If you can express pi as a ratio of circumference and diameter, how can I justify that pi is irrational?
I don't know if this sounds like a crazy man talking, but I think it has to do with the fact that a circle is a regular polygon(I hope the terminology is correct) with an infinite number of sides. This makes the circumference being a never-ending irrational number, and when we divide this number by the diametre we get pi, which also will be an irrational number. Don't trust me completely on this one though, as I'm not sure about it. I think it's right though.
Using Google you will find many proofs that pi is irrational:
Proof that
Niven : A simple proof that $\pi$ is irrational
(Proving that it's transcendental is harder).
yes it's true that you can express pi as the ratio of the circle's area and diamter, but the area and diameter are not completely accurate. In the end, in order to solve for the three variables...we need to know at least two. and the third variable is only as accurate as the other two.