[morokhovets]

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.

[adrian_b]

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.)

[ogogmad]

> 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.

[niknoble]

> 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).

[boppo1]

>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?

[morokhovets]

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

[marginalia_nu]

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

  R = 
     1 0 
     0 1
 
  I = 
     0 1
    -1 0

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.

[sdenton4]

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...

[adrian_b]

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.

[morokhovets]

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

[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.

[iamcreasy]

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

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

[Upitor]

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.

[iamcreasy]

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.