To properly handle the mathematics involved in multiplying and dividing negative numbers you need a two dimensional vector system that is made up of of a real and imaginary component.
The reason this second axis is called imaginary is due to the fact that it was created solely for the sake of multiplying and dividing negative numbers.
If you desire to learn more about the subject here is a good place to start. Any advanced engineer would be lost or incompetent without a thorough understanding of complex numbers (i.e. i^2 = -1).
http://www.cs.bham.ac.uk/~ibs/quantum/complex.pdf