Example 1.2.8: Conjecture for multiplying geometrically 
Let z=1+i, w=1i, and v=i. Compute and
visualize z*w, v*z, v*w,
z^{2}, w^{2}, and
v^{2}. Try to come up with a conjecture how to visualize
multiplication of two general complex numbers.
Context

In the set of pictures below we multiply the red
and blue vectors together and display the
result as a green vector.



z (blue), w (red) and z*w (green) 
z (blue), v (red) and z*v (green) 
v (blue), w (red) and v*w (green) 
Now we square the blue vector (i.e. multiply it
with itself) and display the result as a green vector.



z (blue) and z^{2} (green) 
v (blue) and v^{2} (green) 
w (blue) and w^{2} (green) 
To come up with a conjecture is left to you. As a hint, ignore the lenghts
(at first) and focus on the angles the vectors make with the xaxis.
Once you think you got a handle on the angles, figure out the lenghts.
Interactive Complex Analysis, ver. 1.0.0
(c) 20062007, Bert G. Wachsmuth
Page last modified: May 29, 2007