How can you prove that the square root of two is transcendental?

Answer 1

You can't, because it isn't. The square root of 2 is irrational, but that doesn't make it transcendental. The square root of any positive integer is ALGEBRAIC - and transcendental means "not algebraic".In this case, the square root of 2 is a root of the polynomial equation x squared - 2 = 0; therefore it is algebraic.

Answer 2

You cannot because it is not. The square root of 2 is the solution to the polynomial equation x^2 - 2 = 0 and so it is a plain irrational number, not a transcendental.