# 1.6 Review of Complex Numbers

Any complex number z can be expressed in rectangular form as

z=x+j y,(1.35)

where x and y are the real (R e) and imaginary (I m) parts of z, respectively, and j=−1. That is,

x=R e(z),y=I m(z).(1.36)

Alternatively, z may be cast in polar form as

z=|z|e j θ=|z|θ(1.37)

where |z| is the magnitude of z, θ is its phase angle, and θ is a useful shorthand representation for e j θ. Applying Euler’s identity,

we can convert z from polar form, as in Eq. (1.37), into rectangular form,

z=|z|e j θ=|z|cos θ+j|z|sin θ.(1.39)

This leads to the relations

The two forms are illustrated graphically in Fig. 1-18. When using Eq. (1.41), care should be taken to ensure that θ is in the proper quadrant. Also note that, since |z| is a positive quantity, only the positive root in Eq. (1.41) is applicable. This is denoted by the + sign above the square-root sign.

# Figure 1-18 Relation between rectangular and polar representations of a complex number z=x+j y=|z|e j θ.

Figure 1 18 Full Alternative Text

The complex conjugate of z, denoted with a star superscript (or asterisk), is obtained by replacing j (wherever it appears) with −j, so that

The magnitude |z| is equal to the positive square root of the product of z and its complex conjugate:

We now highlight some of the properties of complex algebra that will be encountered in future chapters.

Equality: If two complex numbers z 1 and z 2 are given by

z 1=x 1+j y 1=|z 1|e j θ 1,(1.44)
z 2=x 2+j y 2=|z 2|e j θ 2,(1.45)

then z 1=z 2 if and only if x 1=x 2 and y 1=y 2 or, equivalently, |z 1|=|z 2| and θ 1=θ 2.

Addition:

z 1+z 2=(x 1+x 2)+j(y 1+y 2).(1.46)

Multiplication:

z 1 z 2=(x 1+j y 1)(x 2+j y 2)=(x 1 x 2−y 1 y 2)+j(x 1 y 2+x 2 y 1), (1.47a)

or

z 1 z 2=|z 1|e j θ 1⋅|z 2|e j θ 2=|z 1||z 2|e j(θ 1+θ 2)=|z 1||z 2|[cos(θ 1+θ 2)+j sin(θ 1+θ 2)]. (1.47b)

Division: For z 2≠0,

z 1 z 2=x 1+j y 1 x 2+j y 2=(x 1+j y 1)(x 2+j y 2)⋅(x 2−j y 2)(x 2−j y 2)=(x 1 x 2+y 1 y 2)+j(x 2 y 1−x 1 y 2)x 2 2+y 2 2, (1.48a)

or

z 1 z 2=|z 1|e j θ 1|z 2|e j θ 2=|z 1||z 2|e j(θ 1−θ 2)=|z 1||z 2|[cos(θ 1−θ 2)+j sin(θ 1−θ 2)]. (1.48b)

Powers: For any positive integer n,

z n=(|z|e j θ)n=|z|n e j n θ=|z|n(cos n θ+j sin n θ), (1.49)
z 1/2=±|z|1/2 e j θ/2=±|z|1/2[cos(θ/2)+j sin(θ/2)]. (1.50)

Useful Relations:

−1=e j π=e−j π=1 180°,j=e j π/2=1 90°, (1.51)
−j=−e j π/2=e−j π/2=1−90°,(1.52)
j=(e j π/2)1/2=±e j π/4=±(1+j)2,(1.53)
−j=±e−j π/4=±(1−j)2.(1.54)

# Example 1-3: Working with Complex Numbers

Given two complex numbers

V=3−j 4,I=−(2+j 3),

(a) express V and I in polar form, and find (b) V I, (c) V I∗, (d) V/I, and (e) I.

Solution:

(a) |V|=V V∗+=(3−j 4)(3+j 4)+=9+16+=5,θ V=tan−1(−4/3)=−53.1°,V=|V|e j θ V=5 e−j 53.1°=5−53.1°,|I|=2 2+3 2+=13+=3.61.

Since I=(−2−j 3) is in the third quadrant in the complex plane (Fig. 1-19),

θ I=180°+tan−1 3 2=236.3°,I=3.61 236.3°.

(b) V I=5 e−j 53.1°×3.61 e j 236.3°=18.03 e j(236.3°−53.1°)=18.03 e j 183.2°.

(c) V I∗=5 e−j 53.1°×3.61 e−j 236.3°=18.03 e−j 289.4°=18.03 e j 70.6°.

(d) V I=5 e−j 53.1°3.61 e j 236.3°=1.39 e−j 289.4°=1.39 e j 70.6°.

(e) I=3.61 e j 236.3°=±3.61 e j 236.3°/2=±1.90 e j 118.15°.

# Figure 1-19 Complex numbers V and I in the complex plane (Example 1-3).

Figure 1 19 Full Alternative Text