How do you prove that the square root of thirty is irrational? - Answers

The proof is by the method of reductio ad absurdum. We start by assuming that sqrt(30) is rational. That means that it can be expressed in the form p/q where p and q are co-prime integers. Thus sqrt(30) = p/q. This can be simplified to 30*q^2 = p^2 Now 2 divides the left hand side (LHS) so it must divide the right hand side (RHS). That is, 2 must divide p^2 and since 2 is a prime, 2 must divide p. That is p = 2*r for some integer r. Then substituting for p gives, 30*q^2 = (2*r)^2 = 4*r^2 Dividing both sides by 2 gives 15*q^2 = 2*r^2. But now 2 divides the RHS so it must divide the LHS. That is, 2 must divide 15*q^2. Since 2 does not divide 15, 2 must divide q^2 and then, since 2 is a prime, 2 must divide q. But then we have 2 dividing p as well as q which contradicts the requirement that p and q are co-prime. The contradiction implies that sqrt(30) cannot be rational.