6:19
> As complex numbers are to real numbers, quaternions are to complex numbers. It's like a way to build up even further. [...] Real numbers are one-dimensional. Complex numbers are two-dimensional. [...] For three dimensions there is no natural number system, but for four dimensions there is and it looks like this.
In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
You might be right, I don't know. Could you explain a bit more how you mean?
If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.
In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.