Engineering Circuit Analysis: International Student Version, Tenth Edition

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 ...