Example 1.2.4 (c): Complex numbers in the plane

Let u = 3 - 4i and v = i, and w = -2. Draw 1/u, 1/v, 1/w and then describe how 1/z looks in relation to z for arbitrary complex numbers.
Context Context
u=3-4i (blue) and 1/u=1/5(3+4i) (red) v=i (blue) and 1/v=-i (red) w=-2 (blue) and 1/w=-1/2

It is hard to generalize from these pictures. But we can do the following:

1/z = 1/z / = / (z ) = / |z|2

Now the direction of 1/z is the same as (reflected around the -axis), while the length is

|1/z| = ||/ |z|2 = 1/|z|

Thus, in general 1/z is the vector z reflected around the x-axis but with lenght 1/|z|. To confirm, let's look at z and 1/z for 1+i and i-1

1+i (blue) and 1/(1+i) (red) i-1 (blue) and 1/(i-1) (red)

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