[jordigh]

I didn't mean my attitude to be dismissive. It's just that truly a lot of people respond this way. "I don't understand it at all." And when you ask what don't they understand, they're unable to say it. Witness for example how exacube seems to have vanished and will probably never tell us what she or he did not understand.

I think what happens is that people are so overwhelmed with unfamiliar ideas when they encounter a proof like this that they just grind to a halt, curl up into a ball, and scream how much they hate it all and don't understand a bit of it. We have at least a couple of other people in this thread who have expressed their hatred of calculus. Starting from that it seems pretty hopeless to try to explain to them this proof.

    Yet I find this proof unsatisfying because it doesn't demonstrate
    clearly, to me, which particular properties of pi it is using that
    bring about the contradiction.

Only one: that it's a root of sin(x). The proof actually works for any nonzero root of sin(x).

In fact, that's a great definition of pi: the least positive root of sin. It's a much easier definition to work with than ratio of circumference to diameter (how do you define cirumference? What is length? What is a curve?)

[jameshart]

well sine has to come from lengths of curves, doesn't it? Because the geometric definition of sine is a function from angles to numbers, and to get pi in there you need to introduce radians as a way to measure angles - otherwise I could argue that the roots of sin are 0, 180, 360, etc. - and none of them are particularly irrational.

[jordigh]

No. There's the analytic definition of sin that does not mention curves or lengths at all: the sine is the unique function that satisfies the following differential equation and initial value problem:

    s''(x) + s(x) = 0
    s(0)  = 0
    s'(0) = 1

It's a nifty way to define sine purely by its differential properties. Of course, it requires some work to show that differential equations have a solution and that this particular solution is sine (i.e. has the properties you want a sine to have), but once you do that work, it's pretty easy to prove things such as sin^2(x) + cos^2(x) = 1.

On the other hand, starting from the geometric definitions (and building the framework for that, such as arclength, which really requires calculus), it takes a longer route to get to the calculus of sine. Historically this was the route, but we have found shortcuts since then.

[retbull]

Dude I needed you when I was working through my Calc 2 stuff. I didn't ever see that definition as a route to use. Way nicer than memorizing the unit circle which eventually translated into "knowing" the answers to the derivatives and integrals.

[jameshart]

Interesting - I was sort of wondering if you could get sine just from simple harmonic motion, and that's basically what those equations do - the first is equivalent to saying s''(x) = -s(x), so acceleration is opposite to and linearly proportional to (well, equal to in this simple case) displacement. That's a good point.