Imaginary Numbers

Why are there Imaginary Numbers? They have no purpose, they cannot be measured, and yet sometimes I find them in a problem that has to do with measuring something. Therefore creating a no answer situation, which can be maddening. I understand that we need negitive numbers, so if someone is in debt they can see how deep they are in. But Imaginary Numbers? I mean come one! The whole idea of it is pure stupidity. Someone give me a logical reason for Imaginary Numbers, please!

They are quite useful, actually.

They make solving linear systems of differential equations much easier.  RLC circuits are generally solved using complex analysis (complex impedances, etc).

Have fun integrating cos(ax)/(x^2 + 1) from 0 -> infinity without complex numbers.  If you know about residues, this is easy.

They also makes fourier transforms more compact, and less messy.
They also allows all polynomials to have at least 1 root.