Prove that 2 7 is an irrational number.

Suppose 2 7 is a rational number. Let 2 7 = r , where, r is a rational number. ⇒ 2 r = 7 Since, r is a rational number, then 2 r is a rational number, then 7 is also a rational number. But we know that 7 is an irrational number. So, our assumption is wrong. Hence, 2 7 is an irrational number.

Prove that 2 7 is an irrational number.

Suppose 2 7 is a rational number. Let 2 7 = r , where, r is a rational number. ⇒ 2 r = 7 Since, r is a rational number, then 2 r is a rational number, then 7 is also a rational number. But we know that 7 is an irrational number. So, our assumption is wrong. Hence, 2 7 is an irrational number.

Prove that following numbers are irrationals: 3 2 5

Suppose 3 2 5 is a rational number. Let 3 2 5 = r , where r is a rational number ⇒ 3 2 r = 5 Since, r is a rational number, then 3 2 r is a rational number, then 5 is a rational number. But we know that 5 is an irrational number. Thus, our assumption is wrong. Hence, 3 2 5 is an irrational number.

Prove the given number as an irrational: 4 + 2

Suppose, 4 + 2 is a rational number. Let, 4 + 2 = r , where r is a rational number. ⇒ 2 = r - 4 Since, r is a rational number, then r - 4 must be a rational number, then 2 is also a rational number. But we know that 2 is an irrational number. So our assumption is wrong. Hence, 4 + 2 is an irrational number.

Prove the following number as an irrational: 5 2

Suppose 5 2 is a rational number. Let 5 2 = r , where r is a rational number. ⇒ 2 = r 5 Since, r is a rational number, then r 5 is a rational number, then 2 is also a rational number. But we know that 2 is an irrational number. So, our assumption is wrong. Hence, 5 2 is an irrational number.

Show that 2 - 3 is an irrational number.

Suppose 2 - 3 is a rational number. i.e., 2 - 3 = r ⇒ 2 - r = 3 Since, r is a rational number, then 2 - r is also a rational number, then 3 is also a rational number. But we know that 3 is an irrational number. So, our assumption is wrong. Hence, 2 - 3 is an irrational number.

Show that 3 + 2 is an irrational number.

Suppose 3 + 2 is a rational number. Let 3 + 2 = r , where r is a rational number. ⇒ 2 = r - 3 Since, r is a rational number, then r - 3 is a rational number, then 2 is a rational number. But we know that 2 is an irrational number. So, our assumption is wrong. Hence, 3 + 2 is an irrational number.

Prove that 4 - 5 2 is an irrational number.

Suppose 4 - 5 2 is a rational number. Let, 4 - 5 2 = r , where r is a rational number. ⇒ 4 - r = 5 2 ⇒ 4 - r 5 = 2 Since r is a rational number, 4 - r 5 is a rational number, then 2 is also a rational number. But, we know that 2 is an irrational number. So, our assumption is wrong. Hence, 4 - 5 2 is an irrational number.

Show that 5 - 2 3 is an irrational number.

Let us assume on the contrary that 5 - 2 3 is a rational number. Then, there exist co-prime positive integers a and b such that, 5 - 2 3 = a b ; b ≠ 0 ⇒ 5 - a b = 2 3 . ⇒ 5 b - a 2 b = 3 . Since, 5 b - a 2 b is a rational number, then 3 is also a rational number. But, we know that 3 is an irrational number. So, our assumption is wrong. Hence, 5 - 2 3 is an irrational number. (proved)

Prove that 2 3 - 1 is an irrational number.

Let us assume on the contrary that 2 3 - 1 is a rational number. Then, there exist co-prime positive integers a and b , such that: 2 3 - 1 = a b ; b ≠ 0 , where a , b are co-prime numbers. ⇒ 2 3 = a b + 1 . ⇒ 3 = a + b 2 b . Since, a + b 2 b is a rational number, then 3 is also a rational number. But we know that, 3 is an irrational number. So, our assumption is wrong. Hence, 2 3 - 1 is an irrational number.