Can someone explain this to me?

[Daisydog]

Why is it that e^(i*pi) = -1? I know how to prove it mathematically but I don't understand what it really means. Is there some way to visualize this or intuitively understand why it ought to be true? Thanks.

[uDevil]

Maybe this will help: e^(i*Some_Angle) is a point on the unit circle in the complex plane. So it's easy to see
<font class="small">Code:</font><hr /><pre>
Some_Angle e^(i*Some_Angle)
0 1
pi/2 0
pi -1
-pi -1
3pi/2 0</pre><hr />

[bigpooch]

e^(iz) = cos (z) + i sin (z) is a periodic function on the
complex plane. This function maps the reals to the unit
circle on the complex plane, so when z = pi (radians), the
result e^(i*pi) = -1.

Also, note that you can check that the first equation is
the case by just looking at the three Taylor series about
the origin. You can extend the definitions of sin z, cos z
and e^z to the domain of complex numbers and it turns out
that all of these functions are analytic ("everywhere
differentiable on the complex plane").