A few Pi questions

So I've been reading a book on the history of math(s). It says that Pi is transcendental.

What I understand by this is that pi cannot be represented algebraically.

Now in the proof that they gave for this was squaring the circle which to me seemed like an immensely circular argument.

And if pi is transcendental why do we have this equation for it?
http://upload.wikimedia.org/math/3/5/5/3554907526d7a18c2c48124688827272.png

While I'm at it the proof by contradiction for root two's irrationality seems a bit convenient. It's like if they didn't set up the part about a/b being in it's simplest form the whole proof couldn't be done.

Anyway, TEAR MY ARGUMENT TO PIECES AND ENLIGHTEN ME!

{ a_n ∪ S }

Series aren't algebraic.

I'd just like to say that the fact that the circle cannot be squared does not mean that it can't be represented algebraically.

For example, you can't construct the cube root of 2 with a straight line and a compass, and you can prove this with a little Galois theory, but the cube root of 2 is certainly algebraic. Using a straight line and compass basically allows you to extend your number system by repeatedly adding in the roots of a quadratic equation, you'll never get most algebraic numbers.

The fact that pi is not algebraic does, however, imply that you can't square the circle. The reverse implication does not hold though.

Also, for the square root of 2 being irrational part, the whole part of a/b being in it's simplest form is just a way to simplify the proof to avoid using induction or the well ordering principle (which is what you'd have to use to show that any rational a/b can be written in simplest form). In any case, unless you doubt that all rational numbers a/b have a simplest form, it isn't anything to worry about.

pi = 3.2 exactly

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