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?
 A: This was already mentioned by Rahul but I think it deserves an answer in its own right.  Digital signal processing of 1d (sound) and 2d (images) real data would take incredible amounts of time and would be much harder to understand if it weren't for the discrete Fourier transform and its fast implementations. This field is very real and complex numbers play a major role in it.
A: The trouble with maths is that, just like in the case of a living organism, all its various apparently unrelated parts are in reality interconnected. For instance, Ramanujan's prime-counting function, belonging to the field of number theory, turned out to be ultimately wrong because, in a veiled or hidden manner, it was equivalent to saying that the Riemann zeta function does not possess any complex zeroes: which, as it happens, is false. He thought that it would always predict the exact number of primes lesser than a given number, and that any error, were it to even exist, would be at worst bounded. Turns out he was wrong on both counts. Which, of course, does not mean that it cannot be used as a very good approximation, but the precision and certainty for which he was aiming proved in the end to be untouchable. And that's just one random example among many about the surprising way in which the various fields of math eventually reveal themselves to be tied together. Hope this helps.
A: One basic example is with eigenvalues and eigenvectors of matrices. Often real matrices are not diagonalisable over $\mathbb{R}$ because they have imaginary eigenvalues, wnad knowing things about these eigenvalues can tell us a lot about the transformation that the matrix represents. The obvious example is the $2D$ rotation matrix $\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta &\cos\theta \end{pmatrix} $ with eigenvalues $e^{\pm i\theta}$ which tell us the angle of rotation that this real matrix gives us. Admittedly a simple example but I'm sure there are plenty more.
On other result that comes to mind is in quantum mechanics! A big area of science right now, it deals with complex wave functions like you wouldn't believe (or maybe you would, it seems like you've done enough maths to have taken a course or two in quantum mechanics!) A lot of problems have complex solutions, and certainly the relation of $e^{i\theta}$ and trig is used to no end, particularly in solving second order differential equations (which the Schrödinger equation frequently reduces to).  
Probably the biggest way that the complex results are translated back to the real world is that the probability of finding a wavefunction in a given region is the integral over that region if it's magnitude squared. The complex wave function is reduced to a real integral to give us a probability, which is certainly a real world result! 
A lot of interesting solutions, known as steady stationary states of the Schrödinger equation, give us wavefunctions where the time dependence looks like $e^{\frac{iE_nt}{\hbar}}$. Here $E_n$ is the energy of the state and $\hbar$ is Planck's (reduced) constant. The point is, the magnitude of these solutions is independent of time. This means that if a particle has this wavefunction, then we know exactly what it's energy is for all time. Further, since the Schrödinger equation is linear, we can superpose solutions to get more solutions, and in fact these steady states form a basis, so we can find the wavefunction for any particle as a combination of these stationary states.
A: I find the use of complex numbers extremely helpful in problems of plane elementary geometry, in particular when there are symmetries present which have to be exploited. 
In the "complex coordinate" $z$ of a point both real coordinates are encoded, you have the full vector algebra of the plane at your disposal, rotations about angles like $90^\circ$ or $120^\circ$ are obtained essentially for free, and on, and on.
A: 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.
