 . 5 < 3 . 6 < 15 Hence, 3 . 5 and 3 . 6 are the rational numbers between 2 3 and 15 .

Insert an irrational number between 5 and 7 .

We have to write two irrational numbers between 5 and 7 . Consider the squares of 5 and 7 . 5 2 = 5 and 7 2 = 7 As 5 < 6 < 7 , it follows that 5 < 6 < 7 . Therefore, 6 lie between 5 and 7 . Hence, an irrational number between 5 and 7 is 6 . Note: [Since infinitely many irrational numbers lie between two distinct irrational numbers, 6 is not the only irrational numbers between 5 and 7 .]