Suppose we let:

Then:

Therefore:

Now we take the real parts:

Using throughout:

We know that:

This is clearly irrational, therefore:

If was rational, would also be rational, which is a contradiction.

Finally:

Using

We get:

If was rational, then would also be rational, and if was rational, then the statement above would also be rational, we have thus arrived at a contradiction:

is irrational.

I have to go to class someone else do or I'll do it later.