Next: 6.4 Square Roots Up: 6. The Complex Numbers Previous: 6.2 Geometrical Representation   Index

# 6.3 Roots of Complex Numbers

I expect from (6.16) that every point in the unit circle has th roots for all , and that in fact

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

6.17   Exercise. The figure below shows the seventeen points .

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

6.18   Exercise. The sixth roots of are the vertices of a regular hexagon having one vertex at . Find these numbers (by geometry or trigonometry) in terms of rational numbers or square roots of rational numbers, and verify by direct calculation that all of them do, in fact, have sixth power equal to .

6.19   Theorem (Polar decomposition.)Let . Then we can write where and . In fact this representation is unique, and

I will call the representation

the polar decomposition of , and I'll call the length of , and I'll call the direction of .

Proof: If where and , then we have

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

6.20   Notation (Direction.) I will refer to any number in as a direction.

6.21   Example. The polar decomposition for is

I recognize from trigonometry that .

6.22   Remark. Let . Let and be the polar decompositions of , respectively, so . Then where and . Hence we have
length of product product of lengths

and
direction of product product of directions.

Next: 6.4 Square Roots Up: 6. The Complex Numbers Previous: 6.2 Geometrical Representation   Index