# My favorite equation

[math]e^{i\pi} + 1 = 0[/math]Yup, Euler's famous equation. It's a favorite of a lot of people's, including mine, as it unites arguably the 5 most important numbers in mathematics.

Why is this true, though? It stems from the fact that:

[math]e^{ix} = \cos(x) + i\sin(x)[/math]

I shall do the proof for it (requires Calculus):

[math]

\begin{eqnarray}

let\ z &=& \cos(x) + i\sin(x) \\

\frac{dz}{dx} &=& -\sin(x) + i\cos(x) \\

&=& i(\cos(x) + \sin(x)) ^* \\

&=& iz \\

\frac{dz}{z} &=& idx \\

\int{\frac{1dz}{z}} &=& \int{idx} \\

\ln{z} + C &=& ix + C \\

\ln{z} &=& C + ix \\

\ln{z} &=& ix ^{**} \\

e^{\ln{z}} &=& e^{ix} \\

z &=& e^{ix} \\

\cos(x) + i\sin(x) &=& e^{ix} \\

QED \\

\\

*: -1 = i^2 \\

**: let\ x = 0 \\

\ln{z} = C \\

\ln(\cos(0) + i\sin(0)) = C \\

\ln{1} = C = 0

\end{eqnarray}

[/math]

There's also a proof using Taylor series, but I really like this one. (And yes I'm posting this as an excuse to try out jsMath TeX stuff.)

There exists a neat way to visualize this as well. Visualize the complex plane as a simple 2D vector space, with real and imaginary components to the vector. Multiplying by the vector

*i*rotates you 90 degrees, and since Pi = 180 degrees, you are rotating one half-circle and thus are facing the negative direction (e^(i*pi) = -1) with no imaginary component.

That might not be a great explanation for the visualization, but if you can see it, you can appreciate it. Eventually I'll polish up my CVector class I wrote in Python and stick it up here, because complex numbers are very useful to represent vectors with.

#### Posted on 2009-11-04 by Jach

**Tags:** math

**Permalink:** https://www.thejach.com/view/id/42

**Trackback URL:** https://www.thejach.com/view/2009/11/my_favorite_equation

Jach
November 06, 2009 07:38:43 PM
Oh crap, my beautiful math proof disappeared! That's what I get for running a global \r replacer as it happens to remove \\ (newline in latex and the command character) as well...