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I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn

I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension. Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant. EDIT: the difference between complex numbers and 2D vectors (as I understand it) is just that you can’t really multiply 2 2D vectors together to get another 2D vector (you need to multiply a scalar times the vector to get a vector), but you CAN multiply two complex numbers together to get another complex number. To do multiplication with 2 vectors you need to introduce new things like the “inner product” (dot product) to produce another vector (as opposed to “outer product” which produces yet ANOTHER kind of thing, a matrix) whereas with complex numbers you can use regular multiplication without introducing new types of multiplication.

tomrod 4y ago
Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.
- Robotbeat 4y ago
  Exactly. You have to introduce these new definitions, which isn’t required for complex numbers.
computerfriend 4y ago
You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
- Robotbeat 4y ago
  You’re right about cross product vs outer product. (I do not know how I got that one wrong…)
lloeki 4y ago
> Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant. I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it. So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.

tomrod 4y ago

Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.

Robotbeat 4y ago
Exactly. You have to introduce these new definitions, which isn’t required for complex numbers.

computerfriend 4y ago

You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.

Robotbeat 4y ago
You’re right about cross product vs outer product. (I do not know how I got that one wrong…)

lloeki 4y ago

> Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant. I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it. So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.