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I cannot agree with calling complex numbers just two dimensional.

Function of a complex variable is very different from a function of two variables. You can say these are two different departments of mathematics.

Real numbers are not algebraically complete, but extending it with 'i' makes it complete. Adding another dimension to go to 'two dimensions' does not do anything like this.

Mathematicians are fascinated with complex numbers because it is THE extension of real numbers that completes them in very important sense but it comes with so many unexpected and fascinating properties.

Quantum phase is not two-dimensional, it is complex and it amazes me much much more than two-dimensionality would.

The complex numbers are 2-dimensional, but their 2 dimensions are not the 2 dimensions of a normal 2-dimensional geometric plane, they are 2 other dimensions.

The 2 dimensions of a geometric plane correspond to the 2 orthogonal translations of the plane.

The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

The multiplication of the complex numbers corresponds to the composition of scalings and plane rotations, which are invertible operations and that is why the set of complex numbers is a commutative field, unlike the set of points of a geometric plane, which does not have such an algebraic structure.

The set of complex numbers can be viewed as a plane, but it must be kept in mind that this plane is a distinct entity from a geometric plane.

(The Cartesian product of a geometric plane with a complex plane forms a geometric algebra with 4 dimensions.)

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

Similar story for other 2D number systems:

For the dual numbers, they express scalings and "Galilean" rotations (i.e. shears).

For the double numbers, they express scalings and "Minkowski" rotations (i.e. Lorentz boosts).

Unfortunately, some of the nice theory of the complex numbers doesn't generalise easily to the dual numbers or double numbers. I'm thinking specifically of complex analysis which is very, very nice, and much nicer than real analysis. But I think these planar number systems have their own intriguing character: For instance, see "screw theory" and "automatic differentiation" for two distinctive applications of the dual numbers.

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

I disagree. When multiplying complex numbers, they are viewed as scalings with rotation like you say. But when adding them, they are viewed as translations. If you only took one of these perspectives, then complex numbers would be very simple and not express anything interesting. Their complexity and power comes from alternating between the scaling with rotation perspective (multiplication) and the translation perspective (addition).

>The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

So, polar coordinates basically?

That's exactly what I was trying to convey, thanks.

They are very much equivalent. You can express complex numbers with a 2x2 matrix of real numbers.

I and R form a basis that spans something that behaves like a complex plane.

You have the expected identities

R^2 = R

IR = I

I^2 = -R

I^3 = -I

I^4 = R

Transposition is complex conjugate. You can put them into exponentials, e^Ix = cos(x) + I sin(x); everything works as you would expect.

It's really inelegant though! To multiply two complex numbers, we need to embed one in a 2x2 matrix while keeping the other as a 2d vector, and then do the matrix vector multiplication. And furthermore, it's not like every 2x2 matrix corresponds to a complex number...

That is not correct.

When complex numbers are represented in the matrix form mentioned by the other poster above, the multiplication of the complex numbers corresponds to the multiplication of the matrices, not to multiplications of vectors by matrices.

The matrices of this form, having just 2 parameters instead of the 4 parameters of a 2x2 matrix, are obviously just a subset of the general 2x2 matrices.

However this subset is closed to matrix addition and multiplication, so it is a field isomorphic to the complex numbers.

This representation of the complex numbers is useful to understand that the multiplication rule for complex numbers is not some random arbitrary rule, but it is the same as the rule for matrix multiplication, which is also not a made-up rule, but it results from function composition, when you compose the functions of a vector expressed by the multiplication of a matrix with the vector.

Actually the construction of the complex numbers goes like this, if you consider the transformations of the 2-dimensional vectors that correspond to a scaling and a rotation of the vector (which are a subset of the linear transformations expressed by multiplication with a non-singular 2x2 matrix of general form), and you compute the matrices that perform such transformations, you arrive at 2x2 matrices with 2 parameters of the form shown by the poster above.

This set of transformations happens to have the algebraic structure of a commutative field. Because the matrices are defined by only 2 parameters, you can simplify the operations by keeping just the 2 parameters and using rules for operations expressed directly with them.

Thus you obtain the standard rules for operations with complex numbers.

To give a simple example - Mandelbrot set is a direct consequence of amazing properties of complex numbers and has nothing to do with two dimension.

Well, it looks great in two dimensions

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.

> 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.

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.

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.

> 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.

> 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.

> 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

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

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.

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.

> I cannot agree with calling complex numbers just two dimensional.

I once read, imaginary number is isomorphic to 2d vectors.

Isomorphic as vector spaces, meaning that their additive structure is the same. But the complex numbers are usually not used as a vector space, but rather as an algebraic field, i.e. considering both their additive and multiplicative structure.

Thank you. But I am guessing I need to know a bit of abstract algebra to understand what you are saying which I don't have.

Historically, the complex numbers were discovered by accident while solving cubic equations using the general formula. The general formula only works in general if you can take the square root of a negative number.

One of the first attempts to provide a geometric meaning to the complex numbers was by John Wallis, and I haven't been able to make much sense of it. I suspect he didn't see it the way we do. Also, there's no indication that in spite of the work Euler did on the complex numbers, that he knew of their geometric meaning. The mystical sounding name "imaginary number" was coined by Descartes in the 1600s at least partly because he didn't have the modern view of them.

Of course, teaching by explaining historical developments is not a common thing in mathematics, and the above facts illustrate why. But you have to be aware that things do start off being mysterious before they're fully understood.

The mystery has been somewhat reawakened with the quaternions and octonions, and some other hypercomplex number systems. And mystery gets some people out of bed, so don't be too hard on it.

> It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions.

Are you talking about matrices?

> Are you talking about matrices?

Yeah, matrices turn out to be a better general representation than adding more ambiguous symbols beyond `i`. Still not the "correct" one by any means, because e.g. they don't represent exotic (non-integer, etc.) dimensions well.

In mathematics in addition to real and complex numbers there are higher dimensional numbers namely hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that. The latter hypercomplex numbers are better at accurately model reality compared to the former real and complex numbers inherent limitations. As an example, electromagnetic waves unlike sound waves has polarization component that can be easily modeled using quaternion numbers compared to just using imaginary numbers. Yet for centuries until now scholars have been reluctant to use quaternion numbers and insisted on conventional imaginary numbers technique for Maxwell's Equations electromagnetic solutions popularized by Heaviside rather than the quaternion techniques being used by Maxwell himself! History is repeating itself as our current scholars are very much reluctant to use hypercomplex numbers, not unlike our predecessors who were reluctant in using imaginary numbers although the advantages of the higher dimensional numbers are plain obvious.

> in addition to real and complex numbers there are higher dimensional numbers namely hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that.

To be precise, no other multi-dimensional numbers beyond octonions (which lack associativity) that follow the behavior of complex numbers. There are indeed an infinite amount of multivector numbers with different algebraic rules.

> hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that

The term "hypercomplex number" refers to more than those examples. Those examples are the normed unital algebras, which are a finite subset of the hypercomplex numbers.

A thousand times this!

Scientists and mathematicians should be embracing cold hard reality instead of wallowing in self-aggrandising mysticism. That's the point of science! Empiricism and rationality is are its distinguishing features. These elevate it above the near-worthless philosophies that it replaced.

The worst offender by far in the mainstream sciences is Quantum Mechanics, which is absolutely infested with woo and self-contradictory nonsense. Its overuse of complex numbers is just the beginning.

Mysterious action at a distance isn't mysterious and isn't action at a distance at all.

The electromagnetic field is made up of continuous waves filling space, there are no point-like photon particles. That was just a mathematical abstraction -- a convenient shorthand term for describing interactions with matter, and shouldn't be taken literally.

Electrons aren't little hard balls, which is why they don't rotate, and this is why spin isn't some huge mystery.

Etc...

At some point you’re stuck with a bunch of equations that work and the only way to make progress is to try to interpret the equations.

> Electrons aren't little hard balls, which is why they don't rotate, and this is why spin isn't some huge mystery.

This doesn’t clarify spin at all. Spin is actually incredibly weird and anyone who writes it off probably doesn’t understand what’s interesting about spin quantization, spin/angular momentum noncommutativity, fermions obeying SU(2) instead of SO(3) rotation characteristics, etc.

Yet, the only instruments we can build pick up particles, and nothing but particles. Maxwell's electromagnetics, and his field waves, are a useful approximation. But as close as we have been able to measure, the Schroedinger equation is an exact description: of waves that resolve to probability of the detection of a particle.

There are things to complain about in our description of the universe with particles: magic numbers that can only be measured, not deduced. But incorrectness isn't one of them.

"Spin" is just a name.

> the only instruments we can build pick up particles

In general, this is not true. As in, there are experiments we can build that demonstrate that the wave nature of light is the true description.

Where people get confused is that practically all instruments are made of matter. Atoms, or at least a plasma or even individual particles, but matter nonetheless.

If "particle behaviours" is a property of matter, and you have only experiments made of matter, then this tells you nothing about the behaviour of light, which is not matter.

This is like basic experimental science 101. Beginner stuff.

Isolate the thing you are testing, and test it, and only it.

This is hard to do, but not entirely impossible.

Try harder to be a scientist and stop reiterating woo that you picked up from textbooks that copied it verbatim from people that died nearly 100 years ago. (The same people that thought that a human looking at an experiment was somehow an essential aspect, leading to a century of stupid people arguing about whether a brain is necessary for physics to occur!)

Nitpick: Anything in reality that can be modeled with complex numbers can be with real numbers. Complex numbers provide incredible convenience, but that's all. There's no phenomenon that requires complex numbers to explain - Schrodinger's Equation included.

There are real, complex, quaternion and octonion numbers. Saying that we can just get by only with real numbers is really being dishonest and in denial, it's just like saying we do not need OS and higher level languages because bare metal and assembly are all that we ever need.

As a rough analogy, certainly we can travel all over the world on foot by walking and swimming, eventually we will get there but in reality we will not. But the more correct analogy is that we certainly cannot go to space on foot, and higher dimensional complex and hypercomplex numbering systems is like our spacecrafts. Ironicaly the famous gimbal lock problem for spacecraft navigation can be easily solved using quaternion as opposed to disaster waiting to happen if we use only complex numbers [1]. Einstein certainly will have never come up with the infamous equation E= mc^2 and corresponding relativity theory if not because of quaternion numbers [2].

[1]Gimbal lock:

https://en.wikipedia.org/wiki/Gimbal_lock

[2]Quaternions in University-Level Physics Considering Special Relativity:

https://arxiv.org/ftp/physics/papers/0308/0308017.pdf

It sounds like you're agreeing with my statement:

> There's no phenomenon that requires complex numbers to explain

I thought the discussion between "mathematics only models the world" vs. alternatives wasn't settled?

(most recently I'm referring to Roger Penrose's views on the "epistemic argument against realism"[0], although I don't fully know where I stand myself)

[0] https://www.wikiwand.com/en/Philosophy_of_mathematics#/Epist...

It is a bit odd that the video really talks up the idea that complex numbers sever algebra from geometry, then without pause goes straight into a geometric interpretation of complex numbers https://youtu.be/cUzklzVXJwo?t=1153 .

It goes to analytic geometry, not really the same.

> "rotate by tau radians"

tau/4 radians?

Thanks!

I've always found the geometric interpretation as easy as useful but felt I was not grasping the whole thing because it was too cute and easy.

Just want to commend you for using tau.

It's the first time time I've seen it outside of tau promotional material and I really hope it catches on!

So, hold on, where I have a point (x,y) with two "real" numbers, if I have (xi, yi) each of those numbers has a 2nd dimension, i.e. 'y part'? So it's sorta like ((x1,y1), (x2,y2)) where the second part is re-oriented about the y-axis?

Forgive me I still struggle with complex numbers, no matter how many videos I watch.

Well, I went through 5 years of electrical engineering which is almost entirely in the realm of complex numbers and I’m struggling to convey my intuition here to you.

I think it’s helpful to understand that complex numbers are two dimensional but they aren’t like an [x,y] where the two dimensions are completely 100% unrelated.

Here’s another way to think about it in a way that is familiar to you that I haven’t seen anyone else post yet.

Imagine natural numbers like 1,2,3… and so on. Now imagine those are just a subset or special case of fractional numbers like 1/2,4/3,… Those fractional (rational) numbers are expressed in terms of two natural numbers.

Now in the past they couldn’t solve all equations because they were frustrated that some numbers couldn’t be expressed as fractions. They called these irrationals. But irrational numbers can be expressed as an infinite sum of rational numbers. This includes numbers like pi and e.

Next you can make all those irrational and rational (real Numbers) to be negative. And then that’s still not the end of it because there’s more gaps! We need complex numbers :)

Then after complex you can go into tensor spaces like real^(3x3) where numbers are more like matrices. Perhaps the easiest to understand is a space of (r,g,b). Each pixel has an x,y coordinate but each point is vector valued (3 natural numbers on the range of 0-255). It should be easy to also visualize a complex number space of the same nature but with different properties.

Perhaps the best equation to understand complex numbers is Euler’s equation

e^(ix) = cos(x) + isin(x) Or if you put in pi you end up with the most beautiful equation ever.

e^(i*pi)-1=0

> The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion

This claim is at least as speculative as any claim that the universe works on some symbolic system.

Let me soften that claim. The probability that we humans are even capable of comprehending the fundamental nature of the universe, if such a thing exists, is approximately 0%.

Why do you think the rules of the universe are complicated? If you use any kind of kolmogorov prior that's not a reasonable thing to assume without evidence.

Yes, 'imaginary' numbers are invented syntax. I think imaginary numbers are just an additional tool tacked on to our existing tools in order to explore new areas. They are a patch written to cover some cases our previous tools failed to cover.

> The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion that only serves the status of the mathematical elite. Mathematics is fundamentally about designing models and abstractions that help us reason about real phenomena with minimal cognitive resources.

Yes, creating abstractions and designing models are aspects of math, but I believe the modern definition of mathematics has expanded to include the study of abstract objects.

Abstractions can describe concepts (concepts that exist in many different places) but are not the same as the concepts. I believe these concepts are discoverable and independent of any notations/models/abstractions we create. So in that sense, I believe mathematics can be discovered.

e.g.

exponential growth/decay (the spread of COVID in an unvaccinated population, bacterial growth, etc.) - invented or discovered?

(https://en.wikipedia.org/wiki/Exponential_growth#Examples)

fractals (pattern of rivers, trees, blood vessels, etc.) - invented or discovered?

I can describe the details of these with tools like 'geometric progression', and 'the Mandelbrot set'. Those are tools aiding my understanding of these concepts, but the concepts themselves certainly seem like they were discovered.

All of mathematics is invented syntax. For every bit of mathematics that is taught, there are a thousand bits somebody explored and, maybe, published, but it wasn't considered beautiful enough to teach. So, the maths that are taught are the bits enough mathematicians liked. Some that are taught were considered ugly by older mathematicians, but were embraced by their students. The really ugly stuff has either fallen away or just not been adopted into canon yet. Physicists have pushed mathematics in directions mathematicians hated to go, but those directions have borne surprising fruit, so mathematicians are beginning to accept them.

Mathematics is a tapestry being woven by living mathematicians guided by what they like.