This document provides an introduction to complex numbers. It discusses how complex numbers were first introduced to solve cubic equations and how their arithmetic rules were developed. It defines complex numbers, discusses representing them in polar form using Euler's formula, and explains De Moivre's theorem for taking complex numbers to powers. The document notes that complex functions have complex domains and ranges and outlines some applications of complex numbers in electrical engineering using Fourier transforms and satellite launches using transformations.

1. SARASWATHY S

2.  INTRODUCTION
 COMPLEX NUMBER
 ROTATIONS OF i
 COMPLEX NUMBER IN POLAR FORM
 EULER FORMULA
 DE MOVIRE’S THEOREM
 COMPLEX FUNCTIONS
 APPLICATIONS

3. INTRODUCTION
 COMPLEX NUMBERS were first introduced by an
Italian mathematician GEROLAMO CARDANO,
during his attempt to solve cubic equations in the
16th century.
 The rules of addition,
subtraction,multiplication,division of complex
number were developed by Italian mathematician
RAFAEL BOMBELI .

4. DEFINITION OF COMPLEX
NUMBER

5. Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Numbers

6. THE ROTATIONS OF i

7. Expressing Complex Numbers
in Polar Form
 THE POLAR FORM OF COMPLEX NUMBER
)

8. EULER FORMULA
 The polar form of a
complex number can be
written as
z=r
=r

9. De Moivre’s Theorem
De Moivre's Theorem shows us how to take complex
numbers to any power easily.
De Moivre's Theorem – Let r(cos F+isin F) be a complex
number and n be any real number. Then
[r(cos F+isin F]n = rn(cos nF+isin nF)
What is this saying?
The resulting r value will be r to the nth power and
the resulting angle will be n times the original angle.

10. Complex Functions
 A complex function is a function whose domain and
range subsets of the complex plane.
and

11. APPLICATIONS
 Complex numbers are
mainly used in Electrical
Engineering
techniques,because
FOURIER
TRANSFORMS are used
in understanding
oscillations and wave
behaviour that occur
both in AC current and
in modulated signals.
 Complex Analysis is used
when we launch a
satellite by
transformation or
mapping