# “Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.

So what practical applications of complex numbers exist and what are the ways in which complex transformation helps address the problem that wasn't immediately addressable?

Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.

PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?

The question: Interesting results easily achieved using complex numbers is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO.

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I am pretty sure this has been asked... – Mariano Suárez-Alvarez Jan 24 at 2:45
– Trevor Wilson Jan 24 at 2:48
I would vote to close as a duplicate of this previous question, but that one's written in a confusing and not very pleasant style. – Rahul Narain Jan 24 at 2:51
I can still duplicate my comment: have you checked en.wikipedia.org/wiki/Complex_number#Applications ? – Rahul Narain Jan 24 at 2:52

Complex numbers are used in electrical engineering all the time, because Fourier transforms are used in understanding oscillations that occur both in alternating current and in signals modulated by electromagnetic waves.

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An example would go a long way for those who don't know this :) – PhD Jan 24 at 2:42
Complex numbers in a sense embody certain aspects of trigonometry. Therefore it is not unexpected for them to arise in situations involving trigonometric functions, such as waves and oscillations mentioned by Michael Hardy. A concrete example of their use is in phasors for example. – EuYu Jan 24 at 2:46
"one begin the current" . . . I presume "one being the current" was meant. – Michael Hardy Jan 24 at 2:52
The letter $j$ rather than $i$ is used in electrical engineering for the imaginary unit. The expression $t\mapsto e^{j\omega t}$ occurs frequently. The real and imaginary parts of that are $\cos(\omega t)$ and $\sin(\omega t)$. Frequency is $\omega$ and $t$ is time. – Michael Hardy Jan 24 at 2:55
Fourier transforms are also used in studying time series in statistics, including the stock market and biorhythms. – Michael Hardy Jan 24 at 2:56
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Electrical engineering with signals.

For example

http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf

Regards

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 +1 Excellent example! What would we do without $i$!? – amWhy May 7 at 0:15 @amWhy: I am actually very surprised how complex math has been used to prove some things, when there were thoughts that there was no connection. There is one famous theorem in this regard, bad sadly my memory fails me to as the name. Thx! – Amzoti May 7 at 0:31

Two-dimensional problems involving Laplace's equation (e.g. heat flow, fluid flow, electrostatics) are often solved using complex analysis, in particular conformal mapping.

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