Is the square root of 6 an irrational number? - Answers

Yes it is. The proof is as follows:We prove the statement by contradiction i.e. Assume that sqrt(6) is a rational number.Then there exist positive integers p and q with gcd(p,q) = 1 such that p/q = sqrt(6).Square both sides: p^2 / q^2 = 6,p^2 = 6q^2.Now as 2 is a divisor of the right-hand side (RHS), it implies that 2 is also a divisor of the left-hand side (LHS).This is only possible if 2 is a factor of p.Let p =2k. Then k is a positive integer as well.Thus, 4k^2 = 6q^2,2k^2 = 3q^2.As 2 is a factor of the LHS, 2 is also a factor of the RHS.But this is only possible if 2 is a factor of q.=> gcd(p,q) >= 2. Contradiction!Thus sqrt(6) is irrational.yes it is