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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
z[email protected]
Phone
Cell-Phone
+8801725431992
Contents
 History.
 Number System.
 Complex numbers.
 Operations.
(Complex Number)
5
History
 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”
(Complex Number)
6
Number System
Imaginary
Numbers
Real
Number
Irrational
Number
Natural
Number
Rational
Number
Whole
Number
Integer
(Complex Number)
7
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.
(Complex Number)
8
Complex Number
When we combine the real and
imaginary number then
complex number is form.
Real
Number
Imaginary
Number
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.
(Complex Number)
10
Complex Numbers
Complex number convert our visualization into physical things.
(Complex Number)
11
COMPLEX NUMBERS
A complex number is a number consisting
of a Real and Imaginary part.
It can be written in the form
i  1
COMPLEX NUMBERS
Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real numbers

x  1
2
x  1
i  1
i  1
2
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
z x y
2
2
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
ADDITION OF COMPLEX NUMBERS
(a  bi)  (c  di)  (a  c)  (b  d )i
Imaginary Axis
z2
EXAMPLE
( 2  3i )  (1  5i )
 ( 2  1)  (3  5)i
 3  8i
z sum
z1
z2
Real Axis
SUBTRACTION OF COMPLEX
NUMBERS
(a  bi)  (c  di)  (a  c)  (b  d )i
Imaginary Axis
Example
( 2  3i )  (1  5i )
 ( 2  1)  (3  5)i
 1  2i
z1
z 2
z diff
z2
z 2
Real Axis
MULTIPLICATION OF COMPLEX
NUMBERS
(a  bi)(c  di)  (ac  bd )  (ad  bc)i
Example
( 2  3i )(1  5i )
 ( 2  15)  (10  3)i
 13  13i
DIVISION OF A COMPLEX
NUMBERS
a  bi 
c  di 

a  bi  c  di 


c  di  c  di 
ac  adi  bci  bdi

2
2
c d
2
ac  bd  bc  ad i

2
2
c d
EXAMPLE
6  7 i 
1  2i 

6  7i   1  2i 
1  2i  1  2i 
6  12i  7 i  14i

12  2 2
20  5i

5
2
6  14  5i

1 4
20 5i


5
5
 4 i
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
The polar form of a complex number can be rewritten as
z  r (cos   j sin  )  x  jy
 re j
This leads to the complex exponential
function :
z  x  jy
e z  e x  jy  e x e jy
 e cos y  j sin y 
x
Expressing Complex Number
in Polar Form
x  r cos
y  r sin 
So any complex number, x + iy,
can be written in
polar form:
x  yi  r cos   r sin i
Example
A complex number, z = 1 - j
has a magnitude
| z | (12  12 )  2
and argument :
 1
 

z  tan    2n     2n  rad
 1 
 4

1
Hence its principal argument is :
Arg z  

4
rad
Hence in polar form :
z  2e
j

4



 2  cos  j sin 
4
4

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 mathematics
Solving diff eqs with function of complex roots
 Cauchy's integral formula
 Calculus of residues
 In Electric circuits
to solve electric circuits
How complex numbers can be applied to
“The Real World”???
 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,
REFERENCES..
 Wikipedia.com
 Howstuffworks.com
 Advanced Engineering
Mathematics
 Complex Analysis