Below is a proof showing that the sum of a rational number and an irrational number is an irrational number. let a be a rational number and b be an irrational number. assume that a + b = x and that x is rational. then b = x – a = x + (–a). x + (–a) is rational because however, it was stated that b is an irrational number. this is a contradiction. therefore, the assumption that x is rational in the equation a + b = x must be incorrect, and x should be an irrational number. in conclusion, the sum of a rational number and an irrational number is irrational.