"Strong typing vs. strong testing" [OT]

[Peter Nilsson]

Keith Thompson <ks... at mib.org> wrote:
> The radian is defined as a ratio of lengths. That ratio
> is the same regardless of the size of the circle.  The
> choice of 1/(2*pi) of the circumference isn't arbitrary
> at all; there are sound mathematical reasons for it.

Yes, but what is pi then?

> Mathematicians could have chosen to set the full
> circumference to 1, for example, but then a lot of
> computations would contain additional multiplications
> and/or divisions by 2*pi.

The formula: circumference = 2 x pi x radius is taught
in primary schools, yet it's actually a very difficult
formula to prove! Most attempts by students collapse
because they assume the formula in advance (pi is the
ratio of circle semi-circumference to its radius.)

Perhaps the most elegant approach is to define arctan x
as the integral from 0 to x of dz/(1 + z^2). The trig
functions and relationships, and their application to
normal geometry, can all be defined and derived from
that alone without any reference to pi. Pi is simply
the constant 4 times arctan 1.

--
Peter