A complex number z can be defined as:

where x and y are real numbers and i satisfies the equation i2 = -1.  The real numbers x and y are called the real and imaginary parts of the complex number z.

Complex numbers are often presented as a point or a vector on the two-dimensional complex plane (sometimes called an Argand diagram) shown in Figure 1.  The coordinates of the complex number z are the real part x = Re(z) and the imaginary part y = Im(z).  The magnitude or modulus of the complex number z is the length of the vector from the origin to the point (x,y):

The angle or argument of z is the angle q between the z vector and the real axis:

where q is measured in radians and is positive for a counterclockwise rotation from the positive real axis.  The real and imaginary coordinates can be derived from the magnitude and angle using the following:

Thus a complex number can be written in many different forms.  The familiar rectangular form is:

A common polar form is:

Through the Euler formula, a complex number may also be expressed in terms of a complex exponential, in what is sometimes referred to as “phasor” form:

Figure 1. Complex z plane

References

Greenberg, Michael D. (1998), Advanced Engineering Mathematics, Prentice-Hall (New Jersey).

Ogata, Katsuhiko (1998), System Dynamics (Third Edition), Prentice-Hill (New Jersey).

Spiegel, Murray R. (1964), Schaum’s Outline of Theory and Problems of Complex Variables, McGraw-Hill (New York).