What are complex numbers for?

Despite their name, complex numbers are used in many areas to make it simpler to treat periodic or rotating systems in two dimensions.

In particular, these are used to greatly simplify two theoretical areas of great interest in nuclear engineering, namely quantum mechanics and signal analysis, although they are not strictly necessary in either.

Handling quantum mechanics without complex numbers is so untapped as to be completely unknown even to experienced physics experts.

Resorting to complex numbers is an easy way to encode the algebra underlying the structure of the theory. The use of exponential notation like the one below:

Where r is the modulus and θ is the phase of the complex number, makes it straightforward to calculate the probability associated with each state. Calculating the nuclear properties of matter without complex numbers, by analytical calculations or computer simulations, is such an inconvenient and neglected approach as to be virtually impossible.

Signal analysis and control theory, used to design the electronic and automatic control systems, are also greatly simplified by the use of complex numbers…