But from this it would follow that both and are rational, which is not the case.
Whether the square root of a natural number is rational or not is determined by wether or not its (essentially) unique prime factor decomposition consists only of even powers of prime numbers or not. In the former case its square root is a natural number, in the latter case its square root is irrational. So are all irrational, whereas are natural numbers.Out of curiosity, why is Sqrt connected to Sqrt + Sqrt other than 15=3x5 ? What is the connection?