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Complex Algebra Review
Dr. V. Këpuska
Complex Algebra Elements
 Definitions:
j  1
R : Set of all Real Numbers
Ι : Set of all Imaginary Numbers
C : Set of all Complex Numbers
If x,y  R then z  x  jy

C
Cartezian form
of a complex
number

Note: Real numbers can be thought of as complex numbers with imaginary
part equal to zero.
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Complex Algebra Elements
If x  0  z  jy  I
If y  0  z  x  R
If z  x  jy then we define
x  Rez  Real part of z
y  Imz Imaginary part of z
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Euler’s Identity
e
j
 cos   j sin 
j
 j
e e
cos  
j
e  cos   j sin 
2

j
 j
 j
e e
e  cos   j sin 
cos  
2j
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Polar Form of Complex Numbers
z  re

j
r R
r  0
  (- ,  ] radians
z r
Magnitude of z
arg z  z  
Angle (or argument) of z
Magnitude of a complex number z is a generalization of the absolute value
function/operator for real numbers. It is buy definition always non-negative.
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Polar Form of Complex Numbers
 Conversion between polar and
rectangular (Cartesian) forms.
z  re j  x  jy
r cos   jsin    x  jy
rcos   jrsin    x  jy
r  x2  y2 
 x  rcos  



 
1  y  
 y  rsin     tan  
 x 

 For z=0+j0; called “complex zero” one can not define
arg(0+j0). Why?
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Geometric Representation of
Complex Numbers.
Im
z
Im{z}
Q2
Axis of
Imaginaries
Q1
Axis of
Reals

Re{z}
Q3
Re
Q4
Complex or
Gaussian plane
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Geometric Representation of
Complex Numbers.
Condition 1
Q1 or Q2
Arg{z} ≥ 0
Q3 or Q4
Arg{z} ≤ 0
Q1 or Q4
Re{z} ≥ 0
Q2 or Q3
Re{z} ≤ 0
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Condition 2
Im{z} ≥ 0
Im
Q2
Im{z}
Complex
Number in
Quadrant
Im{z} ≤ 0
Axis of
Imaginarie
s z
Q1
Axis of
Reals

Re{z}
Q3
Re
Q4
Complex or
Gaussian plane
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Example
Im
z1
1
z2
-2
-1
z3
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Re
-1
z1  2
z1  1  j1  {
3
z1 
4
z2  2
z 2  2  j 0  {
z 2  
z3  2
z3  1  j1  {
3
z 3  
4
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Conjugation of Complex Numbers
 Definition: If z = x+jy ∈ C then z* = x-jy is called
the “Complex Conjugate” number of z.
 Example: If z=ej (polar form) then what is z* also
in polar form?
z  re j  r cos   jr sin   
z   r cos   jr sin  
 r cos   jr sin   
 sin      sin  
 cos    cos 
 r cos    jr sin     re  j
If z=rej then z*=re-j
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Geometric Representation of
Conjugate Numbers
 If z=rej then z*=re-j
Im
z
y

x
Re
-
-y
Complex or
Gaussian plane
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z*
11
Complex Number Operations
 Extension of Operations for Real
Numbers
 When adding/subtracting complex
numbers it is most convenient to use
Cartesian form.
 When multiplying/dividing complex
numbers it is most convenient to use
Polar form.
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Addition/Subtraction of Complex
Numbers
Let
z1  x1  jy1 , & z 2  x2  jy2
then
z1  z 2   x1  x2   j  y1  y2 
Thus :
Rez1  z 2   Rez1  Rez 2 
Imz1  z 2   Imz1  Imz 2 
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Multiplication/Division of Complex
Numbers
Olso
Let
z1 r1e j1  r1  j1  j 2

  e e
j 2
z 2 r2 e
 r2 
z1  r1e j1 & z 2  r2 e j 2
then
z1 z 2  r1e j1 r2 e j 2  r1r2 e j1 e j 2
z1 z 2  r1r2 e j 1  2 
Therefore :
z1  r1  j 1  2 
  e
z 2  r2 
Therefore :
z1 z 2  z1 z 2
z1
z1

z2
z2
 z1 z 2   z1  z 2
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 z1 
   z1  z 2
 z2 
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Useful Identities
 z ∈ C,  ∈ R & n ∈ Z (integer set)
1)
z  z
 
3)
Re z   Rez
5)
z1 z2   z1 z2
7)
z1  z2   z1  z2
8)
10)
zz   z
Rez   Rez
12)
z   z
14)
zn  z
n
 n
n 
16)
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2
z   z 
z   z
2)
 
Im z    Imz


 z1 
z1
   
z2
 z2 
4)
6)
z 
 
z
Imz   Imz
 0 if   0 
13) z     z  z  

0


if



nz   nz
15)
9)
11)
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Useful Identities
 Example: z = +j0
 =2 then arg(2)=0
 =-2 then arg(-2)=
Im
j

z
-2
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-1
0
1
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Re
16
Silly Examples and Tricks
Im
e j 0  cos0   j sin 0   1  j 0  1
e
e
e
j

2
j
3
j
2
e j 2
 
 
 cos   j sin    0  j1  j
2
2
 cos   j sin    1  j 0  1
j
 /2
-1 3/2 0
 3 
 3 
 cos   j sin    0  j  1   j
 2 
 2 
 cos2   j sin 2   1  j 0  1
j  0 j  j  e
j

j 2  jj  e 2 e

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j

2
j

2
 e j  1
j0  1
j1  j
j 2  1
j3   j
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Re
-j
j4  1
j5  j
j 6  1
j7   j
j8  1
j9  j
j10  1
j11   j
j12  1
j13  j
j14  1
j15   j
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Division Example
 Division of two complex numbers in
rectangular form.
z1  x1  jy1 , z2  x1  jy1
z1 x1  jy1 x1  jy1  x2  jy2  x1 x2  y1 y2   j x2 y1  x1 y2 



z2 x2  jy2 x2  jy2  x2  jy2 
x 22  y 22

 



z 2
z2
z1 x1 x2  y1 y2  x2 y1  x1 y2 

j 2 2
2
2
z2
x2  y2
x2  y2

 
 
 



z 
Re  1 
 z2 
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

z 
Im  1 
 z2 
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Roots of Unity
 Regard the equation:
zN-1=0, where z ∈ C & N ∈ Z+ (i.e. N>0)
 The fundamental theorem of algebra
(Gauss) states that an Nth degree algebraic
equation has N roots (not necessarily
distinct).
st
3

z

1
z

1
(
1
root ) 
1
 Example:


nd
3
3
(2 root )
 N=3; z -1=0  z =1 ⇒  z 2  ?
rd
z ?

(
3
root
)
 3

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Roots of Unity
 zN-1=0 has roots , k=0,1,..,N-1, where
k
e
k

e
 The roots of
j
j
2
N
2k
N
, k  0,1,..., N  1
are called Nth roots of unity.
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Roots of Unity
 Verification:
2k
j
N
N


e
  1  0  e j 2k  1  0




Applying Eulers Identity
e j 2k  cos2k   j sin 2k 
cos2k   j sin 2k   1  0
cos2k   j sin 2k   1  j 0
cos2k   1

 wich is true for k  0,1,..., N  1
sin 2k   0
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Geometric Representation
N 3
 e
2
2 1
j
3
j
2 2
3
1
e
e
2
j
3
j
4
3
k 0
Im
k 1
k 2
-1
J2
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j1
J1
2/3
 e
1
2 0
3
4/3
0  e
j
J0
0
1
Re
-j1
22
Important Observations
1.
Magnitude of each root are equal to 1. Thus, the Nth roots of
unity are located on the unit circle. (Unit circle is a circle on the
complex plane with radius of 1).
|e
2.
2k
N
| 1, k
The difference in angle between two consecutive roots is 2/N.
k 1  k  
3.
j
k 1

1


 e
k

j
2
N

2
N
 Q.E.D
The roots, if complex, appear in complex-conjugate pairs. For
example for N=3, (J1)*=J2. In general the following property
holds: JN-k=(Jk)*
2  N  k 
2N
2k
2k
2k
2k *
j
j
j
j
j
j


*
 N  k  e N  e N e N  e j 2 e N  1e N   e N   k 


 
 N  k  k
*
 
*
For N  3 & k  1  31  2  1  2  *
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