To prove :- root p + root q is an irrational number .
Since in the given proof , we proved that root q is rational , then how root p + root q can be irrtaional .
Please explain ... !! Thank you experts .
(This proof only valids when p and q both are non-perfect square number.)Let root p + root q is rational.Consider 2 co-prime integers a and b such that root p + root q = a/b.=》a/b - root q = root p =》(a/b - root q)^2 = root p^2 = p =》(a^2/b^2 + q - 2a×rootq/b ) = p.=》(a^2/b^2 + q - p) × b/2a = root q =》root q = (a^2/b^2 + q - p) × b/2a.Now, root q is irrational as q is non-perfect square number, but RHS value of root q above is rational.This is a contradiction which has arisen by assuming root p + root q is rational.This contradiction shows the irrationality of root p +root q.(If it would be given that either of p and q is perfect square, then, either of root q and root q will be rational.then, In root p + root q , one term (root p or root q) will be irrational and other will be rational.Then, we can easily prove its irrationality as the sum of a rational and irrational is always irational.)