How are complex numbers useful to real number mathematics? Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful?
Of course, the obviously applications are:


*

*contour integration

*understand radius of convergence of power series

*algebra with $\exp(ix)$ instead of $\sin(x)$


No need to elaborate on these ones :) I'd be interested in some more suggestions!
In a way this question is asking how to show the advantage of complex numbers for real number mathematics of (scientifc) everyday problems. Ideally these examples should provide a considerable insight and not just reformulation.
EDIT: These examples are the most real world I could come up with. I could imagine an engineer doing work that leads to some real world product in a few months, might need integrals or sine/cosine. Basically I'm looking for a examples that can be shown to a large audience of laymen for the work they already do. Examples like quantum mechanics are hard to justify, because due to many-particle problems QM rarely makes any useful predictions (where experiments aren't needed anyway). Anything closer to application?

I have used complex numbers to solve real life problems:
- Digital Signal Processing, Control Engineering: Z-Transform.
- AC Circuits: Phasors. This is a handful of applications broadly labeled under load-flow studies and resonant frequency devices (with electric devices modeled into resistors, inductors, capacitors at AC steady state).
- Analog Computers and Control Engineering: Laplace Transform.
Not sure if it falls into Complex Numbers, but since it has (x,y) form
- CNC programming: scaling, rotating coordinates.
- Rotating Dynamic Balancers.
... maybe some more but I can't recall.
