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– Cudos to π

Cudos to π

03/14/2015

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

3.141592653589793238462643383279502884197169399375105820974944592307816406

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.

Complex Exponentiation

So what is e to the power z, if 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 Cos and Sin functions, we can verify that

Exp[[x1, x2]] = Exp[x1] * [Cos[x2], Sin[x2]]

And therefore

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]] * 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.


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