>>14368482>Are they even realYes and no.
No in that they are orthogonal to the reals.
Yes in that there are real phenomena that are described by imaginary numbers.
No again in that we are often implicitly "typecasting" other objects into imaginary numbers, and in these cases, it can be conceptually cleaner to just avoid this casting.
For example:
Numbers have no direction.
Vectors are directed line segments.
Bivectors are oriented plane segments (put the vectors head to tail and follow them to determine clockwise or anticlockwise).
Trivectors are oriented volume segments (same idea but harder to visualize).
Etc, etc.
Most people never encounter Bivectors, Trivectors etc explicitly, but they use them implicitly all the time. Bivectors in 2D are imaginary numbers. They're identical in every way. If you are in a vector heavy context and imaginary numbers are popping up, they're usually bivectors in disguise, and thinking about them as bivectors can often make some things clearer geometrically.
The cross product in 3D is the same thing. The thing it returns is technically not a vector, since it has some goofy behaviours that other vectors don't have. Physicists use "pseudo-vector" to describe this, but just like imaginary numbers, they are really just a misunderstanding of bivectors. The cross product should actually return a bivector, but because a vector and bivector in 3D both have 3 components, people got them confused early in the development of vector algebra, and we are stuck with their mistake. This is why the cross product only works in 3D. The idea of rotations being "about an axis" is also a 3D only phenomena. Rotation should really be thought of as "in a plane" since this holds in all dimensions. In 2D rotations are about a point, in 3D they are about an axis, in 4D they are about a plane etc. It's always either in a plane, or about an N-2 dimensional thing normal to the plane in N dimensions.
