No, that would allow pi to be any number. 0/0 ain’t 0, or anything else.
oops.
I’d never thought of the degenerate case - does that mean that the rule should be, “the ratio of the circumference to the diameter of any circle is equal to pi, providing the radius is greater than zero” ? Or should we not consider a point a circle of radius zero? Or is this a proof that in fact by letting radius -> 0, 0/0 = pi ?? (Juuust kidding on the last part!)
Here’s Dr. Math’s take on an attempt to use Euler’s formula to prove that pi = 0.
Probably best to do it as the limit as r goes to 0.
ultrafilter, your proof that 0 is not equal to -1 is an impressive bit of mathematical tomfoolery. I’m still not sold on your implicit assumption that if something is an element of a set, then it is in that set (kidding!), but I’ll let it slide for now. 
I think we need a definition of pi that does not depend on Calculus. How is pi defined nowadays?
The “proof” beginning with e[sup]2pi[/sup] = 1 is equivalent to this “proof”:
sin(2p) = 0
sin(0) = 0
sin(2p) = sin(0)
Take the arcsin of both sides:
2p = 0
Divide by 2:
p = 0
It’s the ratio of the circumference of a circle to its diameter, same as it always has been. Not sure I understand why you don’t like the calculus of it.
Because now, just about everything is defined in terms of set theory. I would think that something as straightforward as pi would have gotten there too, but of course you’re right, it hasn’t.
Well, keep in mind that real numbers are defined in terms of set theory. Once you’ve got a set-theoretic entity that has all the same properties as the reals, every theorem that applies to the reals applies to these entities as well.
IOW, [symbol]p[/symbol] is a Dedekind cut, but it’s the one that corresponds to a particular notion from calculus.
Well yes of course it all goes back to sets eventually. But I was hoping you knew what I meant. Something more direct.
To define pi in the modern formal spirit, you can define sin in terms of power series
sin(x) = x - x^3/3! + x^5/5! - . . .
and then define pi to be the smallest root of sin in the positive real numbers.
… but be aware that even the smallest root of sin can make baby Jesus cry.
<unabashed hijack since the question has already been very well answered>If [symbol]p[/symbol] is defined as the ratio of the circumference of a circle to its diameter, is there any simple geometrical interpretation of e ? Why do I think this has something to do with the Golden Section or logarithmic spirals? </>
A simple interpretation? I doubt it. Maybe you could come up with a contrived one. The fact is that pi is the number of geometry, and e is the number of calculus.
bwhahaha
I think e is defined as:
the limit of (1+1/n)^n
but what the hell do I know
That is one way of getting e, but that’s not the definition. I think there are a couple ways you could define e:
- The value of x such that the integral of 1/t from 1 to x is equal to 1.
OR
- The sum over n from zero to infinity of x[sup]n[/sup]/n! evaluated at x = 1.
I think the second definition would be preferred these days.
Is there some reason you don’t say it like this:
- The sum over n from zero to infinity of 1/n!
The whole bit with x seems like an unnecessary complication. LifeWillFall’s definition of e is the only one I’ve ever seen (for instance, equation 1 on this MathWorld page). However, it’s probably better to define the function exp(x) differently, and then show that it’s equivalent to e[sup]x[/sup].
I wasn’t thinking clearly. Let’s use the simpler form instead.