The Irrationality of 2

Question

I am sorry it is not 'research level'. A quick answer will do. When I attempt using the Square root of 2 method to prove the rationality of Square root of 4 according to how it was done in a book, 2 became Irrational. I think it has to do with primality but all the proofs I have seen didn't mention that. Is there something I am missing?

Answer 1

The technique to show that $\sqrt{2}$ is irrational is as follows:

Suppose that $\sqrt{2} = \frac a b$, and square to get $2b^2 = a^2$. Conclude that $2 | a^2$, so therefore, $2$ must be a divisor of $a$ (question for the reader: Why does $2 | a^2 \implies 2 | a$?). Proceed to show that $2 | b$, a contradiction.

Now when we try the same technique with $\sqrt{4}$, we conclude that $4 b^2 = a^2$, so $4 | a^2$. Now, however, this does not imply that $4 | a$, so the proof breaks down. (An easy example is $a = 2$).

This is similar to what Prism stated in the comments.