Have you seen this argument? If sqrt(2) = p/q is in lowest terms, then also 2/1 = p^2/q^2 is in lowest terms. Since lowest terms is unique, p^2 = 2 and q^2 = 1. Thus sqrt(2) is the integer p. But 1^2 is too small, 2^2 is too big and all the resta re even bigger, so this is false. So sqrt(2) is not rational. does anyone have a shorter one?