1
A Review of Complex Numbers
Ilya Pollak
ECE 301 Signals and Systems Section 2, Fall 2010
Purdue University
A complex number is represented in the form
z = x + jy,
where x and y are real numbers satisfying the usual rules of addition and multiplication,
and the symbol j, called the imaginary unit, has the property
j 2 = −1.
The numbers x and y are called the real and imaginary part of z, respectively, and are
denoted by
x = ℜ(z), y = ℑ(z).
We say that z is real if y = 0, while it is purely imaginary if x = 0.
Example 0.1. The complex number z = 3 + 2j has real part 3 and imaginary part 2,
while the real number 5 can be viewed as the complex number z = 5 + 0j whose real part
is 5 and imaginary part is 0.
Geometrically, complex numbers can be represented as vectors in the plane (Fig. 1).
We will call the xy-plane, when viewed in this manner, the complex plane, with the
x-axis designated as the real axis, and the y-axis as the imaginary axis. We designate
the complex number zero as the origin. Thus,
x + jy = 0 means x = y = 0.
In addition, since two points in the plane are the same if and only if both their x- and
y-coordinates agree, we can define equality of two complex numbers as follows:
x1 + jy1 = x2 + jy2 means x1 = x2 and y1 = y2 .
ℑ
y
"
"
"
R "
"
"
θ
"
"
x
z
ℜ
Figure 1. A complex number z can be represented in Cartesian coordinates (x, y) or polar coordinates
(R, θ).
2
Thus, we see that a single statement about the equality of two complex quantities
actually contains two real equations.
Definition 0.1 (Complex Arithmetic). Let z1 = x1 + jy1 and z2 = x2 + jy2 . Then we
define:
(a) z1 ± z2 = (x1 ± x2 ) + j(y1 ± y2 );
(b) z1 z2 = (x1 x2 − y1 y2 ) + j(x1 y2 + x2 y1 );
(c) for z2 6= 0, w =
z1
z2
is the complex number for which z1 = z2 w.
Note that, instead of the Cartesian coordinates x and y, we could use polar coordinates to represent points in the plane. The polar coordinates are radial distance R and
angle θ, as illustrated in Fig. 1. The relationship between the two sets of coordinates
is:
x = R cos θ,
y = R sin θ,
p
x2 + y 2 = |z|,
R =
y θ = arctan
.
x
Note that R is called the modulus, or the absolute value of z, and it alternatively
denoted |z|. Thus, the polar representation is:
z = |z| cos θ + j|z| sin θ = |z|(cos θ + j sin θ).
Definition 0.2 (Complex Exponential Function). The complex exponential function,
denoted by ez , or exp(z), is defined by
ez = ex+jy = ex (cos y + j sin y).
In particular, if x = 0, we have Euler’s equation:
ejy = cos y + j sin y.
Comparing this with the terms in the polar representation of a complex variable, we
see that any complex variable can be written as:
z = |z|ejθ .
3
z = z1 z2
|z| = |z1 | · |z2 |
ℑ
θ = θ1 + θ2
z
y
" 1
"
θ2 "
""
"
θ1
"
"
ℜ
x
Figure 2. Multiplication of two complex numbers z1 = |z1 |ejθ1 and z2 = |z2 |ejθ2 (z2 is not shown).
The result is z = |z|ejθ with |z| = |z1 | · |z2 | and θ = θ1 + θ2 .
Properties of Complex Exponentials.
1 jθ
(e + e−jθ ),
2
1 jθ
sin θ =
(e − e−jθ ),
2j
|ejθ | = 1,
cos θ =
ez1 ez2
e−z
ez+2πjn
= ez1 +z2 ,
1
= z,
e
= ex (cos(y + 2πn) + j sin(y + 2πn))
= ex (cos y + j sin y) = ez , for any integer n.
DT complex exponential functions whose frequencies differ by 2π are thus identical:
ej(ω+2π)n = ejωn+2πjn = ejωn .
We have seen examples of this phenomenon before, when we discussed DT sinusoids.
It follows from the multiplication rule that
z1 z2 = |z1 |ejθ1 |z2 |ejθ2 = |z1 ||z2 |ej(θ1 +θ2 ) .
Therefore, in order to multiply two complex numbers,
• add the angles;
• multiply the absolute values.
Multiplication of two complex numbers is illustrated in Fig. 2.
Definition 0.3 (Complex Conjugate). If z = x + jy, then the complex conjugate of z
is z ∗ = x − jy (sometimes also denoted z̄).
4
ℑ
2 C 2
C z = 2j
ℑ
z = x + jy
y
"
"
"
"
"
"
"
"
b
ℜ
x
b
b
b
b
b
b
b z ∗ = x − jy
−y
1
z = 1 + j, |z| =
√
2
θ = π/4
ℜ
1
@
@ 1/z = 1/2 − j/2
@
@
@
@ z∗ = 1 − j
−1
(a)
(b)
Figure 3. (a) Complex number z and its complex conjugate z ∗ . (b) Illustrations to Example 0.2.
This definition is illustrated in Fig. 3(a). Note that, if z = |z|ejθ , then z ∗ = |z|e−jθ .
Here are some other useful identities involving complex conjugates:
ℜ(z)
=
ℑ(z)
=
|z|
=
1
(z + z ∗ ),
2
1
(z − z ∗ ),
2j
√
zz ∗ ,
=
z,
∗ ∗
(z )
∗
z = z ⇔ z is real,
(z1 + z2 )∗
=
z1∗ + z2∗ ,
(z1 z2 )∗
=
z1∗ z2∗ .
Example 0.2. Let us compute the various quantities defined above for z = 1 + j.
1. z ∗ = 1 − j.
√
√
2. |z| = √12 + 12p= 2. Alternatively,
p
√
|z| = zz ∗ = (1 + j)(1 − j) = 1 + j − j − j 2 = 2.
3. ℜ(z) = ℑ(z) = 1.
4. Polar representation: z =
√
2(cos π4 + j sin π4 ) =
√
π
2ej 4 .
5
5. To compute z 2 , square the absolute value and double the angle:
π
z 2 = 2(cos π2 + j sin π2 ) = 2j = 2ej 2 .
The same answer is obtained from the Cartesian representation:
(1 + j)(1 + j) = 1 + 2j + j 2 = 1 + 2j − 1 = 2j.
6. To compute 1/z, multiply both the numerator and the denominator by z ∗ :
z∗
1−j
1−j
1 j
1
= ∗ =
=
= − .
z
zz
(1 + j)(1 − j)
2
2 2
Alternatively, use the polar representation:
1
z
π
1
1
= √ e−j 4
j π4
2
2e
√
√ !
π 2
2
1
1
π
= √ cos −
+ j sin −
=√
−j
4
4
2
2
2
2
1 j
− .
=
2 2
2
We can check to make sure that (1/z) · z = 1: 21 − 2j (1 + j) = 12 − 2j + 2j − j2 = 1.
=
√
These computations are illustrated in Fig. 3(b).