What are complex numbers good for?

[lukephysics]

TL;DR Summary

Complex numbers purpose?

I was thinking of investigating field theory because i like reading about quantum interpretations.

What role does complex numbers have in physics? I have a hard time seeing why properties of a point in that field are not just multi dimensional properties on some parameter space? Why start talking about real and imaginary stuff?

I come from machine learning so I understand high dimensional notions. I also come from writing 3d engines from scratch so I understand various spaces. eg. camera space, world space, object space, and how to transform and operate in all of these high dimensional regions.

So I cant make a leap to fit in such a thing as complex and real. i remember at uni years ago it was because complex numbers were a spherical coodinate system. which might be handy? but why do we need a spherical coordinate system when discussing parameter space in fields?

Thanks in advance!

 

[Dale]

Complex numbers purpose?

Complex numbers are very good at handling two dimensional rotations and spectra.

Why can’t they just be multi dimensional properties?

Certainly, they could be. Many different mathematical structures are equivalent

 

[lukephysics]

Complex numbers are very good at handling two dimensional rotations and spectra.

i feel like i have deja vu here (but i couldnt find an existing thread by me so sorry if im repeating to someone). so its just a convenience? why cant you talk about theory in terms of matrixes like we do in 3d maths or machine learning, and piss off the complex theory? it sounds so silly to call something real or imaginary. how the f did that happen? what about when we talk in hundreds of dimensions in ML, there is no real or imaginary.

 

[Dale]

why cant you talk about theory in terms of matrixes like we do in 3d maths or machine learning, and piss off the complex theory?

You could. But why would you? What would be the benefit?

 

[Baluncore]

TL;DR Summary: Complex numbers purpose?

I have a hard time seeing why properties of a point in that field have complex numbers? Why can’t they just be multi dimensional properties?

Complex numbers are an early example of a data structure.

They can be treated as separate dimensions, but it is easier to manipulate one vector symbol than many.

Maxwell's 12 original equations, shrank to become only 4 equations, when rewritten by Heaviside as 3D vectors, and then there was light.

 

[lukephysics]

You could. But why would you? What would be the benefit?

because it sounds silly and extraneous (to me, im sure its not but i dont get it yet), and i dont want to learn something if i can avoid it. if something has 10 dimensional configuration or parameter space that sits atop its position space (like a field), what is the real or imaginary part of these 10 parameters? why is complex theory only 2 parameters? this seems wrong (or a non-generalised part of a bigger theory?)

Maxwell's 12 original equations, shrank to become only 4 equations, when rewritten by Heaviside as 3D vectors, and then there was light

yeah thanks that sounds like what my intuition (naive) tells me. i can bypass complex numbers and go to more generalised maths. but as soon as i start to learn about field theory they say the fields are complex numbers at positions in space. lol.

 

[Dale]

it sounds silly … i dont want to learn something if i can avoid it

That isn’t a good reason.

why is complex theory only 2 parameters?

That is all you need to describe amplitude and phase, or rotations and multiplications in 2D.

If you want you can make the replacement $$a + b i \rightarrow \begin{pmatrix}
a & -b \\
b & a
\end{pmatrix}$$ It is equivalent, but wastes space on paper and memory in a computer.

 

[lukephysics]

thanks for you replies. you gave me some good motivation to learn more.

just a quick one out of interest, if complex numbers, quarternions, up to octonions, are used for <=8 dimensional space, what if you come across a more complex space like in ML where you have a billion dimensional space. these tools are not applicable right? how would you do rotations in large dimensional space? just one-dimensional at a time? i assume this only works if your dimensions or transformations are separable(!?)

 

[Baluncore]

how would you do rotations in large dimensional space?

What meaning would a rotation have in that space. Would it not be represented by a matrix multiplication?

 

[Structure seeker]

IMO the complex numbers deserve to be named differently than just 2D vectors because they form a field, in fact a field over which all polynomials can be decomposed as products of linear functions.

 

[lukephysics]

IMO the complex numbers deserve to be named differently than just 2D vectors because they form a field, in fact a field over which all polynomials can be decomposed as products of linear functions.

Just look like a 2d vector to me.

My next job is to learn why the notation isn’t simply z=(a,b). Since a+bi looks silly to me. Maybe it’s algebra, like y=mx+b!?

 

[vanhees71]

You can, of course, reduce the complex numbers to a 2D real vector space, but that's not the point. First of all they rather have very convenient algebraic properties, i.e., they form with the usual sum and product a field, and the polynomials can be decomposed into linear factors.

Second, and I think that's even more important for physics, you can do analysis with them pretty much like you can do analysis with the real numbers. Already "function theory", i.e., the analysis of functions $D \rightarrow \mathbb{C}$ with $D \subseteq \mathbb{C}$ have amazing properties. E.g., a function, which is differentiable in a neighborhood of some $z_0 \in D$ is analytic there. An analytic function, defined along a closed curve, can be analytically continued to the area bounded by the curve. You have the theorem of residues to integrate, and all that.

 

[Dale]

My next job is to learn why the notation isn’t simply z=(a,b).

Well, that definitely won’t work. The product of two vectors is a scalar.

For a change of notation to work, at a minimum it must have the following properties. There must be defined an addition of two objects that produces another object with all the usual properties, including a $0$ object and negatives. There must be defined a multiplication of two objects that produces another object with the usual properties, plus the existence of a $1$ object, and an $i$ object with the property that $i^2=-1$.

Ordinary 2D vectors don’t satisfy those requirements, but the 2x2 matrix representation that I showed above does.

 

[A.T.]

I also come from writing 3d engines from scratch so I understand various spaces. eg. camera space, world space, object space, and how to transform and operate in all of these high dimensional regions.

Many 3D applications use quaternions, which are a form of complex numbers:
https://en.wikipedia.org/wiki/Quaternion#Quaternions_and_the_space_geometry