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Name :
Md. Arifuzzaman
Employee ID
710001113
Designation
Lecturer
Department
Department of Natural Sciences
Faculty
Facult...
 History.
 Number System.
 Complex numbers.
 Operations.
5(Complex Number)
Contents
 Complex numbers were first introduced by G.
Cardano
 R. Bombelli introduced the symbol 𝑖.
 A. Girard called “solutions ...
7(Complex Number)
Number System
Real
Number
Irrational
Number
Rational
Number
Natural
Number
Whole
Number
Integer
Imaginar...
8(Complex Number)
Complex Numbers
• A complex number is a number that can b express in
the form of "a+b𝒊".
• Where a and b...
Complex Number
Real
Number
Imaginary
Number
Complex
Number
When we combine the real and
imaginary number then
complex numb...
10(Complex Number)
Complex Number
• A complex number has a real part and an imaginary part,
But either part can be 0 .
• S...
11(Complex Number)
Complex Numbers
Complex number convert our visualization into physical things.
A complex number is a number consisting
of a Real and Imaginary part.
It can be written in the form
COMPLEX NUMBERS
1i
COMPLEX NUMBERS
 Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real number...
COMPLEX CONJUGATE
 The COMPLEX CONJUGATE of a complex number
z = x + iy, denoted by z* , is given by
z* = x – iy
 The Mo...
COMPLEX NUMBERS
Equal complex numbers
Two complex numbers are equal if their
real parts are equal and their imaginary
part...
idbcadicbia )()()()( 
ADDITION OF COMPLEX NUMBERS
i
ii
)53()12(
)51()32(


i83 
EXAMPLE
Real Axis
Imaginar...
SUBTRACTION OF COMPLEX
NUMBERS
idbcadicbia )()()()( 
i
i
ii
21
)53()12(
)51()32(



Example
Real Axis
Imag...
MULTIPLICATION OF COMPLEX
NUMBERS
ibcadbdacdicbia )()())(( 
i
i
ii
1313
)310()152(
)51)(32(



Example
DIVISION OFACOMPLEX
NUMBERS
 
 dic
bia

  
 
 
 dic
dic
dic
bia






22
2
dc
bdibciadiac



 
...
EXAMPLE
 
 i
i
21
76

  
 
 
 i
i
i
i
21
21
21
76






22
2
21
147126



iii
41
5146



i
5
...
DE MOIVRE'S THEORoM
DE MOIVRE'S THEORM is the theorm which show us
how to take complex number to any power easily.
Euler Formula


j
re
jyxjrz

 )sin(cos
 yjye
eeee
jyxz
x
jyxjyxz
sincos 



This leads to the complex exp...
So any complex number, x + iy,
can be written in
polar form:
Expressing Complex Number
in Polar Form
sinry cosrx 
irry...
A complex number, z = 1 - j
has a magnitude
2)11(|| 22
z
Example
rad2
4
2
1
1
tan 1











 
 
...
EXPRESSING COMPLEX NUMBERS IN POLAR FORM
x = r cos 0 y = r sin 0
Z = r ( cos 0 + i sin 0 )
APPLICATIONS
 Complex numbers has a wide range of
applications in Science, Engineering,
Statistics etc.
Applied mathemati...
 Examples of the application of complex numbers:
1) Electric field and magnetic field.
2) Application in ohms law.
3) In ...
REFERENCES..
 Wikipedia.com
 Howstuffworks.com
 Advanced Engineering
Mathematics
 Complex Analysis
Complex number
Complex number
Complex number
Complex number
Complex number
Complex number
Complex number
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  1. 1. Name : Md. Arifuzzaman Employee ID 710001113 Designation Lecturer Department Department of Natural Sciences Faculty Faculty of Science and Information Technology Personal Webpage http://faculty.daffodilvarsity.edu.bd/profile/ns/arifuzzaman.ht ml E-mail zaman.ns@daffodilvarsity.edu.bd Phone Cell-Phone +8801725431992
  2. 2.  History.  Number System.  Complex numbers.  Operations. 5(Complex Number) Contents
  3. 3.  Complex numbers were first introduced by G. Cardano  R. Bombelli introduced the symbol 𝑖.  A. Girard called “solutions impossible”.  C. F. Gauss called “complex number” 6(Complex Number) History
  4. 4. 7(Complex Number) Number System Real Number Irrational Number Rational Number Natural Number Whole Number Integer Imaginary Numbers
  5. 5. 8(Complex Number) Complex Numbers • A complex number is a number that can b express in the form of "a+b𝒊". • Where a and b are real number and 𝑖 is an imaginary. • In this expression, a is the real part and b is the imaginary part of complex number.
  6. 6. Complex Number Real Number Imaginary Number Complex Number When we combine the real and imaginary number then complex number is form.
  7. 7. 10(Complex Number) Complex Number • A complex number has a real part and an imaginary part, But either part can be 0 . • So, all real number and Imaginary number are also complex number.
  8. 8. 11(Complex Number) Complex Numbers Complex number convert our visualization into physical things.
  9. 9. A complex number is a number consisting of a Real and Imaginary part. It can be written in the form COMPLEX NUMBERS 1i
  10. 10. COMPLEX NUMBERS  Why complex numbers are introduced??? Equations like x2=-1 do not have a solution within the real numbers 12 x 1x 1i 12 i
  11. 11. COMPLEX CONJUGATE  The COMPLEX CONJUGATE of a complex number z = x + iy, denoted by z* , is given by z* = x – iy  The Modulus or absolute value is defined by 22 yxz 
  12. 12. COMPLEX NUMBERS Equal complex numbers Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. If a + bi = c + di, then a = c and b = d
  13. 13. idbcadicbia )()()()(  ADDITION OF COMPLEX NUMBERS i ii )53()12( )51()32(   i83  EXAMPLE Real Axis Imaginary Axis 1z 2z 2z sumz
  14. 14. SUBTRACTION OF COMPLEX NUMBERS idbcadicbia )()()()(  i i ii 21 )53()12( )51()32(    Example Real Axis Imaginary Axis 1z 2z  2z diffz  2z
  15. 15. MULTIPLICATION OF COMPLEX NUMBERS ibcadbdacdicbia )()())((  i i ii 1313 )310()152( )51)(32(    Example
  16. 16. DIVISION OFACOMPLEX NUMBERS    dic bia          dic dic dic bia       22 2 dc bdibciadiac      22 dc iadbcbdac   
  17. 17. EXAMPLE    i i 21 76          i i i i 21 21 21 76       22 2 21 147126    iii 41 5146    i 5 520 i  5 5 5 20 i  i 4
  18. 18. DE MOIVRE'S THEORoM DE MOIVRE'S THEORM is the theorm which show us how to take complex number to any power easily.
  19. 19. Euler Formula   j re jyxjrz   )sin(cos  yjye eeee jyxz x jyxjyxz sincos     This leads to the complex exponential function : The polar form of a complex number can be rewritten as
  20. 20. So any complex number, x + iy, can be written in polar form: Expressing Complex Number in Polar Form sinry cosrx  irryix  sincos 
  21. 21. A complex number, z = 1 - j has a magnitude 2)11(|| 22 z Example rad2 4 2 1 1 tan 1                   nnzand argument : Hence its principal argument is : rad Hence in polar form : 4  zArg         4 sin 4 cos22 4   jez j
  22. 22. EXPRESSING COMPLEX NUMBERS IN POLAR FORM x = r cos 0 y = r sin 0 Z = r ( cos 0 + i sin 0 )
  23. 23. APPLICATIONS  Complex numbers has a wide range of applications in Science, Engineering, Statistics etc. Applied mathematics Solving diff eqs with function of complex roots Cauchy's integral formula Calculus of residues In Electric circuits to solve electric circuits
  24. 24.  Examples of the application of complex numbers: 1) Electric field and magnetic field. 2) Application in ohms law. 3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes 4) A complex number could be used to represent the position of an object in a two dimensional plane, How complex numbers can be applied to “The Real World”???
  25. 25. REFERENCES..  Wikipedia.com  Howstuffworks.com  Advanced Engineering Mathematics  Complex Analysis
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