I’m writing a paper on real world applications of complex numbers and can’t understand it at all. If you have any other real world applications, that would be super helpful too. Thanks!

In: Mathematics

Electrical power can be considered in complex number terms. Google “power factor” for some breakdowns.

Electrical impedance can be expressed on a graph with resistance on the X axis and inductance and capacitance as positive and negative on the y axis. So a certain impedance would be considered to have a real resistance value plus an imaginary inductance/capacitance value.

In industry power factor is very important because if say the power draw of your factory has a large imaginary component, then the power plants have to cope with that. Many industries get penalised if they have a poor power factor. In modern practice, companies will install large capacitor banks in their factories to improve the power factor. Poor power factor is usually because they have massive inductance from big electrical motors, so the capacitor banks compensate for that.

Much of it has to do with the function e^(ix) . e is of course “euler’s number”, 2.71818. Now it may be bizarre to have an exponent with an imaginary number in it, this idea doesn’t work within our standard understanding of exponentiation, but we do a little mathematical wizardry and distance ourselves from the idea of repeated multiplication while looking into the properties e^x has and you get this weird function e^(ix) = cos x + i sin x

Why? How? This can be explained better with calculus but even without there is a general sense in which this is true but that lies more with the fundamentals of what this mysterious number e is. If you are more interested, there is an excellent YouTube series by 3blue1brown in YouTube about this topic (Lockdown math) that he recently released during this quarantine. But take it as a given, e^(ix) = cos x + i sin x

Stepping back, this function can be considered a rotation about the complex plane. What does this mean? Well if you increase x, you always get a complex number with distance 1 from the origin if you Pythagorean theorem the real and imaginary parts, and it seems to “walk around in a circle” in this sense. This turns out to be a useful pattern to keep in mind often.

This function is convenient because it gives us a shorthand way to keep track of both sine and cosine in a single function. There are a lot of things in electronics that can be modeled by sine and cosine, you will commonly see this in “RLC circuits”, resistor inductor capacitor circuits because these tend to be modeled quite nicely by sine and cosine, some parts sine and some parts cosine. These circuits are periodic and have properties that are “circle like,” for instance the current flowing through can be modeled with e^(ix).

This function can be viewed as an exchange in some way, the real “values” being exchanged for imaginary “values,” then back to real “values” and continuing the cycle. For instance in a circuit with a capacitor and inductor, real current one moment flows into a capacitor to in this sense become “imaginary current”, then at a point this capacitor begins to discharge into the inductor to become real current once again.

This concept of rotation just happens to be very common and [even things that don’t seem to be rotations often will happen to be sums of different rotations](https://youtu.be/-qgreAUpPwM). And rotations just have a number of useful calculus properties that just appear in nature.

Complex numbers are not necessary, but they make the math of waves easier.

The main idea is that sinus and cosinus behave a lot like exponentials, but the math of exponentials is very easy, especially when solving differential equations or differentiating any function.

sin'(ax) = acos(ax) and sin”(ax)=-a²sin(ax)

exp'(ax) = aexp(ax) and exp”(ax)=a²exp(ax)

if you replace a by i*a in the second line, it almost looks like you could find a way to use the convenient properties of exponentials to solve trigonometry problems, but it makes no sense to apply exponential to an imaginary number.

exp(ix) = cos(x) +isin(x) is a new definition, that shares with exp(x) most of its convenient properties.

It’s used in any mathematical thing that is wavy, you just put your wavy function in the imaginary part of an exponential, and the math becomes super easy.

I studied physics and don’t now what you call electrical engineering exactly, we used complex numbers to study electromagnetic waves, signal processing with fourrier transforms, the responses of electrical systems to a wave function and the responses of physical systems.

Think of the way we use negative numbers in systems in which there’s an importance to direction, or in other words an importance to if two different objects are “in sync”, i.e, possessing the same sign or “out of sync”, i.e, possessing differing signs.

You can think of a examples of this like banking, in which money you have is positive compared to money you owe which is treated as negative.

Negative numbers help us in this scenario because they allow us to encode the directionality of the money in the numbers themselves, and then treat negative money the same exact way as regular money.

In the same way, the complex numbers allow us to talk about systems in which objects can be in sync in a non binary way – as in not just, completely in sync, or completely out of sync.

An example of this would be to take two identical sine waves, move one of them so that it’s shifted compared to the other, and then observing how their sum behaves.

The waves can be in varying degrees of synchronisation or phase as it’s usually called, and this greatly affects how their sum looks like, they can be totally in phase, and then their sum is just twice of them, or they can be totally out of phase and then their sum is zero, or they can be neither this nor that and then their sum is somewhere in between.

Here’s a quick Desmos visualisation of this to help demonstrate what I mean :

https://www.desmos.com/calculator/sjzhuhiumh

You can interact with the slider to control the size of the parameter a and thus decide how much phase will there be between the sine wave.

Now it turns out that many physical phenomena can be described using this concept of being in varying degrees of phase, and in which case complex numbers can help us in our description because they are defined in a way which allows us to talk about angles between them – essentially the phase between them.

As for actual fields in which they are relevant, many commenters have given in length explanations about their usage in electrical power systems, so I thought I’d mention that complex numbers are also used extensively in quantum physics due to the deep importance of interference in quantum systems, I won’t go into detail about this because I don’t think it can be expressed well in this format but in essence many systems are alike in certain ways to the two sine waves I mentioned earlier.

Sorry if I deviated a bit from your question but I think talking about complex numbers is essentially talking about phase and why it matters.

Hope this helps you in any way with your paper!

Complex numbers have “two” axes, the “real” number and the “imaginary” number. A number otherwise has one “value” and can be represented [on a line](https://i2.wp.com/mathblog.wpengine.com/wp-content/uploads/2017/03/numberlines-thumbnail.jpeg?resize=573%2C247&ssl=1); complex numbers on the other hand have 2 “axes” and can be represented on a plane. This makes them useful for representing vectors, or things that rotate (cycles), or have periodic behavior, or waves (AC electricity for example).

[Article with more details](https://www.electronics-tutorials.ws/accircuits/complex-numbers.html).