complex algebra review
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Complex Algebra Review. Dr. V. K ëpuska. Complex Algebra Elements. Definitions: Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. Complex Algebra Elements. Euler’s Identity. Polar Form of Complex Numbers. - PowerPoint PPT PresentationTRANSCRIPT
Complex Algebra Review
Dr. V. Këpuska
April 21, 2023 Veton Këpuska 2
Complex Algebra Elements Definitions:
Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero.
CR
C
Ι
R
then If
NumbersComplex all ofSet :
NumbersImaginary all ofSet :
Numbers Real all ofSet :
1
numbercomplex a of
formCartezian
jyxzx,y
j
April 21, 2023 Veton Këpuska 3
Complex Algebra Elements
z ofpart Imaginary
z ofpart Real
Im
Re
define then we If
0 If
0 If
zy
zx
jy xz
x zy
jy zx
R
I
April 21, 2023 Veton Këpuska 4
Euler’s Identity
j
ee
ee
je
je
je
jj
jj
j
j
j
2cos
2cos
sincos
sincos
sincos
April 21, 2023 Veton Këpuska 5
Polar Form of Complex Numbers
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.
z of argument)(or Angle z arg
z of Magnitude
radians ],-(
0r
z
rz
r rez j R
April 21, 2023 Veton Këpuska 6
Polar Form of Complex Numbers
Conversion between polar and rectangular (Cartesian) forms.
For z=0+j0; called “complex zero” one can not define arg(0+j0). Why?
x
yyxr
ry
rx
jy xjrr
jy xjr
jy xrez j
1
22
tansin
cos
sincos
sincos
April 21, 2023 Veton Këpuska 7
Geometric Representation of Complex Numbers.
Q1Q2
Q3 Q4
Im
Re
z
Re{z}Im
{z} |z
|
Complex or Gaussian plane
Axis of Reals
Axis of Imaginaries
April 21, 2023 Veton Këpuska 8
Geometric Representation of Complex Numbers.
Q1Q2
Q3 Q4
Im
Re
z
Re{z}
Im{
z}
|z|
Complex or Gaussian plane
Axis of Reals
Axis of Imaginaries
Complex Number in Quadrant
Condition 1 Condition 2
Q1 or Q2 Arg{z} ≥ 0 Im{z} ≥ 0
Q3 or Q4 Arg{z} ≤ 0 Im{z} ≤ 0
Q1 or Q4 Re{z} ≥ 0
Q2 or Q3 Re{z} ≤ 0
April 21, 2023 Veton Këpuska 9
Example
Im
Re
z1 1
-1
-1-2
z2
z3
4
32
11
202
4
32
11
3
3
3
2
22
1
1
1
z
zjz
z
zjz
z
zjz
{
{
{
April 21, 2023 Veton Këpuska 10
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?
j
j
rejrr
jrr
jrrz
jrrrez
sincos
coscos sincos
sinsin sincos
sincos
If z=rej then z*=re-j
April 21, 2023 Veton Këpuska 11
Geometric Representation of Conjugate Numbers
If z=rej then z*=re-j
Im
Re
z
r
Complex or Gaussian plane
-
r
x
y
-yz*
April 21, 2023 Veton Këpuska 12
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.
April 21, 2023 Veton Këpuska 13
Addition/Subtraction of Complex Numbers
2121
2121
212121
222111
III
ReReRe
:Thus
then
& ,
Let
zmzmzzm
zzzz
yyjxxzz
jyxzjyxz
April 21, 2023 Veton Këpuska 14
Multiplication/Division of Complex Numbers
2121
2121
2121
212121
2211
:Therefore
then
&
Let
21
2121
21
zzzz
zzzz
errzz
eerrererzz
erzerz
j
jjjj
jj
212
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
:Therefore
Olso
21
21
2
1
zzz
z
z
z
z
z
er
r
z
z
eer
r
er
er
z
z
j
jjj
j
April 21, 2023 Veton Këpuska 15
Useful Identities
z ∈ C, ∈ R & n ∈ Z (integer set)
nn
nn
zz
znnzzz
zzzzz
zzzz
zzzzz
zzzz
z
z
z
zzzzz
zzzz
zzzz
)16
)15)14
0 if
0 if 0)13)12
ImIm)11ReRe)10
)9)8
)7
)6)5
ImIm)4ReRe)3
)2)1
22121
2
1
2
12121
April 21, 2023 Veton Këpuska 16
Useful Identities
Example: z = +j0 =2 then arg(2)=0 =-2 then arg(-2)=
Im
Re
j
-1-2
z
210
April 21, 2023 Veton Këpuska 17
Silly Examples and Tricks
1012sin2cos
102
3sin
2
3cos
101sincos
102
sin2
cos
1010sin0cos
2
2
3
2
0
jje
jjje
jje
jjje
jje
j
j
j
j
j
Im
Re
j
-1 10
-j
/2
3/2
jjjjjjjj
jjjj
jjjjjjjj
jjjj
151173
141062
13951
12840
1111
1111
1
0
222
2
jjj
j
eeejjj
ejjj
April 21, 2023 Veton Këpuska 18
Division Example
Division of two complex numbers in rectangular form.
2
1
22
2
1
22
22
22
Im
222112
Re
222121
2
1
2221122121
22
22
22
11
22
11
2
1
112111 ,
z
z
z
z
zz
yx
yxyxj
yx
yyxx
z
z
yx
yxyxjyyxx
jyx
jyx
jyx
jyx
jyx
jyx
z
z
jyxzjyxz
April 21, 2023 Veton Këpuska 19
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).
Example: N=3; z3-1=0 z3=1 ⇒
)root 3(?
)root 2(?
)root 1(11
rd3
nd2
st1
3
z
z
zz
April 21, 2023 Veton Këpuska 20
Roots of Unity
zN-1=0 has roots , k=0,1,..,N-1, where
The roots ofare called Nth roots of unity.
Nj
e2
1,...,1,0,2
Nke N
kj
k
k
April 21, 2023 Veton Këpuska 21
Roots of Unity
Verification:
1,...,1,0for trueis wich
02sin
12cos
012sin2cos
012sin2cos
2sin2cos
Identity Eulers Applying
0101
2
22
Nkk
k
jkjk
kjk
kjke
ee
kj
kj
N
N
kj
April 21, 2023 Veton Këpuska 22
J1
Geometric Representation
2
1
01
3
3
4
3
222
3
2
3
121
3
020
kee
kee
ke
N
jj
jj
j
Im
Re1
-j1
-1
J2
0
j1
2/3
4/
3 J0
2/
3
2/3
April 21, 2023 Veton Këpuska 23
Important Observations1. 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).
2. The difference in angle between two consecutive roots is 2/N.
3. 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)*
ke N
kj
,1||2
DEQN
e Nj
k
kkk ..
221
11
*2*1213
*
**222
2222
1&3For
1
kN
eeeeeee
kkN
kN
kj
N
kj
N
kj
jN
kj
N
Nj
N
kNj
kN