Abel's theorem in problems and solutions - School of Mathematics

60 Chapter 2
2.5 The trigonometric form of the complex
numbers
Recall that the angle between the rays OA and OB is defined as the angle
by which one has to turn counterclockwise the ray OA around O in order
to take it over to the ray OB (if the rotation is clockwise, the angle has
the sign ‘minus’). So the angle is not defined uniquely, but up to rotations
by where is any integer.
DEFINITION. Let point O be the origin of the coordinates, and suppose
that the vector OA with coordinates corresponds to the complex
number One calls the argument of the complex number
(denoted by the angle between the positive direction of the axis
and the ray OA (Figure 15) (if then is not defined).
FIGURE 15
Since for a given number the angle is not uniquely defined,
by the expression we mean a multi-valued function taking infinite
values, between which the differences are equal to multiples of
The expression will mean that one of the values of the
argument is equal to
Let and The vector with coordinates
corresponds to the complex number and its length is therefore
equal to Let Thus by the definition of the trigonometric