Prove thatsquare root of 2 is irrational?

Answer 1

You can find several such proofs in the Wikipedia article "Square root of 2". Many of these proofs (or perhaps all, but I didn't check carefully) apply to the square root of ANY positive integer, assuming the integer is not a perfect square.

Answer 2

The proof is by the method of reductio ad absurdum. We start by assuming that sqrt(2) is rational. That means that it can be expressed in the form p/q where p and q are co-prime integers. Thus sqrt(2) = p/q. This can be simplified to 2*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, 2*q^2 = (2*r)^2 = 4*r^2 Dividing both sides by 2 gives q^2 = 2*r^2. But now 2 divides the RHS so it must divide the LHS. That is, 2 must divide q^2 and 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(2) cannot be rational.