# Appendix A. COMPLEX NUMBERS

Complex numbers are typically represented in three forms: exponential, polar, or rectangular. In the exponential form a complex number A is written as

**Equation 1. **

The real quantity *z* is known as the amplitude or magnitude, the real quantity θ is called the *angle* as shown in Fig. 1, and *j* is the imaginary operator

The polar form of a complex number A, which is symbolically equivalent to the exponential form, is written as

**Equation 2. **

and the rectangular representation of a complex number is written as

**Equation 3. **

where *x* is the real part of A and y is the imaginary part of A.

The connection between the various representations of A can be seen via Euler's identity, which is

**Equation 4. **

Fig. 2 illustrates that this function in rectangular form is a complex number with a unit amplitude.

**Figure 1. The exponential form of a complex number.**

**Figure 2. A graphical interpretation ...**

Get *Engineering Circuit Analysis: International Student Version, Tenth Edition* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.