A funny question, about the source of complex number. As the video on http://www.youtube.com/watch?v=2kbM96Jr4nk
It says that:  $1\cdot(-1)=-1$ can represent as $1$ rotated $180^{\circ}$ around the Origin to get $-1$
$$1\cdot(-1)\mapsto 1\quad\text{Rotate}(180 ^{\circ})$$
so it must be:
$$1\cdot\sqrt{-1}\mapsto 1\quad\text{Rotate}(90 ^{\circ})$$
but how to map:
$$(-1\rightarrow\sqrt{-1}\ )\mapsto (180 ^{\circ}\rightarrow90^{\circ}) $$
Is there any principles to map the two transforms, or it's just a basic define as $1+1=2$.  
 A: I'm a little fuzzy on the question, but let's just review using complex numbers to rotate the complex plane.
Multiplication by a fixed complex number creates a transformation of the plane. Explicitly we're talking about the map $x\mapsto cx$ for some fixed $c\in \mathbb{C}$. The transformation will stretch and rotate the plane. If $|c|=1$ there will be no stretching, so we'll restrict our attention to this case.
If you experiment using $c=i$, then you'll find that 
$1\mapsto i$, $i\mapsto -1$, $-1\mapsto -i$ and $-i\mapsto 1$.
You can play a bit and show that this is truly a 90 degree counterclockwise rotation of the complex plane.
In general, the rotation caused by multiplication with a complex number can be seen in the polar form of the complex number. Complex numbers with modulus 1 have the form $e^{i\theta}$, and $\theta$ is going to give the angle of rotation achieved. In the case of $i$, $i=e^{i \pi/2}$, and in the case of $-1$, $-1=e^{i\pi}$
A: This is related to your last question about the matrix representation of complex numbers. Recall that a 2d rotation can be represented by the matrix
$$R_\theta = \pmatrix{\cos \theta&\sin \theta\\-\sin\theta&\cos\theta}$$
Notice that $R_0 = I$ and
$$R_{\pi/2} = J = \pmatrix{0&1\\-1&0} \cong i\\
  R_{\pi} = \pmatrix{-1&0\\0&-1} \cong -1$$
The complex number $z$ "given" by a rotation matrix will always have $|z|=1$.
