[pphysch]

Two-dimensional does not necessarily mean two spatial dimensions. It is dimensionality, or orthogonality, in the most abstract sense. For all practical purposes, complex numbers represent a two-dimensional system.

It is not a coincidence that they arose in 16th century Italy in the context of "completing the square" and related 2D methods/intuitions for solving equations.

[morokhovets]

> For all practical purposes, complex numbers represent a two-dimensional system

Yet operations on complex numbers are not the same as operations on vectors on simple two-dimensional plane. This is my point.

[pphysch]

Complex numbers and (2D) vectors/matrices are different representations of multi-dimensional number systems. For each operation over one, you can find an analogous operation in the other.

You can even find attempts to mix the two representations, like i+j+k vector syntax. But matrices generalize better to higher dimensions and are easier to parse.

[morokhovets]

I agree with you here, but I don't agree on downplaying complex numbers to be just a base vector and orthogonal.

If we take a matrix representation of a complex number it is usually done as a 2x2 matrix of very specific structure. I completely agree that it is easier to work with. But looking at them this way misses very important place of them in the grand scheme of things.

Complex numbers are actually what real numbers really ARE under the hood, we just aren't taught to think this way. 'i' is what real numbers miss to be completed. And you don't need 'j's, 'k's and others.

If your point is that introducing 'i' above traditional real numbers syntax is ugly - I completely agree.

[pphysch]

> Complex numbers are actually what real numbers really ARE under the hood, we just aren't taught to think this way. 'i' is what real numbers miss to be completed. And you don't need 'j's, 'k's and others.

This is an unnecessarily absolute statement. On what basis are you claiming that all number systems are fundamentally two-dimensional, and not one-dimensional, three-dimensional, or some other dimension?

I'm guessing that it is because you spent a lot of time working with mathematics in a 2D context, i.e. on paper or blackboard or screen.

[morokhovets]

> On what basis are you claiming that all number systems are fundamentally two-dimensional, and not one-dimensional, three-dimensional, or some other dimension?

I never said anything like this. I was talking about complex numbers only.

I suggest to stop here, we are talking about two different things. But, if anything, there is a comment in this thread by adrian_b which explains what I mean in more detail.

[ogogmad]

> It is not a coincidence that they arose in 16th century Italy in the context of "completing the square" and related 2D methods/intuitions for solving equations.

That's... not how it happened

[pphysch]

According to the OP, it did. I'm sure that there were many independent inventions of higher dimensional number systems.

[ogogmad]

The complex numbers prefigured other "hypercomplex" number systems by several centuries. And the modern 2-dimensional view of them was only described close to the year 1800. Before that, they were purely algebraic.

The video (I briefly skimmed it) shows that they were invented to solve a problem in algebra. Nobody thought of them as being 2D back then.

I'm not a historian or anything, but your claim is textbook "whig history" -- and as far as I can understood you, I already proved you wrong in a previous comment.

[pphysch]

I am not sure where this aggressive condescension is coming from. You appear to be conflating the inherent multi-dimensionality of complex numbers with their geometric/2D spatial visualization, which came later. I'm pretty sure we are in agreement here.