Example 1.2.8: Conjecture for multiplying geometrically

Let z=1+i, w=1-i, and v=i. Compute and visualize z*w, v*z, v*w, z2, w2, and v2. Try to come up with a conjecture how to visualize multiplication of two general complex numbers.
Context 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 z2 (green) v (blue) and v2 (green) w (blue) and w2 (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 x-axis. Once you think you got a handle on the angles, figure out the lenghts.


Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007