Is there a reason that complex numbers multiplied so readily represent rotations in a plane? Any intuition behind this would help.
[Math] Complex multiplication as rotation
complex numbersrotations
Related Solutions
Stick your thumb of your right hand along the direction of the rotation axis. A positive angle is along the way your fingers point.
Like this
http://electron9.phys.utk.edu/Collisions/rotational_motiondetails.htm
What you are probably looking for is a discussion on Clifford algebras or maybe one tailored to so-called geometric algebras which is pretty much under the same umbrella.
Clifford algebra explains the connection between elements of a special algebra and rotations in a vector space (and more.) In particular, it generalizes what you see for the reals, complexes, and quaternions on $\mathbb R$, $\mathbb R^2$ and $\mathbb R^3$ respectively.
Best Answer
You can represent the complex number $z=x+iy$ by $z=re^{i\theta}$, where $r=\sqrt{x^2+y^2}$ and $\theta$ is the counter-clockwise angle from the positive $x$ axis. So if we multiply two complex numbers together, e.g. $z=re^{i\theta}$ and $w=se^{i\phi}$ we get $$zw = re^{i\theta}se^{i\phi}=(rs)e^{i(\theta+\phi)},$$ so as you can see the resulting complex number has angle $\theta+\phi$ and length $rs$.