Complex multiplication as rotation Is there a reason that complex numbers multiplied so readily represent rotations in a plane?  Any intuition behind this would help.  
 A: 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 $rs$.
A: suppose you extend the field $\mathbb{R}$ by adjoining a root $\alpha$ of the equation:
$$
x^2 + 1 = 0
$$
the extension field $\mathbb{R}(\alpha)$ properly contains $\mathbb{R}$ since an ordered field lacks square roots of elements less than zero.
$\mathbb{R}(\alpha)$ is a 2-dimensional vector space over $\mathbb{R}$, and we may take as a basis the pair $\{1,\alpha\}$. with respect to this basis this basis, multiplication by an element $c+\alpha d$ can be viewed as a linear transformation, with the matrix representation:
$$
a+\alpha b \to \begin{pmatrix} a &-b \\ b &a \end{pmatrix}
$$
the determinant  $D=a^2+b^2$ is positive unless $a=b=0$. so we can find $\theta \in [0,2\pi)$ satisfying:
$$
\cos \theta = aD^{-\frac12} \\
\sin \theta = bD^{-\frac12} 
$$
and the multiplication factorizes into a real multiplication coupled with an anticlockwise rotation through $\theta$
