Chapter II

Complex Numbers, A Geometric View

1. Polar Form

The complex number system may be regarded as a numerical representation of the points in a plane (called the Argand Plane in this context). In the Argand Plane, one selects two points and calls them O (origin) and U (unity). The distance between O and U is chosen as the unit length. Then the location of any other point P in the plane is specified by polar coordinates where r is the distance from O to P and The angle is positive when measured counterclockwise and negative when measured clockwise. When is measured in radians, we will generally indicate that P has as polar coordinates by writing this borrows a notation from Complex Variables courses. (See Supplementary Problem 6.)

For example, if the distance between O and M is two units and the angle (measured counterclockwise) from ray OU to ray OM is we write [See Figure 1a.] Similarly, if the distance between O and N is ½ and the (clockwise) angle from ray OU to ray ON is we write [See Figure 1b.] The origin O has zero as its

r-coordinate and any angle may be chosen as its angle; thus for all real numbers

Figure 1a

Figure 1b