Question

prove that the sum of a rational number and an irrational number is always irrational

Answer 1

sum of rational and irrational number is always an Irrational number as
$\sqrt{2} + 1 = \sqrt{2}$
which is an irrational number not rational

Answer 2

1)To prove the sum of rational and irrational number is always irrational

Let 4/2 be rational number in p/q form----------(1)
let root 2 be irrational number---------(2)

Adding (1)and (2)
4/2+root2==>>2+root2

2root 2 is a irrational number