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I've always enjoyed solving problems in the complex numbers 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
I can still duplicate my comment: have you checked ? – Rahul Jan 24 '13 at 2:52
up vote 15 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

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. – bwv869 Apr 16 '14 at 12:21

Electrical engineering with signals, for example:

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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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Since you mentioned "real world".

The "real world" consists of miniscule particles: protons, electrons, etc. Which are not exactly particles: quantum mechanics says each of them looks like a wave. Normal waves have some "value" or "displacement" or "magnitude" in each point of space.

Magnitude (amplitude) of waves in quantum mechanics are complex! Just imagine, the whole "real world", everything you can see or touch consists of some waves with complex amplitudes!

Complex numbers are used in real world literally EVERYWHERE.

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Quantum mechanics is the best answer to this question because unlike the case of "classical" waves, where complex numbers are simply a convenience, in quantum mechanics, they are unavoidable and carry the entire essence of physics. – orion Dec 24 '14 at 17:49

The other answers nicely cover specific examples of alternating current and wave equations. Basically, wherever you encounter an oscillatory phenomenon of any type, complex numbers are a natural tool to describe them easily and efficiently.

I'd like to add a related point here. The relationship between the exponential function and trigonometric functions is transparent when you use complex numbers. Damped and oscillatory motion are two sides of the same coin: they are solutions of the same (differential) equations with slightly different parameters. Varying a parameter can switch between oscillation and damping, which is related to when solutions of a quadratic equation turn from real to complex. With this example, students can "feel" the emergence of imaginary component when something starts to resonate instead of just fading out. It makes for a nice demonstration in a classroom. It helps to convince that complex numbers are not some made-up constructs but a part of nature just as reals, and make up a much more coherent theory with nicer rules and less exceptions compared to real arithmetics.

Another more dry and technical use is in equation solving in general. For instance, solving for real roots of a real polynomial can be done through complex arithmetics (with complex intermediate results). This still begs a question, where in real life you need to solve a cubic equation (as an example) but that's another story.

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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.

Well that's about what it is. It's just that the applications for complex numbers gets simpler and sometimes more elegant using them. But in fact they are not required, you could do the same thing without them.

You've got answers about electrical engineering, in which the use of complex numbers originates from the same source as in Fourier analysis. Here it's just the relationship between exponential and trigonometrics that's the reason. I'd say that Fourier analysis becomes more elegant using complex numbers and electrical engineering becomes much easier with them, but there's nothing inherent that makes you need them.

Another example is quantum mechanics. Here we have complex valued waves, but the waves themselves is not "observable", that is they will never leave the theory and escape out into reality. They are just used in the calculations, and one could probably formulate the quantum mechanics and still avoid complex numbers, but at the expense of making the theory more complex.

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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

protected by arjafi Oct 29 '14 at 22:23

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