### Chapter 15 - Complex Numbers

The complex numbers are important for several reasons. First, the real numbers are incomplete. Many mathematical expressions such as arcsin (2), ln (−1) and have no meaning over the real numbers and many polynomials cannot be factored over the real numbers. But if we include complex numbers in our number system then these expressions all have values and every polynomial can be factored. The complex numbers complete the number system.

Second, many real number problems have real solutions which cannot be found by using real arithmetic but can be found easily via complex arithmetic. The reason is that complex numbers add another dimension to the number system, thus opening up other routes to the solution.

Finally complex numbers lead to remarkable insights. For example complex numbers show that there is a very close connection between sinusoidal oscillations and exponential decay in a physical system.

This chapter contains the following sections:
• section 15.1 - In this section we introduce complex numbers in rectangular form; how to add, subtract, multiply and divide them, and how to graph them in the complex plane.

• section 15.2 - In this section we talk about complex numbers in polar form and exponential form and how to do arithmetic with them.