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?
Best Answer
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.