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.

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Posted on 2009-11-04 by Jach

Tags: math

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