I am trying to understand more about quaternions and I was watching the series of videos from Norman Wildberger, in particular I am a bit stuck on this video: https://www.youtube.com/watch?v=uRKZnFAR7yw
You can see he associates to a complex number a rotation $\varphi$. I forced myself to do some test with numbers, but assuming I have whatever complex number (i.e. $z=3+2i$) and I want to rotate it about the origin of $45^\circ$ what I tried to do is to use the parametrization of the unit circle displayed in the screenshot and enforced the $b/a = 2t/(1-t^2) = 1$ (the slope/trigonometric tangent) to find the complex number that would make such rotation, but that is going nowhere for me.
Probably I misunderstood what he is doing there but I still not see how I can use all of this theory in practice.
Thanks, Daniele
What Wildberger says is correct, but the rational parametrization he uses is just a convenience. In the case of $\, 45^\circ, \,$ if you use $\, t=-1+\sqrt2 \,$ you get that $$ r_\theta := \frac{1-t^2}{1+t^2} + \frac{2t}{1+t^2}\ i = \frac{\sqrt2}2 (1+i) $$ is the complex number associated with the rotation by $\,45^\circ. \,$ Now, if $\, z = 3 + 2\:i, \,$ then $$ z\, r_\theta = \frac{\sqrt2}2(3 + 2i)(1 + i) = \frac{\sqrt2}2 (1 + 5i). $$
Wildberger is well known for his rational approach to mathematics. In my opinion, it has some merit but it does not solve all problems. This case is an example. I suggest that you continue studying and then you will find that you understand more as you go on. Don't get derailed by temporary confusion.