This post is not directly related to Mathematica. Instead it is about the number π. This is because this week contains the best approximation to π you can get in this millennium (at least if you are using american notation): 3/14/15.
As you have surely learned in school, π is roughly
Here I want to talk about a nice relation to π with the complex numbers. And how you can extract a bunch of trigonometry stuff from it, if you now complex exponentiation.
Very short definition of the complex numbers
A complex number
z can be treated as an ordered pair of two real numbers:
z = [z1, z2]
Two such numbers
z = [z1,z2] and
y = [y1,y2] can be added by adding there components:
z + y = [z1 + y1, z2 + y2]
and multiplied by this strange formula:
z * y = [z1 * y1 - z2 * y2, z1 * y2 + z2 * y1]
Basically we treat the second component as a factor for the strange number
i whose square is defined to be -1.
So what is
e to the power
z is a complex number? Using the Taylor expansion, the e function can be written as
Exp[x_] := 1 + Sum[1/j! * x^j, j]
In this expansion we are only using addition and multiplication. Both are defined for complex numbers, so we use the Taylor expansion as a definition for the complex exponentiation.
Coming to π
If use the Taylor expansion of the
Sin functions, we can verify that
Exp[[x1, x2]] = Exp[x1] * [Cos[x2], Sin[x2]]
Exp[[0, Pi / 2]] = [0, 1]
So we can define π as the double of the second entry of the inverse function:
Log[[0, 1]][] * 2 = Pi
This is nice, because we found π solely by using the complex exponentiation and its inverse: the logarithm. No trigonometry needed (only to find the right value).
So good luck π, for the next millennia.
You can read more about π in Petr Beckmanns A History of Pi.