Signals and Systems

Appendix A

Complex Numbers and Euler’s Formula

A.1 Introduction

Complex numbers allow us to solve equations for which no real solution can be found. Consider, for example, the equation

$$\begin{array}{ll}{x}^2+9=0\hfill & \left(\text{A}.1\right)\hfill \end{array}$$

which cannot be satisfied for any real number. By introducing an imaginary unit1 $j=\sqrt{-1}$ so that j2 = −1, Eqn. (A.1) can be solved to yield

$$x=\mp j3$$

Similarly, the equation

$$\begin{array}{ll}{\left(x+2\right)}^2+9=0\hfill & \left(\text{A}.2\right)\hfill \end{array}$$

has the solutions

$$\begin{array}{ll}x=-2\mp j3\hfill & \left(\text{A}.3\right)\hfill \end{array}$$

A general complex number is in the form

$$\begin{array}{ll}x=a+jb\hfill & \left(\text{A}.4\right)\hfill \end{array}$$

where a and b are real numbers. The values a and b are referred to as the real part and imaginary part of the complex number x respectively. Following notation is used for real and imaginary parts of a complex number:

$$\begin{array}{ll}a\hfill & =Re\left\{x\right\}\hfill \\ b\hfill & =Im\left\{x\right\}\hfill \end{array}$$

For two complex numbers to be equal, both their real ...