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