In particular, each vertex of the regular -gon inscribed in the unit circle and having a vertex at will be an th root of .

Let and . Draw the polygons ---- and ---- on different sets of axes, (i.e. draw segments connecting to , to , , to , and segnents joining to , , to .)

Proof: If where
and , then we have

This shows that , and it then follows that . Since , we see and gives the desired decomposition.

length of product product of lengths

and

direction of product product of directions.