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




u=34i (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 xaxis but with lenght 1/z. To confirm, let's
look at z and 1/z for 1+i and
i1


1+i (blue) and 1/(1+i) (red) 
i1 (blue) and 1/(i1) (red) 
Interactive Complex Analysis, ver. 1.0.0
(c) 20062007, Bert G. Wachsmuth
Page last modified: May 29, 2007