But then it turns out that that definition is not rigorous:

Since we don't have a rigorous definition of a real number it is hard to justify that pi is a real number. Let's discuss what it means to 'represent a quantity'. Say I were to ask you what is the quantity of x, ultimately, you would have to give me an answer which is a complete sentence with a period at the end, in other words, you're explanation of the quantity would have to end. You would then indicate where on the number line the quantity exists. You can't indicate where pi exists on the number line. If you say that it is between 3.14 and 3.15 then you haven't told me where it exists exactly you have only told where it exists roughly. This could go on ad infinitum. Hence, you can't "represent the quantity" of pi.

Admittedly, though in order to give a real rigorous proof that pi is not real, we would need a rigorous definition of real number, something much more precise then represent the quantity and we don't have that yet.

Pi is real, just irrational as well- because you can graph it, it's real- like try graphing i (or the square root of -1), that's when numbers get imaginary-
Oh imaginary numbers.... Gotta love math sometimes :D

This is nonsense. We have perfectly rigorous definitions of real numbers. Axiomatically they are the unique Dedekind-complete ordered field and constructions of the real numbers via Dedekind cuts or Cauchy sequences have been known for over a century. A rigorous definition for π is also easy to give. For example:
[tex]
\pi = 2\int_{-1}^1 \sqrt{1-x^2} \; \mathrm{d}x
[/tex]
So I am confused where you are getting all these wild ideas from since everything I just mentioned is very standard stuff.

I would be interested to know if you could give me a perfectly rigorous definition of perfectly rigorous?

If you cannot, then I'll give you mine and you can analyze my definition. A perfectly rigorous definition starts with a group of indefinable words. The axioms tell us the true and false combination of these indefinables. Every definition is built up from these indefinables. A perfectly rigorous (aka decidedable) sentence is then justified when it can be broken down into a group of sentences which are all composed of indefinables and each sentence has already been declared true but the axioms. For example, say you start with the indefinable words:

x, y, z, R, S, T, &

Then you have a new sentence: gh FG ui

You break that sentence down into:

xRy & zSx & yTx

Those three sentences are already declared true by the axioms. So the new sentence gh FG ui is a perfectly rigorous definition. I seriously doubt there is such a perfect definition as to what Dedkind cuts are. I could be wrong and I'm willing to listen.

You can't put pi on a number line. If you put it between 3.14 and 3.15, that's wrong. So then you try to be more precise and you put it between 3.1415 and 3.1416 that's still wrong. No matter how much you increase your precision, you're still wrong.

You're never gonna be perfect, but with 5 trillion digits, I think you are going to be fine in a practical sense- at that point you start to run out of molecules in the paper to be so precise!

Weird way of formulating things, but yes, you can give a definition of what it means to be perfectly rigorous. Just take any logic book and work through it, you'll see soon enough what mathematicians call perfectly rigorous.

Basically, you are given some collection of well-formed formula's, called axioms. And you are also given inference rules which allow you to go from one well-formed formula to another. A perfectly rigorous proof is now a list of well-formed formulas. Every well-formed formula follows from a previous one by either an axiom or an inference rule.

Giving definitions of real numbers and giving definitions of pi is then perfectly possible.

Also, claiming that pi is not a real number counts as crackpottery and is not allowed on this forum.

Applied math is about what's true for getting something practical accomplished. We want to know what's true in the abstract. You have yet to prove that pi can be represented on a line.

Who knows? EVENTUALLY we may hit an end... But at 5 trillion digits I wouldn't hold your breath... Regardless of its precise location on a number line, it's irrational, but real