TITLE: How does $e^{\pi i}$ equal $-1$
QUESTION [15 upvotes]: I was on xkcd a while back and there was the equation $e^{\pi i}$ which somehow miraculously equals $-1$. So, I put it into Google and it works. So, I tried solving it on my own:
$\begin{align}
e^{\pi i} &= -1 \\
\ln e^{\pi i} &= \ln (-1) \\
\pi i \times \ln e &= \ln (-1) \\
\pi i \times 1 &= \ln (-1) \\
\pi i &= \ln (-1)
\end{align}$
But that's as far as I can get as $\ln (-1)$ gives Error on my calculator. So how does $e^{\pi i} = -1$ work?

REPLY [6 votes]: $\pi$ is 180 degrees.   
-1 is 180 degrees away from 1, around the complex plane's origin.
$e^{0i}$ is 1.  
$e^{\theta i}$ is a point on the unity circle around the complex plane origin determined by the angle $\theta$, in radians. So if $\theta =\pi$, we get -1.
Complex exponentiation with other bases is also a rotation around the unity circle. For instance $5^{\phi i}$ also traces out a circle as we vary $\phi$. However, for such other bases, $\phi$, though an angle, isn't an angle measured in radians, and so $5^{\pi i}$ is not -1.
The correspondence between radians and the coefficient is only in the case of $e$.
$e$ is a special number which is inherently connected to the special number $\pi$ in multiple ways.