Technical Mathematics, Sixth Edition

Chapter 19. Complex Numbers

OBJECTIVES When you have completed this chapter, you should be able to Write complex numbers in rectangular, polar, trigonometric, and exponential forms. Find the sums, differences, products, quotients, and powers of complex numbers. Find components and resultants of vectors using complex numbers. Solve alternating current applications using complex numbers.

Up to now we have avoided square roots of negative numbers, expressions like . We deal with them here by introducing imaginary numbers and complex numbers. We will show that an imaginary number is not imaginary and a complex number is not complicated, as its unfortunate name implies, but actually simplifies many computations.

A complex number can be written in many different forms, each with its own advantages. They are easily manipulated by calculator, which will greatly simplify our work. We will do the usual operations, addition, subtraction, multiplication, and so forth.

A graph of a complex number will show how the different forms of a complex number are related, and why they can be used to represent vectors. We will repeat the operations with vectors we did in our trigonometry chapters, resolving a vector into components, and finding the resultant of several vectors. For example, finding the resultant of the vectors in Fig. 19-1 can be a long process, but we will see later how complex numbers ...