Comment by ralusek | Hacker News Reader

Comment by ralusek | Hacker News Reader

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> Imaginary numbers exist on a dimension perpendicular to the real number line.

Correct me if I'm wrong, but I don't think that imaginary numbers exist in a dimension with any sort of fixed spatial relation to real numbers. Isn't the fact that we chose to graph the imaginary axis perpendicular to the real axis on the complex plane simply arbitrary? Or maybe better described as "a functional way to visually plot the negative root component of a number."

My understanding is that visualizing the imaginary portion of complex numbers on a spatial plane with perpendicular axes also just happened to get a convenient 90 degree rotation by multiplying by i, and rotations like this just happen to show up all over nature. For example, the wave function they mention in the video, where they say it has an "imaginary component," seems like complete bullshit to me. You could write a program with real numbers in order to graph that wave...in fact I guarantee that's how a technical artist visualized it. In fact, it's the same with the Mandelbrot set, which supposedly "exists on the complex plane." No...it doesn't. It just happens that if you describe the arbitrary rotation operation that is performed in order to define whether or not a coordinate falls within the Mandelbrot set or not, the rotation from the way we've chosen to graph the complex plane means the equation can be written very concisely like this.

Every independent variable should be graphed perpendicular. X and Y when graphed are perpendicular. If they are not graphed perpendicular it indicates that a change in X position must change Y (or vice versa) whereas being perpendicular makes it clear that they are independent and you can change one without any thought that you're changing the value of the other.

Complex numbers have 2 independent variables. The real and the imaginary parts. You can change one part without being forced to change the other. So you graph them perpendicular.

It's not really a convention, it's a rule that each independent variable is perpendicular (or else they aren't independent!). Complex numbers have 2 independent variables.

Well yes, if we want to plot any sort of independent values, we have 3 spatial axes we are familiar with and will use them to graph accordingly. But I'm not going to argue that "housing prices therefore exist on an axes perpendicular to square footage." I just plotted the relationship that way.

I think focusing on the perpendicular doesn't make much sense since of course we graph different variables perpendicular.

The real point i think you're making is that complex numbers are just a notational shorthand for multidimensional vectors. And yeah. That's true.

https://xkcd.com/2028/

This is why it's fun to know the history of math.

The complex plane was introduced by Caper Wessel in a paper that was published in 1799 so it would've existed before vector notation.

My guess is by the time vectors got popular the complex notation, and theorems that people had proved which used complex notation, had already stuck. But I'm only a hack math historian so I can definitely be wrong here.

I think it's important to keep in mind that math and science, much like the code base that I am trying my hardest to avoid, is evolved.

There are valid reasons to plot variables that aren't perpendicular. Maybe you want a plot of the radius of a cylinder and its volume, for example. Then a change in the x axis changes the value of the y axis and the axes wouldn't be drawn perpendicular.

Dimension in this sense means degree of freedom, not a spatial dimension. That is, a number on the "imaginary dimension" cannot be represented on the "real dimension". This is similar to how a length does not represent a width or height.

When visualizing the real and imaginary dimensions, it is convenient to represent them using x/y axes. However, some visualizations of complex functions use colour/hue as a way of representing the real or imaginary part of the result.

Dimensional analysis is used in physics, etc. when manipulating fundamental units (time, length, luminosity, etc.). That is, m/s^2 has dimensions length=1 and time=-2.

Right, my point is that length and time don't exist perpendicular to each other simply because there might be some utility in visualizing them that way. The complex plane seems to be generally taught as if imaginary numbers in some capacity exist on this perpendicular axis. Even Veritasium, who is supposed to be teaching deconstruction/first principles approaches to these concepts, is saying this.

The spatial unit vectors for a 3 spatial dimension [x,y,z] vector are i^=[1,0,0], j^=[0,1,0], and k^=[0,0,1]. These are all said to be orthogonal to each other, which is a generalization of perpendicularity [1] to non-spatial vector spaces.

As a lay person who is not familiar with the mathematics would recognize the term perpendicular, using that in the video is fine.

[1] https://en.wikipedia.org/wiki/Orthogonality

That is correct. In the Clifford formulation that is equivalent (and in general a superset) of complex numbers you can choose the dimension or behavior to be whatever you want it to be in the resulting algebra. You can even assign it a particular metric or nullify it. The 'square to -1' that implies quadrature is just a specialization in that case. Many algebraic things of complex numbers like the exponential map still work or have general counterparts.

I disagree, it's not simply arbitrary. Euler's formula eⁱˣ=cos(x)+i⋅sin(x) exists outside of any physical representation of the complex plane. And the fact that we already graphed trigonometic functions lent itself naturally to a corresponding graph of complex functions. As sibling comments have stated, any graphical visual representation of completely independent variables is most efficiently represented with perpendicular axes, so the mapping is inevitable.

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