2

When I was in high school, I asked a physics teacher "Why do we use complex to calculate stuff with electricity?" (something along that line) and he said "Because it just happens to work" without further details on WHY it works.

I'm in CS/IT and am not very well-versed in EE or abstract maths (though I know enough of the basics to make some robots simple.) As I learned about EE basics, I've learnt that complex numbers are VERY useful in EE and complex analysis is a must for students of EE.

I know complex numbers work well for calculating stuff in EE, but WHY does it work? In CS, I've learnt about how logic algebra works and why its maths works and have a pretty good fundamental understanding of it. Complex numbers, however, still seem quite abstract to me. "It just works" doesn't sit quite right with me.

So, how and WHY do imaginary numbers work to calculate stuff in EE?

Edit:

I'm well aware that complex numbers are used in EE like this. However, what I'm asking is, I think, even more mathematically fundamental. Why do complex numbers work here at all?

They only possible answer to this I've seen so far is because of Euler's Formula: \begin{equation} \label{Eq:I:22:9} e^{i\theta}=\cos\theta+i\sin\theta. \end{equation}

Is that it? Is that the sole reason complex numbers are used?

JRE
  • 67,678
  • 8
  • 104
  • 179
ChocolateOverflow
  • 153
  • 1
  • 5
  • why do we use cos and sin in math. why do we multiply or divide or do calculus? it makes it easier. there is no simple/real answer to "why" questions. usually the answer is "just because". or because it works. if we could start over or in a parallel universe with different individuals in history doing different things we could very well have found some other solution that just works. – old_timer Nov 25 '19 at 07:50
  • Why do we continue to do math assuming current flows from + to - now that we know it doesnt? because it just works and changing now would mess everything up. – old_timer Nov 25 '19 at 07:51
  • 1
    It is not that the complex numbers "work" in EE, it is a way to "decouple" current and voltage which is needed for capacitors and inductors. Complex numbers are the most convenient way to describe how these components behave. Without complex numbers, how would you describe how voltage and current across a capacitor behave? It is not impossible to do that without complex numbers but it is more convenient. – Bimpelrekkie Nov 25 '19 at 07:56
  • 2
    @Bimpelrekkie Actually there are some very deep mathematical reasons for the use of complex numbers that have to do with the representations of the general linear group. It has to do with the symmetries of the differential equations that appear in EE. Stuff to look up: Lie groups, Lie algebras, representation theory, Pontryagin duality. – user110971 Nov 25 '19 at 08:15
  • @user110971 That sounds like what I'm looking for. Can you give some (simple) examples? – ChocolateOverflow Nov 25 '19 at 08:18
  • @JohnZhau unfortunately the above mentioned topics are not simple at all. They are graduate level mathematics after all. I may try to write a simplified explanation later, if the question doesn’t get closed. Alternatively you can trust the mathematicians that there are good reasons for it. – user110971 Nov 25 '19 at 08:22
  • Simplistically: Reactive components and sinusoidal variables produce equations that vary in space and time / frequency / phase relative to each other. Complex numbers are a mathematical tool that allows the "numbers" concerned to be manipulated in useful ways that produce numerical results that model real world results. || Some say "God is a mathematician". Better is probably "mathematics is a way of 'explaining' / modelling what God does. [Substitute xxx for God if desired - God is often a better descriptor than alternatives :-)]. – Russell McMahon Nov 25 '19 at 08:26
  • 4
    Complex numbers are a natural for describing things that go round in a circle. Many things in EE, as well as mechanics, and physics in general, go round in a circle, simple harmonic motion. This would have been an answer but for getting here after the question was closed. – Neil_UK Nov 25 '19 at 08:30
  • recommend watching Mike Ossmann's video on the topic for a good "geometric" explanation of complex numbers: https://greatscottgadgets.com/sdr/6/ – vicatcu Nov 25 '19 at 09:31
  • @John Zhau to get familiar with the concepts mentioned by user110971 I recommend reading Abstract Algebra by Tom Judson. It's freely available for download. – Bart Nov 25 '19 at 09:48
  • Well before all the formal proofs, scientists from the time of Newton discovered that many new physics theories or laws could be written in the form of fairly simple differential equations. Mathematicians around the time of Bernoulli and Euler found (with sloppy but effective proofs) that the complex exponential was an elegant solution to these equations. – hotpaw2 Nov 25 '19 at 12:24
  • Also look up “The Unreasonable Effectiveness of Mathematics...” https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences – hotpaw2 Nov 25 '19 at 12:27

1 Answers1

5

The reason why you get complex exponential functions in EE is because the differential equations that appear have certain symmetries. These symmetries are encapsulated in the general linear group (GL). The solution to the differential equations are then the so called representations of the GL.

The way to think about it is that GL contains matrices that are linear transformations. These linear transformations are the symmetries of the differential equations. Then there is a group homomorphism \$\pi\$ that takes each linear transformation and spits out a function that is a solution to the differential equations.

As it happens these functions have the form \$e^{j \omega t}\$, where \$\omega\$ is some number. So we can now reconstruct the solution in the time domain \$f(t)\$ as a sum (integral really) of said functions. Hence you get the Fourier transform.

I’m skipping a lot of detail here. So much so that the above is not an entirely accurate description. However this is a very complicated topic and you need to read a couple of books to grasp it fully. This should give you a basic understanding of where the complex numbers are coming from.

user110971
  • 6,067
  • 1
  • 15
  • 23
  • A friend has gently pushed me for years to learn more on these matters. Can you recommend a starter book, for an EE? – analogsystemsrf Nov 25 '19 at 11:23
  • @analogsystemsrf Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Hall is the standard reference. Be warned that this is a mathematics book for mathematicians. It is very abstract and is full of theorems and proofs. You need a different way of thinking when doing mathematics. You need to be very rigorous as opposed to concluding something is good enough (like we do in engineering). This ends up being important in the end but is not apparent in the beginning. The book is self sufficient otherwise provided you know some calculus. – user110971 Nov 25 '19 at 11:33
  • @analogsystemsrf However as you start getting into the more advanced topics, e.g. harmonic analysis, you’ll find that you don’t even understand the language used. This is because engineers only learn mathematics at the Newtonian level. You need to learn how to write proofs really. A typical math curriculum is something like this: real analysis, complex analysis, topology*, differential geometry (at least Riemannian; maybe symplectic), abstract algebra* (groups, rings*, modules*, algebras* etc.), Lie groups and Lie algebras, partial differential equations. Those marked with * can be skipped. – user110971 Nov 25 '19 at 11:46
  • Anonymous down voter, do you disagree with anything I have written? – user110971 Nov 25 '19 at 13:26
  • To be honest I only understand the Fourier part of your answer and hardly any of the symmetry but I'm still just a 2nd year. May be I'll ```accept``` this in a few years. – ChocolateOverflow Nov 25 '19 at 13:36
  • @JohnZhau Fair enough. – user110971 Nov 25 '19 at 13:37
  • User... don’t worry -1 means they can’t understand it. John.. X+jY where Y are imaginary values for qty. for stored energy components such as L,C values or in mechanical, harmonic springs and rotating mass, X is the real part that transfers energy but can be resistance, voltage or current and Y stores energy and releases it. – Tony Stewart EE75 Nov 25 '19 at 13:49