Question

Prove that the product of a nonzero rational number together with an irrational number is an irrational number.

Answer 1

Each time they assume the sum is rational; however, upon rearranging the terms of their equation, they get a contradiction (that an irrational number is equal to a rational number). Since the assumption that the sum of a rational and irrational number is rationalleads to a contradiction, the sum must be irrational.