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15.2.1 Complex Numbers and Vectors

It is also possible to represent a point on a two dimensional plane in terms of polar coordinates. Polar coordinates are given in terms of:

1. the length, , of the (directional) line from the origin to the point, and
2. the counterclockwise angle, , that the line makes with the reference axis, namely the positive horizontal axis.

Thus, a complex number, represented by the pair in polar coordinates, can be written in rectangular coordinates as:

Thus, the real and imaginary parts, and , in terms of and are:

Since, can also be written as: As we shall soon see, this exponential form is convenient for multiplication and division.

Given rectangular coordinates and , we can determine and as follows. We know:

so,

Observe that the length, , is the square root of . The length is called the magnitude of the vector, and the angle is called the angle or phase angle of the vector.

As we have seen, addition and subtraction of complex numbers is easy to perform in rectangular coordinates. On the other hand, multiplication and division of two complex numbers in rectangular coordinates is not so easy. Conversely, it is easy to perform multiplication and division in polar coordinates. Given that two numbers are:

It is easy to see that:

From this analysis, we can implement complex numbers in polar coordinates as shown in Figure 15.11 together with functions for multiplication and division in polar coordinates. It is also important to be able to convert back and forth between rectangular and polar coordinates. It is easy to write the necessary conversion routines to convert complex numbers in rectangular coordinates to polar coordinates, and vice versa - they are shown in Figure 15.12.

The function polar_to_rect() is quite straight forward; rect_to_polar() uses the arc tangent function atan() defined in the standard library. This function returns an angle in the range to , thus we need to adjust the angle when the real part is zero and when it is negative. If the real part is zero, the angle is if the imaginary part is positive, and if it is negative. Next, if the real part is negative, the angle must be incremented by . Since we use many standard library trigonometric functions, the file math.h must be included at the head of computil.c and we must link the math library when the program is compiled.

These functions provide a useful library for processing with complex numbers. Let us now make use of them in two application programs.