>>11428210>c/D of a circleThat begs the question. How do you define c? The lengths of straight line segments are easy enough, same with polygons, but how do you define the length of a curve which is not a line segment nor made up of line segments?
>Your definition of pi assumes pi = 3.14It actually doesn't. It might seem that way because I used the function sinx, so let me rephrase my definition. I assume we can both agree that you can parametrize the circle by some continuous function f(t):[0,1]->R^2. I define the length of such a curve by first subdividing the interval [0,1] into points t_0 = 0 < t_1 < ... < t_n = 1. Then I compute length of the polygonal approximating curve of f (which we can all agree on) by the formula
. Intuitively, the more points I have in the subdivision and the closer they are together, the better the approximation becomes. For some curves, called rectifiable curves, which includes the circle, the length so computed approaches some definite value. I call the length of the curve f(t) to be the least upper bound of the approximating lengths among all possible subdivisions of the interval [0,1]. That's how most mathematicians define length.
Using this definition, you can derive the fact that the length of a circle is an irrational number close to 3.14 that we call pi.
Note that if in this definition you change the formula from
to
you get that the length of the circle is 4. However there are many reasons not to take this this definition of length, one of which is that it gives completely different values of length of regular line segments than the usual metric.
