This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.
The most fundamental new idea in the study of complex numbers is
the “imaginary number”
jj. This imaginary number is defined to be the square
root of
-
1
-1:
j
2
=
-
1
.
j
2
=
-
1
.
(2)The imaginary number jj is used to build complex numbers xx and yy in the following way:
We say that the complex number zz has “real part” xx and “imaginary part” yy:
z
=
Re
[
z
]
+
j
Im
[
z
]
z
=
Re
[
z
]
+
j
Im
[
z
]
(4)
Re
[
z
]
=
x
;
Im
[
z
]
=
y
.
Re
[
z
]
=
x
;
Im
[
z
]
=
y
.
(5)In MATLAB, the variable xx is denoted by real(z), and the variable yy is
denoted by imag(z). In communication theory, xx is called the “in-phase”
component of zz, and yy is called the “quadrature” component. We call z=z=x+jyx+jy the Cartesian representation of zz, with real component xx and imaginary component yy. We say that the Cartesian pair (x,y)(x,y)codes the complex
number zz.
We may plot the complex number zz on the plane as in Figure 1. We
call the horizontal axis the “real axis” and the vertical axis the “imaginary
axis.” The plane is called the “complex plane.” The radius and angle of the
line to the point z=x+jyz=x+jy are
r
=
x
2
+
y
2
r
=
x
2
+
y
2
(6)
θ
=
tan
-
1
(
y
x
)
.
θ
=
tan
-
1
(
y
x
)
.
(7)See Figure 1. In MATLAB, rr is denoted by abs(z), and θθ is denoted by angle(z).
The original Cartesian representation is obtained from the radius rr and
angle θθ as follows:
y
=
r
s
i
n
θ
.
y
=
r
s
i
n
θ
.
(9)The complex number zz may therefore be written as
z
=
x
+
j
y
=
r
cos
θ+jrsin θ
=
r
(
cos
θ
+
j
s
i
n
θ
)
.
z
=
x
+
j
y
=
r
cos
θ+jrsin θ
=
r
(
cos
θ
+
j
s
i
n
θ
)
.
(10)The complex number cosθ+jsinθcosθ+jsinθ is, itself, a number that may be represented
on the complex plane and coded with the Cartesian pair (cosθ,sin θ)
(cosθ,sinθ). This is
illustrated in Figure 2. The radius and angle to the point z=cosθ+jsin θz=cosθ+jsinθ
are 1 and θθ. Can you see why?
The complex number cosθ+jsinθcosθ+jsinθ is of such fundamental importance
to our study of complex numbers that we give it the special symbol ejθejθ :
ejθ=cosθ+jsinθ.ejθ=cosθ+jsinθ.
(11)As illustrated in Figure 2, the complex number ejθejθ has radius 1 and angle
θθ. With the symbol ejθejθ, we may write the complex number zz as
We call z=rejθz=rejθ a polar representation for the complex number zz. We say that
the polar pair r∠θr∠θcodes the complex number zz. In this polar representation,
we define |z|=r|z|=r to be the magnitude of zz and arg(z)=θarg(z)=θ to be the angle, or
phase, of zz:
a
r
g
(
z
)
=
θ
.
a
r
g
(
z
)
=
θ
.
(14)With these definitions of magnitude and phase, we can write the complex
number zz as
z=|z|ejarg(z).z=|z|ejarg(z).
(15)Let's summarize our ways of writing the complex number z and record
the corresponding geometric codes:
z
=
x+jy
=
r
e
j
θ
=
|
z
|
e
j
arg
(
z
)
.
↓
↓
(
x
,
y
)
r
∠
θ
z
=
x+jy
=
r
e
j
θ
=
|
z
|
e
j
arg
(
z
)
.
↓
↓
(
x
,
y
)
r
∠
θ
(16)In "Roots of Quadratic Equations" we show that the definition
ejθ=cosθ+jsinθejθ=cosθ+jsinθ is more than
symbolic. We show, in fact, that
ejθejθ is just the familiar function
e
x
e
x
evaluated
at the imaginary argument x=jθx=jθ. We call ejθejθ a “complex exponential,”
meaning that it is an exponential with an imaginary argument.
Prove (j)2n=(-1)n(j)2n=(-1)n and
(j)2n+1=(-1)nj(j)2n+1=(-1)nj. Evaluate j3,j4,j5j3,j4,j5.
Prove ej[(π/2)+m2π]=j,ej[(3π/2)+m2π]=-j,ej(0+m2π)=1ej[(π/2)+m2π]=j,ej[(3π/2)+m2π]=-j,ej(0+m2π)=1,
and ej(π+m2π)=-1ej(π+m2π)=-1. Plot these identities on the complex plane. (Assume mm is an integer.)
Find the polar representation z=rejθz=rejθ for each of the following
complex numbers:
- z=1+j0z=1+j0;
- z=0+j1z=0+j1;
- z=1+j1z=1+j1;
- z=-1-j1z=-1-j1.
Plot the points on the complex plane.
Find the Cartesian representation z=x+jyz=x+jy for each of the
following complex numbers:
- z=2ejπ/2z=2ejπ/2 ;
- z=2ejπ/4z=2ejπ/4;
- z=ej3π/4z=ej3π/4 ;
- z=2ej3π/2z=2ej3π/2.
Plot the points on the complex plane.
The following geometric codes represent complex numbers. Decode each by writing down the corresponding complex number z:
- (0.7,-0.1)z=(0.7,-0.1)z= ?
- (-1.0,0.5)z=(-1.0,0.5)z= ?
- 1.6∠π/8z=1.6∠π/8z=?
- 0.4∠7π/8z=0.4∠7π/8z=?
Show that
Im[jz]= Re [z]Im[jz]= Re [z] and Re [-jz]= Im [z] Re [-jz]= Im [z].
Demo 1.1 (MATLAB). Run the following MATLAB program in order to
compute and plot the complex number ejθejθ for θ=i2π/360,i=1,2,...,360θ=i2π/360,i=1,2,...,360:
j=sqrt(-1)
n=360
for i=1:n,circle(i)=exp(j*2*pi*i/n);end;
axis('square')
plot(circle)
Replace the explicit for loop of line 3 by the implicit loop
circle=exp(j*2*pi*[1:n]/n);
to speed up the calculation. You can see from
Figure 3 that the complex number
ejθejθ,
evaluated at angles
θ=2π/360,2(2π/360),...θ=2π/360,2(2π/360),..., turns out complex
numbers that lie at angle
θθ and radius 1. We say that
ejθejθ is a complex number
that lies on the unit circle. We will have much more to say about the unit
circle in Chapter 2.
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book as a "required primary textbook." This text attempts to lower the barriers for students that take courses such as Principles of Electrical Engineering, […]"