Prove that the product α β and the quotient α β of a rational number α ≠ 0 and an irrational number β is an irrational number.

Given that Rational number α ≠ 0 and an Irrational number β . Let α β = p q ; q ≠ 0 Here p , q are integers. Let α = m n ; m , n are integers. ⇒ m n β = p q ⇒ β = p n m q p n , m q are integers. ⇒ β is rational number So our assumption is wrong. ⇒ α β is irrational. Let α β = p q ; q ≠ 0 Here p , q are integers. Let α = m n ; m , n are integers. ⇒ m n β = p q ⇒ β = m q p n p n , m q are integers. ⇒ β is rational number So our assumption is wrong. ⇒ α β is irrational.

Find all rational values of x at which y = x 2 + x + 3 is a rational number.

Given that y = x 2 + x + 3 x 2 + 2 . x . 1 2 + 3 = x 2 + 2 1 2 x + 1 2 2 + 3 - 1 2 2 = x + 1 2 2 + 11 2 2 y = x + 1 2 2 + 11 2 2 To get rational number x + 1 2 2 + 11 2 2 must be a perfect square. It will be a perfect square if x + 1 2 2 = 5 4 , 14 4 , 25 4 , - - - - So there will be infinitely many rational values of x which gives y as rational number.

Prove that 2 is an irrational number.

We need to prove, 2 is an irrational number. ∴ 2 can be expressed in the form p q , where p and q are both integers and co-primes, i.e., There is no common factor in p and q except 1 q ≠ 0 . ∴ 2 = p q ⇒ p = 2 q ⇒ p 2 = 2 q 2 ⇒ p 2 is an even number ⇒ p is an even number. So, let p = 2 m , where m is an integer ⇒ p 2 = 2 m 2 ⇒ 2 q 2 = 4 m 2 ∵ p 2 = 2 q 2 ⇒ q 2 = 2 m 2 ⇒ q 2 is an even number ⇒ q is an even number. Thus, p and q are both even numbers. So, there exists 2 as the common factor of both p and q . But, we have let that there is no common factor in p and q , except 1 . This is a contradiction. ∴ 2 is not a rational number. Hence, 2 is an irrational number.

Prove that 3 + 2 is an irrational number.

Let us assume that 3 + 2 is a rational number. Now, let 3 + 2 = a b , where a , b are co-prime numbers and b ≠ 0 ⇒ 2 = a b - 3 = a - 3 b b Since a , b are integers, then a - 3 b b is also a rational number ⇒ 2 is a rational number But, we know that, 2 is an irrational number. This shows our initial assumption was incorrect. Therefore, 3 + 2 is an irrational number.