If you're having math problems, I feel bad for you, son...

[Joseph8th]

27. Wellll....

... "So pi has this divine transcendent existence, but where is that? Where does it exist?"

That's not at all what I'm saying even sort of. That's why the distinction between empirical and logical is necessary, and is reflected in math depts everywhere: applied vs pure math. But there's no such thing as the "decimal numbers", as you put it. That's one way to represent a number. There are ways to represent every number, but the most convenient is set theory via countable sets with algebraic extensions.

What I was saying is that we can construct a circle, and measure it's circumference and diameter and so get an approximation of pi, or throw darts and get an approximation, and both of these is empirical (so it exists 'in nature' not as a real thing, but as a property of relationships). That's one way to go about it. But we can consider a pure circle, as well (not drawn imperfectly or susceptible to imperfect measurement) and by that alone mathematically (logically) deduce pi. It is exactly precise, because it is never evaluated.

To get to pi itself, just assume it. Then any bijective function you plug pi into will treat it as a unique point on the real number line, somewhere sandwiched between (always) an infinite number of other irrationals and an infinite number of rationals. If you decided to play Zeno's game, you can play it forever with any point in the reals, including the point pi, by just assuming as much (since one point is the same as any other).

So there's not just nothing special about pi, there's infinitely nothing special about pi. Like I said in another post, there's only a couple named irrational numbers: pi, e, and phi -- and at least two of these have ugly spiritual mumbo-jumbo attached to them by believer types, going back to Pythagoras. The only reason they have names is that they're important to geometers (and from there to physicists, who use lots of geometry) because they appear in nature. So do other relationships that have no names, but are maybe rational numbers. Is C god-magic? What about these two numbers, which I just named for the first time, each of which is equally irrational and transcendent:

Plurp = pi + phi + e
Blurp = pi * phi * e

They have inverses! Plurp - Plurp = 0 and Blurp/Blurp = 1! That's just a few. We could permute these three elements in linear combinations with various operations and bijective funtions to get to any other point in the real number line. So pi's not special. Just named.

If the reals don't make sense, that's probably a good sign. They don't make sense, because as you mentioned we like smooth curves. The deal is Euclid (the ancient Greek geometry) didn't define point, line or plane. These are called undefined objects. We get to just assume them, we name them and label them, then we manipulate them in various ways, these things we think of as numbers.

Not just for science. Lot's of times just for the sheer beauty of it. Number Theory is called the "Queen of Mathematics", and there is no Nobel Prize for us. Fame and glory, baby. Fame and glory.

-- edit --
I take it back. The exponential map (e) is special b/c it's fundamental to both trig and calculus and a host of other areas. For instance your most basic differential equation:

y' = -y

To get the general solution y(t) do

dy/y = -1dt

then integrate both sides wrt to time t

log y = -t + c

We want y(t) so apply exp function to both sides to get

y(t) = e^{-t}.

The solutions y(t) --> 0 exponentially fast as t --> infinity.