This slide describes on the historical events, representation types and application of the complex number.

1. Representation of Complex Number
Presentation on

2. What is complex number?
Complex number is a union of Real numbers and
Imaginary numbers.

3. Venn Diagram
Imaginary Number
0 + yi
Real Number
x + 0i
Complex Number
x + yi

4. 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 𝟖𝟏 − 𝟏𝟏𝟒 which was impractical in the then period.

5. Historical events
o In 1545, Girolamo Cardano solved the equation x(10-x) =
40, finding the answer to be 5 ± −15 . He used to dislike
imaginary number and called them as “mental tortures”.

6. 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.

7. Historical events
o In 1777, Euler made the symbol i stands for √-1, which
made it a little easier to understand complex number.

8. How did we get i ?
𝒙 𝟐
+1=0
𝒙 𝟐
= −1
∴ 𝒙 = −1
This 𝒙 is actually is the i we’re searching for.
So,
𝒊 = −1

9. Graphical representation
Graphical Representation
Polar form Cartesian form
The graphical representation or pictorial
representation of complex number is implemented in
Argand diagram.

10. Polar representation
Im
Re
X
Y
0
-Y
𝜃
r
= r𝒆𝒊𝜽= rcos𝜽 + i rsin𝜽Z

11. Cartesian representation
Im
Re
X
Y
0
-Y
= x +i y
(x,y)
x
y
Z

12. Let’s plot a number
X
Y
0
-Y
3+2i
(3,2)
3
2

13. Applications
o To find impedance of an RLC circuit:
𝑍 = 𝑅 + 𝑗 𝑋
V

14. Applications
o To find AC voltage:
𝑉 = 𝑉0 𝑒 𝑗𝜔𝑡
V

15. Applications
o To do signal analysis:
Image After FFT
𝑋 𝑘 = ∑𝑥 𝑛 𝑒−2𝜋𝑖𝑘𝑛/𝑁

16. 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

17. Applications
Other fields of application in a nut shell:
 Relativity
 Fluid dynamics
 Quantum mechanics
 Improper integrals
 Control theory

18. Any Query?

19. Thanks!