How can I practice Jean-Robert Argand idea of the rotation of a square root of -1 I am studying complex numbers and I really need an intuition on how they work.
I found the following video of Mathematician named Adrien Douady 

https://www.youtube.com/watch?v=2kbM96Jr4nk

He explains complex numbers in a algebraic geometry way, as he mentions using his protractor and turning it by 180 degrees to represent multiplication by (-1). But then at 4:26 he mentiond a mathematician Robert Argand http://en.wikipedia.org/wiki/Jean-Robert_Argand, and that he had a great Idea. "He said to himself, since multiplying by -1 is a 180 degree rotation, the square root is 1/2 of 180 degree rotation. How is this? how can I practice this? from what philophy of mathematics does this come from? 
Thank and forgive me if I sound to dumb.
 A: For vectors on the plane, the mapping $v\mapsto c\cdot v$ is zooming the vector by scalar $c$, if $c\in\Bbb R$. 
When we compose two such maps, $f:=v\mapsto c\cdot v$ and $g:=v\mapsto d\cdot v$, we have that $g\circ f=v\mapsto d\cdot c\cdot v$. 
In particular, if $c=d$, we get $v\mapsto c^2\cdot v$.
The mapping  $s:v\mapsto -v$  can be indeed viewed as the rotation by $180^\circ$. Let us find its square root(s)!
Both rotations $r_{1,2}$ by $\pm90^\circ$ are such that $r_1\circ r_1=s=r_2\circ r_2$.
This can be set in a correspondence with the complex imaginary units $\pm i$.

This way, complex numbers can be identified with 'zooming-and-rotating' transformations of the plane: 


*

*The complex number $1$ corresponds to the identity, i.e. zoom by $1$ and rotate by $0^\circ$.

*$0$ corresponds to the degenerated zooming by $0$, i.e. the mapping $v\mapsto 0$

*A positive real number $c$ corresponds to the zooming by $c$ (i.e. $v\mapsto c\cdot v$) and no rotation (i.e. rotating by $0^\circ$)

*A negative real number $-c$ (with $c>0$) corresponds to the zooming by $c$ and rotating by $180^\circ$.

*The imaginary unit $i$ corresponds to rotation by $+90^\circ$ (and no zooming, i.e. zoom by $1$).


Addition of complex numbers corresponds to 'pointwise addition' of these transformation:
$$(f+g):=v\mapsto f(v)+g(v)$$
(There's an important point hidden here: try to prove that $f+g$ is again a 'zooming-and-rotating' transformation.)
Multiplication of complex numbers corresponds to composition of these transformation.
