Department of Natural Sciences
Faculty of Science and Information Technology
Complex numbers were ﬁrst introduced by G.
R. Bombelli introduced the symbol 𝑖.
A. Girard called “solutions impossible”.
C. F. Gauss called “complex number”
• 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.
When we combine the real and
imaginary number then
complex number is form.
• 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 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
Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real numbers
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
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
SUBTRACTION OF COMPLEX
MULTIPLICATION OF COMPLEX
DE MOIVRE'S THEORoM
DE MOIVRE'S THEORM is the theorm which show us
how to take complex number to any power easily.
This leads to the complex exponential
The polar form of a complex number can be rewritten as
So any complex number, x + iy,
can be written in
Expressing Complex Number
in Polar Form
A complex number, z = 1 - j
has a magnitude
nnzand argument :
Hence its principal argument is : rad
Hence in polar form :
EXPRESSING COMPLEX NUMBERS IN POLAR FORM
x = r cos 0 y = r sin 0
Z = r ( cos 0 + i sin 0 )
Complex numbers has a wide range of
applications in Science, Engineering,
Solving diff eqs with function of complex roots
Cauchy's integral formula
Calculus of residues
In Electric circuits
to solve electric circuits
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
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”???