The complex numbers
z
1
z
1
and
z
2
z
2
are
added according to the rule
z
1
+
z
2
=
(
x
1
+
j
y
1
)
+
(
x
2
+
j
y
2
)
=
(
x
1
+
x
2
)
+
j
(
y
1
+
y
2
)
.
z
1
+
z
2
=
(
x
1
+
j
y
1
)
+
(
x
2
+
j
y
2
)
=
(
x
1
+
x
2
)
+
j
(
y
1
+
y
2
)
.
(1)We say that the real parts add and the imaginary parts add. As illustrated
in Figure 1, the complex number z1+z2z1+z2 is computed from a “parallelogram
rule,” wherein z1+z2z1+z2 lies on the node of a parallelogram formed from
z1
z1
and
z
2
z
2
.
Let
z1=r1ejθ1z1=r1ejθ1 and z2=r2ejθ2z2=r2ejθ2. Find a polar formula z3=z3=r3ejθ3r3ejθ3 for z3=z1+z2z3=z1+z2 that involves only the variables r1,r2,θ1r1,r2,θ1, and
θ
2
θ
2
. The formula for
r
3
r
3
is the “law of cosines.”
The product of z1 and z2 is
z
1
z
2
=
(
x
1
+
j
y
1
)
(
x
2
+
j
y
2
)
=
(
x
1
x
2
-
y
1
y
2
)
+
j
(
y
1
x
2
+
x
1
y
2
)
.
z
1
z
2
=
(
x
1
+
j
y
1
)
(
x
2
+
j
y
2
)
=
(
x
1
x
2
-
y
1
y
2
)
+
j
(
y
1
x
2
+
x
1
y
2
)
.
(2)If the polar representations for
z
1
z
1
and
z
2
z
2
are used, then the product may be
written as
z
1
z
2
=
r
1
e
j
θ
1
r
2
e
j
θ
2
=
(
r
1
cos
θ1+
j
r
1
sin
θ
1
)
(
r
2
cos
θ
2
+
j
r
2
sin
θ
2
)
=
(
r
1
c
o
s
θ
1
r
2
c
o
s
θ
2
-
r
1
s
i
n
θ
1
r
2
s
i
n
θ
2
)
+
j
(
r
1
s
i
n
θ
1
r
2
c
o
s
θ
2
+
r
1
c
o
s
θ
1
r
2
s
i
n
θ
2
)
=
r
1
r
2
cos
(
θ
1
+
θ
2
)
+
j
r
1
r
2
sin
(
θ
1
+
θ
2
)
=
r
1
r
2
e
j
(
θ
1
+
θ
2
)
.
z
1
z
2
=
r
1
e
j
θ
1
r
2
e
j
θ
2
=
(
r
1
cos
θ1+
j
r
1
sin
θ
1
)
(
r
2
cos
θ
2
+
j
r
2
sin
θ
2
)
=
(
r
1
c
o
s
θ
1
r
2
c
o
s
θ
2
-
r
1
s
i
n
θ
1
r
2
s
i
n
θ
2
)
+
j
(
r
1
s
i
n
θ
1
r
2
c
o
s
θ
2
+
r
1
c
o
s
θ
1
r
2
s
i
n
θ
2
)
=
r
1
r
2
cos
(
θ
1
+
θ
2
)
+
j
r
1
r
2
sin
(
θ
1
+
θ
2
)
=
r
1
r
2
e
j
(
θ
1
+
θ
2
)
.
(3)We say that the magnitudes multiply and the angles add. As illustrated in
Figure 2, the product z1z2z1z2 lies at the angle (θ1+θ2)(θ1+θ2).
Rotation. There is a special case of complex multiplication that will
become very important in our study of phasors in the chapter on Phasors. When
z
1
z
1
is the
complex number
z1=r1ejθ1z1=r1ejθ1 and
z
2
z
2
is the complex number z2=ejθ2z2=ejθ2, then
the product of
z
1
z
1
and
z
2
z
2
is
z1z2=z1ejθ2=r1ej(θ1+θ2).z1z2=z1ejθ2=r1ej(θ1+θ2).
(4)As illustrated in Figure 3, z1z2z1z2 is just a rotation of
z
1
z
1
through the angle
θ
2
θ
2
.
Begin with the complex number z1=x+jy=rejθz1=x+jy=rejθ. Compute
the complex number z2=jz1z2=jz1 in its Cartesian and polar forms. The complex
number
z
2
z
2
is sometimes called perp(z1)(z1). Explain why by writing perp(z1)(z1) as
z1ejθ2z1ejθ2. What is
θ
2
θ
2
? Repeat this problem for z3=-jz1z3=-jz1.
Powers. If the complex number
z
1
z
1
multiplies itself NN times, then the
result is
(z1)N=r1NejNθ1.(z1)N=r1NejNθ1.
(5)This result may be proved with a simple induction argument. Assume z1k=r1kejkθ1z1k=r1kejkθ1. (The assumption is true for k=1.k=1.) Then use the recursion z1k+1=z1kz1=r1k+1ej(k+1)θ1z1k+1=z1kz1=r1k+1ej(k+1)θ1. Iterate this recursion (or induction) until k+1=Nk+1=N.
Can you see that, as n ranges from n=1,...,Nn=1,...,N, the angle of z§ranges from
θ1 to 2θ1,...2θ1,..., to Nθ1Nθ1 and the radius ranges from r1 to r12,...r12,..., to r1Nr1N ? This
result is explored more fully in Problem 1.19.
Complex Conjugate. Corresponding to every complex number z=z=
x+jy=rejθx+jy=rejθ is the complex conjugate
z*=x-jy=re-jθ.z*=x-jy=re-jθ.
(6)The complex number zz and its complex conjugate are illustrated in Figure 4.
The recipe for finding complex conjugates is to “change jto-jjto-j. This changes
the sign of the imaginary part of the complex number.
Magnitude Squared. The product of z and its complex conjugate is
called the magnitude squared of z and is denoted by |z|2|z|2 :
|z|2=z*z=(x-jy)(x+jy)=x2+y2=re-jθrejθ=r2.|z|2=z*z=(x-jy)(x+jy)=x2+y2=re-jθrejθ=r2.
(7)Note that |z|=r|z|=r is the radius, or magnitude, that we defined in "Geometry of Complex Numbers".
Write
z
*
z
*
as z*=zwz*=zw. Find
ww in its Cartesian and polar forms.
Prove that angle (z2z1*)=θ2-θ1(z2z1*)=θ2-θ1.
Show that the real and imaginary parts of z=x+jyz=x+jy may be written as
Re
[
z
]
=
1
2
(
z
+
z
*
)
Re
[
z
]
=
1
2
(
z
+
z
*
)
(8)
Im
[
z
]
=
2
j
¯
(
z
-
z
*
)
.
Im
[
z
]
=
2
j
¯
(
z
-
z
*
)
.
(9)
Commutativity, Associativity, and Distributivity. The complex numbers commute, associate, and distribute under addition and multiplication
as follows:
z
1
+
z
2
=
z
2
+
z
1
z
1
z
2
=
z
2
z
1
z
1
+
z
2
=
z
2
+
z
1
z
1
z
2
=
z
2
z
1
(10)
(
z
1
+
z
2
)
+
z
3
=
z
1
+
(
z
2
+
z
3
)
z
1
(
z
2
z
3
)
=
(
z
1
z
2
)
z
3
z
1
(
z
2
+
z
3
)
=
z
1
z
2
+
z
1
z
3
.
(
z
1
+
z
2
)
+
z
3
=
z
1
+
(
z
2
+
z
3
)
z
1
(
z
2
z
3
)
=
(
z
1
z
2
)
z
3
z
1
(
z
2
+
z
3
)
=
z
1
z
2
+
z
1
z
3
.
(11)Identities and Inverses. In the field of complex numbers, the complex number 0+j00+j0 (denoted by 0) plays the role of an additive identity, and
the complex number 1+j01+j0 (denoted by 1) plays the role of a multiplicative
identity:
z
+
0
=
z
=
0
+
z
z
1
=
z
=
1
z
.
z
+
0
=
z
=
0
+
z
z
1
=
z
=
1
z
.
(12)
In this field, the complex number -z=-x+j(-y)-z=-x+j(-y) is the additive inverse
of z, and the complex number z-1=xx2+y2+j(-yx2+y2)z-1=xx2+y2+j(-yx2+y2) is the multiplicative
inverse:
z
+
(
-
z
)
=
0
z
z
-
1
=
1
.
z
+
(
-
z
)
=
0
z
z
-
1
=
1
.
(13)
Show that the additive inverse of z=rejθz=rejθ may be written as
rej(θ+π).rej(θ+π).
Show that the multiplicative inverse of
z
z may be written as
z
-
1
=
1
z
*
z
z
*
=
1
x
2
+
y
2
(
x
-
jy
)
.
z
-
1
=
1
z
*
z
z
*
=
1
x
2
+
y
2
(
x
-
jy
)
.
(14)
Show that
z*zz*z is real. Show that
z-1z-1 may also be written as
z
-
1
=
r
-
1
e
-
j
θ
.
z
-
1
=
r
-
1
e
-
j
θ
.
(15)
Plot
zz and
z-1z-1 for a representative
zz.
Find z-1z-1 when z=1+j1z=1+j1.
Prove (z-1)*=(z*)-1=r-1ejθ=1z*zz(z-1)*=(z*)-1=r-1ejθ=1z*zz. Plot zz and (z-1)*(z-1)*
for a representative zz.
Find all of the complex numbers
zz with the property that
jz=-z*jz=-z*. Illustrate these complex numbers on the complex plane.
Demo 1.2 (MATLAB). Create and run the following script file (name it Complex Numbers)
clear, clg
j=sqrt(-1)
z1=1+j*.5,z2=2+j*1.5
z3=z1+z2,z4=z1*z2
z5=conj(z1),z6=j*z2
avis([-4 4 -4 4]),axis('square'),plot(z1,'0')
hold on
plot(z2,'0'),plot(z3,'+'),plot(z4,'*'),
plot(z2,'0'),plot(z3,'+'),plot(z4,'*'),
plot(z5,'x'),plot(z6,'x')
With the help of Appendix 1, you should be able to annotate each line of this
program. View your graphics display to verify the rules for add, multiply,
conjugate, and perp. See Figure 5.
Prove that
z
0
=
1
z
0
=1.
(MATLAB) Choose
z
1
=
1.05
e
j
2
π
/
16
z
1
=1.05
e
j
2
π
/
16
and
z
2
=
0.95
e
j
2
π
/
16
z
2
=0.95
e
j
2
π
/
16
.
Write a MATLAB program to compute and plot
z
1
n
z
1
n
and
z
2
n
z
2
n
for
n
=
1
,
2
,
...
,
32
.
n=1,2,...,32.
You should observe a figure like Figure 6.
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