Complex Numbers
Complex numbers are some of the most general numbers used in algebra. Any number
that can be expressed in the form a + bi, where a and b are real numbers and
i2 = -1 is a complex number. This may be confusing to anyone unfamiliar with this
definition, since for example, many calculators cannot compute the square root of -1.
This is because most calculators only do real arithmetic. The existence of i is of
fundamental importance to mathematics. Engineers often use the "j" notation, j = i.
Complex numbers are not an abstraction only of use in theoretical mathematics.
Without complex numbers, many polynomial equations would have no solution.
One very simple example is x2 = -1. The solutions to this equation (x =+ i) cannot be
represented by a real number. Complex numbers have many applications in applied
mathematics, physics and engineering.
A complex number can be thought of as a two dimensional vector (a,b), where a is the
real part and b is the imaginary part. The term "imaginary" is an unfortunate misnomer
left over from the 17th century when mathematicians were still uncomfortable with the
concept of complex numbers. The imaginary part is every bit as real as the real part of the
complex number.
Sometimes complex numbers are represented in the standard form a + bi.
Another common representation is the polar or trigonometric form
(a,b) = z = r [cos(θ) + i sin(θ)]
As shown in the following figure
(a=
r cosθ, b= r sinθ)
Imaginary
b
r
θ
Real
a
Polar Representation of the Complex Number a+bi
The magnitude of a complex number is the square root of the sum of the squares of its
real and imaginary parts:
r = ( a, b) = a 2 + b 2
The phase of a complex number is the inverse tangent of the quotient formed by dividing
the imaginary part by the real part
θ = arctan(b / a) = tan −1 (b / a)
where a = r cos(θ) and b = rsin(θ).
The complex conjugate of the complex number (a, b) is (a ,-b). The complex conjugate
is sometimes denoted as (a , b)* where
(a , b)* = (a ,-b)
Complex Arithmetic. The arithmetic of complex numbers is defined as follows:
addition
subtraction
multiplication
division
Powers of i.
(a,b) + (c,d) = (a+c, b+d)
(a,b) - (c,d) = (a-c, b-d)
(a,b) X (c,d) = (ac - bd, ad - bc)
(a,b) / (c,d) = (ac + bd, ad + bc)/(c2+d2) (c & d not both zero)
Some important identities involve the powers of i:
i1 = i5 = i9 = ... = i
i2 = i6 = i10 = ... = -1
i3 = i7 = i11 = ... = -i
i4 = i8 = i12 = ... = +1.
Note that when the powers of i are simplified, they cycle in steps of four.
Powers and Roots of Complex Numbers
Both the nth power and the nth root of a complex number are also complex numbers,
which are best represented in polar form:
nth power [r (cosө + i sinө)]n = r n [cos(nө) + i sin(nө)]
nth root
r 1/n {cos[(ө+2πk)/n] + i sin[(ө+2πk)/n]} k = 0,1,…n-1
As with any polar representation, both the nth power and the nth root are periodic with a
period of 360o. This is a direct consequence of the periodicity of the sine and cosine
functions. That is, suppose ө is the angle (in degrees) that corresponds to the nth root (or
power) of z. Then for any integer, k, the angle ө + k360 also corresponds to the root (or
power).
This ambiguity leads to the definition of a principal nth root of a complex number.
A principal nth root of z = r cosθ + i rsinθ is a root with a polar angle between -180oand
+180o:
-180o < θ < +180o
There are always n unique principal nth roots of a complex number. For example, there
are three principal cube roots of 2i. For this number θ/n = 90o/3 = 30o and the roots are:
21/3 [cos(30o) + i sin(30o)]
21/3 [cos(150o) + i sin(150o))]
21/3 [cos(270o) + i sin(270o)].
All other cube roots are redundant.
Complex Numbers in Communications Engineering
j = i = −1
± jθ
e
= cos θ ± j sin θ
Euler’s Formula:
Proof: Substitute complex arguments into a real-valued Taylor Series expansion (why?):
cos(θ ) = 1
−
θ2
2!
± j sin(θ ) = ± jθ
e ± jθ = 1 ± jθ −
+
m j
θ2
2!
m j
θ4
4!
θ3
± j
3!
θ3
3!
Useful Related Expressions:
−
+
θ4
4!
± j
θ6
θ5
m j
5!
θ5
5!
+ ....
6!
−
θ6
6!
m j
θ7
7!
θ7
7!
+ ....
+ ....
e jθ − e − jθ
sin θ =
2j
e jθ + e − jθ
cos θ =
,
2
Principal Nth roots of a complex number, z = r (cosθ + j sin θ ) -180o < θ < +180o
z 1 / N = [r (cos θ + j sin θ )]
1/ N
⎧ ⎛ θ + 2kπ ⎞
⎛ θ + 2kπ
= r 1 / N ⎨cos⎜
⎟ + j sin⎜
⎝ N
⎩ ⎝ N ⎠
⎞⎫
⎟⎬ for k = 0,1,2,…N-1
⎠⎭
Imaginary
Z-Plane
Nth Roots of
z = rejθ = 1
Polar Circle
r=1
π/8
Real
Principal Nth roots of unity
(N= 16 for this example)
Magnitude and Phase of a Complex Phasor (number)
Let z = re
jθ
, then the magnitude and phase of z are given by
Magnitude[z] =
{Real[z ]}2 + {Imag[z ]}2 = r
⎛ Imag( z ) ⎞
jθ
1
⎟⎟ (in general)
⎜⎜
Phase[z] = Phase[re ] = θ = tan 2−1ARG [Real( z ), Imag( z )] ≠ tan 1−ARG
⎝ Real( z ) ⎠
where -π < θ < π
θ
QUADRANT II
π
θ = Arctan2ARG[Real(z), Imag(z)]
π
2
QUADRANT I
x=
0
2
QUADRANT IV
1
1
Imag(z)
Real(z)
2
π
2
QUADRANT III
π
Two Argument Arctangent function
For z as specified above, note that the two argument arctangent is related to the single
argument arctangent according to:
⎧ −1 ⎛ Imag( z ) ⎞
⎟⎟
⎪ tan 1 ARG ⎜⎜
Real
(
z
)
⎝
⎠
⎪
⎪
⎛
⎞
Imag
(
z
)
1
⎜⎜
⎟⎟ + π
θ = tan 2−1ARG {Real( z ), Imag( z )} = ⎨tan 1−ARG
⎝ Real( z ) ⎠
⎪
⎪ −1 ⎛ Imag( z ) ⎞
⎟⎟ − π
⎪tan 1 ARG ⎜⎜
Real
(
z
)
⎝
⎠
⎩
for z ∈ Quadrant I or IV
for z ∈ Quadrant II
for z ∈ Quadrant III
For example, consider the 4th roots of z = -1 shown in the following figure:
Imaginary
Z-Plane
QUADRANT II
QUADRANT I
[cos(135o),sin (135o)]
[cos(45o),sin(45o)]
Unit Circle
45o
Real
[cos(45o),sin(-45o)]
[cos(-135o),sin (-135o)]
QUADRANT III
QUADRANT IV
4th roots of z=exp(jθ)
(θ=180o and N= 4 for this example)
QUAD I: Atan2arg[cos(45o),sin(45o)] =Atan1arg[sin(45o)/cos(45o)] = 45o
QUAD IV: Atan2arg[cos(-45o),sin(-45o )] =Atan1arg[sin(-45o)/cos(-45o)] = -45o
QUAD II: Atan2arg[cos(135o),sin(135o)] = Atan1arg[sin(135o)/cos(135o)] + 180o = 135o
QUAD III: Atan2arg[cos(-135o),sin(-135o)] = Atan1arg[sin(-135o)/cos(-135o)] - 180o= -135o
jθ
Phase of a purely real number: z = re =r [cos(θ ) + j 0]
jθ
Phase[z] = Phase[re ] ; z is real => sin(θ ) = 0. Two cases:
A in figure below: θ = 0 for z positive, since cos(0) = 1
C in figure below: θ = π for z negative, since cos(π) = −1
(0, +1)
Imaginary
(cos θ, sin θ)
B
Z-Plane
θ
(−1 , 0)
C
(+1 , 0)
A
(0, −1)
Real
Unit Circle
D
Phase Angle (θ) and the Unit Circle
jθ
Phase of a purely imaginary number: z = re =r [0 + j sin(θ )]
jθ
Phase[z] = Phase[re ] ; z is purely complex => cos(θ ) = 0. Two cases:
B in figure above: θ = π/2 for z/j positive, since sin(π/2) = +1
D in figure above: θ = −π/2 for z/j negative, since sin(−π/2) = −1
Phase of the product of two complex numbers is the sum of the phases of the
individual numbers:
jθ1
jθ2
j(θ1+θ2)
Phase[z1z2] = Phase[r1e
r2e ] = Phase[r1 r2e
]= θ1 +θ2 mod (-π, π]
Phase of the quotient of two complex numbers is the difference between the phase of
the numerator and the phase of the denominator:
jθ1
jθ2
j(θ1−θ2)
]= θ1 −θ2 mod (-π, π]
Phase[z1/z2] = Phase[r1e /r2e ] = Phase[r1/r2e
1
Magnitude[1/(a+jb)] =
2
a + b2
Phase[1/(a+jb)] = -tan-1(b/a)
Phase[(jω) n] = n tan-1(∞) = nπ/2;
Phase[(jω) -n] = n tan-1(-∞) = -nπ/2
The product of complex conjugate roots as a quadratic with real coefficients
(s+a+jb) (s+a-jb) = [s2+2as+(a2+b2)]
Good Practice Problems:
Derive the Magnitude and Phase of the following functions of ω and plot both the
Magnitude and Phase functions on the ω axis using the same logarithmic scale
for 0< ω <∞. These are often referred to as Bode plots.
a) jω
c) 1+jω
b) (jω)2
c) (jω)3
d) (1+jω)2
g) (1+j3ω)/(1+j2ω)
d) 1/jω
e) 1/(1+jω)
h) (1+j2ω)/(1+j3ω)
j) jω/[(1+j2ω)(1+j3ω)]
l) exp(j2ω)
n) exp(-j4ω)jω/[(1+j2ω)(1+j3ω)]
e)1/(jω)2
d)1/(jω)3
f) 1/(1+jω)2
i) jω/(1+jω)
m) exp(-j3ω)
o) X(jω) = 2sin(ωT)/ ω
X(jω) = 2sin(ωT)/ω
Magnitude
φ(ω) =
Phase
π
2sin(ωT)/ω
ω
0
−π /T
π /T