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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 '13 at 2:45
    
    
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 Jan 24 '13 at 2:51
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I can still duplicate my comment: have you checked en.wikipedia.org/wiki/Complex_number#Applications ? –  Rahul Jan 24 '13 at 2:52

6 Answers 6

up vote 9 down vote accepted

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 '13 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 '13 at 2:46
    
"one begin the current" . . . I presume "one being the current" was meant. –  Michael Hardy Jan 24 '13 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 '13 at 2:55
    
Fourier transforms are also used in studying time series in statistics, including the stock market and biorhythms. –  Michael Hardy Jan 24 '13 at 2:56

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 '13 at 0:15

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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I was asked this exact question by my wife last night. She was looking for an everyday example of the use of complex numbers to explain to her 8th grade math class (whose knowledge of complex numbers consists of i = SQRT(-1) ).

My response was this:

Imagine an electronic piano. Each key produces a different tone. A volume control changes the amplitude (volume) of all the keys by the same amount. That's how real numbers affect signals.

Now, imagine a filter. It makes some keys sound louder and some keys sound softer, depending on their frequencies. That's complex numbers -- they allow an "extra dimension" of calculation.

(Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for comprehensive testing, they're required to know a real world application of complex numbers, but not the details of how or why. I don't understand this, but that's the way it is)

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8th grade is young for complex numbers. –  LTS Apr 16 at 12:21

Complex analysis (transformation or mapping) is also used when we launch a satellite and here on earth we have $z$-plane but in space we have $w$-plane as well. So to study various factors we use transformation.

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Electrical engineering, fluid dynamics, quantum mechanics, computer graphics, dynamical systems.

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Computer graphics? I know that quaternions are used to represent and compose rotations, but what specific applications of complex numbers are there? (The others you mention could use specifics too: the question asks for applications, not fields having applications). –  Peter Taylor Oct 9 '13 at 7:48

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