I picked up a book by Stephen Abbott called "Understanding Analysis" and it begins talking about rational and irrational numbers then it goes on proving how √2 is irrational. The proof is easy to understand but I wanted to use the same exact proof on a number I knew was rational. Let (p/q)2 be a rational number equal to 4 where p and q have no common factors and q ≠0. (p/q)2 = 4 which implies p2 = 4*q2. From this we can see that p2 is a multiple of 4 and hence so is p. Let p = 4a where a is an integer. 16a2 = 4*q2 , 4a2 = q2 which implies q2 is a multiple of 4 and hence q is also. Question: There for we now know the ratio p/q is not in lowest terms, it has a common factor of 4. So how am i suppose to interpret this? This is the same exact proof that proved √2 is irrational by showing p and q have common factors. Well I just showed you that √4 = p/q where p and q have common factors but √4 = 2 is obviously rational.