representation of complex number
TRANSCRIPT
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!