In terms of complex numbers, the isometries of the plane are either of the form
or of the form
for some complex numbers  and  with |ω| = 1. This is easy to prove: if  and  and if one defines
then  is an isometry, , and . It is then easy to see that g is either the identity or the conjugation, and the statement being proved follows from this and from the fact that .
This is obviously related to the previous classification of plane isometries, since:
* functions of the type  are translations;
* functions of the type  are rotations (when |ω| = 1);
* the conjugation is a reflection.
Note that a rotation about complex point p is obtained by complex arithmetic with
where the last expression shows the mapping equivalent to rotation at 0 and a translation.
Therefore, given direct isometry  one can solve  to obtain  as the center for an equivalent rotation, provided that , that is, provided the direct isometry is not a pure translation. As stated by Cederberg, "A direct isometry is either a rotation or a translation."