What is the use/need of complex numbers in real life if they are imaginary?
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They are actually essential to modern physics and engineering.
“Imaginary” is just a name that appeared historically when these ‘weird’ (at the time) numbers were invented, and now we’re stuck with it. But complex numbers appear in fundamental physics, e.g. within quantum [wave functions](https://en.wikipedia.org/wiki/Wave_function), mathematical objects that are used along with [Schrödinger’s equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation) which describes the behavior of a quantum object through time. It’s really weird that “non-real” quantities appear when trying to describe an actual, entirely real object, but it’s the most natural way to express these equations.
Linear algebra can be used to replace complex numbers with matrices, but :
* matrix multiplication works in such a convoluted way (in relation to usual number multiplication) that it does not feel as if you’re actually manipulating the ‘true’ physics
* the matrices that are used in these kinds of expressions actually *are* (in a [mathematical kind of way](https://en.wikipedia.org/wiki/Isomorphism)) complex numbers anyway.
Complex numbers are also widely use to simplify calculations when it *is* possible to do without them, but doing so would make things much more complicated. For instance, a signal `s(t)`, which depends on time, can be expressed as `s(t)=R×e^(iθ)` where `R` is a real *amplitude* and `θ` is a real *phase*, usually between 0 and 2π.
If you represent `s(t)` as an arrow, the `e^(iθ)` term is a complex phase component that describes which direction the arrow is pointing, whereas `R` describes its length. You can now make it so θ depends on time, and now `s(t)=R×e^i(θ(t))` is a signal whose amplitude does not vary, but whose phase does. Which, for instance, is the behavior of electric and magnetic fields in most (arguably simplified) cases!
This is also useful because taking the derivative of a signal with respect to time can be expressed with complex numbers as simply multiplying it by `iω`, where ω is the signal’s pulse (in rad/s). This hugely simplifies calculations. In order to retrieve the ‘real’ signal, you then simply take the real part of your complex signal.
However, taking the real part of a wave function (in quantum mechanics such as mentioned earlier) does not ‘mean’ anything, which is why I used that as an example of complex numbers appearing in a fundamental way.
In my electrical engineering classes, we had to do the math for things like capacitors and inductors in a circuit without using complex numbers. One problem took about 30 minutes to solve using two whiteboards and involved multiple calculus equations. Doing the same problem using complex numbers took about 30 seconds and only required algebra.
Practically anything electrical except the most primitive circuits are using complex numbers as their foundation. Especially anything that has a rotating field or a frequency attached to it, such as cell phones, radios, motors, and so forth.
The worst thing they could’ve ever done for this mathematical concept is calling it imaginary numbers. Something like “rotational number” or “directional number” or something else would have helped.
Math isn’t just about numbers, it’s about relationships between things. The real numbers are one way that things can be related to one another, and the complex numbers are another.
In the real world, electronic circuits that use alternating current are one system where complex numbers are useful, and quantum mechanics is another. In general, complex numbers show up in any situation where you can imagine rotation as a useful metaphor, such as a changing system that goes from positive to zero to negative to zero over time.
Video games! Complex numbers map really well to rotations!
See, when you describe rotations in 3d, its easy to use spherical coordinates:
an example would be:
spin to face your target, (y rotation)
raise your arm to aim a gun, (x rotation)
rotate your gun so you look like a gangster. (z rotation)
That works really well if your making a ground based shooter.
But what if you are making a space based game?
You would have no frame of reference to rotate the ship by.
This is where you use something called a transform matrix
its a 4 by 4 grid of numbers that accurately describe the rotation/scale/position of something.
However, theres a lot of extra information there thats not needed if you just want to control the rotation of a spaceship. And when you are making computer games, efficiency and space matter.
Instead, you would use something called a quaternion.
A quaternion is a number that has:
1 part real and 3 imaginary parts.
Where Complex numbers have the definition i*i = -1
then you add in j*j = -1, and k*k = -1, and most importantly: i*j*k = -1
This creates an extended imaginary system, that is ideal for working with rotations in 3D video games.
Most space, and some underwater games will use quaternion, and a lot of game engines use quaternions under the hood for other stuff
Let me try to put it in simple terms. You have a straight path. Forward is one direction that you call normal or positive. To go backwards, you call it negative and you basically turn 180º and go the opposite directio.
If we’re talking numbers, if you add 5 meters you go forward, and if you add negative 5 what you’re doing is adding 5 but 180º rotated, so negative. You are walking 5 meters but backwards. So you can think of the – sign as short for 180º degrees rotated.
This is consistent. If you apply – twice it is 180+180 = 360=~0 so its back to the same direction. That would be subtracting negative 5 -> 10- (-5)=10+5=15
Ok, now where it is interesting is i=square root of -1 means i*i = -1. If you do it 4 times, (i*i) times (i*i)= – times – = + You just did a 360. Funny that. Replace i for 90 and you got 90+90+90+90=360=~0
So you can say i is a 90º rotation just as you can say – is a 180º rotation, and it rotates stuff 90º. Instead of going back and forward, you’re going sideways.
This makes it funny in that you can think of – (minus) as ii (i times i) or 180 as 2×90
Complex numbers are complex because now you can either think of it as describing stuff on a plane, or rotated. They work really well to describe stuff that is rotating or cycling.
You can even write them as an absolute value with a rotation. 5 rotated 90 is 5*i, 5 rotated 180 is minus 5.
Are they imaginary? Like all other numbers, they are symbols with which you describe something -> and that something may be real, or another mathematician’s dream.
I’m studying electrical engineering so my ideas are from this pov. Imagine three sinusoidal waves like [this](https://www.johndcook.com/sum_phase_shifted_sines.png). The mathematical description it’s kind of annoying, they are different in “size”, some are bigger than the others and they aren’t even in the same place aka same phase. If you wanna know the relationship between the three of them needs a lot of work. Imaginary numbers allows us to describe this waves with their basic characteristics, also how big are they and where are they at the beginning (phase at t=0). Then you have waves that have magnitude 3 or 4, and at the time 0 one has a phase 0 and the other one it’s clearly not by 0 by the initial time. Expressing this on imaginary numbers it’s really easy and then you can know how is the relationship between them. For example, if you know the phase between a Tension (voltage) and a Current you can know exactly which component you have there, for example a Capacitor. Imaginary numbers are really handy with this problems because they are “built” with a magnitud, a sinus and a cosinus, so every imaginary number have this 3 characteristics, which are really handy. Also you can know what happens with different frequencies. If you take a sinus wave on time and you wanna know what happens with different frequencies it’s kind of annoying. If you take imaginary numbers you can easily know that, for example a wave will disappear with a extremely big frequency, or the other case, that a system is really unstable because for a given frequency the magnitude go nuts. Imaginary numbers are definitely superior for all this problems because they are really simple: multiplying two imaginary numbers is really easy (you multiply their “radius”/magnitud and sum their phase/angles) in comparison to trigonometry (multiplying sinus with cosinus is a pain in the ass). In that example, It’s also a lot cleaner to read the results of these multiplication, you can know exactly how that resultant wave is
Electricity consumption is measured in watts (real power). However, the rating of the equipment is measured in kVA (complex power). So, from a numerical standpoint, watts is like your real number component and kVA is your complex number. The imaginary component is called reactive power and measured in kVAR (volt-ampere reactive).
In general, you want your electricity load to require as low reactive power as possible because this means you’re equipment will be cheaper and you are essentially charging the same amount of real power consumption. This property is called power factor, ratio of real power over complex/apparent power. High power factor means your reactive power load is low.
Utility companies incentivise industrial consumers to increase their power factor with discounts and there are actually businesses whose main service is to do exactly that.
In electrical engineering they are very useful. Because of complex numbers you do not need to solve differential equations.
There are Alternating Current circuits in which the voltage is represented as an imaginary number. 300 imaginary volts can kill you just as dead as 300 real ones.
“imaginary” is just a naming convention.
All numbers are imaginary. They’re an abstract concept that reflects reality in one way or another. You can’t hold a number in your hand. Numbers are a property of things – and that’s what makes them useful.
And even then, only positive integers have any simple, intuitive relation to actual objects. Fractions, negative numbers, even zero are all things we made up because they were useful concepts, but they’re all in one way or another removed from the simple concept of numbers like three or five thousand.
Complex numbers are no different, they’re just even less intuitive to most people than, say, fractions because they’re taught later and are used for more abstract calculations.
But make no mistake – even mathematicians at one point strongly objected even to the IDEA of irrational numbers, of negative numbers, even of zero. Today it seems absurd that zero would be a tricky idea to wrap one’s head around, but it used to be the case even for people who dealt with mathematics in-depth.
As to where they’re actually useful – in many situations where you need to consider two values that are related but can’t be directly added to one another, or where you want to express something as numbers but you have more than one axis to work on, complex numbers are a useful tool. It can be complex stuff like physics, but it can be straightforward stuff like measuring angles/directions for navigation.
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