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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.

Community
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PhD
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    I am pretty sure this has been asked... – Mariano Suárez-Álvarez Jan 24 '13 at 02:45
  • See https://en.wikipedia.org/wiki/Electrical_impedance#Complex_impedance – Trevor Wilson Jan 24 '13 at 02:48
  • I would vote to close as a duplicate of [this previous question](http://math.stackexchange.com/questions/244240/how-are-complex-numbers-useful-to-real-number-mathematics), but that one's written in a confusing and not very pleasant style. –  Jan 24 '13 at 02:51
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    I can still duplicate my comment: have you checked http://en.wikipedia.org/wiki/Complex_number#Applications ? –  Jan 24 '13 at 02:52
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    Mathematicians use complex numbers because we are too cool for regular vectors. https://xkcd.com/2028/ – Gerry Myerson Aug 04 '18 at 09:41
  • Note that if $x$ and $y$ are complex, $xy$ can be thought of as scaling $y$ by the absolute value of $x$ and rotating by it's argument. Obviously, complex addition corresponds to translation. Also, complex inversion, ie $1/x$, corresponds to a circle inversion + reflection. All of this means that complex numbers often prove useful in understanding problems involving 2D geometry. – Eben Kadile Aug 07 '18 at 19:38
  • related: https://math.stackexchange.com/questions/3244132/what-is-a-simple-physical-situation-where-complex-numbers-emerge-naturally/3244234#3244234 – Ethan Bolker Jun 03 '19 at 14:24
  • There are plenty of applications of complex numbers, but from what I have seen they are typically used to simplify solving a math equation, and the end result is still a real number. Or in some cases (lile quantum) a 2d vectors are represented with complex numbers, but could be represented with 2d vectors. The point is these equations could be solved without introducing complex numbers, but sometimes the math feels easier to do with conplex numbers. Note: introducing a complex number by taking the squareroot of -1 is an invertable operation. This is the key that allows the math to work out. – Alex Amato Jun 05 '21 at 20:13

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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.

Michael Hardy
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    An example would go a long way for those who don't know this :) – PhD Jan 24 '13 at 02:42
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    Complex numbers in a sense [embody](http://en.wikipedia.org/wiki/Euler%27s_formula) 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](http://en.wikipedia.org/wiki/Phasor) for example. – EuYu Jan 24 '13 at 02:46
  • "one begin the current" . . . I presume "one being the current" was meant. – Michael Hardy Jan 24 '13 at 02:52
  • 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 02:55
  • Fourier transforms are also used in studying time series in statistics, including the stock market and biorhythms. – Michael Hardy Jan 24 '13 at 02:56
  • Typical example: http://personal.ph.surrey.ac.uk/~phs3sd/word_files/loudspkr99a.doc – Robert Israel Jan 24 '13 at 08:31
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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)

Kamil Jarosz
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Peter
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    8th grade is young for complex numbers. – user85798 Apr 16 '14 at 12:21
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    This seems more like an example of a 2D vector space. The defining feature of complex numbers is their algebra, but I understand that's more difficult to explain is an "everyday example". – sfmiller940 Nov 11 '19 at 18:38
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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.

lesnik
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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
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    Protons and electrons are particles, not waves. Are you familiar with Quantum Electrodynamics? You are referring to the fields maintained by charges particles. – Mark Viola Jun 05 '18 at 23:04
  • @MarkViola You got me, I am not familiar with Quantum Electrodynamics. Does it really state that protons, electrons, neutrons are particles, not waves? If they are particles, does each of them have strictly defined coordinates and speed at any moment? – lesnik Jun 06 '18 at 07:16
  • @lesnik QED describes electrons,.which are particles. From the uncertainty principle, there is no way to simultaneously observe both a particles location and velocity. – Mark Viola Jun 06 '18 at 14:06
  • @MarkViola "Usual" quantum mechanics tells us that a "particle" may be in a very well defined state (like an electron in atom), but still having no strictly defined coordinates. Not only that it's impossible to measure the coordinates - the "particle" just doesn't have them. Particle without coordinates is not actually a particle, it's something else. Is situation the same in Quantum Electrodynamics? Does it really states that particles do have coordinates that's just impossible to measure? – lesnik Jun 06 '18 at 15:06
  • In "usual" quantum mechanics what we get is a function assigning a point in spacetime to a complex number - the square magnitude of this complex number is the probability density of finding the particle in that region of spacetime. To account for different kinds of states these complex functions are abstracted to vectors in a Hilbert space, and observables correspond to certain operators on this space. By an argument involving these operators we can show that it is impossible to measure two "conjugate" observables such that the product of their certainty is below a certain lower bound. – Eben Kadile Jul 17 '18 at 20:43
  • I might add that QED, and more broadly quantum field theory, is more complicated and relies heavily on the use of Feynman diagrams. I agree that quantum theory is the best example for a use of complex numbers, but we can't say for certain that they are absolutely necessary, and even if they were the universe isn't really "using" them so it seems inappropriate to say that they are used "literally everywhere." – Eben Kadile Jul 17 '18 at 20:47
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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.

orion
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  • I know this is an old post, but an example I came across (solving a cubic equation) the other day involved "cubic spline interpolation", where to display data (like an oscilloscope) with fewer data points than pixels on the screen, for smoothness, one can use cubic spline interpolation to be able to "zoom in" on certain features of an oscilloscope trace. – cowboydan Aug 30 '16 at 16:25
  • I want to endorse orion's post, in that complex numbers show us that the exponential and trigonometric functions are really the *same* family of functions! This insight, like any mathematical ephiphany, suddenly has applications to lots of areas, such as Fourier analysis, digital signals and many of the other examples mentioned elsewhere. – KenWSmith Feb 06 '18 at 19:28
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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.

skyking
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    Sum up: use complex numbers to not make things complex. – Hi-Angel Jan 25 '17 at 08:45
  • You should mention that linear filter design really doesn't exist without complex numbers in the complex plain. – rrogers Aug 07 '18 at 19:43
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    @rrogers I'm not sure about that, I'm rather confident that's **not** the case. The whole point with my answer is that complex numbers are only for simplifying the applications - that there's actually no application that absolutely require them. After all since complex numbers are constructed from real numbers (in a relatively simple way too) everything that can be done with them could be done without them as well. Why that shouldn't apply to linear filter design I can't see. – skyking Aug 08 '18 at 05:36
  • Well I come from the analog background. At one time I had more analog filter design books than the number of filters I designed :) Have a look at: https://en.wikipedia.org/wiki/Electronic_filter – rrogers Aug 08 '18 at 14:14
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Electrical engineering with signals, for example:

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

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

Robert Israel
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Where are complex number used in the real world: iIn almost anything involving waves. Some examples are in cameras forming images, in x-ray crystallography used to determine the structure of molecules such as proteins, in MRI and CT scanners used in hospitals, in various forms of spectroscopy used to identify molecules and in lasers used to understand and predict their behaviour.

porphyrin
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I am afraid your conditions on what real utility of a mathematical object means is so strict as to render all applications of mathematics ridiculous. Indeed, such a restriction soon leads us into philosophising about what is useful and what is not, what is real and what is not.

Why should the compactness with which the complex numbers unify some elementary relationships (and so effect easier methods of mathematical analysis) not be counted as a real world application, since, as others have pointed out (and you yourself probably knew) they are used in the study of such periodic phenomena as the analysis of signals -- which has applications in the systems designed by engineers to convert between such signals? For that matter why should the mechanical engineer who analyses vibrations not count it a real world application when it makes him able to predict in advance the effects of the vibration of his system on a structure?

So, I hope that you can see that the idea of what is useful, real, or applicable is very flexible indeed.

Finally, let me comment on what I think the issue usually is with people finding it hard to see the practical utility (whatever that means elsewhere) of the complex number system: the positive real numbers are a natural model of the concept of quantity or magnitude, and that was the first reason for the invention of numbers (hence the negative reals and complex numbers were seen as not real, fictitious, etc. for a long time, even as recently as the nineteenth century, towards the end of which they generally came to be accepted in most places). But gradually, our idea of number evolved to include things that are not quantities (at least not in the usual sense of the term -- the discovery of antimatter is very, very recent, compared to the timeline of mathematical history) in the original understanding of the term -- eventually with the discovery of other division algebras, which began with the quaternions of Hamilton, the concept of number became nebulous and is today best left as primitive. In summary, the utility of a mathematical object should be judged in light of its nature and what it can model appropriately (which may be far removed from the ordinary experience of the common man); this nevertheless does not detract from its utility. The following is a personal opinion, but I think mathematics, no matter how rarefied and strange, is a kind of reflection of the world. I think the world is bizarre and strange, and that we have barely begun to understand and control it.

Allawonder
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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.

YYC
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vandna rathor srivastava
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