Representation of Complex Number

Presentation on

What is complex number?

Complex number is a union of Real numbers and Imaginary numbers.

Venn Diagram

Imaginary Number0 + yi

Real Numberx + 0i

Complex Numberx + yi

Historical events

o In 50 A.D. Heron of Alexandria studied the volume of an

impossible section of a pyramid. He calculated the value

as which was impractical in the then period.

Historical events

o In 1545, Girolamo Cardano solved the equation x(10-x) =

40, finding the answer to be He used to dislike

imaginary number and called them as “mental tortures”.

Historical events

o In 1637, Rene Descartes came up with the standard form

for complex numbers which is a + bi. However, he didn’t

like complex numbers either.

Historical events

o In 1777, Euler made the symbol i stands for √-1, which

made it a little easier to understand complex number.

How did we get i ?

+1=0

𝒙𝟐=  −1

∴ =

This is actually is the i we’re searching for.So,

=

Graphical representation

Graphical Representation

Polar form Cartesian form

The graphical representation or pictorial representation of complex number is implemented

in Argand diagram.

Polar representation

Im

ReX

Y

0

-Y

𝜃

r= r= rcos+ i rsinZ

Cartesian representation

Im

ReX

Y

0

-Y

= x +i y(x,y)

x

y

Z

Let’s plot a number

X

Y

0

-Y

3+2i

(3,2)

3

2

Applications

o To find impedance of an RLC circuit:

𝑍=𝑅+ 𝑗 𝑋

V

Applications

o To find AC voltage:

𝑉=𝑉 0𝑒 𝑗 𝜔𝑡

V

Applications

o To do signal analysis:

Image After FFT

𝑋𝑘=∑𝑥𝑛𝑒− 2𝜋 𝑖𝑘𝑛 /𝑁

Applications

o To do signal sampling:

E-mu Emax

This has a button called “Transform multiply” This is a fancy name of convolution Convolution is performed using DFT method

Applications

Other fields of application in a nut shell:

Relativity Fluid dynamics Quantum mechanics Improper integrals Control theory

Any Query?

Thanks!