2. 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
3.
4.
5. History.
Number System.
Complex numbers.
Operations.
5(Complex Number)
Contents
6. 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
8. 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.
10. 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.
12. A complex number is a number consisting
of a Real and Imaginary part.
It can be written in the form
COMPLEX NUMBERS
1i
13. COMPLEX NUMBERS
Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real numbers
12
x
1x
1i
12
i
14. 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
15. 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
20. 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
21. DE MOIVRE'S THEORoM
DE MOIVRE'S THEORM is the theorm which show us
how to take complex number to any power easily.
22. 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
23. So any complex number, x + iy,
can be written in
polar form:
Expressing Complex Number
in Polar Form
sinry cosrx
irryix sincos
24. 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
28. 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
29. 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”???