Originally Posted by
topsquark True but it's usefulness is contained in what's called Euler's identity $\displaystyle e^{i \theta } = cos( \theta ) + i~sin( \theta )$ in conjuction with de Moivre's theorem: $\displaystyle (cos( \theta ) + i~sin( \theta ) )^n = cos( n \theta ) + i~ sin( n \theta )$. This has all sorts of uses in Geometry and (what I know best) Physics where complex numbers are used. Many properties in Physics have some sort of "attenuation" properties and complex numbers can often be used to describe the phenomena. These identities make it easy to introduce such.
-Dan